PLoS Computational Biology

Dong, Xin Y.;Liu, Renjie;Seroski, Dillon T.;Hudalla, Gregory A.;Hall, Carol K.

doi: 10.1371/journal.pcbi.1011685pmid: 38048311

Introduction Peptides have been extensively used as building blocks for supramolecular biomaterials in applications ranging from drug delivery and tissue engineering, to biosensors [1,2]. Peptide-based hydrogels are appealing due to their biocompatibility, biodegradability, and low toxicity. Peptide-based biomaterials can be formed from a single component (“self-assembly”) or through a blend of components (“co-assembly”). Here we describe a system that can be made via selective co-assembly, which occurs when peptides A and B co-assemble in solution, but remain in a random-coil state when separated. Selective co-assembly allows for control of the assembly pathway and, in turn, enables precise formation of nanofiber structures, resulting in a hydrogel with predictable and uniform properties. CATCH (Co-Assembly Tags based on CHarge complementarity) are binary systems of oppositely charged synthetic peptides that selectively co-assemble into β-sheet nanofibers [3]. Charge complementarity drives CATCH peptide co-assembly; attraction between oppositely-charged peptides promotes cooperative co-assembly, while repulsion between like-charged peptides discourages self-assembly. CATCH peptides are cationic and anionic variants of Q11[QQKFQFQFEQQ]. The alternating motif of hydrophobic and hydrophilic residues in Q11 is a common feature in self-assembling peptides and is preserved in the CATCH peptides [4] The original pair of CATCH peptides reported were: CATCH(4+), (Ac-QQKFKFKFKQQ-Am) and CATCH(6−), (Ac-EQEFEFEFEQE-Am), where the number and sign denote the overall charge of the peptide as measured by the number of (positively-charged) lysine (K) or (negatively-charged) glutamic acid (E) residues [4]. CATCH peptides have been used successfully to immobilize functional proteins within macroscopic hydrogels [4]. The effect of the total charge on the co-assembly of pairs of CATCH peptides was determined by Seroski et. al who investigated CATCH(2+/2-), (4+/4-), and (6+/6-) peptide systems. They found that increasing the number of charged residues within each peptide results in an increased rate of co-assembly [5]. However, these studies did not explore the effects of replacing the type of cationic or anionic residues within CATCH peptides on their co-assembly. Here we study the effect of sidechain type on CATCH co-assembly [4]. We substitute negatively-charged aspartic acid residues (D) for the negatively-charged glutamic acid residues (E) in CATCH(6+/6-) pairs. Aspartic acid (D) is a convenient substitute for glutamic acid (E), as it is one methylene group shorter (Fig 1A). The positively-charged amino acid residue, lysine, remains the same. We will refer to the CATCH(6+/6-) mixture with glutamic acid as CATCH(6K+/6E-) (KQKFKFKFKQK/EQEFEFEFEQE), and the mixture with aspartic acid residues as CATCH(6K+/6D-) (KQKFKFKFKQK/DQDFDFDFDQD). Experiments performed by Liu et. al show that CATCH(6K+/6E-) and CATCH(6K+/6D-) form hydrogels with different structural and mechanical properties. Cryogenic scanning electron microscopy and conventional transmission electron microscopy (TEM) (Fig 2) show that CATCH(6K+/6E-) nanofibers are randomly entangled, whereas CATCH(6K+/6D-) form multi-layer stacks of aligned fibrils [6]. We hypothesize that the mismatched lengths of the charged residues in CATCH(6K+/6D-) create an incentive for two bilayers to stack together (Fig 1B and 1C). Thioflavin T analyses show that CATCH(6K+/6E-) assembles at a faster rate than CATCH(6K+/6D-), suggesting a difference in interaction strength between the (K+/E-) pair and the (K+/D-) pair. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. (A) Schematic of CATCH peptide sequence and sidechain structure for CATCH(6K+) in blue, (6E-) in red and (6D-) in orange, (B) Front view of CATCH(6K+/6E-) and CATCH(6K+/6D-) fibril showing two stacked bilayer starting structures built in PACKMOL and rendered in Chimera.[10,11] Sidechain structures are represented using sticks and colored based on the schematic from (A). Backbones are represented using black arrows and are directed into or out of the page. (C) Side view of CATCH(6K+/6E-) and CATCH(6K+/6D-) system. https://doi.org/10.1371/journal.pcbi.1011685.g001 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Morphology of CATCH(6K+/6E-) and CATCH(6K+/6D-) co-assemblies. (A) Cryogenic TEM micrographs of CATCH(6K+/6E-) and CATCH(6K+/6D-) in the sol state (1 mM total peptide). (B) Cryogenic SEM micrographs of CATCH(6K+/6E-) and CATCH(6K+/6D-) in the gel state (12 mM total peptide). https://doi.org/10.1371/journal.pcbi.1011685.g002 The aim of this work is to determine the effect of sidechain type on peptide co-assembly and how this corresponds to the nanofiber structures and morphologies observed in experiments. A computational approach is taken. Atomistic molecular dynamics simulation is used to analyze sidechain-sidechain interactions in detail as this affords a closer look (higher resolution) than can be obtained in biophysical experiments or in coarse-grained simulations. Atomistic simulations of a single bilayer and of two stacked bilayers (two bilayers stacked upon one another) are performed and analyzed for each CATCH(6K+/6E+) and CATCH(6K+/6D-) system. Single bilayers are also simulated to predict the fibril structure for each peptide pair. The two stacked bilayer simulations are used to quantify the sidechain-sidechain interactions between charged residues that sit between the bilayers. We also perform simulations of single monomeric peptides to determine the native monomeric state for CATCH(6K+), (6D-), and (6E-). A coarse-grained simulations approach, discontinuous molecular dynamics (DMD) with the PRIME20 forcefield, is used to investigate assembly kinetics and pathway for large CATCH(6K+/6E-) systems and CATCH(6K+/6D-) systems starting from random-coil conformations. DMD/PRIME20 simulations allow access to timescales that are not available through traditional atomistic molecular dynamics, and spatial resolution that is not accessible through biophysical measurements. Highlights of our results include the following: Single bilayer atomistic simulations show that CATCH(6K+/6E-) has a more pronounced twist than CATCH(6K+/6D-), providing a possible explanation for the experimentally-observed differences in nanofiber thickness between the two. Atomistic simulations of the two stacked bilayers show weaker van der Waals and electrostatic interactions between charged residues (on the second and third layer) for CATCH(6K+/6E-) than for CATCH(6K+/6D-). Atomistic simulations of the two separated bilayers show fewer number of contacts between charged residues (on the second and third layer) for CATCH(6K+/6E-) than for CATCH(6K+/6D-). Analysis of DMD/PRIME20 results show that CATCH(6K+/6E-) monomers co-assemble at a faster rate than CATCH(6K+/6D-) monomers, in agreement with experimental thioflavin T analyses. Discordant helical segments found in the CATCH(6E-) single peptide atomistic REMD simulation offer an additional possible explanation for the fast co-assembly observed for CATCH(6K+/6E-). Visualization of CATCH(6K+/6D-) DMD results shows that some β-barrel intermediates undergo a β-barrel-to-β-sheet conformation change during β-sheet assembly. Overall, our results suggest that the anionic sidechain composition in CATCH(6K+/6E-) results in random entanglement of nanofibers, while the anionic sidechain composition in CATCH(6K+/6D-) results in multi-layer stacks of aligned fibrils. Methods Explicit-solvent atomistic molecular dynamics simulation Explicit-solvent atomistic MD simulations at T = 310 K are carried out in the canonical ensemble using the AMBER package with the AMBER ff14SB force field [7] to quantify the sidechain-sidechain interactions between CATCH peptide pairs for CATCH(6K+/6E-) and for CATCH(6K+/6D-). Temperature is maintained using the Langevin thermostat [8]. The SHAKE algorithm is used to maintain bond length constraints on bonds involving hydrogens [9]. Four different atomistic simulation configurations were built: (1) two stacked bilayers—two bilayers stacked upon one another, (2) two separated bilayers—two bilayers separated by a distance of ~13Å measured from the surface of each bilayer, (3) a single bilayer, and (4) a single peptide. The AMBER tLEaP program was used to build the peptide sequence; the N-terminal was capped with an acetyl group and the C-terminal was capped with a methyl group. Phi-psi angles were modified to conform to an antiparallel β-strand using Chimera [10]. PACKMOL was used to arrange peptides to create a single bilayer, the two stacked bilayers, and the two separated bilayers (Figs 3 and 4) [11]. The single bilayer was built with 12 peptides in each in-register antiparallel β-sheet. The two stacked bilayers and the two separated bilayers models consist of four in-register antiparallel β-sheet layers, with 12 peptides in each layer stacked on top of one another. In all bilayer systems, the neighboring β-strands were spaced ~5 Å apart and the β-sheets within a bilayer were spaced ~13Å apart (to promote hydrophobic interactions). The inter-strand spacing and antiparallel orientation of our model is validated by previous PITHIRDS-CT and FTIR work [12]. Each CATCH bilayer structure was solvated in a periodic truncated octahedral box containing TIP3P water with a 12 Å buffer [13]. Single peptide simulations were started in an extended conformation with phi-psi angles of -180 and 180° respectively. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Snapshots of (A-B) CATCH(6K+/6E-) and (C-D) CATCH(6K+/6D-) bilayers before and after 200 ns of simulation. Final structure of CATCH(6K+/6E-) and CATCH(6K+/6D-) have an average twist of -3.55 and -2.22° between neighboring peptides, respectively. https://doi.org/10.1371/journal.pcbi.1011685.g003 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Contact map for (A) CATCH(6K+/6E-) and (B) CATCH(6K+/6D-). Contacts are counted for all atoms in each single bilayer system and grouped by residue. Contacts between two atoms were determined using a distance cutoff of 7Å. Values reported are averaged over three independent MD simulations. https://doi.org/10.1371/journal.pcbi.1011685.g004 Three independent simulations were run for the single bilayer systems, the two stacked bilayers, and the two separated bilayers. Each system was subjected to thermal annealing steps prior to the production run. The protocol for our atomistic MD simulations was as follows: (1) a 1000-step energy minimization using the steepest descent method was performed on the solvent molecules with the peptide structure constrained by a force of 500 kcal/mol. (2) A 2500-step energy minimization was performed on all atoms in the system. (3) Systems were brought up to 310 K through a series of heating stages over the course of 50 ps. (4) Thermal annealing was performed in the following steps: heat from 310 K to 400 K, equilibrate at 400 K, heat from 400 K to 500 K, equilibrate at 500 K, cool from 500 K to 310 K. Each step was performed for 100 ps. (5) 200 ns production runs were conducted in the NPT ensemble at 310 K. Root-mean-squared deviation (RMSD) analysis was performed on the last 10 ns of each simulation trajectory using the output structure from the minimization step to verify that the trajectory had reached an equilibrated state. All-atom implicit-solvent temperature REMD simulations for a single CATCH(6K+), (6E-), and (6D-) peptide were carried out using the AMBER package. Peptides were parameterized with the ff14SB forcefield [7]. For an exchange probability of ~0.25, with temperatures ranging from 310 to 600 K, eight replicas were generated for each CATCH peptide [14]. Chirality restraints were generated for each system to avoid a chirality inversion. Langevin dynamics was employed for temperature control [8]. Constraints were maintained using the SHAKE algorithm [9]. Prior to simulation, each system underwent energy minimization using the steepest descent algorithm for 500 cycles. Each system was then equilibrated at the desired temperature for 200 ps. Exchange attempts were made every 2 ps between adjacent replicas. Exchanges are accepted or rejected based on a Metropolis acceptance criterion that satisfies the detailed balance. Each simulation had a total of 100,000 exchange attempts for a total simulation time of 200 ns. Secondary structure content for REMD simulation results were calculated using the DSSP algorithm [15,16]. The implicit-solvent molecular mechanics/Generalized Born Surface Area (MM/GBSA) [17] approach was used to analyze the last 5 ns of the simulation trajectories to calculate the interaction energy between charged residues. MMGBSA is typically used in drug design to determine the binding affinity between a ligand and receptor by calculating the binding free energy (ΔGbinding). For the MMGBSA analysis, we define the top bilayer to be the “ligand,” the bottom bilayer to be the “receptor,” and the overall structure to be the “complex." Here we neglect entropy and focus on the interaction energy, as our system is quite large compared to the small molecules typically modeled. We calculated the van der Waals and electrostatic interaction energies between charged residues on the second and third β-sheets that result when the two bilayers stack together and the exposed charged residues between the bilayers interact. In this case, VDW and ELE are defined to be the difference in the van der Waals and electrostatic energies, respectively, between two bilayers before and after they stack together. The linear interaction energy (LIE) approach [18] was used to calculate the VDW interaction energies between the sidechains of intra-sheet and inter-sheet neighboring charged residues for the last 5 ns of the simulation trajectories. Similar to MMGBSA, it is typically used to predict the binding affinity of protein-ligand complexes. We define one group of residues to be the “ligand” and the other group to be the “receptor.” Here we utilize LIE to calculate the VDW interactions between two specific groups of sidechain atoms. Coarse-grained DMD simulations Implicit-solvent discontinuous molecular dynamics (DMD) simulations are carried out using the PRIME20 forcefield—designed specifically for modeling peptide aggregation. In PRIME20, amino acids are modeled by four spheres: three backbone spheres NH, Cα, and CO and one sidechain sphere R. Each sidechain sphere on each amino acid has a distinct size (effective van der Waals radius) and a distinct geometric structure (R-NH, R-Cα, and R-CO bond lengths). The two major non-bonded interactions in PRIME20 are directional hydrogen bonding interactions between backbone NH and CO spheres, and (non-directional) interactions between two sidechain R spheres. Both are modeled as square-well interactions. Polar, charge–charge, and hydrophobic interactions between amino acid sidechains are described using a combination of 210 different square-well widths and 19 different square-well depths [19]. All other interactions are modeled using a hard-sphere potential. Hard-sphere diameters, square-well widths, and square-well depths were determined by Cheon et. al using a perceptron learning algorithm [19]. A detailed description of the geometric and energetic parameters of the PRIME20 model is provided in earlier work [19–21]. Three independent simulations of CATCH(6K+/6E-) and CATCH(6K+/6D-) were run. Each system contained a total of 200 peptides—100 positively-charged peptides and 100 negatively-charged peptides—randomly distributed in a cubic box with a side length of 321 Å for a peptide concentration of 20 mM. All simulations were carried out for approximately 16 μs in the canonical ensemble. The Andersen thermostat was employed to maintain the simulation at a constant reduced temperature T* of 0.18, roughly corresponding to 296 K [22,23]. The reduced temperature in our system is defined as T* = kBT/εHB, where εHB = 12.47 kJ/mol is the hydrogen bonding energy [22]. Elements of graph theory are used to determine the rate of oligomerization and fibril formation [12]. Each oligomer cluster is defined as a network of peptides that are connected through a combination of hydrophobic and/or hydrogen bonding interactions. The fibril is considered to be the final β-sheet formed at the end of the simulation. A pair of peptides is considered “connected” if one of two conditions is met: (1) there are at least five hydrogen bonds between the pair of peptides, or (2) there are at least two hydrophobic interactions. For our system of CATCH peptides, these conditions are considered sufficient to accurately track the formation of oligomers and fibril formation over the course of a simulation. Data generated from DMD and MD simulations are provided on Dryad [24]. Explicit-solvent atomistic molecular dynamics simulation Explicit-solvent atomistic MD simulations at T = 310 K are carried out in the canonical ensemble using the AMBER package with the AMBER ff14SB force field [7] to quantify the sidechain-sidechain interactions between CATCH peptide pairs for CATCH(6K+/6E-) and for CATCH(6K+/6D-). Temperature is maintained using the Langevin thermostat [8]. The SHAKE algorithm is used to maintain bond length constraints on bonds involving hydrogens [9]. Four different atomistic simulation configurations were built: (1) two stacked bilayers—two bilayers stacked upon one another, (2) two separated bilayers—two bilayers separated by a distance of ~13Å measured from the surface of each bilayer, (3) a single bilayer, and (4) a single peptide. The AMBER tLEaP program was used to build the peptide sequence; the N-terminal was capped with an acetyl group and the C-terminal was capped with a methyl group. Phi-psi angles were modified to conform to an antiparallel β-strand using Chimera [10]. PACKMOL was used to arrange peptides to create a single bilayer, the two stacked bilayers, and the two separated bilayers (Figs 3 and 4) [11]. The single bilayer was built with 12 peptides in each in-register antiparallel β-sheet. The two stacked bilayers and the two separated bilayers models consist of four in-register antiparallel β-sheet layers, with 12 peptides in each layer stacked on top of one another. In all bilayer systems, the neighboring β-strands were spaced ~5 Å apart and the β-sheets within a bilayer were spaced ~13Å apart (to promote hydrophobic interactions). The inter-strand spacing and antiparallel orientation of our model is validated by previous PITHIRDS-CT and FTIR work [12]. Each CATCH bilayer structure was solvated in a periodic truncated octahedral box containing TIP3P water with a 12 Å buffer [13]. Single peptide simulations were started in an extended conformation with phi-psi angles of -180 and 180° respectively. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Snapshots of (A-B) CATCH(6K+/6E-) and (C-D) CATCH(6K+/6D-) bilayers before and after 200 ns of simulation. Final structure of CATCH(6K+/6E-) and CATCH(6K+/6D-) have an average twist of -3.55 and -2.22° between neighboring peptides, respectively. https://doi.org/10.1371/journal.pcbi.1011685.g003 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Contact map for (A) CATCH(6K+/6E-) and (B) CATCH(6K+/6D-). Contacts are counted for all atoms in each single bilayer system and grouped by residue. Contacts between two atoms were determined using a distance cutoff of 7Å. Values reported are averaged over three independent MD simulations. https://doi.org/10.1371/journal.pcbi.1011685.g004 Three independent simulations were run for the single bilayer systems, the two stacked bilayers, and the two separated bilayers. Each system was subjected to thermal annealing steps prior to the production run. The protocol for our atomistic MD simulations was as follows: (1) a 1000-step energy minimization using the steepest descent method was performed on the solvent molecules with the peptide structure constrained by a force of 500 kcal/mol. (2) A 2500-step energy minimization was performed on all atoms in the system. (3) Systems were brought up to 310 K through a series of heating stages over the course of 50 ps. (4) Thermal annealing was performed in the following steps: heat from 310 K to 400 K, equilibrate at 400 K, heat from 400 K to 500 K, equilibrate at 500 K, cool from 500 K to 310 K. Each step was performed for 100 ps. (5) 200 ns production runs were conducted in the NPT ensemble at 310 K. Root-mean-squared deviation (RMSD) analysis was performed on the last 10 ns of each simulation trajectory using the output structure from the minimization step to verify that the trajectory had reached an equilibrated state. All-atom implicit-solvent temperature REMD simulations for a single CATCH(6K+), (6E-), and (6D-) peptide were carried out using the AMBER package. Peptides were parameterized with the ff14SB forcefield [7]. For an exchange probability of ~0.25, with temperatures ranging from 310 to 600 K, eight replicas were generated for each CATCH peptide [14]. Chirality restraints were generated for each system to avoid a chirality inversion. Langevin dynamics was employed for temperature control [8]. Constraints were maintained using the SHAKE algorithm [9]. Prior to simulation, each system underwent energy minimization using the steepest descent algorithm for 500 cycles. Each system was then equilibrated at the desired temperature for 200 ps. Exchange attempts were made every 2 ps between adjacent replicas. Exchanges are accepted or rejected based on a Metropolis acceptance criterion that satisfies the detailed balance. Each simulation had a total of 100,000 exchange attempts for a total simulation time of 200 ns. Secondary structure content for REMD simulation results were calculated using the DSSP algorithm [15,16]. The implicit-solvent molecular mechanics/Generalized Born Surface Area (MM/GBSA) [17] approach was used to analyze the last 5 ns of the simulation trajectories to calculate the interaction energy between charged residues. MMGBSA is typically used in drug design to determine the binding affinity between a ligand and receptor by calculating the binding free energy (ΔGbinding). For the MMGBSA analysis, we define the top bilayer to be the “ligand,” the bottom bilayer to be the “receptor,” and the overall structure to be the “complex." Here we neglect entropy and focus on the interaction energy, as our system is quite large compared to the small molecules typically modeled. We calculated the van der Waals and electrostatic interaction energies between charged residues on the second and third β-sheets that result when the two bilayers stack together and the exposed charged residues between the bilayers interact. In this case, VDW and ELE are defined to be the difference in the van der Waals and electrostatic energies, respectively, between two bilayers before and after they stack together. The linear interaction energy (LIE) approach [18] was used to calculate the VDW interaction energies between the sidechains of intra-sheet and inter-sheet neighboring charged residues for the last 5 ns of the simulation trajectories. Similar to MMGBSA, it is typically used to predict the binding affinity of protein-ligand complexes. We define one group of residues to be the “ligand” and the other group to be the “receptor.” Here we utilize LIE to calculate the VDW interactions between two specific groups of sidechain atoms. Coarse-grained DMD simulations Implicit-solvent discontinuous molecular dynamics (DMD) simulations are carried out using the PRIME20 forcefield—designed specifically for modeling peptide aggregation. In PRIME20, amino acids are modeled by four spheres: three backbone spheres NH, Cα, and CO and one sidechain sphere R. Each sidechain sphere on each amino acid has a distinct size (effective van der Waals radius) and a distinct geometric structure (R-NH, R-Cα, and R-CO bond lengths). The two major non-bonded interactions in PRIME20 are directional hydrogen bonding interactions between backbone NH and CO spheres, and (non-directional) interactions between two sidechain R spheres. Both are modeled as square-well interactions. Polar, charge–charge, and hydrophobic interactions between amino acid sidechains are described using a combination of 210 different square-well widths and 19 different square-well depths [19]. All other interactions are modeled using a hard-sphere potential. Hard-sphere diameters, square-well widths, and square-well depths were determined by Cheon et. al using a perceptron learning algorithm [19]. A detailed description of the geometric and energetic parameters of the PRIME20 model is provided in earlier work [19–21]. Three independent simulations of CATCH(6K+/6E-) and CATCH(6K+/6D-) were run. Each system contained a total of 200 peptides—100 positively-charged peptides and 100 negatively-charged peptides—randomly distributed in a cubic box with a side length of 321 Å for a peptide concentration of 20 mM. All simulations were carried out for approximately 16 μs in the canonical ensemble. The Andersen thermostat was employed to maintain the simulation at a constant reduced temperature T* of 0.18, roughly corresponding to 296 K [22,23]. The reduced temperature in our system is defined as T* = kBT/εHB, where εHB = 12.47 kJ/mol is the hydrogen bonding energy [22]. Elements of graph theory are used to determine the rate of oligomerization and fibril formation [12]. Each oligomer cluster is defined as a network of peptides that are connected through a combination of hydrophobic and/or hydrogen bonding interactions. The fibril is considered to be the final β-sheet formed at the end of the simulation. A pair of peptides is considered “connected” if one of two conditions is met: (1) there are at least five hydrogen bonds between the pair of peptides, or (2) there are at least two hydrophobic interactions. For our system of CATCH peptides, these conditions are considered sufficient to accurately track the formation of oligomers and fibril formation over the course of a simulation. Data generated from DMD and MD simulations are provided on Dryad [24]. Dryad DOI 10.5061/dryad.5mkkwh7bp. Results and discussion Analysis of single CATCH bilayer geometry and intra-sheet sidechain-sidechain interactions Three independent explicit-solvent atomistic MD simulations for a single CATCH(6K+/6E-) bilayer structure and for a single CATCH(6K+/6D-) bilayer structure were carried out to determine the bilayer geometry and intra-sheet sidechain-sidechain interactions (Fig 3A–3D). The initial configuration for each CATCH mixture was an “ideal” structure with 12 in-register antiparallel peptides in each layer. Over the course of the 200 ns simulation, the sidechains in each structure relaxed into a more realistic geometry. The longer anionic sidechain in CATCH(6K+/6E-) leads to a more pronounced left-handed bilayer twist in CATCH(6K+/6E-) than in the CATCH(6K+/6D-) bilayer (Fig 3B and 3D). Here, we define the twist in terms of the angle between two neighboring β-strands. This angle was measured by calculating the line of best fit for the set of Cα atoms in each peptide, defining that line to be a vector with end points at the N- and the C-terminal Cα atoms, and then measuring the angle between the two resulting vectors. The CATCH(6K+/6E-) simulations resulted in a twisted bilayer, while the CATCH(6K+/6D-) simulations resulted in a relatively flat bilayer. The average angle of twist between the nearest neighbor peptides for CATCH(6K+/6E-) and CATCH(6K+/6D-) were -3.55°and -2.22°, respectively. The left-handed β-sheet twist (denoted by the negative sign) observed in the CATCH(6K+/6E-) β-sheet bilayer is inherent to antiparallel β-sheets. This twisting phenomenon can be attributed to the chirality of the amino acids, the inter-strand backbone hydrogen bonding, and inter-strand sidechain interactions [25–31]. Over the course of the last 5 ns of the MD simulation, CATCH(6K+/6E-) and CATCH(6K+/6D-) possess roughly the same amount of backbone hydrogen bonds and salt bridge interactions. Salt bridges are defined to be interactions between two oppositely charged groups containing at least two heavy atoms within hydrogen bonding distance of each other. For simplicity, we define salt bridges to be between any oxygen on the carboxylate group (in D or E) and any hydrogen on the ammonium groups (in K) that satisfy a distance cutoff of 3Å and an angle cutoff of 135°. Salt bridges in our simulations had an average length of ~2.8 Å and an average bonding angle of ~156°. Salt bridge interactions were calculated using the LIE approach (Table 1); VDW and ELE interactions were calculated between the atoms on lysine’s ammonium group and the atoms glutamic acid and aspartic acid’s carboxylic acid group. CATCH(6K+/6E-) had less favorable VDW interactions than (6K+/6D-), but more favorable ELE interactions; CATCH(6K+/6E-) and (6K+/6D-) had ELE interactions of -5402 and -4410 kcal/mol, respectively. However, when considering the charged residues as a whole (excluding the backbone atoms), CATCH(6K+/6E-) had more favorable VDW interactions than (6K+/6D-), -125.9 vs. -85.8 kcal/mol. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Summary of hydrogen bonding and LIE analysis for CATCH single bilayer structures. Hydrogen bonds and salt bridges were calculated using geometric criteria: an angle cutoff of 135° and a distance cutoff of 3.0Å. Salt bridges were defined to be between the hydrogens on lysine’s ammonium group and the oxygens on glutamic acid or on aspartic acid’s α-carboxylic acid group. Salt bridge VDW and ELE interactions were calculated between the atoms on the lysine’s ammonium group and the atoms on glutamic acid and aspartic acid’s carboxylic acid group. VDW interactions between charged residues are calculated using the LIE approach and exclude backbone atoms. Values listed are averaged over three independent simulations. https://doi.org/10.1371/journal.pcbi.1011685.t001 The difference in β-sheet twisting for the CATCH(6K+/6E-) and CATCH(6K+/6D-) single bilayers may be explained by how each CATCH pair organizes its β-sheet structure to maximize hydrogen bonding and salt bridge interactions. In CATCH(6K+/6E-), the sidechains of lysine (K) and glutamic acid (E) are of similar length, facilitating a salt bridge interaction between the charged groups on the ends of the charged residues. In CATCH(6K+/6D-), the sidechains of lysine (K) and aspartic acid (D) are of mismatched length. Given that the salt bridge interactions in both CATCH systems have similar geometry in terms of bond length and bond angle, it is likely that CATCH(6K+/6D-) accommodates the mismatched sidechain lengths by hindering its backbone from forming an inherent left-handed twist. The relationship between sidechain length and β-sheet conformation observed in the CATCH system is consistent with a previous quantitative and experimental studies that have shown that β-strands containing glutamic acid have a greater propensity for twisting than those containing aspartic acid [32–34]. To further interrogate the interactions within the single bilayer structure, we calculated the number of contacts between residues within the structure and the overall VDW interactions between the cationic sidechains and the anionic sidechains. Contacts are defined to be any two atoms within 7Å of one another. The greatest number of contacts for both CATCH(6K+/6E-) and (6K+/6D-) was between the phenylalanine residues, 45,149 and 45,163 contacts, respectively (Fig 4). This is expected as the phenylalanine residues make up the hydrophobic core of the bilayer structures. The second greatest number of contacts in each system was between the cationic sidechains and the anionic sidechains. CATCH(6K+/6E-) had 39,552 contacts and a VDW interaction of -125.9 kcal/mol between the lysine and glutamic acid sidechains; (6K+/6D-) had 30,543 contacts and a VDW interaction of -85.8 kcal/mol between the lysine and aspartic acid sidechains. The additional methylene group in glutamic acid (E) in the CATCH(6K+/6E-) pair leads to more contacts and stronger inter-strand interactions between charged residues than in the CATCH(6K+/6D-) pair. The stronger interaction for the K/E sidechain pair compared to the K/D sidechain pair is consistent with pair correlations calculated by Wouters et al. and isothermal titration calorimetry experiments performed by Petrauskas et al. [35,36] The relationship between strong inter-strand interactions and the β-sheet twisting observed in CATCH(6K+/6E-) is consistent with previous experimental studies that suggest that increased interactions between intra-sheet neighboring sidechains is related to increased twisting [26–28]. Evaluation of face-to-face interactions between the two stacked CATCH bilayers Three independent explicit-solvent atomistic MD simulations were carried out for a CATCH(6K+/6E-) mixture and for a CATCH(6K+/6D-) mixture, with each arranged in the two-stacked bilayer configuration. Each simulation started from a pre-formed “ideal” structure (Fig 5A–5D). Intra-sheet β-strands were spaced ~5 Å apart (measuring from the backbone center) to promote backbone hydrogen bonding between neighboring β-strands. The initial bilayers for both CATCH(6K+/6E-) and CATCH(6K+/6D-) were spaced ~3 Å apart, measuring between the end of the sidechains facing inward on the second and third β-sheet. As the simulation progressed, the van der Waals and electrostatic interactions between the charged residues drove the bilayers closer together and stabilized the structure, as expected. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Snapshots of (A-B) CATCH(6K+/6E-) and (C-D) CATCH(6K+/6D-) two stacked bilayers before and after 200 ns of simulation. Distances between the second and third layer of each structure are indicated. https://doi.org/10.1371/journal.pcbi.1011685.g005 Atomistic simulations of the two stacked bilayer systems showed tighter packing between the second and third layer for CATCH(6K+/6D-) than for CATCH(6K+/6E-). The distance between the second and third layer was measured by fitting the backbone atoms of each sheet to a plane and measuring the distance between the centroids of the two fitted planes. The average distance between the second and third β-sheet for CATCH(6K+/6E-) and for CATCH(6K+/6D-) changed over the course of the simulation from an initial distance of 14.8 and 12.7 Å to a final distance of 13.2 and 11.5 Å, respectively. Thus, the CATCH(6K+/6E-) bilayers were further apart than the CATCH(6K+/6E-) bilayers by roughly 2 Å. The longer anionic sidechain (E) in CATCH(6K+/6E-) acts as a physical barrier in the stacking process, while the shorter anionic sidechain (D) in CATCH(6K+/6D-) allows the charged residues to interdigitate in an alternating manner, bringing them closer together. In CATCH(6K+/6D-) the complementary charged residues between the second and third layer create a tight steric zipper that draws the sheets closer together [37,38]. Atomistic MD simulations of the two stacked bilayer CATCH systems show that CATCH(6K+/6D-) bilayers have more favorable face-to-face interactions than CATCH(6K+/6E-) bilayers (Table 2). MMGBSA analysis is used to calculate the van der Waals (VDW) and electrostatic (ELE) interaction energies between the two CATCH bilayers as a result of stacking (Table 2). Both the VDW and ELE interactions are more negative in CATCH(6K+/6D-) than in CATCH(6K+/6E-), suggesting that CATCH(6K+/6D-) has more favorable interactions between the two bilayers and a greater energy incentive for bilayer stacking than CATCH(6K+/6E-). LIE analysis is used to calculate the pairwise VDW interactions between the sidechains of the exposed charged residues (excluding the backbone atoms) on the second and third layer. CATCH(6K+/6D-) experiences a VDW interaction that is 2-fold stronger than for CATCH(6K+/6E-), -49.8 vs -20.7 kcal/mol, in agreement with the MMGBSA results (Table 2). The contrast in interaction energies between the two CATCH systems can be explained by the physical arrangement of the stacked bilayers. The two stacked bilayer simulations have shown that CATCH(6K+/6D-) can stack more closely together than CATCH(6K+/6E-), facilitating the VDW and ELE interactions between the charged residues on the second and third sheets. The single bilayer simulations have shown that the intra-sheet lysine and glutamic acid residues in CATCH(6K+/6E-) have stronger interactions than the intra-sheet lysine and aspartic acid residues in CATCH(6K+/6D-), -125.9 vs. -85.8 kcal/mol, respectively. For CATCH(6K+/6E-) there is not as large of an incentive to interact with sidechains on the opposing bilayer face as for CATCH 6K+/6D-). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Summary of MMGBSA and LIE analysis for CATCH two stacked bilayer structures. MMGBSA values for VDW and ELE energies are calculated by considering the interactions between the top bilayer and the bottom bilayer. LIE values for VDW energies are calculated by considering only the sidechain-sidechain interactions between the charged residues on the top bilayer and the bottom bilayer. Values listed are averaged over three independent simulations. https://doi.org/10.1371/journal.pcbi.1011685.t002 The difference in bilayer structure and face-to-face interactions between CATCH(6K+/6E-) and CATCH(6K+/6D-) provides a possible explanation for the difference in their experimentally-observed nanofiber structures. Cryogenic EM and TEM images show that CATCH(6K+/6E-) forms randomly entangled and tortuous nanofibers with short persistence lengths, whereas CATCH(6K+/6D-) forms aligned bundles of nanofibers with long persistence lengths that tend to appear as multi-layer stacks (Fig 2). The single bilayer atomistic simulations predict that CATCH(6K+/6E-) tends to form more twisted β-sheets than CATCH(6K+/6D-). As the β-sheets stack together, they must either untwist or twist together, both of which incur some energy cost [39,40]. However, for sheets with weak face-to-face attraction, i.e. CATCH(6K+/6E-), twisting comes at a lower cost. Generally, as fibril twisting decreases, the likelihood of β-sheet stacking and fibril thickness growth increases [40]. For CATCH(6K+/6E-), the twisted bilayer structure and weak face-to-face interactions are consistent with the formation of thin randomly entangled nanofibers which are not favored to stack or align. In contrast, for CATCH(6K+/6D-), the combination of a flat bilayer structure and strong face-to-face interactions between charged residues are consistent with the formation of multi-layer fibril bundles. The combination of strong intra-sheet interactions and weak face-to-face interactions leads to a twisted structure for CATCH(6K+/6E-), while the combination of weak intra-sheet interactions and strong face-to-face interactions lead to a flat structure for CATCH(6K+/6D-). This phenomenon has also been observed in atomistic MD simulations of short self-assembling peptides [39]. Quantification of CATCH bilayer stacking Three independent explicit-solvent atomistic MD simulations were carried out for a CATCH(6K+/6E-) mixture and for a CATCH(6K+/6D-) mixture, with each arranged in the two separated bilayer configuration. The two separated bilayer configurations are essentially the aforementioned two stacked bilayer configurations with an additional 10Å of space between the bilayers, for a total spacing of ~13Å between the edges of each bilayer. The additional spacing allowed the bilayers to have more choice in their stacking arrangement. CATCH(6K+/6D-) had a greater number of contacts between charged residues on the 2nd and 3rd layer than CATCH(6K+/6E-), suggesting that CATCH(6K+/6D-) fibrils are more likely to be well-aligned than CATCH(6K+/6E-). CATCH(6K+/6D-) had 25,185 contacts between charged residues and an average twist angle of -2.2° between neighboring strands. CATCH(6K+/6E-) had 17,652 contacts between charged residues and an average twist angle of -2.8° between neighboring strands. Overall, our simulations suggest that the flatter bilayer observed for CATCH(6K+/6D-) leads to well-aligned fibrils. In-silico assessment of CATCH co-assembly pathway and co-assembly kinetics DMD/PRIME20 simulations of CATCH systems were carried out to determine their co-assembly pathway and co-assembly kinetics. Three independent systems of equimolar mixtures of CATCH(6K+/6E-) and of CATCH(6K+/6D-) containing 200 peptides, starting from random-coil configurations, were simulated for 500 billion collisions (~16 μs) at T* = 0.18 (296 K). Simulation snapshots at 0, 8, and 16 μs were taken to examine their assembly pathways (Fig 6A and 6B). At t = 0 μs, the peptides in the CATCH(6K+/6E-) and CATCH(6K+/6D-) systems are in random-coil conformations and are randomly arranged in the box. For CATCH(6K+/6E-), at t = 8 μs, nearly all the peptides have assembled into either a β-sheet structure or an ordered oligomer. At 16 μs, CATCH(6K+/6E-) has fully assembled into β-sheet structures and off-pathway β-barrels. For CATCH(6K+/6D-) at t = 8 μs, we see formation of a single β-sheet, multiple oligomers, and many free peptides still remaining. At t = 16 μs, we observed elongation of the previously-mentioned β-sheet, however there are still free peptides remaining. ThT assays have demonstrated that CATCH(6K/6E) assembles faster than CATCH(6K/6D) [6]. Given the timescale (~16 μs) of our simulations and the difference in assembly kinetics between CATCH(6K+/6E-) and CATCH(6K+/6D-), it is not unexpected to observe some “free” peptides in the CATCH(6K+/6D-) system. For both CATCH systems we observe β-sheet growth through monomer addition or through the interaction of two small-ordered structures. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. DMD Snapshots of (A) CATCH(6K+/6E-) and (B) CATCH(6K+/6D-) over the course of a 16 μs DMD simulation. Cationic peptides containing lysine are represented in teal. Anionic peptides containing aspartic acid are represented in orange, while anionic peptides containing glutamic acid are represented in red. (C) Chronological snapshots of oligomer growth, conformation change, and elongation of a β-barrel in the CATCH(6K+/6D-) simulation. https://doi.org/10.1371/journal.pcbi.1011685.g006 Comparison of the hydrogen bond formation rates between CATCH(6K+/6E-) and CATCH(6K+/6D-) suggests that CATCH(6K+/6E-) fibrillizes at a faster rate than CATCH(6K+/6D-) (Fig 7A). The difference in assembly rates can be attributed to the difference in the anionic residue types. ThT analyses showed that CATCH(6K+/6E-) and CATCH(6K+/6D-) are both capable of assembling into β-sheet structures, however, CATCH(6K+/6E-) assembles at a significantly faster rate than CATCH(6K+/6D-) [6]. In experiments, this is likely related to the greater interaction strength between (K+/E-) pair than the (K+/D-) pair [35,36]. Due to its longer length, the glutamic acid sidechain in CATCH(6K+/6E-) has a greater range of interaction than the aspartic acid sidechain in CATCH(6K+/6D-), increasing the likelihood of finding the complementary lysine residue and forming backbone hydrogen bonds soon afterwards. By analyzing the rates of cluster formation and growth in CATCH(6K+/6E-) and CATCH(6K+/6D-) over time, we observe that CATCH(6K+/6E-) has a higher rate of β-sheet assembly than CATCH(6K+/6D-), and a greater depletion rate of free peptides, in agreement with the hydrogen bond kinetics (Fig 7) and ThT analyses. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. (A) Quantitative assessment of hydrogen bond formation over DMD simulation. (B) Analysis of free peptide depletion (orange), oligomerization (purple), and fibrillization (black). https://doi.org/10.1371/journal.pcbi.1011685.g007 Independent implicit-solvent atomistic REMD simulations of a single peptide were carried out for CATCH(6K+), (6E-), and (6D-) to explore the conformational space of CATCH free peptides in solution. Each peptide started in an extended conformation and was simulated for 200 ns at temperatures ranging from 310 to 600K. The DSSP (Define Secondary Structure of Proteins) algorithm was used to determine the average secondary content of each residue for trajectories at 310K (Table 3). Here, the term helix refers to 3–10, alpha, and pi helices. For convenient comparison, the sum of the averages for each peptide are also provided. CATCH(6K+), (6E-), and (6D-) all had helix, bend, turn, and coil conformations. Helices were found in all CATCH peptides between residues 3 and 9. CATCH(6E-) had the greatest total amount of helical content compared to CATCH(6K+) and (6D-), 6.27 vs 4.42 and 5.24 respectively. Notably, CATCH(6K+) had the most bend conformations compared to CATCH(6E-) and (6D-); CATCH(6K+) had a total bend conformation of 1.25, while CATCH(6E-) and (6D-) had a total bend conformation of 0.43 and 0.62, respectively. CATCH(6K+) and (6D-) had more coil content than CATCH(6E-), 2.39 vs. 1.63, respectively. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Summary of DSSP analysis for each CATCH single peptide REMD simulation. Average secondary content over all frames for each residue are reported. Helix is the sum of the averages for 3–10, alpha, and pi helices. The total value is the sum of the averages over each CATCH single peptide. https://doi.org/10.1371/journal.pcbi.1011685.t003 The difference in discordant helix content between CATCH(K+), (6E-), and (6D-) observed in atomistic REMD simulations provides additional insight into experimental co-assembly kinetics. A discordant helix is defined to be a helical segment in a peptide strand that has a tendency to form β-strands [41,42]. Our predicted conformations for CATCH(6E-) and CATCH(6D-) are in agreement with previously-observed conformations for aspartic acid oligomer and glutamic acid oligomer in simulations performed by Hunkler et. al. Their simulations showed that glutamic acid (E) oligomers tend to form stable α-helical structures, while aspartic acid (D) oligomers are intrinsically disordered [43]. In addition, glutamic acid has been frequently found in protein sequences with a propensity for helical structures [34,44,45]. The amount of α-helix/-beta-strand-discordance in CATCH(6E-) may facilitate β-sheet co-assembly. Kallberg et al. found that multiple amyloidogenic peptides that are predicted to form β-strands contain α-helices, and that fibril formation is lost upon removal or mutation of the α-helix-forming segments [41]. In a study on amyloid β-protein (Aβ), Fezoui and Teplow found that a partially-folded helix-containing conformer is an intermediate in Aβ fibril assembly, and that helix stabilization may facilitate fibril formation [46]. However, they also found that if a helix is sufficiently stabilized, it can resist structural reorganization and inhibit fibril formation [46,47]. The greater proportion of discordant helix observed in CATCH(6E-) than in (6D-) may promote faster β-sheet assembly in CATCH(6K+/6E-) than in (6K+/6D-). More recent studies of Aβ aggregation suggest that a β-hairpin intermediate promotes dimer formation through the intermolecular β-bridges [48–51]. CATCH peptides are too short to properly organize into a β-hairpin. However, the bend conformation observed in CATCH(6K+) may play a similar role to the β-hairpin conformation observed in Aβ by acting as a partially stable intermediate that facilitates peptide-peptide interactions. β-barrels observed in DMD simulations of CATCH peptides DMD/PRIME20 simulations also reveal on-pathway and off-pathway oligomers, including β-barrels in CATCH peptide co-assembly. In both CATCH(6K+/6E-) and CATCH(6K+/6D-) systems, β-barrels and ordered oligomers formed in addition to β-sheet structures. A β-barrel is a β-sheet that wraps around to form a cylindrical structure, with the last and first β-strands connected by backbone hydrogen bonds. β-barrels ranging in size from 6 to 8 peptides (Fig 6A and 6B) formed relatively early in the simulations for both CATCH(6K+/6E-) and CATCH(6K+/6D-). Nearly all the β-barrels that formed remained in a β-barrel structure throughout the course of the simulation—stabilized by hydrogen bonds and hydrophobic interactions between phenylalanine residues. The β-barrels that were not sufficiently stabilized by their intramolecular interactions unraveled to form β-sheets. DMD simulations capture β-barrel-to-β-sheet transitions in both CATCH systems. Fig 6C shows a 6-mer β-barrel intermediate in the CATCH(6K+/6D-) system that eventually seeds the final 42-mer β-sheet structure. The DMD snapshots show the CATCH (6K+/6D-) β-barrel shifting into an elliptical shape, flattening out, and opening at the ends. The β-sheet then grows through a combination of monomer addition and interactions with other small ordered-structures containing β-strands, ultimately forming the final 42-mer β-sheet structure. β-barrel intermediates have also been observed by Sun et. al in atomistic DMD simulations of hIAPP19–29 and its S20G mutant, hIAPP22–28, Aβ16–22, and the α-synuclein NACore. Each of these peptides ultimately self-assembled into cross-β aggregates [52]. β-barrel oligomers have garnered a lot of interest in the field of neurodegenerative diseases as a source of toxicity and a possible therapeutic target [53]. Understanding what factors contribute to the formation of β-barrel oligomers can aid in designing new therapeutics. Our results suggest that the combination of charge complementarity and alternation of hydrophilic and hydrophobic residues used to design CATCH peptides may also promote the formation of β-barrel structures, similar to those found in amyloidogenic peptides and in agreement with conclusions drawn by Shao et. al on previous CATCH simulations [12]. Although CATCH peptides produce on-pathway and off-pathway β-barrels, our research provides insight into designing stable β-barrels with potential applications such as single-molecule sensors or DNA sequencing [54,55]. Analysis of single CATCH bilayer geometry and intra-sheet sidechain-sidechain interactions Three independent explicit-solvent atomistic MD simulations for a single CATCH(6K+/6E-) bilayer structure and for a single CATCH(6K+/6D-) bilayer structure were carried out to determine the bilayer geometry and intra-sheet sidechain-sidechain interactions (Fig 3A–3D). The initial configuration for each CATCH mixture was an “ideal” structure with 12 in-register antiparallel peptides in each layer. Over the course of the 200 ns simulation, the sidechains in each structure relaxed into a more realistic geometry. The longer anionic sidechain in CATCH(6K+/6E-) leads to a more pronounced left-handed bilayer twist in CATCH(6K+/6E-) than in the CATCH(6K+/6D-) bilayer (Fig 3B and 3D). Here, we define the twist in terms of the angle between two neighboring β-strands. This angle was measured by calculating the line of best fit for the set of Cα atoms in each peptide, defining that line to be a vector with end points at the N- and the C-terminal Cα atoms, and then measuring the angle between the two resulting vectors. The CATCH(6K+/6E-) simulations resulted in a twisted bilayer, while the CATCH(6K+/6D-) simulations resulted in a relatively flat bilayer. The average angle of twist between the nearest neighbor peptides for CATCH(6K+/6E-) and CATCH(6K+/6D-) were -3.55°and -2.22°, respectively. The left-handed β-sheet twist (denoted by the negative sign) observed in the CATCH(6K+/6E-) β-sheet bilayer is inherent to antiparallel β-sheets. This twisting phenomenon can be attributed to the chirality of the amino acids, the inter-strand backbone hydrogen bonding, and inter-strand sidechain interactions [25–31]. Over the course of the last 5 ns of the MD simulation, CATCH(6K+/6E-) and CATCH(6K+/6D-) possess roughly the same amount of backbone hydrogen bonds and salt bridge interactions. Salt bridges are defined to be interactions between two oppositely charged groups containing at least two heavy atoms within hydrogen bonding distance of each other. For simplicity, we define salt bridges to be between any oxygen on the carboxylate group (in D or E) and any hydrogen on the ammonium groups (in K) that satisfy a distance cutoff of 3Å and an angle cutoff of 135°. Salt bridges in our simulations had an average length of ~2.8 Å and an average bonding angle of ~156°. Salt bridge interactions were calculated using the LIE approach (Table 1); VDW and ELE interactions were calculated between the atoms on lysine’s ammonium group and the atoms glutamic acid and aspartic acid’s carboxylic acid group. CATCH(6K+/6E-) had less favorable VDW interactions than (6K+/6D-), but more favorable ELE interactions; CATCH(6K+/6E-) and (6K+/6D-) had ELE interactions of -5402 and -4410 kcal/mol, respectively. However, when considering the charged residues as a whole (excluding the backbone atoms), CATCH(6K+/6E-) had more favorable VDW interactions than (6K+/6D-), -125.9 vs. -85.8 kcal/mol. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Summary of hydrogen bonding and LIE analysis for CATCH single bilayer structures. Hydrogen bonds and salt bridges were calculated using geometric criteria: an angle cutoff of 135° and a distance cutoff of 3.0Å. Salt bridges were defined to be between the hydrogens on lysine’s ammonium group and the oxygens on glutamic acid or on aspartic acid’s α-carboxylic acid group. Salt bridge VDW and ELE interactions were calculated between the atoms on the lysine’s ammonium group and the atoms on glutamic acid and aspartic acid’s carboxylic acid group. VDW interactions between charged residues are calculated using the LIE approach and exclude backbone atoms. Values listed are averaged over three independent simulations. https://doi.org/10.1371/journal.pcbi.1011685.t001 The difference in β-sheet twisting for the CATCH(6K+/6E-) and CATCH(6K+/6D-) single bilayers may be explained by how each CATCH pair organizes its β-sheet structure to maximize hydrogen bonding and salt bridge interactions. In CATCH(6K+/6E-), the sidechains of lysine (K) and glutamic acid (E) are of similar length, facilitating a salt bridge interaction between the charged groups on the ends of the charged residues. In CATCH(6K+/6D-), the sidechains of lysine (K) and aspartic acid (D) are of mismatched length. Given that the salt bridge interactions in both CATCH systems have similar geometry in terms of bond length and bond angle, it is likely that CATCH(6K+/6D-) accommodates the mismatched sidechain lengths by hindering its backbone from forming an inherent left-handed twist. The relationship between sidechain length and β-sheet conformation observed in the CATCH system is consistent with a previous quantitative and experimental studies that have shown that β-strands containing glutamic acid have a greater propensity for twisting than those containing aspartic acid [32–34]. To further interrogate the interactions within the single bilayer structure, we calculated the number of contacts between residues within the structure and the overall VDW interactions between the cationic sidechains and the anionic sidechains. Contacts are defined to be any two atoms within 7Å of one another. The greatest number of contacts for both CATCH(6K+/6E-) and (6K+/6D-) was between the phenylalanine residues, 45,149 and 45,163 contacts, respectively (Fig 4). This is expected as the phenylalanine residues make up the hydrophobic core of the bilayer structures. The second greatest number of contacts in each system was between the cationic sidechains and the anionic sidechains. CATCH(6K+/6E-) had 39,552 contacts and a VDW interaction of -125.9 kcal/mol between the lysine and glutamic acid sidechains; (6K+/6D-) had 30,543 contacts and a VDW interaction of -85.8 kcal/mol between the lysine and aspartic acid sidechains. The additional methylene group in glutamic acid (E) in the CATCH(6K+/6E-) pair leads to more contacts and stronger inter-strand interactions between charged residues than in the CATCH(6K+/6D-) pair. The stronger interaction for the K/E sidechain pair compared to the K/D sidechain pair is consistent with pair correlations calculated by Wouters et al. and isothermal titration calorimetry experiments performed by Petrauskas et al. [35,36] The relationship between strong inter-strand interactions and the β-sheet twisting observed in CATCH(6K+/6E-) is consistent with previous experimental studies that suggest that increased interactions between intra-sheet neighboring sidechains is related to increased twisting [26–28]. Evaluation of face-to-face interactions between the two stacked CATCH bilayers Three independent explicit-solvent atomistic MD simulations were carried out for a CATCH(6K+/6E-) mixture and for a CATCH(6K+/6D-) mixture, with each arranged in the two-stacked bilayer configuration. Each simulation started from a pre-formed “ideal” structure (Fig 5A–5D). Intra-sheet β-strands were spaced ~5 Å apart (measuring from the backbone center) to promote backbone hydrogen bonding between neighboring β-strands. The initial bilayers for both CATCH(6K+/6E-) and CATCH(6K+/6D-) were spaced ~3 Å apart, measuring between the end of the sidechains facing inward on the second and third β-sheet. As the simulation progressed, the van der Waals and electrostatic interactions between the charged residues drove the bilayers closer together and stabilized the structure, as expected. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Snapshots of (A-B) CATCH(6K+/6E-) and (C-D) CATCH(6K+/6D-) two stacked bilayers before and after 200 ns of simulation. Distances between the second and third layer of each structure are indicated. https://doi.org/10.1371/journal.pcbi.1011685.g005 Atomistic simulations of the two stacked bilayer systems showed tighter packing between the second and third layer for CATCH(6K+/6D-) than for CATCH(6K+/6E-). The distance between the second and third layer was measured by fitting the backbone atoms of each sheet to a plane and measuring the distance between the centroids of the two fitted planes. The average distance between the second and third β-sheet for CATCH(6K+/6E-) and for CATCH(6K+/6D-) changed over the course of the simulation from an initial distance of 14.8 and 12.7 Å to a final distance of 13.2 and 11.5 Å, respectively. Thus, the CATCH(6K+/6E-) bilayers were further apart than the CATCH(6K+/6E-) bilayers by roughly 2 Å. The longer anionic sidechain (E) in CATCH(6K+/6E-) acts as a physical barrier in the stacking process, while the shorter anionic sidechain (D) in CATCH(6K+/6D-) allows the charged residues to interdigitate in an alternating manner, bringing them closer together. In CATCH(6K+/6D-) the complementary charged residues between the second and third layer create a tight steric zipper that draws the sheets closer together [37,38]. Atomistic MD simulations of the two stacked bilayer CATCH systems show that CATCH(6K+/6D-) bilayers have more favorable face-to-face interactions than CATCH(6K+/6E-) bilayers (Table 2). MMGBSA analysis is used to calculate the van der Waals (VDW) and electrostatic (ELE) interaction energies between the two CATCH bilayers as a result of stacking (Table 2). Both the VDW and ELE interactions are more negative in CATCH(6K+/6D-) than in CATCH(6K+/6E-), suggesting that CATCH(6K+/6D-) has more favorable interactions between the two bilayers and a greater energy incentive for bilayer stacking than CATCH(6K+/6E-). LIE analysis is used to calculate the pairwise VDW interactions between the sidechains of the exposed charged residues (excluding the backbone atoms) on the second and third layer. CATCH(6K+/6D-) experiences a VDW interaction that is 2-fold stronger than for CATCH(6K+/6E-), -49.8 vs -20.7 kcal/mol, in agreement with the MMGBSA results (Table 2). The contrast in interaction energies between the two CATCH systems can be explained by the physical arrangement of the stacked bilayers. The two stacked bilayer simulations have shown that CATCH(6K+/6D-) can stack more closely together than CATCH(6K+/6E-), facilitating the VDW and ELE interactions between the charged residues on the second and third sheets. The single bilayer simulations have shown that the intra-sheet lysine and glutamic acid residues in CATCH(6K+/6E-) have stronger interactions than the intra-sheet lysine and aspartic acid residues in CATCH(6K+/6D-), -125.9 vs. -85.8 kcal/mol, respectively. For CATCH(6K+/6E-) there is not as large of an incentive to interact with sidechains on the opposing bilayer face as for CATCH 6K+/6D-). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Summary of MMGBSA and LIE analysis for CATCH two stacked bilayer structures. MMGBSA values for VDW and ELE energies are calculated by considering the interactions between the top bilayer and the bottom bilayer. LIE values for VDW energies are calculated by considering only the sidechain-sidechain interactions between the charged residues on the top bilayer and the bottom bilayer. Values listed are averaged over three independent simulations. https://doi.org/10.1371/journal.pcbi.1011685.t002 The difference in bilayer structure and face-to-face interactions between CATCH(6K+/6E-) and CATCH(6K+/6D-) provides a possible explanation for the difference in their experimentally-observed nanofiber structures. Cryogenic EM and TEM images show that CATCH(6K+/6E-) forms randomly entangled and tortuous nanofibers with short persistence lengths, whereas CATCH(6K+/6D-) forms aligned bundles of nanofibers with long persistence lengths that tend to appear as multi-layer stacks (Fig 2). The single bilayer atomistic simulations predict that CATCH(6K+/6E-) tends to form more twisted β-sheets than CATCH(6K+/6D-). As the β-sheets stack together, they must either untwist or twist together, both of which incur some energy cost [39,40]. However, for sheets with weak face-to-face attraction, i.e. CATCH(6K+/6E-), twisting comes at a lower cost. Generally, as fibril twisting decreases, the likelihood of β-sheet stacking and fibril thickness growth increases [40]. For CATCH(6K+/6E-), the twisted bilayer structure and weak face-to-face interactions are consistent with the formation of thin randomly entangled nanofibers which are not favored to stack or align. In contrast, for CATCH(6K+/6D-), the combination of a flat bilayer structure and strong face-to-face interactions between charged residues are consistent with the formation of multi-layer fibril bundles. The combination of strong intra-sheet interactions and weak face-to-face interactions leads to a twisted structure for CATCH(6K+/6E-), while the combination of weak intra-sheet interactions and strong face-to-face interactions lead to a flat structure for CATCH(6K+/6D-). This phenomenon has also been observed in atomistic MD simulations of short self-assembling peptides [39]. Quantification of CATCH bilayer stacking Three independent explicit-solvent atomistic MD simulations were carried out for a CATCH(6K+/6E-) mixture and for a CATCH(6K+/6D-) mixture, with each arranged in the two separated bilayer configuration. The two separated bilayer configurations are essentially the aforementioned two stacked bilayer configurations with an additional 10Å of space between the bilayers, for a total spacing of ~13Å between the edges of each bilayer. The additional spacing allowed the bilayers to have more choice in their stacking arrangement. CATCH(6K+/6D-) had a greater number of contacts between charged residues on the 2nd and 3rd layer than CATCH(6K+/6E-), suggesting that CATCH(6K+/6D-) fibrils are more likely to be well-aligned than CATCH(6K+/6E-). CATCH(6K+/6D-) had 25,185 contacts between charged residues and an average twist angle of -2.2° between neighboring strands. CATCH(6K+/6E-) had 17,652 contacts between charged residues and an average twist angle of -2.8° between neighboring strands. Overall, our simulations suggest that the flatter bilayer observed for CATCH(6K+/6D-) leads to well-aligned fibrils. In-silico assessment of CATCH co-assembly pathway and co-assembly kinetics DMD/PRIME20 simulations of CATCH systems were carried out to determine their co-assembly pathway and co-assembly kinetics. Three independent systems of equimolar mixtures of CATCH(6K+/6E-) and of CATCH(6K+/6D-) containing 200 peptides, starting from random-coil configurations, were simulated for 500 billion collisions (~16 μs) at T* = 0.18 (296 K). Simulation snapshots at 0, 8, and 16 μs were taken to examine their assembly pathways (Fig 6A and 6B). At t = 0 μs, the peptides in the CATCH(6K+/6E-) and CATCH(6K+/6D-) systems are in random-coil conformations and are randomly arranged in the box. For CATCH(6K+/6E-), at t = 8 μs, nearly all the peptides have assembled into either a β-sheet structure or an ordered oligomer. At 16 μs, CATCH(6K+/6E-) has fully assembled into β-sheet structures and off-pathway β-barrels. For CATCH(6K+/6D-) at t = 8 μs, we see formation of a single β-sheet, multiple oligomers, and many free peptides still remaining. At t = 16 μs, we observed elongation of the previously-mentioned β-sheet, however there are still free peptides remaining. ThT assays have demonstrated that CATCH(6K/6E) assembles faster than CATCH(6K/6D) [6]. Given the timescale (~16 μs) of our simulations and the difference in assembly kinetics between CATCH(6K+/6E-) and CATCH(6K+/6D-), it is not unexpected to observe some “free” peptides in the CATCH(6K+/6D-) system. For both CATCH systems we observe β-sheet growth through monomer addition or through the interaction of two small-ordered structures. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. DMD Snapshots of (A) CATCH(6K+/6E-) and (B) CATCH(6K+/6D-) over the course of a 16 μs DMD simulation. Cationic peptides containing lysine are represented in teal. Anionic peptides containing aspartic acid are represented in orange, while anionic peptides containing glutamic acid are represented in red. (C) Chronological snapshots of oligomer growth, conformation change, and elongation of a β-barrel in the CATCH(6K+/6D-) simulation. https://doi.org/10.1371/journal.pcbi.1011685.g006 Comparison of the hydrogen bond formation rates between CATCH(6K+/6E-) and CATCH(6K+/6D-) suggests that CATCH(6K+/6E-) fibrillizes at a faster rate than CATCH(6K+/6D-) (Fig 7A). The difference in assembly rates can be attributed to the difference in the anionic residue types. ThT analyses showed that CATCH(6K+/6E-) and CATCH(6K+/6D-) are both capable of assembling into β-sheet structures, however, CATCH(6K+/6E-) assembles at a significantly faster rate than CATCH(6K+/6D-) [6]. In experiments, this is likely related to the greater interaction strength between (K+/E-) pair than the (K+/D-) pair [35,36]. Due to its longer length, the glutamic acid sidechain in CATCH(6K+/6E-) has a greater range of interaction than the aspartic acid sidechain in CATCH(6K+/6D-), increasing the likelihood of finding the complementary lysine residue and forming backbone hydrogen bonds soon afterwards. By analyzing the rates of cluster formation and growth in CATCH(6K+/6E-) and CATCH(6K+/6D-) over time, we observe that CATCH(6K+/6E-) has a higher rate of β-sheet assembly than CATCH(6K+/6D-), and a greater depletion rate of free peptides, in agreement with the hydrogen bond kinetics (Fig 7) and ThT analyses. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. (A) Quantitative assessment of hydrogen bond formation over DMD simulation. (B) Analysis of free peptide depletion (orange), oligomerization (purple), and fibrillization (black). https://doi.org/10.1371/journal.pcbi.1011685.g007 Independent implicit-solvent atomistic REMD simulations of a single peptide were carried out for CATCH(6K+), (6E-), and (6D-) to explore the conformational space of CATCH free peptides in solution. Each peptide started in an extended conformation and was simulated for 200 ns at temperatures ranging from 310 to 600K. The DSSP (Define Secondary Structure of Proteins) algorithm was used to determine the average secondary content of each residue for trajectories at 310K (Table 3). Here, the term helix refers to 3–10, alpha, and pi helices. For convenient comparison, the sum of the averages for each peptide are also provided. CATCH(6K+), (6E-), and (6D-) all had helix, bend, turn, and coil conformations. Helices were found in all CATCH peptides between residues 3 and 9. CATCH(6E-) had the greatest total amount of helical content compared to CATCH(6K+) and (6D-), 6.27 vs 4.42 and 5.24 respectively. Notably, CATCH(6K+) had the most bend conformations compared to CATCH(6E-) and (6D-); CATCH(6K+) had a total bend conformation of 1.25, while CATCH(6E-) and (6D-) had a total bend conformation of 0.43 and 0.62, respectively. CATCH(6K+) and (6D-) had more coil content than CATCH(6E-), 2.39 vs. 1.63, respectively. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Summary of DSSP analysis for each CATCH single peptide REMD simulation. Average secondary content over all frames for each residue are reported. Helix is the sum of the averages for 3–10, alpha, and pi helices. The total value is the sum of the averages over each CATCH single peptide. https://doi.org/10.1371/journal.pcbi.1011685.t003 The difference in discordant helix content between CATCH(K+), (6E-), and (6D-) observed in atomistic REMD simulations provides additional insight into experimental co-assembly kinetics. A discordant helix is defined to be a helical segment in a peptide strand that has a tendency to form β-strands [41,42]. Our predicted conformations for CATCH(6E-) and CATCH(6D-) are in agreement with previously-observed conformations for aspartic acid oligomer and glutamic acid oligomer in simulations performed by Hunkler et. al. Their simulations showed that glutamic acid (E) oligomers tend to form stable α-helical structures, while aspartic acid (D) oligomers are intrinsically disordered [43]. In addition, glutamic acid has been frequently found in protein sequences with a propensity for helical structures [34,44,45]. The amount of α-helix/-beta-strand-discordance in CATCH(6E-) may facilitate β-sheet co-assembly. Kallberg et al. found that multiple amyloidogenic peptides that are predicted to form β-strands contain α-helices, and that fibril formation is lost upon removal or mutation of the α-helix-forming segments [41]. In a study on amyloid β-protein (Aβ), Fezoui and Teplow found that a partially-folded helix-containing conformer is an intermediate in Aβ fibril assembly, and that helix stabilization may facilitate fibril formation [46]. However, they also found that if a helix is sufficiently stabilized, it can resist structural reorganization and inhibit fibril formation [46,47]. The greater proportion of discordant helix observed in CATCH(6E-) than in (6D-) may promote faster β-sheet assembly in CATCH(6K+/6E-) than in (6K+/6D-). More recent studies of Aβ aggregation suggest that a β-hairpin intermediate promotes dimer formation through the intermolecular β-bridges [48–51]. CATCH peptides are too short to properly organize into a β-hairpin. However, the bend conformation observed in CATCH(6K+) may play a similar role to the β-hairpin conformation observed in Aβ by acting as a partially stable intermediate that facilitates peptide-peptide interactions. β-barrels observed in DMD simulations of CATCH peptides DMD/PRIME20 simulations also reveal on-pathway and off-pathway oligomers, including β-barrels in CATCH peptide co-assembly. In both CATCH(6K+/6E-) and CATCH(6K+/6D-) systems, β-barrels and ordered oligomers formed in addition to β-sheet structures. A β-barrel is a β-sheet that wraps around to form a cylindrical structure, with the last and first β-strands connected by backbone hydrogen bonds. β-barrels ranging in size from 6 to 8 peptides (Fig 6A and 6B) formed relatively early in the simulations for both CATCH(6K+/6E-) and CATCH(6K+/6D-). Nearly all the β-barrels that formed remained in a β-barrel structure throughout the course of the simulation—stabilized by hydrogen bonds and hydrophobic interactions between phenylalanine residues. The β-barrels that were not sufficiently stabilized by their intramolecular interactions unraveled to form β-sheets. DMD simulations capture β-barrel-to-β-sheet transitions in both CATCH systems. Fig 6C shows a 6-mer β-barrel intermediate in the CATCH(6K+/6D-) system that eventually seeds the final 42-mer β-sheet structure. The DMD snapshots show the CATCH (6K+/6D-) β-barrel shifting into an elliptical shape, flattening out, and opening at the ends. The β-sheet then grows through a combination of monomer addition and interactions with other small ordered-structures containing β-strands, ultimately forming the final 42-mer β-sheet structure. β-barrel intermediates have also been observed by Sun et. al in atomistic DMD simulations of hIAPP19–29 and its S20G mutant, hIAPP22–28, Aβ16–22, and the α-synuclein NACore. Each of these peptides ultimately self-assembled into cross-β aggregates [52]. β-barrel oligomers have garnered a lot of interest in the field of neurodegenerative diseases as a source of toxicity and a possible therapeutic target [53]. Understanding what factors contribute to the formation of β-barrel oligomers can aid in designing new therapeutics. Our results suggest that the combination of charge complementarity and alternation of hydrophilic and hydrophobic residues used to design CATCH peptides may also promote the formation of β-barrel structures, similar to those found in amyloidogenic peptides and in agreement with conclusions drawn by Shao et. al on previous CATCH simulations [12]. Although CATCH peptides produce on-pathway and off-pathway β-barrels, our research provides insight into designing stable β-barrels with potential applications such as single-molecule sensors or DNA sequencing [54,55]. Conclusion In conclusion, we investigated the charged residue-residue interactions and the assembly pathway for two CATCH(+/-) pairs: CATCH(6K+/6E-) and CATCH(6K+/6D-). Although glutamic acid (E) only differs from aspartic acid (D) by one methylene group, previous studies have shown that this small difference in composition can lead to significant changes in structure at fibril and bulk material scale [45,56]. Our in silico results demonstrate that sidechain type plays a significant role in peptide co-assembly, and in turn, may affect the resulting fibril structure and morphology. The difference in the CATCH(6K+/6E-) and CATCH(6K+/6D-) bilayer structures and face-to-face interactions observed in simulations provides a possible explanation for the difference in their nanofiber structures. In experiments CATCH(6K+/6E-) forms randomly entangled nanofibers, while CATCH(6K+/6D-) forms multi-layer stacks of aligned fibrils. Atomistic molecular dynamics simulations of single bilayers predict that CATCH(6K+/6E-) adopts a twisted structure, while CATCH(6K+/6D-) adopts a relatively flat β-sheet structure. The inherent twist in the CATCH(6K+/6E-) bilayer is due to the chirality of amino acids and strong inter-strand sidechain-sidechain interactions. However, the CATCH(6K+/6D-) bilayer is restricted from twisting to accommodate the backbone hydrogen bonding and salt bridge interactions between neighboring charged groups. Atomistic molecular dynamics simulations of two bilayers stacked on top of one another show that CATCH(6K+/6E-) has weaker face-to-face interactions between the two bilayers than CATCH(6K+/6D-). These results are further corroborated with MD simulations of two separated bilayers, that show that CATCH(6K+/6E-) has fewer number of contacts between the two bilayers than CATCH(6K+/6D-). For CATCH(6K+/6E-), the twisted bilayer structure and weak face-to-face interactions lead to formation of thin randomly entangled nanofibers. While for CATCH(6K+/6D-), the combination of a flat bilayer structure and strong face-to-face interactions between charged residues leads to formation of multi-layer fibril bundles. Discontinuous molecular dynamics simulations with the PRIME20 forcefield reveal the β-sheet co-assembly pathway for CATCH(6K+/6E-) and CATCH(6K+/6D-). DMD/PRIME20 simulation results show that CATCH(6K+/6E-) co-assembles into β-sheet structures at a faster rate than CATCH(6K+/6D-), further substantiating Thioflavin T results showing faster assembly kinetics for CATCH(6K+/6E-) than for CATCH(6K+/6D-) [6]. The discordant helix observed in atomistic single peptide REMD simulations for CATCH(6E-) provides an additional possible explanation for the fast assembly kinetics observed experimentally for CATCH(6K+/6E-). Previous studies of discordant helices in Aβ peptides have shown that the presence of a helical component can lead to faster self-assembly than in the absence of the helical component [46]. However, further experimental and computational investigation on CATCH(6K+/6E-) and (6K+/6D-) dimerization and energy barriers for coil-to-β-sheet transitions are required to better understand CATCH co-assembly kinetics. We acknowledge that while the implicit solvent DMD and REMD simulations allow for longer simulation timescales, implicit solvent simulations by nature neglect the effect of solvent environment and counter-ion environment. The work presented focuses on sidechain-sidechain interactions and bilayer geometry. A future investigation on the competition between a loss in conformational entropy and a gain in counter-ion and solvent release entropy is necessary to determine the thermodynamic pathway for CATCH coassembly. In addition to β-sheet formation, we also observed β-barrel intermediates in both CATCH(6K+/6E-) and CATCH(6K+/6D-) DMD simulations, similar to those found in atomistic MD simulations of amyloidogenic peptides by Sun et. al. [52] In future work, we hope to extend our investigation and explore the free-energy surface of CATCH intermediates to gain a better understand of CATCH coassembly. Understanding how sidechain composition relates to assembly kinetics and ultimately to mechanical properties expands the bioengineer’s toolkit for peptide design. CATCH(+/-) peptides are cationic and anionic variants of Q11[QQKFQFQFEQQ]. The alternating motif of hydrophobic and hydrophilic residues is a common theme in self-assembling peptides and is maintained in the CATCH system. CATCH peptides are similar to intrinsically disordered region (IDR) sequences in that both are typically made up of charged residues [57]. The charged residues hinder two CATCH peptides of the same charge from self-assembling. However, CATCH peptides contain the hydrophobic residues which IDR sequences generally lack; the hydrophobic residues form the hydrophobic core of CATCH bilayers, creating an ordered structure with potential for fibril growth. There is still a daunting number of sequence mutations within the CATCH system to explore—and with each mutation, its effect on fibril structure. We envision that through a combination of computational and experimental collaboration, we can design β-strand structures (whether it be β-barrels or β-sheets) with precise organization, and subsequently, biomaterials with uniform properties. Supporting information S1 Fig. Final snapshots of CATCH(6K+/6E-) and CATCH(6K+/6E-) separated bilayer simulations after 200 ns MD simulation. (A) Top row shows side views for three independent simulations of CATCH(6K+/6E-) separated bilayer simulations. (B) Bottom row shows side views for three independent simulations of CATCH(6K+/6E-) separated bilayer simulations. https://doi.org/10.1371/journal.pcbi.1011685.s001 (TIF)

Fang, Zeming;Bloem, Ilona M.;Olsson, Catherine;Ma, Wei Ji;Winawer, Jonathan

doi: 10.1371/journal.pcbi.1011704pmid: 38150484

1. Introduction Primary visual cortex (“V1”) has served as a testing ground for studying physiology, anatomy, brain development, and neuroimaging. There has been considerable success in developing general model forms that capture many of the encoding properties of V1 neurons reasonably well over a range of stimulus conditions, including the normalized energy model of V1 complex cells [1]. This type of model, like many others [reviewed in chapter 6 of 2,3], includes a linear filter as the first stage, i.e., a weighted sum of the stimulus intensity over space and time. In a second stage, the outputs of the filter are squared and summed across nearby spatial locations or across phase [4–6]. If the outputs are summed across a pair of linear filters tuned to the same frequency, orientation, and location, but differing in phase by 90 deg, it is called an energy model. In the third stage, the response of each neuron is normalized (divisively suppressed) by the second-stage outputs of the nearby neural population [1,7]. This effectively adjusts the gain based on the contrast energy in the image patch. There is substantial evidence that each of these three operations–linear filtering, energy, and normalization–contributes to the responses of V1 neurons [reviewed by 8]. The normalized contrast energy model, though initially developed to explain the outputs of single neurons, has also been successfully applied to functional MRI data in human visual cortex. First, a contrast energy model without normalization, applied to voxels in V1, V2, and V3, was used to predict BOLD responses (encoding) and to infer the viewed images from the BOLD responses (decoding) [9]. Subsequent work showed that incorporating a normalization-like non-linearity improved model accuracy when testing stimuli that varied substantially in size [10] or pattern [11], and that normalization could account for the BOLD contrast response function for gratings with and without masking stimuli at other orientations [12]. These models have also shown good prediction accuracy for similar stimulus sets used in human intracranial electrode recordings of visual cortex [13,14]. The normalized contrast energy model, although successful at accounting for responses to a range of stimuli, nonetheless fails to explain some phenomena. There is some evidence that the standard model fit to artificial stimuli generalizes poorly to natural images [15–17] [but see also 18]. Even testing with simple patterns, early studies of V1 and extrastriate electrophysiology showed that some cells had tuning properties differing from simple or complex cells, called “hypercomplex” cells, many of which were associated with “end-stopping” [19,20]. Recent V1 two-photon calcium recordings included a large stimulus set and found that many cells, with or without end-stopping, were surprisingly sparsely tuned, often sensitive to complex patterns such as crosses or composite features [21]. It is unlikely that the standard energy model would predict the kind of tuning they observed, although they did not fit this model to their data. Other studies with single-unit electrophysiology found that a normalization model could be successful but only if the normalization was flexible, such that its strength depended on statistical dependencies in the image [22]. In closed-loop experiments in which models are used in real time to find the optimal stimulus for mouse V1 cells, the most effective stimuli were often quite complex, differing from simple Gabor patterns [23]. In human fMRI studies, the responses to natural/complex images in V1 also appear to be influenced by statistical dependencies and image context [24,25], factors unlikely to influence the predictions of the normalized energy model. Even in relatively simple artificial images with a fixed amount of total contrast energy, the BOLD response is lower when there is a single orientation compared to when there are two divergent orientations [26,27]. In visual areas beyond V1, the normalized energy model is expected to be incomplete, as circuits in these areas contribute new computations. There are no widely adopted encoding models for these areas analogous to the normalized energy model for V1, but there has been some success in modeling patterns in the extrastriate responses by incorporating higher-order statistical dependencies of the modeled V1 outputs [28–30], higher-order statistical dependencies learned from natural image statistics [31], or sensitivity to second-order contrast [11]. There has also been progress in predicting V4 and IT responses from deep convolutional networks [32–34]. Here, using evidence from human functional MRI, we show that the classical normalized energy model fails to account for the relative responses to two classes of stimuli: straight, parallel, band-passed contours (gratings), and curved, band-passed contours (snakes) (Fig 1). The snakes elicit fMRI responses that are about twice as large as the gratings, yet traditional energy models, including normalized energy models, predict responses that are about the same. This is a large model failure which, in conjunction with the other failures of the simple normalization model described above, motivated us to implement a model in which the normalization is tuned, meaning that the normalization pool for a given neural channel has the same orientation tuning as the channel being normalized. We also developed and implemented a computational model that achieves tuned normalization in a different way, in which responses are normalized not by the sum of the contrast energy, but by the anisotropy (standard deviation) in contrast energy computed across orientation channels. Both models account for the differences in responses to snakes vs gratings, supporting the proposal that normalization depends on the spatial arrangement of image features, and not just the total amount of contrast energy. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Example straight-line and curved-line stimuli from our experiments. We observe that in human V1, V2, and V3, stimuli with long straight lines (gratings) reliably evoke a smaller fMRI response than similar stimuli with curved lines. https://doi.org/10.1371/journal.pcbi.1011704.g001 2. Results 2.1. fMRI BOLD responses to snakes are larger than to gratings We first consider an observation about fMRI responses to two classes of simple, grayscale, band-passed, static images. For one class, the stimuli contain several curved contours, which we refer to throughout as snakes. For the other class, the stimuli contain several straight, parallel contours, which we refer to as gratings. We refer to these classes together as the target stimuli. The surprising observation is that for V1, V2 and V3, the fMRI responses are substantially larger for the snakes than for gratings (Fig 2). The responses to the gratings, irrespective of density, are only about as high as the lowest response to the snakes. We confirm this pattern with three additional fMRI data sets, which also show larger responses to snakes than gratings in V1, V2 and V3 (Figs A1-A4 in S5 Appendix). To check whether this pattern of results is specific to fMRI, we replot published intracranial data from Hermes et al., 2019 [13], in which human subjects with ECoG recordings viewed a similar set of stimuli (Fig 2, bottom panel). The ECoG data, plotted as the percent increase in broadband power over baseline, show the same general effect as the fMRI data: The responses to snakes are much larger than to gratings, irrespective of texture density, indicating that the effect is not limited to the fMRI BOLD response. (See Hermes et al., 2019 [13] for methodological details.) Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. FMRI responses are larger for curved patterns than for straight patterns. The mean fMRI responses within a visual area are plotted for V1, V2, and V3 in Data set 2 of Table A in S1 Appendix. For both curved stimuli (snakes, dark bars) and straight stimuli (gratings, light bars), data are plotted in order of increasing texture density from left to right. Examples of the 5 densities are shown above. Error bars are the standard deviation of the mean, bootstrapped across fMRI runs. Responses are larger for the snakes than gratings. The same effect is also observed for Data sets 1, 3 and 4 of Table A in S1 Appendix (Fig A in S5 Appendix). Bottom panel: For comparison, we replot data from intracranial recordings (ECoG) from Hermes et al 2019 [13], which also show larger responses for snakes than for gratings. The full set of stimuli is shown in S2 Appendix. Data plotted from the function s4_visualize(’figure 2’) in the GitHub code repository. https://doi.org/10.1371/journal.pcbi.1011704.g002 In the next four sections, we describe the four models that are fit to the data. For tractability, we fit each model’s parameters to the aggregate (average) data within a visual area, rather than to each voxel individually. All the stimuli are texture-like, meaning that they have similar properties across the whole image aperture, and for each stimulus class, nine different exemplars were shown per 3-s trial. The different exemplars have the same higher-order statistics but vary in their precise spatial distributions. Hence, model variables, such as contrast energy, would have similar values for spatially localized portions of the image (as one would compute for an individual voxel) as for the whole image (as we compute to model the aggregate response of a visual area). For this reason, we did not include model parameters for the spatial location or spatial extent of the receptive fields for individual voxels. We first describe models fit to the target stimuli. We then summarize fits to the larger set of stimuli viewed by the subjects. All four models consist of three components, filtering, spatial pooling, and a power-law nonlinearity. In the filtering stage, we computed the contrast energy of the stimuli at 8 orientations and 4 spatial frequencies (see Methods for details). Although stimuli were designed to have power concentrated close to 3 cycles per degree, there is some spillover to lower and higher frequencies, which is why we use multiple spatial frequency bands in our models. The spatial pooling stage pools the contrast energy to yield a total contrast energy. In some models, the pooling stage also includes divisive normalization. The output of the pooling is a scalar, which is then passed through a power-law nonlinearity to predict BOLD amplitude in units of percent signal change. The power-law nonlinearity achieves compressive spatial summation [10]. All models have the same filtering (first stage) and output non-linearity (third stage). They differ in the spatial pooling stage. 2.2. The larger response to snakes is not captured by a simple contrast energy model A standard contrast energy model pools the contrast energy by simply summing it across space, orientations, and spatial frequencies to give a total contrast energy. It predicts that responses should increase with both stimulus contrast and with density of the pattern. This model does not predict a larger response to the snakes than to gratings, contrary to the data (Fig 3). In fact, the cross-validated variance explained is low (V1, V2) or even negative (V3) in the example data, meaning that the model prediction is less accurate than it would have been if it simply predicted the mean response across all stimuli. (The data are cross-validated, which is why the variance explained can be negative). In short, the contrast energy model provides a poor fit to the fMRI data in V1-V3 for these classes of stimuli. It is also a poor fit to the target stimuli in the other three data sets (Table A in S3 Appendix and Fig A1 in S5 Appendix). This does not mean that contrast energy models are always poor fits to fMRI responses in V1-V3. For example, when stimuli vary in how contrast energy is distributed across space, a contrast energy model can capture a lot of the variance in the responses across images, as shown by Kay et al., 2013 [9]. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. The contrast energy model does not account for V1-V3 responses to snakes and gratings. (A) Schematic representation of the contrast energy model. For simplicity, we show E summed across spatial frequency bands. (B) Mean fMRI responses in V1, V2, and V3 to snake and grating stimuli that vary in density and contrast (Data set 2). Bars: mean and standard error of the responses. Dark bars represent snake stimuli and light bars represent grating stimuli. Each group of stimuli is arranged in increasing order of either density or contrast. Dots: cross-validated predictions from the contrast energy model. See Fig A1 in S5 Appendix for fits to all 4 data sets. See S4 Appendix for model parameters. Data and model fits plotted using the function s4_visualize(’figure 3’) in the code repository. https://doi.org/10.1371/journal.pcbi.1011704.g003 2.3. The larger response to snakes is not captured by an untuned normalization model We then add divisive normalization to the model. After computing contrast energy, we normalize the outputs by dividing the contrast energy at each pixel by the contrast energy of a normalization pool (Fig 4, upper panel). The normalization pool includes nearby locations, all spatial frequencies, and all orientations, giving it the name untuned normalization model (Fig 4, lower left panel). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Two divisive normalization models. (A) Schematic of divisive normalization models. (B) The weights used to calculate normalization for the untuned (left) and tuned (right) models, plotted using the function s4_visualize(’figure 4’) in the code repository. https://doi.org/10.1371/journal.pcbi.1011704.g004 The untuned normalization model and the contrast energy model make similar predictions and explain a similar proportion of the variance in the data. The normalization model results in more saturation at high contrast, as expected from divisive normalization [7]. This is especially evident in V2 and V3 at the highest stimulus contrast. The reason that the normalization model and the contrast energy model have a similar overall pattern of predictions is that the power law output nonlinearity, included in all models, can partially mimic the effects of normalization [10, section “Relationship to Divisive Normalization”]. Like the contrast energy model, the untuned normalization model predicts a similar BOLD amplitude for snakes and gratings, thereby failing to account for the data (Fig 5, upper panel). The model is not sensitive to heterogeneity across orientations because the normalization pool equally weights all orientation channels. Therefore, the output does not depend on whether the contrast energy is concentrated in one orientation channel, as in the gratings, or spread across many channels, as in the snakes. The untuned normalization model’s failure to account for the greater response to snakes is reflected by low variance explained for the target stimuli in each of the four data sets (Table A in S3 Appendix and Fig A2 in S5 Appendix). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. The tuned normalization model is much more accurate than the untuned normalization model. The mean fMRI responses from V1, V2, and V3 are replotted from Fig 3. The dots are the cross-validated predictions from the untuned normalization model (above) or the tuned normalization model (below). Data and model fits plotted from the function s4_visualize(’figure 5A’) and s4_visualize(’figure 5B’) in the code repository. See Figs A2-A3 in S5 Appendix for fits to all 4 data sets. https://doi.org/10.1371/journal.pcbi.1011704.g005 2.4. The larger response to snakes is captured by an orientation-tuned normalization model The untuned normalization model implements a surround suppression that includes energy at all orientations. Findings from electrophysiology [35–39], psychophysics [38,40–42], neuroimaging [26,27,43–45], and theory [46], however, suggest that surround suppression is orientation-tuned. For example, Cavanaugh, Bair, and Movshon, 2002 [35] reported that the response of a neuron to a stimulus at its preferred orientation in its receptive field is suppressed more when the surrounding region contains contrast at the same orientation compared to different orientations. Because our grating stimuli have contrast energy concentrated at a single orientation, and the snake stimuli do not, one might surmise that an orientation-tuned normalization model would show greater suppression for the gratings, where the RF centers and surrounds will have matched orientations, than for the snakes, where the orientations are more likely to differ between center and surround. If so, this could then account for our observed effect. The untuned normalization model is the same as the tuned model except for one difference: At each pixel in the oriented contrast energy images, the tuned model normalizes the contrast energy across nearby locations only at the preferred orientation (orientation-tuned surround) (Fig 4, lower right panel, diagonals). Within a single location (i.e., at each pixel), the normalization is untuned (off-diagonals in the same panel), also called cross-orientation suppression [47]. The orientation-tuned normalization model captures the large and systematic differences in response amplitude between gratings and snakes (Fig 5, lower panel). In the example data set, the BOLD amplitude and the orientation-tuned normalization model predictions for the gratings are about half of those to the snakes, for both the density and contrast manipulations, and for all three visual areas. In addition to capturing this difference in the means between the two stimulus classes, the model also captures the difference in slope. As the density or contrast increases, the model predicts steeper slopes for snakes than gratings. These patterns in the model fits are found across the four data sets (Fig A3 in S5 Appendix). The orientation-tuned normalization model is more accurate than the previous two models in all cases (Table A in S3 Appendix, four data sets and three ROIs). This result holds up for different size surrounds–what mattered was whether the surround was tuned or not tuned, not its size. The indifference to the size of the surround almost certainly reflects the properties of the stimulus set, not neural tuning: the stimuli are all textures, with similar properties across the image. Had the stimuli varied systematically across location, the size of the surround would likely have had a large effect on model accuracy. Note that the data sets sometimes show a negative slope with increasing density (e.g., V3, gratings varying in density). The model is unable to capture this effect. The study in which these sparse stimuli were first used [11] showed that a second-order contrast model could account for the decreased response with increasing sparsity, as the sparser stimuli have more second-order contrast. It is possible that the orientation-tuned normalization model would also be able to do so if it included spatial receptive fields per voxel that were small relative to the image. 2.5. Normalization by orientation anisotropy The large advantage in prediction accuracy of the tuned over the untuned normalization model supports the idea that suppression is feature-specific. As with any model, we chose a specific instantiation of a more general idea, namely feature-specific suppression. The specific instantiation entailed a minimal change from the untuned normalization model, requiring only a change in normalization weights, and builds on the tradition of feedforward, filter-based models. Feature-specific tuning can also be implemented in other ways, for example based on more abstract ideas like predictability or redundancy in the image. There is some evidence for models like these [22,48]. We implemented a second method of achieving orientation-tuned normalization, in which normalization was proportional to orientation anisotropy (Fig 6, Normalization by orientation anisotropy, “NOA”). Normalization in this model is most pronounced when an image patch has a single orientation, without a specification in terms of a match between center and surround (See 3.3 What is the tuning in orientation-tuned normalization?). Specifically, in the normalization by orientation anisotropy model, the contrast energy is normalized by the standard deviation across the outputs of the orientation channels. This normalization by anisotropy model applies greater normalization when the contrast energy is concentrated in a single orientation channel, resulting in a lower response for gratings. There is no explicit representation of centers, surrounds, or feature matching in the normalization pool. This implementation is consistent with the idea that responses are reduced by the amount of redundancy in the image. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. The normalization by orientation anisotropy model also accounts for the responses in V1-V3. (A) Schematic of the normalization by orientation anisotropy model. (B) The mean fMRI responses from V1, V2, and V3 are replotted from Fig 3. The red dots are the cross-validated predictions from the normalization by orientation anisotropy model. Data and model fits plotted from the function s4_visualize(’figure 6’) in the code repository. See Fig A1 in S5 Appendix for fits to all 4 data sets. https://doi.org/10.1371/journal.pcbi.1011704.g006 The normalization by anisotropy model exhibits similar predictions to the orientation-tuned normalization model, capturing the larger response to the snakes (Fig 6). Both models predict that the responses to snakes are about double the responses to gratings, similar to the data. It also predicts a higher slope for the snakes than the gratings, both as a function of density and contrast. The success of these models validates the idea that normalization depends on how contrast energy is distributed across orientations, not just on the overall contrast energy. 2.6. The two normalization models that are sensitive to orientation accurately predict responses to a wide range of stimuli Both models without orientation sensitivity fail to account for the higher responses to snakes than gratings (Fig 7), and have low variance explained (Fig 8). Models that include orientation sensitivity capture the higher responses to snakes (Fig 7), and fit the data accurately (Fig 8). This suggests that sensitivity to orientation should be incorporated into normalization models of visual cortex. The accuracy differences across models are not due to the number of free parameters: the untuned normalization model has the same number of free parameters (three) as the tuned normalization model and the anisotropy model. Moreover, the prediction accuracy was computed using cross-validation, so that having more parameters does not necessarily lead to better predictions. The tuned normalization model has a numerically higher accuracy than the anisotropy model for nearly all data sets in all conditions (S3 Appendix), but the advantage of the two models with orientation sensitivity dwarfs the slight difference between these two models. Interestingly, for the two untuned models, prediction accuracy declines from V1 to V2 to V3, whereas for the two tuned models, accuracy increases (target stimuli) or stays flat (all stimuli) from V1 to V2/V3 (black lines in Fig 8).This pattern is consistent with the notion that along the visual hierarchy, neural responses become increasingly sensitive to statistical regularities, such as similarity in features across the image (see Discussion section 3.4). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. The orientation dependent models account for the higher responses to snakes than gratings. For each visual area, V1-V3, we averaged the response to snakes and to gratings across stimuli in the target set and across the 4 data sets to compute the ratio of snakes to gratings: mean(snakes) / mean(gratings). We computed this value for each data set and plotted the average and standard error across the four data sets. In the data, the response to snakes is about double to gratings. This is matched in the two normalization models that are sensitive to orientation, but not the other models. Data and model fits plotted from the function s4_visualize(’figure 7’) in the code repository. CE = contrast energy; DN = untuned normalization; OTN = orientation-tuned normalization; NOA = normalization by anisotropy. https://doi.org/10.1371/journal.pcbi.1011704.g007 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 8. Cross-validated variance explained for 4 models across all ROIs and data sets. (A) R2 for the target data set, comprising 18 stimuli for data set 1 (DS1) and data set 2 (DS2) and 17 for data set 3 (DS3), data set (DS4). The number of fitted model parameters (degrees of freedom) is indicated in parentheses for each model type. The four bars in each group correspond to data sets 1–4. The black horizontal lines are the means across the 4 data sets. (B) Same as panel A, but for a larger set of stimuli (50 for data set 1; 48 for data set 2; 39 for data set 3 and 4). The R2 values are also reported in S3 Appendix. Data plotted from the function s4_visualize(’figure 8’) in the code repository. Abbreviations as in Fig 7. https://doi.org/10.1371/journal.pcbi.1011704.g008 Because the two orientation-sensitive model were motivated by the need to explain the greater response to snakes than gratings, it is important to test the models on other stimuli as well. We refit all 4 models to the full data sets, which consisted of 50 (data set 1), 48 (data set 2) and 39 (data set 3, data set 4) stimuli, spanning a variety of texture types. In addition to the snakes and gratings, there are textures we refer to as noise bars, waves, plaids, and circular (S2 Appendix and Table C in S1 Appendix). Just as with the target stimuli, across the full sets of stimuli, the tuned normalization model and the anisotropy model made accurate predictions, explaining 63%-77% and 49%-66% of the cross-validated variance in V1-V3 for the example data set (Fig 9). These two models also provide good fits to the other three data sets, shown in S5 Appendix. The fits to the larger stimulus sets, like the fits to the target stimuli alone, capture the observation about the two stimulus classes, meaning a larger predicted response for snakes than gratings. The two models also accurately predict lower responses to waves (one dominant orientation) than noise bars (many orientations). The two models also predict increasing response amplitudes from gratings (one orientation) to plaids (two orientations) to circular (16 orientations), as evident in stimulus sets 3 and 4 (Figs D3-D4 and E3-E4 in S5 Appendix). This pattern of predictions matches the data. The untuned models do not differ in their predictions for these three stimulus categories. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 9. The orientation dependent models fit a wide variety of stimuli. The mean fMRI responses from V1, V2, and V3 are shown for the full set of stimuli from data set 2 (See S2 Appendix). The data for snakes and gratings are replotted from Figs 3, 5 and 6, and again shown in dark and light gray. An even lighter gray is used for all other stimulus classes. The red and yellow dots are the cross-validated predictions from the orientation-tuned normalization model and the normalization by orientation anisotropy model. See Figs B-E in S5 Appendix for similar plots for data sets 2–4. Data and model fits plotted from the function s4_visualize(‘figure 9‘) in the code repository. https://doi.org/10.1371/journal.pcbi.1011704.g009 Across ROIs and data sets, the orientation-tuned normalization model accounts for the highest variance, with R2 ranging from 55% to 77% (Fig 8, bottom; S3 Appendix, the third row). The anisotropy model ranks second in all cases, substantially outperforming the two baseline models. Similar to the pattern with the target stimuli, when fitting to all stimuli the two untuned models show substantially decreasing accuracy from V1 to V2 to V3. The tuned normalization model and the anisotropy model decrease only slightly in accuracy from V1 to V2 to V3, meaning that the advantage for the tuned models is largest in extrastriate areas. 2.1. fMRI BOLD responses to snakes are larger than to gratings We first consider an observation about fMRI responses to two classes of simple, grayscale, band-passed, static images. For one class, the stimuli contain several curved contours, which we refer to throughout as snakes. For the other class, the stimuli contain several straight, parallel contours, which we refer to as gratings. We refer to these classes together as the target stimuli. The surprising observation is that for V1, V2 and V3, the fMRI responses are substantially larger for the snakes than for gratings (Fig 2). The responses to the gratings, irrespective of density, are only about as high as the lowest response to the snakes. We confirm this pattern with three additional fMRI data sets, which also show larger responses to snakes than gratings in V1, V2 and V3 (Figs A1-A4 in S5 Appendix). To check whether this pattern of results is specific to fMRI, we replot published intracranial data from Hermes et al., 2019 [13], in which human subjects with ECoG recordings viewed a similar set of stimuli (Fig 2, bottom panel). The ECoG data, plotted as the percent increase in broadband power over baseline, show the same general effect as the fMRI data: The responses to snakes are much larger than to gratings, irrespective of texture density, indicating that the effect is not limited to the fMRI BOLD response. (See Hermes et al., 2019 [13] for methodological details.) Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. FMRI responses are larger for curved patterns than for straight patterns. The mean fMRI responses within a visual area are plotted for V1, V2, and V3 in Data set 2 of Table A in S1 Appendix. For both curved stimuli (snakes, dark bars) and straight stimuli (gratings, light bars), data are plotted in order of increasing texture density from left to right. Examples of the 5 densities are shown above. Error bars are the standard deviation of the mean, bootstrapped across fMRI runs. Responses are larger for the snakes than gratings. The same effect is also observed for Data sets 1, 3 and 4 of Table A in S1 Appendix (Fig A in S5 Appendix). Bottom panel: For comparison, we replot data from intracranial recordings (ECoG) from Hermes et al 2019 [13], which also show larger responses for snakes than for gratings. The full set of stimuli is shown in S2 Appendix. Data plotted from the function s4_visualize(’figure 2’) in the GitHub code repository. https://doi.org/10.1371/journal.pcbi.1011704.g002 In the next four sections, we describe the four models that are fit to the data. For tractability, we fit each model’s parameters to the aggregate (average) data within a visual area, rather than to each voxel individually. All the stimuli are texture-like, meaning that they have similar properties across the whole image aperture, and for each stimulus class, nine different exemplars were shown per 3-s trial. The different exemplars have the same higher-order statistics but vary in their precise spatial distributions. Hence, model variables, such as contrast energy, would have similar values for spatially localized portions of the image (as one would compute for an individual voxel) as for the whole image (as we compute to model the aggregate response of a visual area). For this reason, we did not include model parameters for the spatial location or spatial extent of the receptive fields for individual voxels. We first describe models fit to the target stimuli. We then summarize fits to the larger set of stimuli viewed by the subjects. All four models consist of three components, filtering, spatial pooling, and a power-law nonlinearity. In the filtering stage, we computed the contrast energy of the stimuli at 8 orientations and 4 spatial frequencies (see Methods for details). Although stimuli were designed to have power concentrated close to 3 cycles per degree, there is some spillover to lower and higher frequencies, which is why we use multiple spatial frequency bands in our models. The spatial pooling stage pools the contrast energy to yield a total contrast energy. In some models, the pooling stage also includes divisive normalization. The output of the pooling is a scalar, which is then passed through a power-law nonlinearity to predict BOLD amplitude in units of percent signal change. The power-law nonlinearity achieves compressive spatial summation [10]. All models have the same filtering (first stage) and output non-linearity (third stage). They differ in the spatial pooling stage. 2.2. The larger response to snakes is not captured by a simple contrast energy model A standard contrast energy model pools the contrast energy by simply summing it across space, orientations, and spatial frequencies to give a total contrast energy. It predicts that responses should increase with both stimulus contrast and with density of the pattern. This model does not predict a larger response to the snakes than to gratings, contrary to the data (Fig 3). In fact, the cross-validated variance explained is low (V1, V2) or even negative (V3) in the example data, meaning that the model prediction is less accurate than it would have been if it simply predicted the mean response across all stimuli. (The data are cross-validated, which is why the variance explained can be negative). In short, the contrast energy model provides a poor fit to the fMRI data in V1-V3 for these classes of stimuli. It is also a poor fit to the target stimuli in the other three data sets (Table A in S3 Appendix and Fig A1 in S5 Appendix). This does not mean that contrast energy models are always poor fits to fMRI responses in V1-V3. For example, when stimuli vary in how contrast energy is distributed across space, a contrast energy model can capture a lot of the variance in the responses across images, as shown by Kay et al., 2013 [9]. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. The contrast energy model does not account for V1-V3 responses to snakes and gratings. (A) Schematic representation of the contrast energy model. For simplicity, we show E summed across spatial frequency bands. (B) Mean fMRI responses in V1, V2, and V3 to snake and grating stimuli that vary in density and contrast (Data set 2). Bars: mean and standard error of the responses. Dark bars represent snake stimuli and light bars represent grating stimuli. Each group of stimuli is arranged in increasing order of either density or contrast. Dots: cross-validated predictions from the contrast energy model. See Fig A1 in S5 Appendix for fits to all 4 data sets. See S4 Appendix for model parameters. Data and model fits plotted using the function s4_visualize(’figure 3’) in the code repository. https://doi.org/10.1371/journal.pcbi.1011704.g003 2.3. The larger response to snakes is not captured by an untuned normalization model We then add divisive normalization to the model. After computing contrast energy, we normalize the outputs by dividing the contrast energy at each pixel by the contrast energy of a normalization pool (Fig 4, upper panel). The normalization pool includes nearby locations, all spatial frequencies, and all orientations, giving it the name untuned normalization model (Fig 4, lower left panel). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Two divisive normalization models. (A) Schematic of divisive normalization models. (B) The weights used to calculate normalization for the untuned (left) and tuned (right) models, plotted using the function s4_visualize(’figure 4’) in the code repository. https://doi.org/10.1371/journal.pcbi.1011704.g004 The untuned normalization model and the contrast energy model make similar predictions and explain a similar proportion of the variance in the data. The normalization model results in more saturation at high contrast, as expected from divisive normalization [7]. This is especially evident in V2 and V3 at the highest stimulus contrast. The reason that the normalization model and the contrast energy model have a similar overall pattern of predictions is that the power law output nonlinearity, included in all models, can partially mimic the effects of normalization [10, section “Relationship to Divisive Normalization”]. Like the contrast energy model, the untuned normalization model predicts a similar BOLD amplitude for snakes and gratings, thereby failing to account for the data (Fig 5, upper panel). The model is not sensitive to heterogeneity across orientations because the normalization pool equally weights all orientation channels. Therefore, the output does not depend on whether the contrast energy is concentrated in one orientation channel, as in the gratings, or spread across many channels, as in the snakes. The untuned normalization model’s failure to account for the greater response to snakes is reflected by low variance explained for the target stimuli in each of the four data sets (Table A in S3 Appendix and Fig A2 in S5 Appendix). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. The tuned normalization model is much more accurate than the untuned normalization model. The mean fMRI responses from V1, V2, and V3 are replotted from Fig 3. The dots are the cross-validated predictions from the untuned normalization model (above) or the tuned normalization model (below). Data and model fits plotted from the function s4_visualize(’figure 5A’) and s4_visualize(’figure 5B’) in the code repository. See Figs A2-A3 in S5 Appendix for fits to all 4 data sets. https://doi.org/10.1371/journal.pcbi.1011704.g005 2.4. The larger response to snakes is captured by an orientation-tuned normalization model The untuned normalization model implements a surround suppression that includes energy at all orientations. Findings from electrophysiology [35–39], psychophysics [38,40–42], neuroimaging [26,27,43–45], and theory [46], however, suggest that surround suppression is orientation-tuned. For example, Cavanaugh, Bair, and Movshon, 2002 [35] reported that the response of a neuron to a stimulus at its preferred orientation in its receptive field is suppressed more when the surrounding region contains contrast at the same orientation compared to different orientations. Because our grating stimuli have contrast energy concentrated at a single orientation, and the snake stimuli do not, one might surmise that an orientation-tuned normalization model would show greater suppression for the gratings, where the RF centers and surrounds will have matched orientations, than for the snakes, where the orientations are more likely to differ between center and surround. If so, this could then account for our observed effect. The untuned normalization model is the same as the tuned model except for one difference: At each pixel in the oriented contrast energy images, the tuned model normalizes the contrast energy across nearby locations only at the preferred orientation (orientation-tuned surround) (Fig 4, lower right panel, diagonals). Within a single location (i.e., at each pixel), the normalization is untuned (off-diagonals in the same panel), also called cross-orientation suppression [47]. The orientation-tuned normalization model captures the large and systematic differences in response amplitude between gratings and snakes (Fig 5, lower panel). In the example data set, the BOLD amplitude and the orientation-tuned normalization model predictions for the gratings are about half of those to the snakes, for both the density and contrast manipulations, and for all three visual areas. In addition to capturing this difference in the means between the two stimulus classes, the model also captures the difference in slope. As the density or contrast increases, the model predicts steeper slopes for snakes than gratings. These patterns in the model fits are found across the four data sets (Fig A3 in S5 Appendix). The orientation-tuned normalization model is more accurate than the previous two models in all cases (Table A in S3 Appendix, four data sets and three ROIs). This result holds up for different size surrounds–what mattered was whether the surround was tuned or not tuned, not its size. The indifference to the size of the surround almost certainly reflects the properties of the stimulus set, not neural tuning: the stimuli are all textures, with similar properties across the image. Had the stimuli varied systematically across location, the size of the surround would likely have had a large effect on model accuracy. Note that the data sets sometimes show a negative slope with increasing density (e.g., V3, gratings varying in density). The model is unable to capture this effect. The study in which these sparse stimuli were first used [11] showed that a second-order contrast model could account for the decreased response with increasing sparsity, as the sparser stimuli have more second-order contrast. It is possible that the orientation-tuned normalization model would also be able to do so if it included spatial receptive fields per voxel that were small relative to the image. 2.5. Normalization by orientation anisotropy The large advantage in prediction accuracy of the tuned over the untuned normalization model supports the idea that suppression is feature-specific. As with any model, we chose a specific instantiation of a more general idea, namely feature-specific suppression. The specific instantiation entailed a minimal change from the untuned normalization model, requiring only a change in normalization weights, and builds on the tradition of feedforward, filter-based models. Feature-specific tuning can also be implemented in other ways, for example based on more abstract ideas like predictability or redundancy in the image. There is some evidence for models like these [22,48]. We implemented a second method of achieving orientation-tuned normalization, in which normalization was proportional to orientation anisotropy (Fig 6, Normalization by orientation anisotropy, “NOA”). Normalization in this model is most pronounced when an image patch has a single orientation, without a specification in terms of a match between center and surround (See 3.3 What is the tuning in orientation-tuned normalization?). Specifically, in the normalization by orientation anisotropy model, the contrast energy is normalized by the standard deviation across the outputs of the orientation channels. This normalization by anisotropy model applies greater normalization when the contrast energy is concentrated in a single orientation channel, resulting in a lower response for gratings. There is no explicit representation of centers, surrounds, or feature matching in the normalization pool. This implementation is consistent with the idea that responses are reduced by the amount of redundancy in the image. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. The normalization by orientation anisotropy model also accounts for the responses in V1-V3. (A) Schematic of the normalization by orientation anisotropy model. (B) The mean fMRI responses from V1, V2, and V3 are replotted from Fig 3. The red dots are the cross-validated predictions from the normalization by orientation anisotropy model. Data and model fits plotted from the function s4_visualize(’figure 6’) in the code repository. See Fig A1 in S5 Appendix for fits to all 4 data sets. https://doi.org/10.1371/journal.pcbi.1011704.g006 The normalization by anisotropy model exhibits similar predictions to the orientation-tuned normalization model, capturing the larger response to the snakes (Fig 6). Both models predict that the responses to snakes are about double the responses to gratings, similar to the data. It also predicts a higher slope for the snakes than the gratings, both as a function of density and contrast. The success of these models validates the idea that normalization depends on how contrast energy is distributed across orientations, not just on the overall contrast energy. 2.6. The two normalization models that are sensitive to orientation accurately predict responses to a wide range of stimuli Both models without orientation sensitivity fail to account for the higher responses to snakes than gratings (Fig 7), and have low variance explained (Fig 8). Models that include orientation sensitivity capture the higher responses to snakes (Fig 7), and fit the data accurately (Fig 8). This suggests that sensitivity to orientation should be incorporated into normalization models of visual cortex. The accuracy differences across models are not due to the number of free parameters: the untuned normalization model has the same number of free parameters (three) as the tuned normalization model and the anisotropy model. Moreover, the prediction accuracy was computed using cross-validation, so that having more parameters does not necessarily lead to better predictions. The tuned normalization model has a numerically higher accuracy than the anisotropy model for nearly all data sets in all conditions (S3 Appendix), but the advantage of the two models with orientation sensitivity dwarfs the slight difference between these two models. Interestingly, for the two untuned models, prediction accuracy declines from V1 to V2 to V3, whereas for the two tuned models, accuracy increases (target stimuli) or stays flat (all stimuli) from V1 to V2/V3 (black lines in Fig 8).This pattern is consistent with the notion that along the visual hierarchy, neural responses become increasingly sensitive to statistical regularities, such as similarity in features across the image (see Discussion section 3.4). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. The orientation dependent models account for the higher responses to snakes than gratings. For each visual area, V1-V3, we averaged the response to snakes and to gratings across stimuli in the target set and across the 4 data sets to compute the ratio of snakes to gratings: mean(snakes) / mean(gratings). We computed this value for each data set and plotted the average and standard error across the four data sets. In the data, the response to snakes is about double to gratings. This is matched in the two normalization models that are sensitive to orientation, but not the other models. Data and model fits plotted from the function s4_visualize(’figure 7’) in the code repository. CE = contrast energy; DN = untuned normalization; OTN = orientation-tuned normalization; NOA = normalization by anisotropy. https://doi.org/10.1371/journal.pcbi.1011704.g007 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 8. Cross-validated variance explained for 4 models across all ROIs and data sets. (A) R2 for the target data set, comprising 18 stimuli for data set 1 (DS1) and data set 2 (DS2) and 17 for data set 3 (DS3), data set (DS4). The number of fitted model parameters (degrees of freedom) is indicated in parentheses for each model type. The four bars in each group correspond to data sets 1–4. The black horizontal lines are the means across the 4 data sets. (B) Same as panel A, but for a larger set of stimuli (50 for data set 1; 48 for data set 2; 39 for data set 3 and 4). The R2 values are also reported in S3 Appendix. Data plotted from the function s4_visualize(’figure 8’) in the code repository. Abbreviations as in Fig 7. https://doi.org/10.1371/journal.pcbi.1011704.g008 Because the two orientation-sensitive model were motivated by the need to explain the greater response to snakes than gratings, it is important to test the models on other stimuli as well. We refit all 4 models to the full data sets, which consisted of 50 (data set 1), 48 (data set 2) and 39 (data set 3, data set 4) stimuli, spanning a variety of texture types. In addition to the snakes and gratings, there are textures we refer to as noise bars, waves, plaids, and circular (S2 Appendix and Table C in S1 Appendix). Just as with the target stimuli, across the full sets of stimuli, the tuned normalization model and the anisotropy model made accurate predictions, explaining 63%-77% and 49%-66% of the cross-validated variance in V1-V3 for the example data set (Fig 9). These two models also provide good fits to the other three data sets, shown in S5 Appendix. The fits to the larger stimulus sets, like the fits to the target stimuli alone, capture the observation about the two stimulus classes, meaning a larger predicted response for snakes than gratings. The two models also accurately predict lower responses to waves (one dominant orientation) than noise bars (many orientations). The two models also predict increasing response amplitudes from gratings (one orientation) to plaids (two orientations) to circular (16 orientations), as evident in stimulus sets 3 and 4 (Figs D3-D4 and E3-E4 in S5 Appendix). This pattern of predictions matches the data. The untuned models do not differ in their predictions for these three stimulus categories. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 9. The orientation dependent models fit a wide variety of stimuli. The mean fMRI responses from V1, V2, and V3 are shown for the full set of stimuli from data set 2 (See S2 Appendix). The data for snakes and gratings are replotted from Figs 3, 5 and 6, and again shown in dark and light gray. An even lighter gray is used for all other stimulus classes. The red and yellow dots are the cross-validated predictions from the orientation-tuned normalization model and the normalization by orientation anisotropy model. See Figs B-E in S5 Appendix for similar plots for data sets 2–4. Data and model fits plotted from the function s4_visualize(‘figure 9‘) in the code repository. https://doi.org/10.1371/journal.pcbi.1011704.g009 Across ROIs and data sets, the orientation-tuned normalization model accounts for the highest variance, with R2 ranging from 55% to 77% (Fig 8, bottom; S3 Appendix, the third row). The anisotropy model ranks second in all cases, substantially outperforming the two baseline models. Similar to the pattern with the target stimuli, when fitting to all stimuli the two untuned models show substantially decreasing accuracy from V1 to V2 to V3. The tuned normalization model and the anisotropy model decrease only slightly in accuracy from V1 to V2 to V3, meaning that the advantage for the tuned models is largest in extrastriate areas. 3. Discussion 3.1 Why do some models fail to account for the much larger response to snakes? We began with the observation that the BOLD response is about twice as large for patterns with curved contours (snakes) as for similar stimuli with straight, parallel contours (gratings). The difference in the responses is not a peculiarity of the BOLD signal, as a similar pattern was also observed in human intracranial measures of the field potential (broadband power from ECoG electrodes). The contrast energy model without normalization and the contrast energy model with untuned normalization did not predict large differences between the responses to the two stimulus classes. Both models pool contrast energy over space and orientation without an orientation-specific normalization or other orientation specific non-linearity. If two images have the same total contrast energy, then the way the energy is distributed across orientation channels will not matter, either for the measure of total energy (contrast energy model) or for the amount of normalization (untuned normalization model). The implementation of the orientation-tuned normalization model was motivated in part from electrophysiology results showing that in V1, surround suppressive fields tend to be tuned to orientations close to the RF center’s preferred orientation [35]. Psychophysical experiments also show interactions between a target and surround that depend on matched orientation [e.g., 42]. While there is a lot of evidence from electrophysiology, fMRI, and psychophysics in support of feature tuning in normalization, here we explicitly compared image-computable models with and without feature-tuned normalization. Implementing the model and fitting it to data enabled us to assess whether (1) it quantitatively accounts for two-fold difference in response to snakes vs gratings (2) whether it also provides good fits to a wide range of other stimuli that it was not explicitly implemented for and (3) whether it can capture other more subtle effects, like the difference in the slope of the contrast response function between snakes and gratings. In all three cases, the answer was yes. We also implemented a second model with tuned normalization (the normalization by orientation anisotropy model), and this model also provided an affirmative answer to these questions, confirming the importance of including orientation dependence in the normalization computation. We discuss similarities (3.2) and differences (3.3) between the two models below. 3.2 Model behavior: Models with orientation dependent normalization capture the differences in both mean and in slope between snakes and gratings The two models with orientation dependent normalization predict larger outputs, on average, for snakes than gratings, even when the contrast energy of the stimuli is approximately matched. This is due to how the normalization is computed. The grating stimuli elicit large outputs in the orientation channels that are matched to the stimulus, moderate outputs in adjacent orientation channels, and little to no response in other channels. As a result, there is high anisotropy (standard deviation across channel outputs), resulting in more suppression in the normalization by orientation anisotropy model. There is also high suppression from the surround in the orientation-tuned normalization model. The two models were implemented to capture the difference in mean between the two stimulus classes, so perhaps it is not so surprising that they do so. The two models, unlike the contrast energy model and the untuned normalization model, also accurately predict steeper slopes for snakes than gratings (with respect to both contrast and density), a pattern that the models were not explicitly motivated to capture. They predict the difference in slope because at low total contrast energy for the image, there is little normalization, and hence the response to a snake and a grating stimulus will be comparable. This explanation applies to both low contrast stimuli and low-density stimuli, because in both cases the summed contrast energy is low, meaning the normalization term in the denominator is small. Hence, little normalization at low contrast is expected from the model, and is also confirmed by empirical measures of spatial summation and surround suppression at different contrast levels [49,50]. The more nuanced prediction from these models is that for stimuli with high total contrast energy (high stimulus contrast and high density), there is a lot of normalization for gratings and much less for snakes, resulting in a more pronounced difference in predicted response. The difference in predicted responses at high contrast but not at low contrast causes a difference in slope. For the two untuned models, normalization increases with contrast energy, but it increases similarly for the snakes and gratings, hence predicting similar slopes. Interestingly, the data and the two models with orientation dependent normalization also show a greater slope for “noise bar” stimuli than “waves” (Figs B3-B4 and C3-C4 in S5 Appendix). The noise bars, like the snakes, have many orientations in first order contrast (but unlike the snakes, have only one dominant orientation for second-order contrast). The waves are the complement, with one dominant orientation for first order contrast (like gratings) but many orientations for second-order contrast. The pattern in the data and model predictions is that the stimuli with many orientations (noise bars) increase more steeply as a function of contrast than the stimuli with a narrower range of orientations (waves), supporting the observations made with snakes and gratings, but adding the further nuance that what seems to matter most is the orientation distribution (wide vs narrow) for first order rather than second-order contrast. The difference in slopes between snakes and gratings in the data is related to, but not identical to, results observed for single unit V1 cells. For a typical V1 cell, the contrast response function has a higher slope for preferred than for non-preferred stimuli [51–53], a pattern also predicted by normalization models [7]. The pattern is predicted because of the difference in the numerator, which is high for preferred stimuli and low for non-preferred. The denominator is about the same for the two stimuli, proportional to the total contrast energy in the stimulus. We attribute the difference in slope between snakes vs gratings to a difference in the denominator of the normalization equation, which is high for gratings, low for snakes. The numerator is about the same for the two classes, proportional to the total contrast energy. In both cases–single unit responses to non-preferred stimuli, and population responses to grating stimuli–the reduced slope is due to a large amount of normalization relative to the driven response. 3.3 What is the tuning in orientation-tuned normalization? We implemented two models with orientation dependent normalization. Both capture the same tendency for more normalization for homogeneous stimuli like gratings. They differ in implementation, however. The tuned normalization model differs only slightly from more typical normalization models: the only difference is that the normalization weights happen to conform to a specific pattern, such that cells with similar feature tuning (here, orientation) with nearby receptive fields have high weights, and cells with different feature tuning have low weights. As a result, surround suppression is most effective according to this model when the orientation of a surrounding region is matched to the preferred orientation of a cell. This is justified by findings from single unit data in macaque V1 that “the surround influence was always suppressive when the surround grating was at the neuron’s preferred orientation” [35]. The normalization by orientation anisotropy model is a larger departure from the standard normalization model, since the computation of anisotropy is not a simple weighting of the outputs of nearby cells. Unlike the tuned normalization model, it has the greatest suppressive effect when the features of a stimulus are anisotropic (like a grating) irrespective of whether the stimulus orientation matches the center tuning of a cell. Interestingly, there is also empirical support for this pattern from the same study by Cavanaugh et al., 2002 [35]: specifically, evidence for “the tuning of the surround being dependent to some degree on the stimulus used in the center—suppression was often stronger for a given center stimulus when the parameters of the surround grating matched the parameters of the center grating even when the center grating was not itself of the optimal direction or orientation.” This pattern has been found in multiple studies, indicating that for macaque and cat V1 cells, surround suppression is often maximal when the surround and center stimulus orientation match, independent of the orientation preference of the cell [37,54,55]. Put another way, the tuning of the surround changes as the center orientation or direction changes. As shown by simulation, our normalization by anisotropy model can capture this observation from single units, but our tuned normalization model cannot (Fig 10). In this regard, the Normalization by orientation anisotropy model is more like Coen-Cagli et al’s [22] model of single-cell data than is the Orientation-tuned surround model. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 10. Center surround interactions in single units and computational models. (A) Data replotted from three example cells from macaque V1 [35]. The thin lines are tuning curves for drifting gratings in the cell’s RF center, with 0 deg indicating the preferred direction. The thick lines and open circles show responses to stimuli including a center and surround. The center direction is indicated by the large black dot, and the surround direction is indicated by the values in the x-axis. The important observation is that the greatest suppression (indicated by the asterisks) occurs when the surround direction matches the center direction, not when the surround direction matches the center’s preferred direction. (B) Simulations for the orientation-tuned normalization model for stimuli similar to those in panel A (but static rather than drifting). Because the surround suppression is matched to the center’s preferred tuning, the largest suppression occurs at 0 deg, differing from the single unit data. (C) Simulations for the normalization by orientation anisotropy model for the same stimuli as in panel B. Here, the greatest suppression is when the surround orientation matches the center orientation, similar to the single unit data. Data and simulations plotted from the function s4_visualize(‘figure 10‘) in the code repository. https://doi.org/10.1371/journal.pcbi.1011704.g010 The two models we implemented leave us in an unusual position. We have one model that has slightly higher prediction accuracy and is more in line with standard normalization models (orientation-tuned normalization), and a second model with slightly lower (but still high) prediction accuracy, but a closer fit to some data from single units (normalization by orientation anisotropy). The simplest conclusion is that a contrast energy model incorporating some form of orientation-dependent normalization greatly outperforms models that do not have tuned normalization. Given the diversity of cells in visual cortex, it is not likely that a single, simplified model will be sufficient to capture the behavior of all cells or cell populations for all stimuli. A neural network model with recurrence, temporal dynamics, and with feedforward, feedback and lateral connections might provide insight into the specific ways in which the surround modulation emerges [54,56], particularly given evidence that normalization changes as the response to a stimuli unfolds [57–60]. Interestingly, the denominator in the normalization by anisotropy model (the normalizer) is almost the same as the numerator of an orientation variance model used to predict the amplitude of gamma oscillations measured with ECoG electrodes in human subjects [13]. The fact that the same term is used in the numerator to predict gamma oscillations and the denominator to predict BOLD is consistent with the empirical observation that many of the stimuli that are most effective for driving large gamma oscillations (high contrast luminance gratings) are relatively ineffective for eliciting BOLD signals [61,62], and multiunit action potentials [63,64]. The link may be that gamma oscillations, rather than being the fundamental mechanism for perception and long-range cortical communication [65,66], are rather a result of the normalization process [13,67]. An open question is whether gamma oscillations might be predicted as accurately, or perhaps more accurately, by the normalization pool used in our orientation-tuned normalization model than the normalization pool of the normalization by anisotropy model. 3.4 Orientation dependent normalization and image statistics The normalization by orientation anisotropy model and the orientation-tuned model showed larger advantages over untuned models for V2 and V3 compared to V1. This pattern is consistent with findings from electrophysiology and computational modeling of behavior. First, evidence from the non-human primate visual system suggests that the orientation tuning of surround suppression in V1 arises in large part from feedback from extrastriate areas, especially for the more spatially distant effects of surround suppression [55,68,69]. If the tuned suppressive effects depend on computations in extrastriate regions, then these regions may also exhibit more tuned suppression than V1. More generally, tuned suppression, either in the form of our tuned normalization model or the anisotropy model, reflects sensitivity to higher-order image statistics (correlations over space between filter outputs). Models of extrastriate neural responses, especially V2, also tend to be more sensitive to higher-order image statistics than to contrast energy per se, such as models based explicitly on texture statistics [28–30,70,71], or models with multiple subunits that may exhibit heterogenous feature tuning [72–74]. We consider one specific way that our normalization by anisotropy model might be linked to the V2 models proposed by Simoncelli and colleagues. Suppose the V1 population computes contrast energy localized in orientation and space. (For simplicity, we ignore spatial frequency, as we did here experimentally by band-pass filtering our stimuli.) We then suppose that the V2 cells compute various weighted sums of the V1 outputs. Specifically, we assume that, for each spatial location, the weights among V2 cells form a Fourier basis set on the V1 outputs (across orientation). If these weights are arranged in pairs (similar to the odd and even V1 filters), and the outputs are squared and summed across phase (similar to the V1 energy model), then then the summed V2 population output will be proportional to the variance in the V1 response across orientation (David Heeger, personal communication). If this population output is the normalization pool for V2, then we get a normalization term like that in the anisotropy model, which normalizes by the standard deviation across orientation channels. A V2 model based on these principles would also need a similar term in the numerator, which we do not include. Hence the link is only to the denominator of the model. 3.5 Are typical contrast energy V1 models missing something important? No model is complete. The standard normalized contrast energy model of V1, when fit with appropriate parameters, can capture substantial variance in V1 responses [75], but not all. Our results indicate that the model fails for at least some relatively simple stimulus classes, and that the failure can be large. But whether a more complex model is needed, such as a gated normalization model [22] or either of the orientation dependent normalization models we implemented, will depend on the stimulus set tested and the purpose of modeling. It is reasonably likely that the computations in both the anisotropy and the tuned normalization models are more connected to computations in extrastriate visual maps than in V1, but may also influence the response in V1 via feedback. [55,76]. More generally, since the development of divisive normalization models in the early 1990s, there has not yet been a generally agreed-upon description of exactly which cell populations contribute to normalization, and with what weights. Some attempts have been made, by assuming efficient coding of natural images [46,77] or by fitting model parameters to neural data [78]. And while there is a large literature on divisive normalization, including its tuning, many issues remain unresolved, including whether normalization within receptive field centers is orientation-tuned (in addition to the extra-classical surround being orientation-tuned) [78]. In this sense, a standard model of V1, with parameters set, that is downloadable and executable on arbitrary input images, does not yet really exist. Both the anisotropy model and the tuned normalization model we presented make some advance but also have some important limits, most notably the lack of spatial receptive fields, as well as a lack of sensitivity to second-order contrast. Hence, they do not supersede other models, but rather provide compact computational summaries of response patterns that are not well captured by other models. Integrating better computational tools for validation [79], model-based stimulus development [80,81], high-quality standardized data sets [82], and theory [83], may offer the best path toward more complete understanding of neural circuits in visual cortex. 3.1 Why do some models fail to account for the much larger response to snakes? We began with the observation that the BOLD response is about twice as large for patterns with curved contours (snakes) as for similar stimuli with straight, parallel contours (gratings). The difference in the responses is not a peculiarity of the BOLD signal, as a similar pattern was also observed in human intracranial measures of the field potential (broadband power from ECoG electrodes). The contrast energy model without normalization and the contrast energy model with untuned normalization did not predict large differences between the responses to the two stimulus classes. Both models pool contrast energy over space and orientation without an orientation-specific normalization or other orientation specific non-linearity. If two images have the same total contrast energy, then the way the energy is distributed across orientation channels will not matter, either for the measure of total energy (contrast energy model) or for the amount of normalization (untuned normalization model). The implementation of the orientation-tuned normalization model was motivated in part from electrophysiology results showing that in V1, surround suppressive fields tend to be tuned to orientations close to the RF center’s preferred orientation [35]. Psychophysical experiments also show interactions between a target and surround that depend on matched orientation [e.g., 42]. While there is a lot of evidence from electrophysiology, fMRI, and psychophysics in support of feature tuning in normalization, here we explicitly compared image-computable models with and without feature-tuned normalization. Implementing the model and fitting it to data enabled us to assess whether (1) it quantitatively accounts for two-fold difference in response to snakes vs gratings (2) whether it also provides good fits to a wide range of other stimuli that it was not explicitly implemented for and (3) whether it can capture other more subtle effects, like the difference in the slope of the contrast response function between snakes and gratings. In all three cases, the answer was yes. We also implemented a second model with tuned normalization (the normalization by orientation anisotropy model), and this model also provided an affirmative answer to these questions, confirming the importance of including orientation dependence in the normalization computation. We discuss similarities (3.2) and differences (3.3) between the two models below. 3.2 Model behavior: Models with orientation dependent normalization capture the differences in both mean and in slope between snakes and gratings The two models with orientation dependent normalization predict larger outputs, on average, for snakes than gratings, even when the contrast energy of the stimuli is approximately matched. This is due to how the normalization is computed. The grating stimuli elicit large outputs in the orientation channels that are matched to the stimulus, moderate outputs in adjacent orientation channels, and little to no response in other channels. As a result, there is high anisotropy (standard deviation across channel outputs), resulting in more suppression in the normalization by orientation anisotropy model. There is also high suppression from the surround in the orientation-tuned normalization model. The two models were implemented to capture the difference in mean between the two stimulus classes, so perhaps it is not so surprising that they do so. The two models, unlike the contrast energy model and the untuned normalization model, also accurately predict steeper slopes for snakes than gratings (with respect to both contrast and density), a pattern that the models were not explicitly motivated to capture. They predict the difference in slope because at low total contrast energy for the image, there is little normalization, and hence the response to a snake and a grating stimulus will be comparable. This explanation applies to both low contrast stimuli and low-density stimuli, because in both cases the summed contrast energy is low, meaning the normalization term in the denominator is small. Hence, little normalization at low contrast is expected from the model, and is also confirmed by empirical measures of spatial summation and surround suppression at different contrast levels [49,50]. The more nuanced prediction from these models is that for stimuli with high total contrast energy (high stimulus contrast and high density), there is a lot of normalization for gratings and much less for snakes, resulting in a more pronounced difference in predicted response. The difference in predicted responses at high contrast but not at low contrast causes a difference in slope. For the two untuned models, normalization increases with contrast energy, but it increases similarly for the snakes and gratings, hence predicting similar slopes. Interestingly, the data and the two models with orientation dependent normalization also show a greater slope for “noise bar” stimuli than “waves” (Figs B3-B4 and C3-C4 in S5 Appendix). The noise bars, like the snakes, have many orientations in first order contrast (but unlike the snakes, have only one dominant orientation for second-order contrast). The waves are the complement, with one dominant orientation for first order contrast (like gratings) but many orientations for second-order contrast. The pattern in the data and model predictions is that the stimuli with many orientations (noise bars) increase more steeply as a function of contrast than the stimuli with a narrower range of orientations (waves), supporting the observations made with snakes and gratings, but adding the further nuance that what seems to matter most is the orientation distribution (wide vs narrow) for first order rather than second-order contrast. The difference in slopes between snakes and gratings in the data is related to, but not identical to, results observed for single unit V1 cells. For a typical V1 cell, the contrast response function has a higher slope for preferred than for non-preferred stimuli [51–53], a pattern also predicted by normalization models [7]. The pattern is predicted because of the difference in the numerator, which is high for preferred stimuli and low for non-preferred. The denominator is about the same for the two stimuli, proportional to the total contrast energy in the stimulus. We attribute the difference in slope between snakes vs gratings to a difference in the denominator of the normalization equation, which is high for gratings, low for snakes. The numerator is about the same for the two classes, proportional to the total contrast energy. In both cases–single unit responses to non-preferred stimuli, and population responses to grating stimuli–the reduced slope is due to a large amount of normalization relative to the driven response. 3.3 What is the tuning in orientation-tuned normalization? We implemented two models with orientation dependent normalization. Both capture the same tendency for more normalization for homogeneous stimuli like gratings. They differ in implementation, however. The tuned normalization model differs only slightly from more typical normalization models: the only difference is that the normalization weights happen to conform to a specific pattern, such that cells with similar feature tuning (here, orientation) with nearby receptive fields have high weights, and cells with different feature tuning have low weights. As a result, surround suppression is most effective according to this model when the orientation of a surrounding region is matched to the preferred orientation of a cell. This is justified by findings from single unit data in macaque V1 that “the surround influence was always suppressive when the surround grating was at the neuron’s preferred orientation” [35]. The normalization by orientation anisotropy model is a larger departure from the standard normalization model, since the computation of anisotropy is not a simple weighting of the outputs of nearby cells. Unlike the tuned normalization model, it has the greatest suppressive effect when the features of a stimulus are anisotropic (like a grating) irrespective of whether the stimulus orientation matches the center tuning of a cell. Interestingly, there is also empirical support for this pattern from the same study by Cavanaugh et al., 2002 [35]: specifically, evidence for “the tuning of the surround being dependent to some degree on the stimulus used in the center—suppression was often stronger for a given center stimulus when the parameters of the surround grating matched the parameters of the center grating even when the center grating was not itself of the optimal direction or orientation.” This pattern has been found in multiple studies, indicating that for macaque and cat V1 cells, surround suppression is often maximal when the surround and center stimulus orientation match, independent of the orientation preference of the cell [37,54,55]. Put another way, the tuning of the surround changes as the center orientation or direction changes. As shown by simulation, our normalization by anisotropy model can capture this observation from single units, but our tuned normalization model cannot (Fig 10). In this regard, the Normalization by orientation anisotropy model is more like Coen-Cagli et al’s [22] model of single-cell data than is the Orientation-tuned surround model. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 10. Center surround interactions in single units and computational models. (A) Data replotted from three example cells from macaque V1 [35]. The thin lines are tuning curves for drifting gratings in the cell’s RF center, with 0 deg indicating the preferred direction. The thick lines and open circles show responses to stimuli including a center and surround. The center direction is indicated by the large black dot, and the surround direction is indicated by the values in the x-axis. The important observation is that the greatest suppression (indicated by the asterisks) occurs when the surround direction matches the center direction, not when the surround direction matches the center’s preferred direction. (B) Simulations for the orientation-tuned normalization model for stimuli similar to those in panel A (but static rather than drifting). Because the surround suppression is matched to the center’s preferred tuning, the largest suppression occurs at 0 deg, differing from the single unit data. (C) Simulations for the normalization by orientation anisotropy model for the same stimuli as in panel B. Here, the greatest suppression is when the surround orientation matches the center orientation, similar to the single unit data. Data and simulations plotted from the function s4_visualize(‘figure 10‘) in the code repository. https://doi.org/10.1371/journal.pcbi.1011704.g010 The two models we implemented leave us in an unusual position. We have one model that has slightly higher prediction accuracy and is more in line with standard normalization models (orientation-tuned normalization), and a second model with slightly lower (but still high) prediction accuracy, but a closer fit to some data from single units (normalization by orientation anisotropy). The simplest conclusion is that a contrast energy model incorporating some form of orientation-dependent normalization greatly outperforms models that do not have tuned normalization. Given the diversity of cells in visual cortex, it is not likely that a single, simplified model will be sufficient to capture the behavior of all cells or cell populations for all stimuli. A neural network model with recurrence, temporal dynamics, and with feedforward, feedback and lateral connections might provide insight into the specific ways in which the surround modulation emerges [54,56], particularly given evidence that normalization changes as the response to a stimuli unfolds [57–60]. Interestingly, the denominator in the normalization by anisotropy model (the normalizer) is almost the same as the numerator of an orientation variance model used to predict the amplitude of gamma oscillations measured with ECoG electrodes in human subjects [13]. The fact that the same term is used in the numerator to predict gamma oscillations and the denominator to predict BOLD is consistent with the empirical observation that many of the stimuli that are most effective for driving large gamma oscillations (high contrast luminance gratings) are relatively ineffective for eliciting BOLD signals [61,62], and multiunit action potentials [63,64]. The link may be that gamma oscillations, rather than being the fundamental mechanism for perception and long-range cortical communication [65,66], are rather a result of the normalization process [13,67]. An open question is whether gamma oscillations might be predicted as accurately, or perhaps more accurately, by the normalization pool used in our orientation-tuned normalization model than the normalization pool of the normalization by anisotropy model. 3.4 Orientation dependent normalization and image statistics The normalization by orientation anisotropy model and the orientation-tuned model showed larger advantages over untuned models for V2 and V3 compared to V1. This pattern is consistent with findings from electrophysiology and computational modeling of behavior. First, evidence from the non-human primate visual system suggests that the orientation tuning of surround suppression in V1 arises in large part from feedback from extrastriate areas, especially for the more spatially distant effects of surround suppression [55,68,69]. If the tuned suppressive effects depend on computations in extrastriate regions, then these regions may also exhibit more tuned suppression than V1. More generally, tuned suppression, either in the form of our tuned normalization model or the anisotropy model, reflects sensitivity to higher-order image statistics (correlations over space between filter outputs). Models of extrastriate neural responses, especially V2, also tend to be more sensitive to higher-order image statistics than to contrast energy per se, such as models based explicitly on texture statistics [28–30,70,71], or models with multiple subunits that may exhibit heterogenous feature tuning [72–74]. We consider one specific way that our normalization by anisotropy model might be linked to the V2 models proposed by Simoncelli and colleagues. Suppose the V1 population computes contrast energy localized in orientation and space. (For simplicity, we ignore spatial frequency, as we did here experimentally by band-pass filtering our stimuli.) We then suppose that the V2 cells compute various weighted sums of the V1 outputs. Specifically, we assume that, for each spatial location, the weights among V2 cells form a Fourier basis set on the V1 outputs (across orientation). If these weights are arranged in pairs (similar to the odd and even V1 filters), and the outputs are squared and summed across phase (similar to the V1 energy model), then then the summed V2 population output will be proportional to the variance in the V1 response across orientation (David Heeger, personal communication). If this population output is the normalization pool for V2, then we get a normalization term like that in the anisotropy model, which normalizes by the standard deviation across orientation channels. A V2 model based on these principles would also need a similar term in the numerator, which we do not include. Hence the link is only to the denominator of the model. 3.5 Are typical contrast energy V1 models missing something important? No model is complete. The standard normalized contrast energy model of V1, when fit with appropriate parameters, can capture substantial variance in V1 responses [75], but not all. Our results indicate that the model fails for at least some relatively simple stimulus classes, and that the failure can be large. But whether a more complex model is needed, such as a gated normalization model [22] or either of the orientation dependent normalization models we implemented, will depend on the stimulus set tested and the purpose of modeling. It is reasonably likely that the computations in both the anisotropy and the tuned normalization models are more connected to computations in extrastriate visual maps than in V1, but may also influence the response in V1 via feedback. [55,76]. More generally, since the development of divisive normalization models in the early 1990s, there has not yet been a generally agreed-upon description of exactly which cell populations contribute to normalization, and with what weights. Some attempts have been made, by assuming efficient coding of natural images [46,77] or by fitting model parameters to neural data [78]. And while there is a large literature on divisive normalization, including its tuning, many issues remain unresolved, including whether normalization within receptive field centers is orientation-tuned (in addition to the extra-classical surround being orientation-tuned) [78]. In this sense, a standard model of V1, with parameters set, that is downloadable and executable on arbitrary input images, does not yet really exist. Both the anisotropy model and the tuned normalization model we presented make some advance but also have some important limits, most notably the lack of spatial receptive fields, as well as a lack of sensitivity to second-order contrast. Hence, they do not supersede other models, but rather provide compact computational summaries of response patterns that are not well captured by other models. Integrating better computational tools for validation [79], model-based stimulus development [80,81], high-quality standardized data sets [82], and theory [83], may offer the best path toward more complete understanding of neural circuits in visual cortex. 4. Methods 4.1 Ethics statement Participants provided written informed consent. The experimental protocol was in compliance with the safety guidelines for MRI research and was approved by the University Committee on Activities involving Human Subjects at New York University (IRB-FY2016-363). We analyzed and modeled four fMRI data sets for this paper. Data sets 1 and 2 were collected at NYU. Data sets 3 and 4 are re-analyzed from a previous paper [11], for which the fMRI data and stimuli are freely available online (http://kendrickkay.net/socmodel/). 4.2 Participants The two NYU participants were both experienced MRI subjects (female; 24, 29 yo). Data were collected at NYU’s Center for Brain Imaging. The experimental protocol was approved by the University Committee on Activities Involving Human Subjects, and informed written consent was obtained from the participants before the study. Both participants had corrected-to-normal vision. The subjects participated in two separate scanning sessions, one for retinotopic mapping and one for the main study on encoding of textures. 4.3 Stimuli Publicly accessible links to the stimuli, the names of the stimulus classes, and the correspondence between our naming convention and those in the Kay et al., 2013 [11] paper are described in Tables A, B, and C in S1 Appendix. The total spectral power of all images can be visualized using function visualizeStimulusPowerSpectrum in the code repository and is shown for target stimuli from Dataset 2 of Fig A in S1 Appendix. 4.3.1 Stimuli for data sets 3 and 4. Data sets 3 and 4 correspond to subject 1 and subject 2, respectively, in Kay et al., 2013 [11]. The stimuli for the two subjects were the same and are referred to as “Stimulus set 2” on the website (http://kendrickkay.net/socmodel/). The publicly available stimulus set includes 156 stimuli, 39 of which were used for this paper (Tables B and C in S1 Appendix). The reason we use only a subset of the stimuli is that we modeled how visual areas respond to textures ignoring retinotopic preference, and many of the stimuli used in the original paper varied systematically over space in order to map spatial receptive fields. We used only the large-field textures, i.e., the subset of stimuli whose patterns were similar across the whole circular aperture. The 39 stimuli are organized into 7 groups, each of which we describe with two terms, one term for the type of texture and one term for the way in which the stimuli within the group vary. For example, GRATINGS (contrast) are stimuli which come from the grating family and vary from low to high contrast. GRATINGS (density) stimuli come from the same family but have uniform contrast and vary in the spacing between the contours. The correspondence between how we refer to the stimuli and how Kay et al., 2013 [11] referred to them is in Table C in S1 Appendix. Below we describe the general stimulus characteristics and the 7 specific classes used for this paper. Most of the text is duplicated from Kay et al., 2013 [11] (p. 11), indicated by italics. General stimulus characteristics. Stimuli were constructed at a resolution of 256 pixels × 256 pixels and were upsampled to 800 pixels × 800 pixels for display purposes. All stimuli were presented within a circular aperture filling the height of the display; the rest of the display was filled with neutral gray. The outer 0.5 deg of the circular aperture was smoothly blended into the background using a half-cosine function. Stimuli consisted of grayscale images restricted to a band-pass range of spatial frequencies centered at 3 cycles per degree. To enforce this restriction, a custom band-pass filter was used in the generation of some of the stimuli. The filter was a zero-mean isotropic 2D Difference-of-Gaussians filter whose amplitude spectrum peaks at 3 cycles per degree and drops to half-maximum at 1.4 and 4.7 cycles per degree. Restricting the spatial frequency content of the stimuli avoids the complications of building multiscale models and helps constrain the scope of the modeling endeavor. Even with the spatial frequency restriction, it is possible to construct a rich diversity of stimuli including objects and other naturalistic stimuli…. Each stimulus consisted of nine distinct images that were presented in quick succession. The purpose of this design was to take advantage of the slow dynamics of the BOLD response and average over stimulus dimensions of no interest (e.g., using sinusoidal gratings differing in phase to average over phase). A key motivating observation for this paper is that the response to gratings was lower than to curved stimuli. Four groups of stimuli, two groups of gratings and two groups of snakes, were studied first and are referred to throughout the paper as target stimuli. These are described first. Target stimuli. SNAKES (contrast, 10 stimuli). Kay et al. refers to these stimuli as noise patterns: Noise patterns were created by low-pass filtering white noise at a cutoff frequency of 0.5 cycles per degree, thresholding the result, performing edge detection using derivative filters, inverting image polarity such that edges are black, and applying the custom band-pass filter (described previously). We generated nine distinct noise patterns and scaled the contrast of the patterns to fill the full luminance range. [The contrast stimuli were then] constructed by varying the contrast of the noise patterns… . Ten different contrast levels were used: 1%, 2%, 3%, 4%, 6%, 9%, 14%, 21%, 32%, and 50%. These contrast levels are relative to the contrast of the patterns used in SPACE [not used in this study], which is taken to be 100%. SNAKES (density, 5 stimuli). These stimuli used the same type of noise patterns as SPACE [not used here] but varied the amount of separation between contours. We generated noise patterns using cutoff frequencies of 2.8, 1.6, 0.9, 0.5, and 0.3 cycles per degree, and numbered these from 1 (smallest separation) to 5 (largest separation). The noise patterns used in SPACE correspond to separation 4; thus, we only constructed stimuli for the remaining separations 1, 2, 3, and 5. The noise patterns occupied the full stimulus extent (no aperture masking) GRATINGS (contrast, 4 stimuli). These stimuli consisted of horizontal sinusoidal gratings at 2%, 4%, 9%, and 20% Michelson contrast. The spatial frequency of the gratings was fixed at 3 cycles per degree. GRATINGS (density, 5 stimuli). The highest density stimulus in this group is similar to the horizontally oriented stimulus in GRATINGS (orientation), i.e., similar to a horizontal high-contrast grating, but it is not precisely a sinusoidal grating. It is made by convolving equally spaced horizontal lines with the custom band-pass filter (described previously). When the gratings are spaced appropriately (⅓ deg spacing) and filtered by a band-pass filter centered at 3 cycles per deg, the result is close to a sinusoidal grating at 3 cycles per deg. When the spacing is larger, there are several parallel band-pass contours with uniform gray between them. The spacing between parallel lines for the 5 stimuli varied in powers of 2, as 1/3 deg × 1,2,4,8, or 16, from densest to sparsest. Because the grating and snakes stimuli were both constructed by convolving lines with the same band-pass filter, they have some similar properties. They differ in that the lines here were straight whereas the lines used for constructing the snakes stimuli were curved. Additional stimuli. GRATINGS (orientation, 8 stimuli). These stimuli consisted of full-contrast sinusoidal gratings at eight different orientations. The spatial frequency of the gratings was fixed at 3 cycles per degree. Each [of the 9 exemplars per stimulus] consisted of gratings with the same orientation but nine different phases (equally spaced from 0 to 2π). PLAID (contrast, 4 stimuli). These stimuli consisted of plaids at 2%, 4%, 9%, and 20% contrast (defined below). Each condition comprised nine plaids, and each plaid was constructed as the sum of a horizontal and a vertical sinusoidal grating (spatial frequency 3 cycles per degree, random phase). The plaids were scaled in contrast to match the root-mean-square (RMS) contrast of the GRATING stimuli. For example, the plaids in the 9% condition were scaled such that the average RMS contrast of the plaids is identical to the average RMS contrast of the gratings in the 9% GRATING stimulus. CIRCULAR (contrast, 4 stimuli). These stimuli were identical to the PLAID stimuli except that sixteen different orientations were used instead of two. 4.3.2 Stimuli for Data set 1. Data set 1 was collected at NYU. The data set was designed to replicate some of the effects observed from data sets 3 and 4 (the greater response to snakes than gratings), but also to extend the measurements to new stimulus classes. The general stimulus characteristics were the same as those used in data sets 3 and 4. However, because the display size differed, the image resolution in pixels also differed (400 × 400 here, vs 800 × 800 above), and there were slight differences in the bandpass filter. The stimulus size in degrees of visual angle was the same (12.5 deg diameter). A total of 50 stimuli were tested. (The numbers below total more than 50 because some stimuli belong to more than one group, as indicated in Table C in S1 Appendix). Target stimuli. GRATINGS (contrast, 5 stimuli). These stimuli are horizontal gratings (but not quite sinusoids), with a similar spatial pattern to the middle stimulus in the GRATINGS (density) stimuli from data sets 3 and 4. They were made by convolving horizontal lines spaced every 1.75 deg with a custom band-pass filter. The images were scaled to yield 5 different contrasts of 3%, 10%, 25%, 50% and 100%. GRATINGS (density; 5 stimuli). These stimuli are similar to horizontal gratings, made by convolving equally spaced horizontal lines with the custom band-pass filter. The 5 stimuli differed in the spacing of the horizontal lines, spaced every 3, 2.5, 1.75, 1, 0.33 deg. The contrast of all stimuli was 25%. The middle stimulus in this sequence was the same as the middle stimulus in the contrast sequence (spacing 1.75 deg, 25% contrast). The highest density (0.33 deg spacing) is close to a sinusoidal grating, as the spacing is the inverse of the peak spatial frequency of the band-pass filter (3 cycles per degree). SNAKES (contrast, 5 stimuli). The spatial pattern is similar to the snakes stimuli in data set 3 and data set 4. Contrasts matched the grating contrasts (3%, 10%, 25%, 50% and 100%). SNAKES (density, 5 stimuli). The spatial pattern is similar to the snakes stimuli in data set 3 and data set 4, but with 5 different densities of the contours. The contrast for all stimuli was 25% (lower than the contrast of the corresponding stimuli in data set 3 and data set 4). The range of densities used here was also lower than the range used in data set 3 and data set 4, with the densest pattern here similar to the middle stimuli in data set 3 and data set 4. Additional stimuli. GRATINGS (orientation, 4 stimuli). These stimuli are the same as the third stimulus in the GRATINGS (density) group (25% contrast, 1.75 deg spacing between contours), except that they are rotated by 0, 45, 90, or 135 deg. Because the 0 deg rotation does not change the image, a new stimulus was not created; for visualization of results, the BOLD measurements and model predictions are plotted for both groups. GRATINGS (cross, 4 stimuli). These stimuli contain horizontal contours similar to two of the stimuli in the GRATINGS (density) sequence, except that they have periodic vertical blank regions which interrupt the contours. For two of the stimuli, the spacing of the horizontal contours matches the densest stimuli in the density sequence (spacing of 0.33 deg) and for two of the stimuli, the spacing matched the middle stimulus in the density sequence (1.75 deg spacing). In all 4 images, the horizontal contours are interrupted by vertical blanks spaced every 1.75 deg. The vertical blanks are either thick (50% duty cycle; 1st and 3rd stimulus) or thin (25% duty cycle; 2nd and 4th stimuli). NOISE BARS (density, 5 stimuli). These stimuli have the same contrast apertures as the GRATINGS (density) stimuli. Specifically, there are horizontal bands containing contrast patterns, spaced the same as the grating stimuli (bands every 3, 2.5, 1.75, 1, or 0.33 deg). These stimuli differ from the gratings in that each band contains band-pass filtered noise, equal in power across orientations, rather than horizontal contours. NOISE BARS (contrast, 5 stimuli). These stimuli are matched in spatial pattern to the middle density of the NOISE BARS (density) stimuli (horizontal lines, spacing 1.75 deg), but scaled in contrast similar to the grating stimuli (3%, 10%, 25%, 50%, 100%). NOISE BARS (orientation, 4 stimuli). The orientation sequence rotated the middle stimulus of the NOISE BARS (density) group (spacing 1.75 deg, contrast 25%) by 0, 45, 90, or 135 deg. WAVES (density, 6 stimuli). These are identical to the snakes (density) stimuli, except that they have been filtered by orientation, such that they only contain power at or near the horizontal. WAVES (contrast, 5 stimuli). These are identical to the snakes (contrast) stimuli, except that they have been filtered by orientation, such that they only contain power at or near the horizontal. WAVES (orientation, 4 stimuli). These are identical to the densest stimulus in the snakes (density) group, except that they have been filtered by orientation, with filter centered at either 0, 45, 90, or 135 deg. 4.3.3 Stimuli for data set 2. Data set 2 was collected at NYU. The stimuli were nearly identical to those in data set 1, differing only in the following ways. First, the stimuli were 50% larger (18.75 × 18.75 deg and 600 × 600 pixels, rather than 12.5 × 12.5 deg and 400 × 400 pixels). The difference in size did not entail a difference in spatial frequency: The spatial frequency was matched between the two data sets (meaning that the stimuli were re-made with a larger aperture rather than by re-scaling). Second, those stimuli which were oriented were oriented vertically rather than horizontally. This applies to all GRATING stimuli, as well as NOISE BARS and WAVES. Third, the WAVES (density) stimuli had only 4 densities rather than 6. We reduced the number of stimuli to slightly shorten the MRI scans. 4.4 MRI The methods for MRI acquisition and preprocessing for data sets 3 and 4 are described in Kay et al., 2013 [11]. In brief, each data set comes from one subject, who viewed a variety of stimuli in an event-related fMRI design. Data set 3 was collected over two scan sessions and each stimulus was presented 6 times. Data set 4 was collected over one scan session and each stimulus was presented 3 times. (Note that both in this paper and the Kay website (http://kendrickkay.net/socmodel/), these two data sets are referred to as data sets 3 and 4. However, the Kay website refers to the stimuli for these two data sets as stimulus set 2 and the subjects themselves as “subject B” and “subject C”. We do not adopt these latter two conventions.) After preprocessing the data (slice-time correction, co-registration, spatial unwarping), a general linear model was applied using the GLMdenoise toolbox [11]. The output of this algorithm includes a coefficient (beta weight) for each stimulus for each voxel solved from the whole fMRI session, as well as 30 bootstrapped estimates of each beta weight (bootstrapping across fMRI runs). The publicly available data (http://kendrickkay.net/socmodel/) are already pre-processed, denoised, and organized by ROI. Specifically, the data we used are in the files called “data set03.mat” and “data set04.mat” (http://kendrickkay.net/socmodel/data set03.mat, http://kendrickkay.net/socmodel/data set04.mat). The data sets are described on the website as “Data set 3 (subject B)” and “Data set 4 (subject C),” respectively. Within the MATLAB files, we used the stored 3D array called “betas” (voxels × stimuli × bootstraps), limited to V1, V2, V3 as indicated in the grouping variables “roi” and “roilabels”, and limited to the 39 stimuli indicated in Table B in S1 Appendix. Visual areas were identified by retinotopic mapping in a separate session. 4.4.1 Acquisition of data sets 1 and 2. Data sets 1 and 2 were acquired in one scanning session each. Each scanning session had 12 fMRI runs of 249 s each (data set 1) or 241.5 s each (data set 2). For each data set, half of the stimuli were assigned to odd fMRI runs and half to even runs, so that each stimulus was shown 6 times in the session. The stimulus events were 3 s long, consisting of 9 alternations between stimulus exemplar and blank, ⅙ s each. Trial onsets were every 7.5 s (so 4.5 s blank between trials). To help estimate the hemodynamic response function, there were 12 s of blank at the beginning and end of each run, as well as 5 additional trials randomly interspersed with no stimulus (meaning that 5 times during the scan, trials were separated by 15 s instead of 7.5 s). Thus, each complete run consisted of either (25 stimuli + 5 blanks) * 7.5 s + 24 s = 249 s (data set 1) or (24 stimuli + 5 blanks) * 7.5 s + 24 s = 241.5 s (data set 2). All MRI data were acquired at New York University Center for Brain Imaging using a Siemens Allegra 3T head-only scanner with a Nova Medical phased array, 8-channel receive surface coil (NMSC072). For each participant, we collected functional images (single shot echo planar images, 1500 ms TR, 30 ms TE, and 72° flip angle). Voxels were 2.0 mm3 isotropic, with 24 slices, with an inplane sampling of 104 × 80 voxels (208 mm A/P × 160 mm L/R). The slice prescription covered most of the occipital lobe, and the posterior part of both the temporal and parietal lobes. Images were corrected for B0 field inhomogeneity using a calibration scan and Center for Brain Imaging algorithms during offline image reconstruction. We also acquired 1 or 2 T1-weighted whole-brain anatomical scans (MPRAGE sequence; 1mm3), as well as a T1-weighted “inplane” image with the same slice prescription as the functional scans. This scan had an inplane resolution of 1.25 × 1.25 mm and a slice thickness of 2.5 mm, and was collected to aid alignment of the functional images to the high-resolution T1 weighted anatomical images. In a separate session, retinotopy scans were collected and analyzed using a pRF model as implemented in the Vistasoft software tool (https://github.com/vistalab/vistasoft). The methods for acquisition and analysis of the retinotopy data are identical to that described by Zhou et al. 2018 [84]. 4.4.2 Data preprocessing and analysis. Data preprocessing. Processing of the fMRI data was identical to that described by Zhou et al. 2018 [84]: We coregistered and segmented the T1 weighted whole-brain anatomical images into gray and white matter voxels using FreeSurfer’s autosegmentation algorithm (http://surfer.nmr.mgh.harvard.edu). Using custom software Vistasoft (https://github.com/vistalab/vistasoft), the functional data were slice-time corrected by resampling the time series in each slice to the center of each 1.5 s volume. Data were then motion-corrected by coregistering all volumes of all scans to the first volume of the first scan. The first 8 volumes (12 s) of each scan were discarded for analysis to allow longitudinal magnetization and stabilized hemodynamic response. GLM. The preprocessed fMRI data were then fit by a general linear model, GLMDenoise [11]. This algorithm denoises the data by projecting out nuisance regressors derived in a data-driven manner, and estimates coefficients for each of the 48 or 50 stimuli for each voxel in the functional images. The algorithm bootstraps the data over fMRI runs. For data sets 1 and 2, we generated 50 bootstraps for data set 1 and 100 bootstraps for data set 2. The publicly available data from Kay et al, 2013 [11], included 30 bootstraps per subject. The algorithm also estimated a hemodynamic impulse response function as a finite impulse response function, with 35 time points (52.5 s) per subject. ROIs. Regions of interest for V1, V2, V3 were delineated manually using the Vistasoft (https://github.com/vistalab/vistasoft) graphical user interface to visualize the results of the pRF models. These methods for identifying these boundaries are well established, as described in many publications [85,86, summarized by 87]. The ROIs for V1, V2 and V3 were identified on the cortical surface and then projected to the functional images. For purposes of data summary and model fitting, we took the average signal from each ROI. We did this by averaging the beta weight across voxels within an ROI separately for each stimulus, after voxel selection (Table A in S1 Appendix). Because noise can be correlated across voxels, but should not be correlated across scans, when we bootstrapped the data, we average across voxels within an ROI for each bootstrap. For the purposes of model fitting, each of the 4 data sets comprised two matrices, one for the means and one for the standard deviation across bootstraps, each of which had a size equal to the number of stimuli by number of ROIs. 4.5 Model equations In the Results, we compared the accuracy of four models fit to the data, three of which are based on existing models or empirical findings–a contrast energy model, a untuned normalization model, and an orientation-tuned normalization model–and one new model, which computes normalization by orientation anisotropy. In this section we describe the computation that comprises each model. All four models consist of three primary steps: (1) computation of oriented contrast energy, (2) pooling across orientation and space, and (3) a power-law nonlinearity. Steps 1 and 3 are identical for all models. Step 2, spatial pooling, varies between models. 1. Contrast energy: We denote by I(x,y) the value of the pre-processed input image at coordinates (x,y). The pre-processing causes the image values to have mean 0 and range from -0.5 to 0.5. The image is projected onto a set of 128 Gabor filters, which comprise 8 orientations θ, spaced every 22.5 deg; 8 spatial frequencies f, with peak spatial frequency log spaced from 0.75 cpd to 6 cpd; and 2 phases ϕ, separated by 90° (i.e., “quadrature”). F(x,y,θ,f,ϕ) indicates the Gabor filter at a spatial location (x,y), orientation θ, spatial frequency f, and phase ϕ. Each filter comprised a cosine or sine function of 4 cycles, windowed by a Gaussian with SD of 1 cycle. The outputs over the two phases are squared and summed to compute the contrast energy and summed across spatial frequencies. Finally, the contrast energy as a function of spatial position (x,y) and orientation θ becomes (1) We convolve the image I and the filter F. The computation of contrast energy has no free parameters. Prior to convolution, stimuli were padded with uniform gray (mean luminance) on all sides by the width of the largest filter. After convolution, all energy images were downsampled to 12 pixels per degree for computational efficiency. We note that for simplicity, we summed over spatial frequency channels with uniform weights. If one were to fit separate parameters for each voxel, then one might expect spatial frequency tuning to vary with eccentricity. Nonetheless, the simplification of uniform weighting is reasonable given that the spatial frequency content of our stimuli is concentrated in a single octave (~2–4 cpd), and fMRI studies of spatial frequency tuning find a wide bandwidth at the voxel level, std of 2.2 octaves, or full width at half max of 5.1 octaves [88]. 2. Spatial pooling. Each model differs in how contrast energy is pooled to yield a scalar value, s: (2) where we use square brackets to indicate a function of a function (also called a functional). We describe the pooling functional Φ, for each model below. 3. Power-law nonlinearity. Finally, the scalar is passed through a power-law nonlinearity to predict the BOLD amplitude r in units of percent signal change: (3) Where g is the gain and α is the exponent parameter. These are free parameters fit to the fMRI data. The power-law nonlinearity is similar to divisive normalization in the case where each unit in a population is normalized by the same pool [10]. 4.5.1 Pooling functional for contrast energy model. In the contrast energy model, the contrast energy is summed over orientations and space to yield a scalar output, s. (4) Nori is the number of orientation channels (always 8) and Npixels is the number of pixels per stimulus in the padded images (3442, 4192, or 3422). There are no free parameters in this pooling functional for contrast energy, so the complete contrast energy model has only two free parameters, g and α, both from the power-law nonlinearity step (Eq 3). 4.5.2 Pooling functional for divisive normalization model. The contrast energy in the untuned divisive normalization model is normalized before it is summed: Each (x,y,θ) element in the energy image is normalized by a weighted sum of elements at (x′,y′,θ′). The weighting is a Gaussian function of distance from location (x,y) and can thus be expressed as a convolution of the contrast energy E, with a Gaussian, G: (5) The standard deviation of G is 4% of the padded image size, which is approximately 1 deg. G is identical across the 8 orientations. The normalized contrast energy is (6) where σ is a parameter to control the strength of normalization. When σ is large, the normalization is low, and the overall expression approximates the contrast energy model. When σ approaches 0, there is strong normalization. We then sum d across space and orientation to result in the scalar, s. (7) The pooling functional for divisive normalization introduces one free parameter, σ. As with the contrast energy model, a power-law nonlinearity is applied to s to predict the BOLD response in percent signal change (Eq 3). Hence the complete model has three free parameters. We note that the complete divisive normalization model has two similar non-linearities, one in the pooling functional (divisive normalization) and one on the final output (power-law). This is consistent with prior work showing that two stages of normalization (a cascade model) improved model accuracy [11]. 4.5.3 Pooling functional for orientation-tuned normalization model. We implement the orientation-tuned normalization (OTN) model identically to the divisive normalization model (Eqs 5–7), except that the contrast energy normalizer Z(x,y,θ) is now orientation-tuned. When the orientation channel of the image and filter matches, θ′ = θ, the 2-D filter of the channel is a 2D Gaussian identical to the untuned normalization (4.4.2). At all other orientations (i.e., θ′ ≠ θ), the filter is a symmetric 2D Gaussian distribution with a much small standard deviation, effectively just one pixel (Fig 4). This is akin to summing two forms of normalization, cross-orientation suppression (same location, other orientations), and an orientation-tuned surround (same orientation, other locations). As with the untuned normalization model, the pooling functional introduces only one free parameter, σ. The complete model, including the power-law non-linearity (Eq 3) has 3 free parameters. 4.5.4 Pooling functional for normalization by orientation anisotropy. In the normalization by orientation anisotropy (NOA) model, the pooling step first sums the contrast energy across space within an orientation band, resulting in one value per orientation band, Eori(θ): (8) Eori indicates the oriented energy. This energy at each orientation is then normalized by the standard deviation across the 8 orientations, and then summed to produce a scalar. (9) Where calculates the standard deviation of the oriented energy and a non-negative parameter w controls the strength of the normalization. When σ is large, the normalization is low, and the overall expression approximates the contrast energy model. When σ approaches 0, there is strong normalization by the standard deviation across orientation channel outputs. Calculating the standard deviation of oriented energy involves a squaring operation. To keep the parameters comparable across different models, we also square the numerator and the parameter σ. The NOA pooling functional has one free parameter, σ. The complete model, including the power-law non-linearity (Eq 3) has 3 free parameters. 4.6 Optimization In each model, we fitted the model free parameters using the MATLAB optimization tool fmincon by minimizing the squared error between the model prediction and the corresponding BOLD amplitude. Because each stimulus consisted of 9 exemplars shown to the subject in rapid succession, the model prediction for each stimulus was obtained by averaging the model predictions across the exemplars. To avoid getting stuck in the local minima of the nonconvex landscape, we ran the optimization algorithm with 40 different parameter initializations. Each initialized value was picked randomly. All parameters were unbounded in the search, minimizing human interference in the fitting. Parameter α is passed through a sigmoid function to ensure its value is between 0 and 1. 4.7 Cross-validation scheme All models were fit using an leave-one-out cross-validation scheme, where n is the number of stimuli. Thus, the BOLD signal prediction for each stimulus was generated by a model fit to all stimuli except that one. Under this scheme, the models are less likely to overfit data sets. 4.8 Accuracy metric The model accuracy was quantified as the percentage of the explained variance (R2) in the human BOLD data by the cross-validated model predictions, (10) where ri represents the BOLD amplitude to the ith stimulus, represents the corresponding model prediction, and is the mean response across stimuli. We can understand this metric as the extra uncertainty reduction brought by the model beyond describing the BOLD data by its mean. 4.1 Ethics statement Participants provided written informed consent. The experimental protocol was in compliance with the safety guidelines for MRI research and was approved by the University Committee on Activities involving Human Subjects at New York University (IRB-FY2016-363). We analyzed and modeled four fMRI data sets for this paper. Data sets 1 and 2 were collected at NYU. Data sets 3 and 4 are re-analyzed from a previous paper [11], for which the fMRI data and stimuli are freely available online (http://kendrickkay.net/socmodel/). 4.2 Participants The two NYU participants were both experienced MRI subjects (female; 24, 29 yo). Data were collected at NYU’s Center for Brain Imaging. The experimental protocol was approved by the University Committee on Activities Involving Human Subjects, and informed written consent was obtained from the participants before the study. Both participants had corrected-to-normal vision. The subjects participated in two separate scanning sessions, one for retinotopic mapping and one for the main study on encoding of textures. 4.3 Stimuli Publicly accessible links to the stimuli, the names of the stimulus classes, and the correspondence between our naming convention and those in the Kay et al., 2013 [11] paper are described in Tables A, B, and C in S1 Appendix. The total spectral power of all images can be visualized using function visualizeStimulusPowerSpectrum in the code repository and is shown for target stimuli from Dataset 2 of Fig A in S1 Appendix. 4.3.1 Stimuli for data sets 3 and 4. Data sets 3 and 4 correspond to subject 1 and subject 2, respectively, in Kay et al., 2013 [11]. The stimuli for the two subjects were the same and are referred to as “Stimulus set 2” on the website (http://kendrickkay.net/socmodel/). The publicly available stimulus set includes 156 stimuli, 39 of which were used for this paper (Tables B and C in S1 Appendix). The reason we use only a subset of the stimuli is that we modeled how visual areas respond to textures ignoring retinotopic preference, and many of the stimuli used in the original paper varied systematically over space in order to map spatial receptive fields. We used only the large-field textures, i.e., the subset of stimuli whose patterns were similar across the whole circular aperture. The 39 stimuli are organized into 7 groups, each of which we describe with two terms, one term for the type of texture and one term for the way in which the stimuli within the group vary. For example, GRATINGS (contrast) are stimuli which come from the grating family and vary from low to high contrast. GRATINGS (density) stimuli come from the same family but have uniform contrast and vary in the spacing between the contours. The correspondence between how we refer to the stimuli and how Kay et al., 2013 [11] referred to them is in Table C in S1 Appendix. Below we describe the general stimulus characteristics and the 7 specific classes used for this paper. Most of the text is duplicated from Kay et al., 2013 [11] (p. 11), indicated by italics. General stimulus characteristics. Stimuli were constructed at a resolution of 256 pixels × 256 pixels and were upsampled to 800 pixels × 800 pixels for display purposes. All stimuli were presented within a circular aperture filling the height of the display; the rest of the display was filled with neutral gray. The outer 0.5 deg of the circular aperture was smoothly blended into the background using a half-cosine function. Stimuli consisted of grayscale images restricted to a band-pass range of spatial frequencies centered at 3 cycles per degree. To enforce this restriction, a custom band-pass filter was used in the generation of some of the stimuli. The filter was a zero-mean isotropic 2D Difference-of-Gaussians filter whose amplitude spectrum peaks at 3 cycles per degree and drops to half-maximum at 1.4 and 4.7 cycles per degree. Restricting the spatial frequency content of the stimuli avoids the complications of building multiscale models and helps constrain the scope of the modeling endeavor. Even with the spatial frequency restriction, it is possible to construct a rich diversity of stimuli including objects and other naturalistic stimuli…. Each stimulus consisted of nine distinct images that were presented in quick succession. The purpose of this design was to take advantage of the slow dynamics of the BOLD response and average over stimulus dimensions of no interest (e.g., using sinusoidal gratings differing in phase to average over phase). A key motivating observation for this paper is that the response to gratings was lower than to curved stimuli. Four groups of stimuli, two groups of gratings and two groups of snakes, were studied first and are referred to throughout the paper as target stimuli. These are described first. Target stimuli. SNAKES (contrast, 10 stimuli). Kay et al. refers to these stimuli as noise patterns: Noise patterns were created by low-pass filtering white noise at a cutoff frequency of 0.5 cycles per degree, thresholding the result, performing edge detection using derivative filters, inverting image polarity such that edges are black, and applying the custom band-pass filter (described previously). We generated nine distinct noise patterns and scaled the contrast of the patterns to fill the full luminance range. [The contrast stimuli were then] constructed by varying the contrast of the noise patterns… . Ten different contrast levels were used: 1%, 2%, 3%, 4%, 6%, 9%, 14%, 21%, 32%, and 50%. These contrast levels are relative to the contrast of the patterns used in SPACE [not used in this study], which is taken to be 100%. SNAKES (density, 5 stimuli). These stimuli used the same type of noise patterns as SPACE [not used here] but varied the amount of separation between contours. We generated noise patterns using cutoff frequencies of 2.8, 1.6, 0.9, 0.5, and 0.3 cycles per degree, and numbered these from 1 (smallest separation) to 5 (largest separation). The noise patterns used in SPACE correspond to separation 4; thus, we only constructed stimuli for the remaining separations 1, 2, 3, and 5. The noise patterns occupied the full stimulus extent (no aperture masking) GRATINGS (contrast, 4 stimuli). These stimuli consisted of horizontal sinusoidal gratings at 2%, 4%, 9%, and 20% Michelson contrast. The spatial frequency of the gratings was fixed at 3 cycles per degree. GRATINGS (density, 5 stimuli). The highest density stimulus in this group is similar to the horizontally oriented stimulus in GRATINGS (orientation), i.e., similar to a horizontal high-contrast grating, but it is not precisely a sinusoidal grating. It is made by convolving equally spaced horizontal lines with the custom band-pass filter (described previously). When the gratings are spaced appropriately (⅓ deg spacing) and filtered by a band-pass filter centered at 3 cycles per deg, the result is close to a sinusoidal grating at 3 cycles per deg. When the spacing is larger, there are several parallel band-pass contours with uniform gray between them. The spacing between parallel lines for the 5 stimuli varied in powers of 2, as 1/3 deg × 1,2,4,8, or 16, from densest to sparsest. Because the grating and snakes stimuli were both constructed by convolving lines with the same band-pass filter, they have some similar properties. They differ in that the lines here were straight whereas the lines used for constructing the snakes stimuli were curved. Additional stimuli. GRATINGS (orientation, 8 stimuli). These stimuli consisted of full-contrast sinusoidal gratings at eight different orientations. The spatial frequency of the gratings was fixed at 3 cycles per degree. Each [of the 9 exemplars per stimulus] consisted of gratings with the same orientation but nine different phases (equally spaced from 0 to 2π). PLAID (contrast, 4 stimuli). These stimuli consisted of plaids at 2%, 4%, 9%, and 20% contrast (defined below). Each condition comprised nine plaids, and each plaid was constructed as the sum of a horizontal and a vertical sinusoidal grating (spatial frequency 3 cycles per degree, random phase). The plaids were scaled in contrast to match the root-mean-square (RMS) contrast of the GRATING stimuli. For example, the plaids in the 9% condition were scaled such that the average RMS contrast of the plaids is identical to the average RMS contrast of the gratings in the 9% GRATING stimulus. CIRCULAR (contrast, 4 stimuli). These stimuli were identical to the PLAID stimuli except that sixteen different orientations were used instead of two. 4.3.2 Stimuli for Data set 1. Data set 1 was collected at NYU. The data set was designed to replicate some of the effects observed from data sets 3 and 4 (the greater response to snakes than gratings), but also to extend the measurements to new stimulus classes. The general stimulus characteristics were the same as those used in data sets 3 and 4. However, because the display size differed, the image resolution in pixels also differed (400 × 400 here, vs 800 × 800 above), and there were slight differences in the bandpass filter. The stimulus size in degrees of visual angle was the same (12.5 deg diameter). A total of 50 stimuli were tested. (The numbers below total more than 50 because some stimuli belong to more than one group, as indicated in Table C in S1 Appendix). Target stimuli. GRATINGS (contrast, 5 stimuli). These stimuli are horizontal gratings (but not quite sinusoids), with a similar spatial pattern to the middle stimulus in the GRATINGS (density) stimuli from data sets 3 and 4. They were made by convolving horizontal lines spaced every 1.75 deg with a custom band-pass filter. The images were scaled to yield 5 different contrasts of 3%, 10%, 25%, 50% and 100%. GRATINGS (density; 5 stimuli). These stimuli are similar to horizontal gratings, made by convolving equally spaced horizontal lines with the custom band-pass filter. The 5 stimuli differed in the spacing of the horizontal lines, spaced every 3, 2.5, 1.75, 1, 0.33 deg. The contrast of all stimuli was 25%. The middle stimulus in this sequence was the same as the middle stimulus in the contrast sequence (spacing 1.75 deg, 25% contrast). The highest density (0.33 deg spacing) is close to a sinusoidal grating, as the spacing is the inverse of the peak spatial frequency of the band-pass filter (3 cycles per degree). SNAKES (contrast, 5 stimuli). The spatial pattern is similar to the snakes stimuli in data set 3 and data set 4. Contrasts matched the grating contrasts (3%, 10%, 25%, 50% and 100%). SNAKES (density, 5 stimuli). The spatial pattern is similar to the snakes stimuli in data set 3 and data set 4, but with 5 different densities of the contours. The contrast for all stimuli was 25% (lower than the contrast of the corresponding stimuli in data set 3 and data set 4). The range of densities used here was also lower than the range used in data set 3 and data set 4, with the densest pattern here similar to the middle stimuli in data set 3 and data set 4. Additional stimuli. GRATINGS (orientation, 4 stimuli). These stimuli are the same as the third stimulus in the GRATINGS (density) group (25% contrast, 1.75 deg spacing between contours), except that they are rotated by 0, 45, 90, or 135 deg. Because the 0 deg rotation does not change the image, a new stimulus was not created; for visualization of results, the BOLD measurements and model predictions are plotted for both groups. GRATINGS (cross, 4 stimuli). These stimuli contain horizontal contours similar to two of the stimuli in the GRATINGS (density) sequence, except that they have periodic vertical blank regions which interrupt the contours. For two of the stimuli, the spacing of the horizontal contours matches the densest stimuli in the density sequence (spacing of 0.33 deg) and for two of the stimuli, the spacing matched the middle stimulus in the density sequence (1.75 deg spacing). In all 4 images, the horizontal contours are interrupted by vertical blanks spaced every 1.75 deg. The vertical blanks are either thick (50% duty cycle; 1st and 3rd stimulus) or thin (25% duty cycle; 2nd and 4th stimuli). NOISE BARS (density, 5 stimuli). These stimuli have the same contrast apertures as the GRATINGS (density) stimuli. Specifically, there are horizontal bands containing contrast patterns, spaced the same as the grating stimuli (bands every 3, 2.5, 1.75, 1, or 0.33 deg). These stimuli differ from the gratings in that each band contains band-pass filtered noise, equal in power across orientations, rather than horizontal contours. NOISE BARS (contrast, 5 stimuli). These stimuli are matched in spatial pattern to the middle density of the NOISE BARS (density) stimuli (horizontal lines, spacing 1.75 deg), but scaled in contrast similar to the grating stimuli (3%, 10%, 25%, 50%, 100%). NOISE BARS (orientation, 4 stimuli). The orientation sequence rotated the middle stimulus of the NOISE BARS (density) group (spacing 1.75 deg, contrast 25%) by 0, 45, 90, or 135 deg. WAVES (density, 6 stimuli). These are identical to the snakes (density) stimuli, except that they have been filtered by orientation, such that they only contain power at or near the horizontal. WAVES (contrast, 5 stimuli). These are identical to the snakes (contrast) stimuli, except that they have been filtered by orientation, such that they only contain power at or near the horizontal. WAVES (orientation, 4 stimuli). These are identical to the densest stimulus in the snakes (density) group, except that they have been filtered by orientation, with filter centered at either 0, 45, 90, or 135 deg. 4.3.3 Stimuli for data set 2. Data set 2 was collected at NYU. The stimuli were nearly identical to those in data set 1, differing only in the following ways. First, the stimuli were 50% larger (18.75 × 18.75 deg and 600 × 600 pixels, rather than 12.5 × 12.5 deg and 400 × 400 pixels). The difference in size did not entail a difference in spatial frequency: The spatial frequency was matched between the two data sets (meaning that the stimuli were re-made with a larger aperture rather than by re-scaling). Second, those stimuli which were oriented were oriented vertically rather than horizontally. This applies to all GRATING stimuli, as well as NOISE BARS and WAVES. Third, the WAVES (density) stimuli had only 4 densities rather than 6. We reduced the number of stimuli to slightly shorten the MRI scans. 4.3.1 Stimuli for data sets 3 and 4. Data sets 3 and 4 correspond to subject 1 and subject 2, respectively, in Kay et al., 2013 [11]. The stimuli for the two subjects were the same and are referred to as “Stimulus set 2” on the website (http://kendrickkay.net/socmodel/). The publicly available stimulus set includes 156 stimuli, 39 of which were used for this paper (Tables B and C in S1 Appendix). The reason we use only a subset of the stimuli is that we modeled how visual areas respond to textures ignoring retinotopic preference, and many of the stimuli used in the original paper varied systematically over space in order to map spatial receptive fields. We used only the large-field textures, i.e., the subset of stimuli whose patterns were similar across the whole circular aperture. The 39 stimuli are organized into 7 groups, each of which we describe with two terms, one term for the type of texture and one term for the way in which the stimuli within the group vary. For example, GRATINGS (contrast) are stimuli which come from the grating family and vary from low to high contrast. GRATINGS (density) stimuli come from the same family but have uniform contrast and vary in the spacing between the contours. The correspondence between how we refer to the stimuli and how Kay et al., 2013 [11] referred to them is in Table C in S1 Appendix. Below we describe the general stimulus characteristics and the 7 specific classes used for this paper. Most of the text is duplicated from Kay et al., 2013 [11] (p. 11), indicated by italics. General stimulus characteristics. Stimuli were constructed at a resolution of 256 pixels × 256 pixels and were upsampled to 800 pixels × 800 pixels for display purposes. All stimuli were presented within a circular aperture filling the height of the display; the rest of the display was filled with neutral gray. The outer 0.5 deg of the circular aperture was smoothly blended into the background using a half-cosine function. Stimuli consisted of grayscale images restricted to a band-pass range of spatial frequencies centered at 3 cycles per degree. To enforce this restriction, a custom band-pass filter was used in the generation of some of the stimuli. The filter was a zero-mean isotropic 2D Difference-of-Gaussians filter whose amplitude spectrum peaks at 3 cycles per degree and drops to half-maximum at 1.4 and 4.7 cycles per degree. Restricting the spatial frequency content of the stimuli avoids the complications of building multiscale models and helps constrain the scope of the modeling endeavor. Even with the spatial frequency restriction, it is possible to construct a rich diversity of stimuli including objects and other naturalistic stimuli…. Each stimulus consisted of nine distinct images that were presented in quick succession. The purpose of this design was to take advantage of the slow dynamics of the BOLD response and average over stimulus dimensions of no interest (e.g., using sinusoidal gratings differing in phase to average over phase). A key motivating observation for this paper is that the response to gratings was lower than to curved stimuli. Four groups of stimuli, two groups of gratings and two groups of snakes, were studied first and are referred to throughout the paper as target stimuli. These are described first. Target stimuli. SNAKES (contrast, 10 stimuli). Kay et al. refers to these stimuli as noise patterns: Noise patterns were created by low-pass filtering white noise at a cutoff frequency of 0.5 cycles per degree, thresholding the result, performing edge detection using derivative filters, inverting image polarity such that edges are black, and applying the custom band-pass filter (described previously). We generated nine distinct noise patterns and scaled the contrast of the patterns to fill the full luminance range. [The contrast stimuli were then] constructed by varying the contrast of the noise patterns… . Ten different contrast levels were used: 1%, 2%, 3%, 4%, 6%, 9%, 14%, 21%, 32%, and 50%. These contrast levels are relative to the contrast of the patterns used in SPACE [not used in this study], which is taken to be 100%. SNAKES (density, 5 stimuli). These stimuli used the same type of noise patterns as SPACE [not used here] but varied the amount of separation between contours. We generated noise patterns using cutoff frequencies of 2.8, 1.6, 0.9, 0.5, and 0.3 cycles per degree, and numbered these from 1 (smallest separation) to 5 (largest separation). The noise patterns used in SPACE correspond to separation 4; thus, we only constructed stimuli for the remaining separations 1, 2, 3, and 5. The noise patterns occupied the full stimulus extent (no aperture masking) GRATINGS (contrast, 4 stimuli). These stimuli consisted of horizontal sinusoidal gratings at 2%, 4%, 9%, and 20% Michelson contrast. The spatial frequency of the gratings was fixed at 3 cycles per degree. GRATINGS (density, 5 stimuli). The highest density stimulus in this group is similar to the horizontally oriented stimulus in GRATINGS (orientation), i.e., similar to a horizontal high-contrast grating, but it is not precisely a sinusoidal grating. It is made by convolving equally spaced horizontal lines with the custom band-pass filter (described previously). When the gratings are spaced appropriately (⅓ deg spacing) and filtered by a band-pass filter centered at 3 cycles per deg, the result is close to a sinusoidal grating at 3 cycles per deg. When the spacing is larger, there are several parallel band-pass contours with uniform gray between them. The spacing between parallel lines for the 5 stimuli varied in powers of 2, as 1/3 deg × 1,2,4,8, or 16, from densest to sparsest. Because the grating and snakes stimuli were both constructed by convolving lines with the same band-pass filter, they have some similar properties. They differ in that the lines here were straight whereas the lines used for constructing the snakes stimuli were curved. Additional stimuli. GRATINGS (orientation, 8 stimuli). These stimuli consisted of full-contrast sinusoidal gratings at eight different orientations. The spatial frequency of the gratings was fixed at 3 cycles per degree. Each [of the 9 exemplars per stimulus] consisted of gratings with the same orientation but nine different phases (equally spaced from 0 to 2π). PLAID (contrast, 4 stimuli). These stimuli consisted of plaids at 2%, 4%, 9%, and 20% contrast (defined below). Each condition comprised nine plaids, and each plaid was constructed as the sum of a horizontal and a vertical sinusoidal grating (spatial frequency 3 cycles per degree, random phase). The plaids were scaled in contrast to match the root-mean-square (RMS) contrast of the GRATING stimuli. For example, the plaids in the 9% condition were scaled such that the average RMS contrast of the plaids is identical to the average RMS contrast of the gratings in the 9% GRATING stimulus. CIRCULAR (contrast, 4 stimuli). These stimuli were identical to the PLAID stimuli except that sixteen different orientations were used instead of two. 4.3.2 Stimuli for Data set 1. Data set 1 was collected at NYU. The data set was designed to replicate some of the effects observed from data sets 3 and 4 (the greater response to snakes than gratings), but also to extend the measurements to new stimulus classes. The general stimulus characteristics were the same as those used in data sets 3 and 4. However, because the display size differed, the image resolution in pixels also differed (400 × 400 here, vs 800 × 800 above), and there were slight differences in the bandpass filter. The stimulus size in degrees of visual angle was the same (12.5 deg diameter). A total of 50 stimuli were tested. (The numbers below total more than 50 because some stimuli belong to more than one group, as indicated in Table C in S1 Appendix). Target stimuli. GRATINGS (contrast, 5 stimuli). These stimuli are horizontal gratings (but not quite sinusoids), with a similar spatial pattern to the middle stimulus in the GRATINGS (density) stimuli from data sets 3 and 4. They were made by convolving horizontal lines spaced every 1.75 deg with a custom band-pass filter. The images were scaled to yield 5 different contrasts of 3%, 10%, 25%, 50% and 100%. GRATINGS (density; 5 stimuli). These stimuli are similar to horizontal gratings, made by convolving equally spaced horizontal lines with the custom band-pass filter. The 5 stimuli differed in the spacing of the horizontal lines, spaced every 3, 2.5, 1.75, 1, 0.33 deg. The contrast of all stimuli was 25%. The middle stimulus in this sequence was the same as the middle stimulus in the contrast sequence (spacing 1.75 deg, 25% contrast). The highest density (0.33 deg spacing) is close to a sinusoidal grating, as the spacing is the inverse of the peak spatial frequency of the band-pass filter (3 cycles per degree). SNAKES (contrast, 5 stimuli). The spatial pattern is similar to the snakes stimuli in data set 3 and data set 4. Contrasts matched the grating contrasts (3%, 10%, 25%, 50% and 100%). SNAKES (density, 5 stimuli). The spatial pattern is similar to the snakes stimuli in data set 3 and data set 4, but with 5 different densities of the contours. The contrast for all stimuli was 25% (lower than the contrast of the corresponding stimuli in data set 3 and data set 4). The range of densities used here was also lower than the range used in data set 3 and data set 4, with the densest pattern here similar to the middle stimuli in data set 3 and data set 4. Additional stimuli. GRATINGS (orientation, 4 stimuli). These stimuli are the same as the third stimulus in the GRATINGS (density) group (25% contrast, 1.75 deg spacing between contours), except that they are rotated by 0, 45, 90, or 135 deg. Because the 0 deg rotation does not change the image, a new stimulus was not created; for visualization of results, the BOLD measurements and model predictions are plotted for both groups. GRATINGS (cross, 4 stimuli). These stimuli contain horizontal contours similar to two of the stimuli in the GRATINGS (density) sequence, except that they have periodic vertical blank regions which interrupt the contours. For two of the stimuli, the spacing of the horizontal contours matches the densest stimuli in the density sequence (spacing of 0.33 deg) and for two of the stimuli, the spacing matched the middle stimulus in the density sequence (1.75 deg spacing). In all 4 images, the horizontal contours are interrupted by vertical blanks spaced every 1.75 deg. The vertical blanks are either thick (50% duty cycle; 1st and 3rd stimulus) or thin (25% duty cycle; 2nd and 4th stimuli). NOISE BARS (density, 5 stimuli). These stimuli have the same contrast apertures as the GRATINGS (density) stimuli. Specifically, there are horizontal bands containing contrast patterns, spaced the same as the grating stimuli (bands every 3, 2.5, 1.75, 1, or 0.33 deg). These stimuli differ from the gratings in that each band contains band-pass filtered noise, equal in power across orientations, rather than horizontal contours. NOISE BARS (contrast, 5 stimuli). These stimuli are matched in spatial pattern to the middle density of the NOISE BARS (density) stimuli (horizontal lines, spacing 1.75 deg), but scaled in contrast similar to the grating stimuli (3%, 10%, 25%, 50%, 100%). NOISE BARS (orientation, 4 stimuli). The orientation sequence rotated the middle stimulus of the NOISE BARS (density) group (spacing 1.75 deg, contrast 25%) by 0, 45, 90, or 135 deg. WAVES (density, 6 stimuli). These are identical to the snakes (density) stimuli, except that they have been filtered by orientation, such that they only contain power at or near the horizontal. WAVES (contrast, 5 stimuli). These are identical to the snakes (contrast) stimuli, except that they have been filtered by orientation, such that they only contain power at or near the horizontal. WAVES (orientation, 4 stimuli). These are identical to the densest stimulus in the snakes (density) group, except that they have been filtered by orientation, with filter centered at either 0, 45, 90, or 135 deg. 4.3.3 Stimuli for data set 2. Data set 2 was collected at NYU. The stimuli were nearly identical to those in data set 1, differing only in the following ways. First, the stimuli were 50% larger (18.75 × 18.75 deg and 600 × 600 pixels, rather than 12.5 × 12.5 deg and 400 × 400 pixels). The difference in size did not entail a difference in spatial frequency: The spatial frequency was matched between the two data sets (meaning that the stimuli were re-made with a larger aperture rather than by re-scaling). Second, those stimuli which were oriented were oriented vertically rather than horizontally. This applies to all GRATING stimuli, as well as NOISE BARS and WAVES. Third, the WAVES (density) stimuli had only 4 densities rather than 6. We reduced the number of stimuli to slightly shorten the MRI scans. 4.4 MRI The methods for MRI acquisition and preprocessing for data sets 3 and 4 are described in Kay et al., 2013 [11]. In brief, each data set comes from one subject, who viewed a variety of stimuli in an event-related fMRI design. Data set 3 was collected over two scan sessions and each stimulus was presented 6 times. Data set 4 was collected over one scan session and each stimulus was presented 3 times. (Note that both in this paper and the Kay website (http://kendrickkay.net/socmodel/), these two data sets are referred to as data sets 3 and 4. However, the Kay website refers to the stimuli for these two data sets as stimulus set 2 and the subjects themselves as “subject B” and “subject C”. We do not adopt these latter two conventions.) After preprocessing the data (slice-time correction, co-registration, spatial unwarping), a general linear model was applied using the GLMdenoise toolbox [11]. The output of this algorithm includes a coefficient (beta weight) for each stimulus for each voxel solved from the whole fMRI session, as well as 30 bootstrapped estimates of each beta weight (bootstrapping across fMRI runs). The publicly available data (http://kendrickkay.net/socmodel/) are already pre-processed, denoised, and organized by ROI. Specifically, the data we used are in the files called “data set03.mat” and “data set04.mat” (http://kendrickkay.net/socmodel/data set03.mat, http://kendrickkay.net/socmodel/data set04.mat). The data sets are described on the website as “Data set 3 (subject B)” and “Data set 4 (subject C),” respectively. Within the MATLAB files, we used the stored 3D array called “betas” (voxels × stimuli × bootstraps), limited to V1, V2, V3 as indicated in the grouping variables “roi” and “roilabels”, and limited to the 39 stimuli indicated in Table B in S1 Appendix. Visual areas were identified by retinotopic mapping in a separate session. 4.4.1 Acquisition of data sets 1 and 2. Data sets 1 and 2 were acquired in one scanning session each. Each scanning session had 12 fMRI runs of 249 s each (data set 1) or 241.5 s each (data set 2). For each data set, half of the stimuli were assigned to odd fMRI runs and half to even runs, so that each stimulus was shown 6 times in the session. The stimulus events were 3 s long, consisting of 9 alternations between stimulus exemplar and blank, ⅙ s each. Trial onsets were every 7.5 s (so 4.5 s blank between trials). To help estimate the hemodynamic response function, there were 12 s of blank at the beginning and end of each run, as well as 5 additional trials randomly interspersed with no stimulus (meaning that 5 times during the scan, trials were separated by 15 s instead of 7.5 s). Thus, each complete run consisted of either (25 stimuli + 5 blanks) * 7.5 s + 24 s = 249 s (data set 1) or (24 stimuli + 5 blanks) * 7.5 s + 24 s = 241.5 s (data set 2). All MRI data were acquired at New York University Center for Brain Imaging using a Siemens Allegra 3T head-only scanner with a Nova Medical phased array, 8-channel receive surface coil (NMSC072). For each participant, we collected functional images (single shot echo planar images, 1500 ms TR, 30 ms TE, and 72° flip angle). Voxels were 2.0 mm3 isotropic, with 24 slices, with an inplane sampling of 104 × 80 voxels (208 mm A/P × 160 mm L/R). The slice prescription covered most of the occipital lobe, and the posterior part of both the temporal and parietal lobes. Images were corrected for B0 field inhomogeneity using a calibration scan and Center for Brain Imaging algorithms during offline image reconstruction. We also acquired 1 or 2 T1-weighted whole-brain anatomical scans (MPRAGE sequence; 1mm3), as well as a T1-weighted “inplane” image with the same slice prescription as the functional scans. This scan had an inplane resolution of 1.25 × 1.25 mm and a slice thickness of 2.5 mm, and was collected to aid alignment of the functional images to the high-resolution T1 weighted anatomical images. In a separate session, retinotopy scans were collected and analyzed using a pRF model as implemented in the Vistasoft software tool (https://github.com/vistalab/vistasoft). The methods for acquisition and analysis of the retinotopy data are identical to that described by Zhou et al. 2018 [84]. 4.4.2 Data preprocessing and analysis. Data preprocessing. Processing of the fMRI data was identical to that described by Zhou et al. 2018 [84]: We coregistered and segmented the T1 weighted whole-brain anatomical images into gray and white matter voxels using FreeSurfer’s autosegmentation algorithm (http://surfer.nmr.mgh.harvard.edu). Using custom software Vistasoft (https://github.com/vistalab/vistasoft), the functional data were slice-time corrected by resampling the time series in each slice to the center of each 1.5 s volume. Data were then motion-corrected by coregistering all volumes of all scans to the first volume of the first scan. The first 8 volumes (12 s) of each scan were discarded for analysis to allow longitudinal magnetization and stabilized hemodynamic response. GLM. The preprocessed fMRI data were then fit by a general linear model, GLMDenoise [11]. This algorithm denoises the data by projecting out nuisance regressors derived in a data-driven manner, and estimates coefficients for each of the 48 or 50 stimuli for each voxel in the functional images. The algorithm bootstraps the data over fMRI runs. For data sets 1 and 2, we generated 50 bootstraps for data set 1 and 100 bootstraps for data set 2. The publicly available data from Kay et al, 2013 [11], included 30 bootstraps per subject. The algorithm also estimated a hemodynamic impulse response function as a finite impulse response function, with 35 time points (52.5 s) per subject. ROIs. Regions of interest for V1, V2, V3 were delineated manually using the Vistasoft (https://github.com/vistalab/vistasoft) graphical user interface to visualize the results of the pRF models. These methods for identifying these boundaries are well established, as described in many publications [85,86, summarized by 87]. The ROIs for V1, V2 and V3 were identified on the cortical surface and then projected to the functional images. For purposes of data summary and model fitting, we took the average signal from each ROI. We did this by averaging the beta weight across voxels within an ROI separately for each stimulus, after voxel selection (Table A in S1 Appendix). Because noise can be correlated across voxels, but should not be correlated across scans, when we bootstrapped the data, we average across voxels within an ROI for each bootstrap. For the purposes of model fitting, each of the 4 data sets comprised two matrices, one for the means and one for the standard deviation across bootstraps, each of which had a size equal to the number of stimuli by number of ROIs. 4.4.1 Acquisition of data sets 1 and 2. Data sets 1 and 2 were acquired in one scanning session each. Each scanning session had 12 fMRI runs of 249 s each (data set 1) or 241.5 s each (data set 2). For each data set, half of the stimuli were assigned to odd fMRI runs and half to even runs, so that each stimulus was shown 6 times in the session. The stimulus events were 3 s long, consisting of 9 alternations between stimulus exemplar and blank, ⅙ s each. Trial onsets were every 7.5 s (so 4.5 s blank between trials). To help estimate the hemodynamic response function, there were 12 s of blank at the beginning and end of each run, as well as 5 additional trials randomly interspersed with no stimulus (meaning that 5 times during the scan, trials were separated by 15 s instead of 7.5 s). Thus, each complete run consisted of either (25 stimuli + 5 blanks) * 7.5 s + 24 s = 249 s (data set 1) or (24 stimuli + 5 blanks) * 7.5 s + 24 s = 241.5 s (data set 2). All MRI data were acquired at New York University Center for Brain Imaging using a Siemens Allegra 3T head-only scanner with a Nova Medical phased array, 8-channel receive surface coil (NMSC072). For each participant, we collected functional images (single shot echo planar images, 1500 ms TR, 30 ms TE, and 72° flip angle). Voxels were 2.0 mm3 isotropic, with 24 slices, with an inplane sampling of 104 × 80 voxels (208 mm A/P × 160 mm L/R). The slice prescription covered most of the occipital lobe, and the posterior part of both the temporal and parietal lobes. Images were corrected for B0 field inhomogeneity using a calibration scan and Center for Brain Imaging algorithms during offline image reconstruction. We also acquired 1 or 2 T1-weighted whole-brain anatomical scans (MPRAGE sequence; 1mm3), as well as a T1-weighted “inplane” image with the same slice prescription as the functional scans. This scan had an inplane resolution of 1.25 × 1.25 mm and a slice thickness of 2.5 mm, and was collected to aid alignment of the functional images to the high-resolution T1 weighted anatomical images. In a separate session, retinotopy scans were collected and analyzed using a pRF model as implemented in the Vistasoft software tool (https://github.com/vistalab/vistasoft). The methods for acquisition and analysis of the retinotopy data are identical to that described by Zhou et al. 2018 [84]. 4.4.2 Data preprocessing and analysis. Data preprocessing. Processing of the fMRI data was identical to that described by Zhou et al. 2018 [84]: We coregistered and segmented the T1 weighted whole-brain anatomical images into gray and white matter voxels using FreeSurfer’s autosegmentation algorithm (http://surfer.nmr.mgh.harvard.edu). Using custom software Vistasoft (https://github.com/vistalab/vistasoft), the functional data were slice-time corrected by resampling the time series in each slice to the center of each 1.5 s volume. Data were then motion-corrected by coregistering all volumes of all scans to the first volume of the first scan. The first 8 volumes (12 s) of each scan were discarded for analysis to allow longitudinal magnetization and stabilized hemodynamic response. GLM. The preprocessed fMRI data were then fit by a general linear model, GLMDenoise [11]. This algorithm denoises the data by projecting out nuisance regressors derived in a data-driven manner, and estimates coefficients for each of the 48 or 50 stimuli for each voxel in the functional images. The algorithm bootstraps the data over fMRI runs. For data sets 1 and 2, we generated 50 bootstraps for data set 1 and 100 bootstraps for data set 2. The publicly available data from Kay et al, 2013 [11], included 30 bootstraps per subject. The algorithm also estimated a hemodynamic impulse response function as a finite impulse response function, with 35 time points (52.5 s) per subject. ROIs. Regions of interest for V1, V2, V3 were delineated manually using the Vistasoft (https://github.com/vistalab/vistasoft) graphical user interface to visualize the results of the pRF models. These methods for identifying these boundaries are well established, as described in many publications [85,86, summarized by 87]. The ROIs for V1, V2 and V3 were identified on the cortical surface and then projected to the functional images. For purposes of data summary and model fitting, we took the average signal from each ROI. We did this by averaging the beta weight across voxels within an ROI separately for each stimulus, after voxel selection (Table A in S1 Appendix). Because noise can be correlated across voxels, but should not be correlated across scans, when we bootstrapped the data, we average across voxels within an ROI for each bootstrap. For the purposes of model fitting, each of the 4 data sets comprised two matrices, one for the means and one for the standard deviation across bootstraps, each of which had a size equal to the number of stimuli by number of ROIs. 4.5 Model equations In the Results, we compared the accuracy of four models fit to the data, three of which are based on existing models or empirical findings–a contrast energy model, a untuned normalization model, and an orientation-tuned normalization model–and one new model, which computes normalization by orientation anisotropy. In this section we describe the computation that comprises each model. All four models consist of three primary steps: (1) computation of oriented contrast energy, (2) pooling across orientation and space, and (3) a power-law nonlinearity. Steps 1 and 3 are identical for all models. Step 2, spatial pooling, varies between models. 1. Contrast energy: We denote by I(x,y) the value of the pre-processed input image at coordinates (x,y). The pre-processing causes the image values to have mean 0 and range from -0.5 to 0.5. The image is projected onto a set of 128 Gabor filters, which comprise 8 orientations θ, spaced every 22.5 deg; 8 spatial frequencies f, with peak spatial frequency log spaced from 0.75 cpd to 6 cpd; and 2 phases ϕ, separated by 90° (i.e., “quadrature”). F(x,y,θ,f,ϕ) indicates the Gabor filter at a spatial location (x,y), orientation θ, spatial frequency f, and phase ϕ. Each filter comprised a cosine or sine function of 4 cycles, windowed by a Gaussian with SD of 1 cycle. The outputs over the two phases are squared and summed to compute the contrast energy and summed across spatial frequencies. Finally, the contrast energy as a function of spatial position (x,y) and orientation θ becomes (1) We convolve the image I and the filter F. The computation of contrast energy has no free parameters. Prior to convolution, stimuli were padded with uniform gray (mean luminance) on all sides by the width of the largest filter. After convolution, all energy images were downsampled to 12 pixels per degree for computational efficiency. We note that for simplicity, we summed over spatial frequency channels with uniform weights. If one were to fit separate parameters for each voxel, then one might expect spatial frequency tuning to vary with eccentricity. Nonetheless, the simplification of uniform weighting is reasonable given that the spatial frequency content of our stimuli is concentrated in a single octave (~2–4 cpd), and fMRI studies of spatial frequency tuning find a wide bandwidth at the voxel level, std of 2.2 octaves, or full width at half max of 5.1 octaves [88]. 2. Spatial pooling. Each model differs in how contrast energy is pooled to yield a scalar value, s: (2) where we use square brackets to indicate a function of a function (also called a functional). We describe the pooling functional Φ, for each model below. 3. Power-law nonlinearity. Finally, the scalar is passed through a power-law nonlinearity to predict the BOLD amplitude r in units of percent signal change: (3) Where g is the gain and α is the exponent parameter. These are free parameters fit to the fMRI data. The power-law nonlinearity is similar to divisive normalization in the case where each unit in a population is normalized by the same pool [10]. 4.5.1 Pooling functional for contrast energy model. In the contrast energy model, the contrast energy is summed over orientations and space to yield a scalar output, s. (4) Nori is the number of orientation channels (always 8) and Npixels is the number of pixels per stimulus in the padded images (3442, 4192, or 3422). There are no free parameters in this pooling functional for contrast energy, so the complete contrast energy model has only two free parameters, g and α, both from the power-law nonlinearity step (Eq 3). 4.5.2 Pooling functional for divisive normalization model. The contrast energy in the untuned divisive normalization model is normalized before it is summed: Each (x,y,θ) element in the energy image is normalized by a weighted sum of elements at (x′,y′,θ′). The weighting is a Gaussian function of distance from location (x,y) and can thus be expressed as a convolution of the contrast energy E, with a Gaussian, G: (5) The standard deviation of G is 4% of the padded image size, which is approximately 1 deg. G is identical across the 8 orientations. The normalized contrast energy is (6) where σ is a parameter to control the strength of normalization. When σ is large, the normalization is low, and the overall expression approximates the contrast energy model. When σ approaches 0, there is strong normalization. We then sum d across space and orientation to result in the scalar, s. (7) The pooling functional for divisive normalization introduces one free parameter, σ. As with the contrast energy model, a power-law nonlinearity is applied to s to predict the BOLD response in percent signal change (Eq 3). Hence the complete model has three free parameters. We note that the complete divisive normalization model has two similar non-linearities, one in the pooling functional (divisive normalization) and one on the final output (power-law). This is consistent with prior work showing that two stages of normalization (a cascade model) improved model accuracy [11]. 4.5.3 Pooling functional for orientation-tuned normalization model. We implement the orientation-tuned normalization (OTN) model identically to the divisive normalization model (Eqs 5–7), except that the contrast energy normalizer Z(x,y,θ) is now orientation-tuned. When the orientation channel of the image and filter matches, θ′ = θ, the 2-D filter of the channel is a 2D Gaussian identical to the untuned normalization (4.4.2). At all other orientations (i.e., θ′ ≠ θ), the filter is a symmetric 2D Gaussian distribution with a much small standard deviation, effectively just one pixel (Fig 4). This is akin to summing two forms of normalization, cross-orientation suppression (same location, other orientations), and an orientation-tuned surround (same orientation, other locations). As with the untuned normalization model, the pooling functional introduces only one free parameter, σ. The complete model, including the power-law non-linearity (Eq 3) has 3 free parameters. 4.5.4 Pooling functional for normalization by orientation anisotropy. In the normalization by orientation anisotropy (NOA) model, the pooling step first sums the contrast energy across space within an orientation band, resulting in one value per orientation band, Eori(θ): (8) Eori indicates the oriented energy. This energy at each orientation is then normalized by the standard deviation across the 8 orientations, and then summed to produce a scalar. (9) Where calculates the standard deviation of the oriented energy and a non-negative parameter w controls the strength of the normalization. When σ is large, the normalization is low, and the overall expression approximates the contrast energy model. When σ approaches 0, there is strong normalization by the standard deviation across orientation channel outputs. Calculating the standard deviation of oriented energy involves a squaring operation. To keep the parameters comparable across different models, we also square the numerator and the parameter σ. The NOA pooling functional has one free parameter, σ. The complete model, including the power-law non-linearity (Eq 3) has 3 free parameters. 4.5.1 Pooling functional for contrast energy model. In the contrast energy model, the contrast energy is summed over orientations and space to yield a scalar output, s. (4) Nori is the number of orientation channels (always 8) and Npixels is the number of pixels per stimulus in the padded images (3442, 4192, or 3422). There are no free parameters in this pooling functional for contrast energy, so the complete contrast energy model has only two free parameters, g and α, both from the power-law nonlinearity step (Eq 3). 4.5.2 Pooling functional for divisive normalization model. The contrast energy in the untuned divisive normalization model is normalized before it is summed: Each (x,y,θ) element in the energy image is normalized by a weighted sum of elements at (x′,y′,θ′). The weighting is a Gaussian function of distance from location (x,y) and can thus be expressed as a convolution of the contrast energy E, with a Gaussian, G: (5) The standard deviation of G is 4% of the padded image size, which is approximately 1 deg. G is identical across the 8 orientations. The normalized contrast energy is (6) where σ is a parameter to control the strength of normalization. When σ is large, the normalization is low, and the overall expression approximates the contrast energy model. When σ approaches 0, there is strong normalization. We then sum d across space and orientation to result in the scalar, s. (7) The pooling functional for divisive normalization introduces one free parameter, σ. As with the contrast energy model, a power-law nonlinearity is applied to s to predict the BOLD response in percent signal change (Eq 3). Hence the complete model has three free parameters. We note that the complete divisive normalization model has two similar non-linearities, one in the pooling functional (divisive normalization) and one on the final output (power-law). This is consistent with prior work showing that two stages of normalization (a cascade model) improved model accuracy [11]. 4.5.3 Pooling functional for orientation-tuned normalization model. We implement the orientation-tuned normalization (OTN) model identically to the divisive normalization model (Eqs 5–7), except that the contrast energy normalizer Z(x,y,θ) is now orientation-tuned. When the orientation channel of the image and filter matches, θ′ = θ, the 2-D filter of the channel is a 2D Gaussian identical to the untuned normalization (4.4.2). At all other orientations (i.e., θ′ ≠ θ), the filter is a symmetric 2D Gaussian distribution with a much small standard deviation, effectively just one pixel (Fig 4). This is akin to summing two forms of normalization, cross-orientation suppression (same location, other orientations), and an orientation-tuned surround (same orientation, other locations). As with the untuned normalization model, the pooling functional introduces only one free parameter, σ. The complete model, including the power-law non-linearity (Eq 3) has 3 free parameters. 4.5.4 Pooling functional for normalization by orientation anisotropy. In the normalization by orientation anisotropy (NOA) model, the pooling step first sums the contrast energy across space within an orientation band, resulting in one value per orientation band, Eori(θ): (8) Eori indicates the oriented energy. This energy at each orientation is then normalized by the standard deviation across the 8 orientations, and then summed to produce a scalar. (9) Where calculates the standard deviation of the oriented energy and a non-negative parameter w controls the strength of the normalization. When σ is large, the normalization is low, and the overall expression approximates the contrast energy model. When σ approaches 0, there is strong normalization by the standard deviation across orientation channel outputs. Calculating the standard deviation of oriented energy involves a squaring operation. To keep the parameters comparable across different models, we also square the numerator and the parameter σ. The NOA pooling functional has one free parameter, σ. The complete model, including the power-law non-linearity (Eq 3) has 3 free parameters. 4.6 Optimization In each model, we fitted the model free parameters using the MATLAB optimization tool fmincon by minimizing the squared error between the model prediction and the corresponding BOLD amplitude. Because each stimulus consisted of 9 exemplars shown to the subject in rapid succession, the model prediction for each stimulus was obtained by averaging the model predictions across the exemplars. To avoid getting stuck in the local minima of the nonconvex landscape, we ran the optimization algorithm with 40 different parameter initializations. Each initialized value was picked randomly. All parameters were unbounded in the search, minimizing human interference in the fitting. Parameter α is passed through a sigmoid function to ensure its value is between 0 and 1. 4.7 Cross-validation scheme All models were fit using an leave-one-out cross-validation scheme, where n is the number of stimuli. Thus, the BOLD signal prediction for each stimulus was generated by a model fit to all stimuli except that one. Under this scheme, the models are less likely to overfit data sets. 4.8 Accuracy metric The model accuracy was quantified as the percentage of the explained variance (R2) in the human BOLD data by the cross-validated model predictions, (10) where ri represents the BOLD amplitude to the ith stimulus, represents the corresponding model prediction, and is the mean response across stimuli. We can understand this metric as the extra uncertainty reduction brought by the model beyond describing the BOLD data by its mean. Supporting information S1 Appendix. Data Set and Stimulus Properties. https://doi.org/10.1371/journal.pcbi.1011704.s001 (PDF) S2 Appendix. Stimulus Images. https://doi.org/10.1371/journal.pcbi.1011704.s002 (PDF) S3 Appendix. Model Variance Explained. https://doi.org/10.1371/journal.pcbi.1011704.s003 (PDF) S4 Appendix. Model Parameter Estimates. https://doi.org/10.1371/journal.pcbi.1011704.s004 (PDF) S5 Appendix. Plots of Model Fits. https://doi.org/10.1371/journal.pcbi.1011704.s005 (PDF)

Hake, Anna;Germann, Anja;de Beer, Corena;Thielen, Alexander;Däumer, Martin;Preiser, Wolfgang;von Briesen, Hagen;Pfeifer, Nico

doi: 10.1371/journal.pcbi.1010355pmid: 38127856

Introduction Without the prospect of a vaccine or a cure within reach, controlling viral replication, either using antiretroviral therapy or by achieving sustainable immune control, remains one of the major pillars for combating the HIV pandemic [1]. The coreceptor usage of HVI-1 may affect the ability to achieve sustainable immune control over the virus. Apart from the CD4 receptor, HIV-1 needs a coreceptor for successful cell entry. Only two coreceptors have clinical relevance, CCR5 and CXCR4 [2]. Depending on their coreceptor usage, HIV-1 isolates are classified into R5-capable variants (CCR5 usage), X4-capable variants (CXCR4 usage), or R5X4-capable variants (CCR5 and CXCR4 usage) [3, 4]. While R5 variants are known to dominate early infection [5], a switch to X4 at later stage of infection occurs in roughly 50% of patients infected with subtype B HIV-1 associated with increased depletion of CD4+ T cells, faster progression to AIDS, and a higher mortality rate [4, 6–8]. In patients infected with subtype C HIV-1, a switch to CXCR4 usage is observed less frequently compared to subtype B [9]. Recent studies suggest that an increase in subtype C X4 variants might emerge with the increasing access to antiretroviral drug treatment and the ongoing evolution of the subtype C HIV epidemic [10]. The importance of accurate determination of coreceptor usage has increased with the approval of entry inhibitor drugs that target the CCR5 coreceptor. A determinant of coreceptor usage is the env protein of HIV-1. Currently, phenotypic [11–14] and genotypic [15–22] tropism assays still have difficulties accurately detecting minority populations of X4-using variants, which might lead to a predominance of X4 usage after treatment with a CCR5 antagonist. In addition, it is not only important to predict the correct coreceptor usage, but it would be of advantage to predict how close the variant is to a coreceptor switch. Though the clinical significance of the coreceptor usage is well studied, the trigger mechanisms behind the coreceptor switch from R5 to X4 variants remain unsolved. The emergence of X4-capable variants is associated with a decrease in N-linked glycosylation of the envelope glycoprotein env of HIV-1 [23]. Glycosylation is a viral mechanism to mask conserved amino acids from antibody recognition, such that X4-capable variants should be more prone to antibody neutralization in theory. For antibody development, B cells have to be activated by CD4+ T cells. Thus, concurrent CD4+ T cell depletion counteracts this mechanism. How X4 variants can emerge with still high CD4+ T cells remains inconclusive. However, this is of great importance, since patients with intermediate to high CD4+ T cells contradict the current typical clinical indicators for a potential coreceptor switch such as low numbers of CD4+ T cells. The potential interplay between HLA adaptation and coreceptor usage has not been explored so far. Viral adaptation to the immune system includes the emergence of viral escape mutations to the host’s individual HLA profile. The central role of HLA molecules is to bind peptides and present them on the cell surface to compatible T cells, which are part of the adaptive immune response. T cells are HLA-restricted, meaning that they recognize only a specific HLA-antigen complex. There are two major HLA classes—HLA class I and HLA class II. HLA class I molecules exist on all nucleated cells and bind to (self and pathogen-derived) antigens degraded from synthesized proteins in the cytosol. The corresponding HLA-antigen complex is recognized by specific CD8+ T cells. HLA class II molecules only occur on professional antigen-presenting cells that are able to uptake pathogens and proteins from extracellular fluid by phagocytosis or endocytosis. Thus, HLA class II molecules bind pathogen-derived antigens degraded from extracellular proteins in the vesicular compartment of the cell. The corresponding HLA:antigen complex is recognized by CD4+ T cells. The emergence of a mutation that hinders the successful building of the HLA-antigen complex, a so-called escape mutation, allows HIV-1 to evade a T cell-mediated immune response [24, 25]. High-throughput technologies have enabled large-scale population studies to identify many HLA-restricted polymorphisms (HLA footprints) and their role on viral control [26–31]. A prominent example is the influence of the HLA-B*27, the HLA-B*57:01 allele, or recently the HLA-B*46 associates (in Asian cohort) on disease progression [32–34]. Determining virus-host adaptation experimentally and computationally on an individual level is challenging due to the extraordinary genetic diversity of both the HLA complex and HIV-1. HLA adaptation models usually focus on viral polymorphisms that likely emerged due to the patient’s HLA profile. This approach requires the general consideration of the extreme large number of possible HLA alleles in the population and viral polymorphisms while modeling the fact that only few HLA alleles have a potential influence on a particular polymorphism. Current computational approaches [35, 36] tackle the complex modeling task by carrying out many rounds of preselection, including the identification of potential HLA-polymorphism candidates on large-scale cohort data and additional greedy feature selection steps to select the HLA alleles per polymorphism within the model, such that potential sites and HLA alleles might get disregarded based on significance threshold values. Since human populations and HIV subtypes display substantial genetic differences, such approaches require a large amount of data for every group of interest. Correcting for potential phylogenetic relatedness of the viral sequences used within the model as proposed by [37] is currently implemented by incorporating a transmission probability that has to be learned in a separate model. While HLA-1 restricted escape mechanisms to CTLs have been studied in detail, only few studies exist that have analyzed the impact of HLA-restricted CD4+ T cell escape polymorphisms [36] on the capability of the host immune responses to control the virus. In total, there is currently no available approach to estimate viral adaptation jointly to HLA class I and class II. Moreover, the available approaches require rather complex training steps to be used on new data. This study aims to improve our understanding of the host immune response with respect to HLA adaptation and coreceptor usage. In particular, we investigate the hypothesis that coreceptor usage is associated with the adaptation of the virus to the host’s HLA system, especially to the HLA class II alleles. We explore the novel possibility that viral adaptation to the HLA class II molecules would mask the virus from recognition by CD4+ T cells, such that no B cells are activated, and, thus, no antibodies are developed despite still high numbers of CD4+ T cells. Escape mutations in the rather conserved p24 protein of HIV-1, which is involved in forming the viral capsid, emerge more likely under substantial fitness cost [38, 39]. Therefore, we estimate viral adaptation to the patient’s HLA profile only based on the p24 protein of the gag gene as done previously [40–45]. This study requires a data set consisting of (1) the envelope protein sequences of the virus for determining the coreceptor usage, (2) the p24 protein for estimating the HLA adaptation, and (3) the HLA class I and II profile of the corresponding host. Chronically-infected HIV-1 patients are more likely to harbor viruses that have accumulated escape mutations to the HLA system due to the longer exposure to the human immune system. In treatment-naïve patients, the viral evolution is not restricted by selection pressure from drug exposure and is more able to mutate towards escape variants with respect to the immune system. Current available data sets often lack HLA class II allele information or have not sequenced the envelope sequence of the virus. Thus, we sequence the viral envelope gene env as well as the viral gag (p24) gene, and genotype the corresponding HLA class I (HLA-A, HLA-B, HLA-C) and II genes (HLA-DRB1, HLA-DQB1, HLA-DPB1) of the host in a new cohort of 312 treatment-naive, subtype C, chronically-infected HIV-1 patients from South Africa. To jointly model HLA class I and class II adaptation, we develop a novel computational approach. In detail, the adaptation of a particular amino acid in a viral sequence to the host HLA profile is inferred using phylogeny-corrected, multinomial, Bayesian generalized linear mixed models (GLMMs). Without the need for an additional model, GLMMs allow to correct for phylogenetic relatedness of the variants directly by modeling the between-subject correlation as a group-level effect. Using a Bayesian setting allows to learn feature importance directly within the model by applying the horseshoe prior on all HLA class I and class II alleles of the data set and without the need for additional preselection steps or a large amount of data. The horseshoe prior is used in sparse model settings to shrink the majority of the coefficients to zero by having the point mass at zero and symmetric fat tails [46]. Materials and methods Ethical statement PBMC and plasma samples from HIV-1 positive donors were provided by Stellenbosch University with the written informed consent of the donors. Sample collection was approved under the following ethical statement “VIROLOGICAL AND IMMUNOLOGICAL CHARACTERIZATION OF CRYOPRESERVED BLOOD AND VIRUS SAMPLES” PROJECT NUMBER: NO7/06/13. Study cohort Patients (male and female) who attended Wellness, Antenatal and HIV Clinics in the Durbanville and Stellenbosch regions of the Western Cape were recruited. Only patients older than 18 years were selected. Most of the patients were assumed to be in the chronic stage of the infection. Inclusion was based on recent diagnosis of HIV-1 infection (within the previous 6 months). In total, samples from 329 HIV-infected individuals were available. Subtype C was confirmed for 317 of the 329 samples using the COMET Tool [47]. Patients on antiretroviral were excluded from the analysis, resulting in a total of 312 patients. Based on CD4+ T cell count, the data set was further reduced to 274 samples (see subsection on Data sets). For each patient, clinical parameters such as sex, age, ethnicity, CD4 count, and viral load were collected. In addition, the HIV-1 genes gag (p24) and env were sequenced and the patients’ HLA I and II genes were genotyped. Data sets We divide the newly sequenced study cohort of 312 samples based on a CD4+ T cell count cutoff of 500 cells/mm3 into a chronic_highCD4 data set (n = 38) and a chronic_lowCD4 count data set (n = 274). High CD4+ T cell count indicates a stronger immune system. Since infection duration is not known for the patients, a high CD4+ T cell count might indicate that the patients have been infected for a shorter time (less chronic). Moreover, a virus is assumed to be less adapted to a host with a strong immune system compared to a host with a weak immune system. Thus, the adaptation model is only trained on the chronic_lowCD4 data set. In addition, we create an artificial data set (random) based on the chronic_lowCD4 data set, where the HLA alleles per HLA gene and haplotype have been randomized 100 times. HLA adaptation for this random data set is predicted with models based on the chronic_lowCD4 data set as well. For further validation of the adaptation model, we estimate HIV-1 adaptation of publicly available cohort of acutely-infected HIV-1 patients (n = 23) from the Los Alamos HIV sequence database (http://www.hiv.lanl.gov). The acute data set comprised the p24 sequence as well as the HLA I information of 23 patients with the following accession numbers GQ275453, GQ275750, GQ275852, GQ275894, KM192425, KM192440, KM192471, KM192536, KM192566, KM192640, KM192653, KM192674, KM192686, KM192702, KM192762, KM192844, KM192856, KM192870, KM192884, KM192912, KM192942, KM192970, KM192998. Since only the HLA I profile was available, we build an adaptation model based only on the HLA I profile for this purpose (n = 274). Molecular methods HIV status was confirmed with a serological test (Architect HIV Ab/Ab Combo, 3rd generation) on serum according to the manufacturer protocol. After the surface staining of PBMCs by incubation with a monoclonal mouse anti-human antibody coupled to fluorescent dyes, the quantification of cells expressing the CD4 antigen was measured by FACS analysis. Acquisition and analysis was performed on FACs flow cytometer using Cell Quest software. HIV-1 deep sequencing was performed using previous described protocols [48]. Analysis of deep sequencing data was performed using an internally-developed analysis pipeline, where sequence reads in the form of FASTQ files were processed and aligned via a multi-step method. HLA genotyping was performed using the following protocol. Genomic DNA was isolated from 200 μl of EDTA-anticoagulated blood using the QIAamp DNA Blood Mini Kit (QIAGEN, Hilden, Germany). Long-range PCR primers amplified the full-length of HLA class I genes (A, B, C) from 5’- to 3’-UTR. Class II genes (DPB1, DQB1, DRB1) were amplified from exon 2 to 3’-UTR. Fragment sizes were estimated to be around 3000 bp for Class I genes and 6000 bp for Class II genes, respectively. The PCR solution contained 1 x Phusion GC buffer (including 1.5 mM MgCl2), 200 μM dNTPs, 1 M Betaine, 8 μg Bovine Serum Albumin (BSA), 0.4 U Phusion Hot Start II High-Fidelity DNA Polymerase (Finnzymes, Vantaa, Finland), 0.5 μM of each primer and 90 ng of DNA in a total volume of 20 μl. After initial denaturation at 98°C for 1 minute, 35 cycles of 98°C for 10 seconds, 65°C for 20 seconds, and 72°C for 4 minutes were performed, followed by a final extension at 72°C for 20 minutes. Agarose gel electrophoresis was used to confirm amplification and correct fragment size as well as to check for non-specific product contamination. The 3 HLA class I and class II amplicons for each individual were pooled and afterwards purified with the Agencourt AMPure XP system (Agentcourt Bioscience, Beverly, MA, United States) according to the manufacturer’s protocol to inactivate unconsumed dNTPs and to eliminate extraneous primers before library preparation. These pooled amplicons then comprised a single sample. Concentrations were measured on a FLUOstar OPTIMA microplate fluorimeter (BMP LABTECH, Ortenberg, Germany) using the Quant-iT PicoGreen assay (Invitrogen, Carlsbad, CA, United States). Sample libraries for NGS were then prepared with the Nextera XT DNA Sample Prep Kit (Illumina, San Diego, CA, United States) according to the manufacturer’s protocol, including distinct DNA fragmentation, end-polishing, and adaptor-ligation steps. Through the adaptor, every sample was finally labeled with a unique identifier sequence. Sequencing was carried out then on the Illumina MiSeq Personal Sequencer (Illumina, San Diego, CA, United States) as described by the manufacturer. Coreceptor prediction Coreceptor usage is predicted using the well-established tool geno2pheno[coreceptor] [17] on the viral envelope sequences. The provided false-positive rate (FPR) corresponds to the confidence with which the sequence is classified as X4-capable. The higher the FPR, the more likely the sequence is not X4-capable, but R5. Viral strains with an FPR cutoff less than 20% are classified as X4-capable, otherwise as R5-capable according to the European Consensus Group on clinical management of HIV-1 tropism testing [49]. Estimating HLA adaptation Assuming independence of all sites in the viral sequence, we define the adaptation of a sequence to its host HLA profile as the adaptation of each frequent single amino acid site in the sequence to the HLA profile. Moreover, though every patient is infected by a quasispecies of viruses, we only consider the consensus sequence as in previous approaches. In order to correct for potential phylogenetic relatedness of the viral sequences used within the model as proposed by [37], we also incorporate the phylogeny of the viral sequences into the model. Thus, our model requires the amino acid sequences of the viral p24 protein, the corresponding host’s HLA I and II alleles, and the phylogeny between the viral sequences for learning the HLA adaptation (training). For each frequent site, we infer a model (HLA model) to estimate the likelihood that the site is under HLA pressure as well as a hypothetical model (baseline model) that computes the likelihood that the site is not under HLA pressure. HLA adaptation of the complete protein is then defined as a function over the product of the per-site likelihood ratios of the HLA model against the baseline model. Each per-site model is built using multinomial Bayesian generalized linear mixed models (GLMMs). In the following, we formalize the per-site model and the final adaptation score. Afterwards, we present the selection process of the frequent sites. Since each per-site model is built using Bayesian GLMMs, we provide a brief introduction to Bayesian GLMMs and their benefit over classical GLMMs and phylogeny-corrected LMMs. In addition, we provide a section on the model specification for each per-site model. Notation. Let S be a random variable representing the set of all possible HIV-1 amino acid sequences of a particular protein of length L. A particular sequence s is a realization of S covering all sites l = 1, …, L of the protein. A particular site sl can be realized by any amino acid (aa). Since we do not have enough power to find an HLA-restricted polymorphism at a very conserved site, we restrict the sites to m frequent single amino acid sites sj with j = 1, …, m, which are defined by sites that vary over the set of all HIV-1 sequences in their amino acid realization. A site is defined as frequent, if the particular amino acid is observed in at least 1% of the sequences. The host immune system is represented by the HLA alleles of the HLA I and HLA II genes. The HLA profile of an individual consists in our case of six (homozygous in all genes) to 12 (heterozygous in all genes) different HLA alleles. Let H represent the set of all possible HLA I and HLA II alleles. A particular HLA profile h is encoded as a binary vector with zeros everywhere, apart from the positions corresponding to the HLA alleles of the HLA profile. Note, thereby homozygosity is not modeled. We model adaptation as the conditional probability that a sequence s occurs under pressure from the host HLA profile similarly to [35]: (1) Assuming independence among sites and relevance of only frequent sites, the conditional probability over the sequence s can be decomposed to the product over the conditional probabilities over all m frequent sites sj (per-site model): (2) Similarly, a hypothetical model estimating the likelihood of the sequence s without any HLA pressure is defined as: (3) The conditional probabilities P(sj|H = h) and P(sj|H = ⌀) for each site are referred to as the HLA model and the baseline model, respectively. Identification and encoding of frequent sites. Since all patients in the study are infected with the subtype C variant of HIV-1, we align all nucleotide sequences to the subtype C consensus sequence using the alignment tool MAFFT (version 7.407) [50]. The subtype C consensus sequence is retrieved using the HIV Sequence Alignments tool from the Los Alamos HIV sequence database (www.hiv.lanl.gov/content/sequence/NEWALIGN/align.html). We correct and translate the nucleotide alignment using the Codon Align Tool from the Los Alamos HIV sequence database (www.hiv.lanl.gov/content/sequence/CodonAlign/codonalign.html). The alignment positions are mapped to the corresponding HXB2 reference gene with Genbank accession ‘AAB50258.1’ (gag) using the alignment tool MAFFT (version 7.407) [50]. Ambiguous amino acids X in the env sequence are not considered and set to NA. Frameshifts and stop codons are disregarded and set to gaps. Each site in the sequence s with at least two frequent (1% prevalence) amino acid variants is selected as potential site sj under HLA pressure. For each frequent site and each hypothesis (HLA and baseline model), a multinomial Bayesian generalized linear mixed model is built, where each frequent amino acid is considered a class, and all non-frequent amino acids are grouped together to an ‘OTHER’ class. Bayesian generalized linear mixed models. We model the conditional probabilities for site adaptation (see Eqs 2 and 3) using separate multinomial Bayesian generalized linear mixed models (GLMMs). GLMMs are tailored for data with non-normal response distributions and dependency structures in the observations by combining the properties of generalized linear models (GLMs) [51, 52] and linear mixed models (LMMs). While GLMs model non-normal response distributions (such as binomial) via link functions of the means (e.g. logistic regression), LMMs enable to model not only population-level effects but also group-level effects assuming dependency structures in the samples. Mathematically, GLMMs have the following form excluding the residuals (ϵ) [53]: (4) where Y is the response variable, β and u the coefficients for the population and group-level effects, respectively, X and Z the corresponding design matrices and g(x) a link function relating the response Y to the linear predictor η. Thus, between-subject correlations, like the phylogenetic relatedness of some viruses, can be modeled as a group-level effect. While y, X and Z are given by the data, β and are unknown and have to be estimated. We use Markov chain Monte-Carlo (MCMC) based Bayesian GLMMs, since they are more robust and accurate in their parameter estimations of the group-level effects in contrast to classical maximum likelihood (ML) and restricted maximum likelihood (REML) methods [54]. In non-Bayesian frameworks, the group-level effect vector u is treated as part of the error term and thus likelihood computation requires the integration over the likelihood of all group-level effects, which might be analytically intractable for complex group-level structures [55]. In Bayesian settings where posterior distributions of the parameters are estimated by combining likelihood and prior distributions, both u and β are treated as parameters, allowing more accurate variance estimates for the group-level effects. We use the MCMC Bayesian GLMM implementation of the R [56] package brms [57] that provides an interface to the STAN software [58]. By implementing Hamiltonian Monte Carlo [59] and the No-U-Turn Sampler (NUTS) [60], Stan allows for faster convergence compared to conventional MCMC methods. Another advantage of Bayesian models is the possibility to include the prior information of the parameters into the model. The prior knowledge that only few HLA alleles have potential influence on a variant site [31, 35] is modeled using the horseshoe prior that has a global parameter τ shrinking most of the coefficients to zero and a local parameter λ, which is a heavy-tailed half-Cauchy (C+(0, 1)) prior, allowing some coefficients to escape the shrinkage [46]. Thus, the the horseshoe prior for the D population level coefficients β = (β1, …, βD) has the following form: (5) In addition, we regularize the horseshoe prior by setting the ratio of the expected number of non-zero coefficients to the expected number of zero coefficients to 10%. All other parameters of the horseshoe prior are set to default. For the remaining coefficients the default priors of the brm function are used (non or very weakly informative priors). Estimating per-site adaptation. For each frequent site, we model an HLA model (see Eq 2) and a baseline model (see Eq 3) using multinomial Bayesian generalized linear mixed models (GLMMs) as implemented by the brms package [57] in R [56]. Both models estimate the probability distribution of each site sj spanning over the space Y of all frequent amino acid variants (and ‘OTHER’ for the non-frequent variants’) conditioned on the potential confounders age, sex, and ethnicity. Age is defined as the interval between sample extraction date and birthday and scaled to mean 0 and variance 1. If missing, months and days are set to the first day and month, respectively. Due to the ambiguous recording of ethnicity groups, samples are assigned to either African, Caucasian, or ‘Other’ ethnicity. Sex is modeled as a binary feature. Though deep sequencing has been performed, we use for this study only the consensus sequences derived using a 10% prevalence cutoff, which is commonly used in the research community [61]. The NGS reads were mapped with a customized version of MinVar [62]. Predicting if a polymorphism is under HLA pressure or not is confounded by the phylogenetic relatedness of the viral sequences. As proposed by [63], the phylogeny of the viral sequences of the subjects is incorporated as group-level effect (1|subject) into the model using the option cov_ranef = list(subject = A). Here, A denotes the computed covariance-matrix of the phylogenetic tree calculated using the vcv.phylo function from the ape package. A phylogenetic tree is constructed based on the nucleotide sequences of the p24 protein from the chronic_lowCD4 data set using the RAxML software (version 8.2.12) [64] under the GTRGAMMA model. Thus, the formula to compute the HLA model taking all HLA alleles H as potential covariates into the model has the following form: (6) in contrast to the baseline model, which estimates the probability that the frequent site is not under HLA pressure: (7) The logistic function is used as a link function. As described in previous sections, the horseshoe prior is used on all population-level effects [46]. Alleles in H are converted to four digit resolution. Alleles with alternative expression (suffix ‘L’, ‘S’, ‘C’, ‘A’, or ‘Q’) are treated separately from the normally expressed allele. The complete call to compute the per-site models using the brms package is provided in the code repository. Calculation of adaptation score. We define the adaptation score, as proposed by [35], as: (8) where the per-site likelihood P(sj|H = h) and P(sj|H = ⌀) are defined by Eqs 2 and 3, respectively. For better interpretation, we also transform the estimated adaptation x = adapt(s, h) using a sigmoidal function g(x) to a range of -1 to 1 [35]: Thereby, a positive adaptation score denotes that the sequence has more likely occurred under HLA pressure than without, and vice versa. Logo computation. The adaptation score can be decomposed into the likelihood ratios per frequent variant sites (see Eq 8). Odds ratios above or below 1 indicate that either the polymorphism at the site sj is more likely to be under HLA pressure, or vice versa. We use this information to provide a visual logo depicting the amino acids that contributed most to the adaptation score. Therefore, only sites with odds ratios differing from 1 (and an offset of 0.01 to account for the variance) are considered. The contribution is scaled by the maximum contribution. In order to use the existing Weblogo 3.0 software to produce the logos [65], we create a pseudo-alignment with 100 sequences with length of the number of important sites. Each position in the alignment represents a polymorphism site. The sequences contain the polymorphism at this position with a frequency equal to the scaled contribution and a gap for the remaining sequences. Thus, the logo is a consensus logo for the pseudo-alignment. Statistical analyses We perform a one-sided Wilcoxon rank-sum test to compare the adaptation scores (i) between different data sets and (ii) with respect to different clinical characteristics. For settings, where the data is paired (random data set—same subjects, R5-FPR analysis—matched CD4 count, heterologous—autologous viruses), a one-sided Wilcoxon signed-rank test is performed. A significance threshold of 0.05 is set for all hypothesis tests. Ethical statement PBMC and plasma samples from HIV-1 positive donors were provided by Stellenbosch University with the written informed consent of the donors. Sample collection was approved under the following ethical statement “VIROLOGICAL AND IMMUNOLOGICAL CHARACTERIZATION OF CRYOPRESERVED BLOOD AND VIRUS SAMPLES” PROJECT NUMBER: NO7/06/13. Study cohort Patients (male and female) who attended Wellness, Antenatal and HIV Clinics in the Durbanville and Stellenbosch regions of the Western Cape were recruited. Only patients older than 18 years were selected. Most of the patients were assumed to be in the chronic stage of the infection. Inclusion was based on recent diagnosis of HIV-1 infection (within the previous 6 months). In total, samples from 329 HIV-infected individuals were available. Subtype C was confirmed for 317 of the 329 samples using the COMET Tool [47]. Patients on antiretroviral were excluded from the analysis, resulting in a total of 312 patients. Based on CD4+ T cell count, the data set was further reduced to 274 samples (see subsection on Data sets). For each patient, clinical parameters such as sex, age, ethnicity, CD4 count, and viral load were collected. In addition, the HIV-1 genes gag (p24) and env were sequenced and the patients’ HLA I and II genes were genotyped. Data sets We divide the newly sequenced study cohort of 312 samples based on a CD4+ T cell count cutoff of 500 cells/mm3 into a chronic_highCD4 data set (n = 38) and a chronic_lowCD4 count data set (n = 274). High CD4+ T cell count indicates a stronger immune system. Since infection duration is not known for the patients, a high CD4+ T cell count might indicate that the patients have been infected for a shorter time (less chronic). Moreover, a virus is assumed to be less adapted to a host with a strong immune system compared to a host with a weak immune system. Thus, the adaptation model is only trained on the chronic_lowCD4 data set. In addition, we create an artificial data set (random) based on the chronic_lowCD4 data set, where the HLA alleles per HLA gene and haplotype have been randomized 100 times. HLA adaptation for this random data set is predicted with models based on the chronic_lowCD4 data set as well. For further validation of the adaptation model, we estimate HIV-1 adaptation of publicly available cohort of acutely-infected HIV-1 patients (n = 23) from the Los Alamos HIV sequence database (http://www.hiv.lanl.gov). The acute data set comprised the p24 sequence as well as the HLA I information of 23 patients with the following accession numbers GQ275453, GQ275750, GQ275852, GQ275894, KM192425, KM192440, KM192471, KM192536, KM192566, KM192640, KM192653, KM192674, KM192686, KM192702, KM192762, KM192844, KM192856, KM192870, KM192884, KM192912, KM192942, KM192970, KM192998. Since only the HLA I profile was available, we build an adaptation model based only on the HLA I profile for this purpose (n = 274). Molecular methods HIV status was confirmed with a serological test (Architect HIV Ab/Ab Combo, 3rd generation) on serum according to the manufacturer protocol. After the surface staining of PBMCs by incubation with a monoclonal mouse anti-human antibody coupled to fluorescent dyes, the quantification of cells expressing the CD4 antigen was measured by FACS analysis. Acquisition and analysis was performed on FACs flow cytometer using Cell Quest software. HIV-1 deep sequencing was performed using previous described protocols [48]. Analysis of deep sequencing data was performed using an internally-developed analysis pipeline, where sequence reads in the form of FASTQ files were processed and aligned via a multi-step method. HLA genotyping was performed using the following protocol. Genomic DNA was isolated from 200 μl of EDTA-anticoagulated blood using the QIAamp DNA Blood Mini Kit (QIAGEN, Hilden, Germany). Long-range PCR primers amplified the full-length of HLA class I genes (A, B, C) from 5’- to 3’-UTR. Class II genes (DPB1, DQB1, DRB1) were amplified from exon 2 to 3’-UTR. Fragment sizes were estimated to be around 3000 bp for Class I genes and 6000 bp for Class II genes, respectively. The PCR solution contained 1 x Phusion GC buffer (including 1.5 mM MgCl2), 200 μM dNTPs, 1 M Betaine, 8 μg Bovine Serum Albumin (BSA), 0.4 U Phusion Hot Start II High-Fidelity DNA Polymerase (Finnzymes, Vantaa, Finland), 0.5 μM of each primer and 90 ng of DNA in a total volume of 20 μl. After initial denaturation at 98°C for 1 minute, 35 cycles of 98°C for 10 seconds, 65°C for 20 seconds, and 72°C for 4 minutes were performed, followed by a final extension at 72°C for 20 minutes. Agarose gel electrophoresis was used to confirm amplification and correct fragment size as well as to check for non-specific product contamination. The 3 HLA class I and class II amplicons for each individual were pooled and afterwards purified with the Agencourt AMPure XP system (Agentcourt Bioscience, Beverly, MA, United States) according to the manufacturer’s protocol to inactivate unconsumed dNTPs and to eliminate extraneous primers before library preparation. These pooled amplicons then comprised a single sample. Concentrations were measured on a FLUOstar OPTIMA microplate fluorimeter (BMP LABTECH, Ortenberg, Germany) using the Quant-iT PicoGreen assay (Invitrogen, Carlsbad, CA, United States). Sample libraries for NGS were then prepared with the Nextera XT DNA Sample Prep Kit (Illumina, San Diego, CA, United States) according to the manufacturer’s protocol, including distinct DNA fragmentation, end-polishing, and adaptor-ligation steps. Through the adaptor, every sample was finally labeled with a unique identifier sequence. Sequencing was carried out then on the Illumina MiSeq Personal Sequencer (Illumina, San Diego, CA, United States) as described by the manufacturer. Coreceptor prediction Coreceptor usage is predicted using the well-established tool geno2pheno[coreceptor] [17] on the viral envelope sequences. The provided false-positive rate (FPR) corresponds to the confidence with which the sequence is classified as X4-capable. The higher the FPR, the more likely the sequence is not X4-capable, but R5. Viral strains with an FPR cutoff less than 20% are classified as X4-capable, otherwise as R5-capable according to the European Consensus Group on clinical management of HIV-1 tropism testing [49]. Estimating HLA adaptation Assuming independence of all sites in the viral sequence, we define the adaptation of a sequence to its host HLA profile as the adaptation of each frequent single amino acid site in the sequence to the HLA profile. Moreover, though every patient is infected by a quasispecies of viruses, we only consider the consensus sequence as in previous approaches. In order to correct for potential phylogenetic relatedness of the viral sequences used within the model as proposed by [37], we also incorporate the phylogeny of the viral sequences into the model. Thus, our model requires the amino acid sequences of the viral p24 protein, the corresponding host’s HLA I and II alleles, and the phylogeny between the viral sequences for learning the HLA adaptation (training). For each frequent site, we infer a model (HLA model) to estimate the likelihood that the site is under HLA pressure as well as a hypothetical model (baseline model) that computes the likelihood that the site is not under HLA pressure. HLA adaptation of the complete protein is then defined as a function over the product of the per-site likelihood ratios of the HLA model against the baseline model. Each per-site model is built using multinomial Bayesian generalized linear mixed models (GLMMs). In the following, we formalize the per-site model and the final adaptation score. Afterwards, we present the selection process of the frequent sites. Since each per-site model is built using Bayesian GLMMs, we provide a brief introduction to Bayesian GLMMs and their benefit over classical GLMMs and phylogeny-corrected LMMs. In addition, we provide a section on the model specification for each per-site model. Notation. Let S be a random variable representing the set of all possible HIV-1 amino acid sequences of a particular protein of length L. A particular sequence s is a realization of S covering all sites l = 1, …, L of the protein. A particular site sl can be realized by any amino acid (aa). Since we do not have enough power to find an HLA-restricted polymorphism at a very conserved site, we restrict the sites to m frequent single amino acid sites sj with j = 1, …, m, which are defined by sites that vary over the set of all HIV-1 sequences in their amino acid realization. A site is defined as frequent, if the particular amino acid is observed in at least 1% of the sequences. The host immune system is represented by the HLA alleles of the HLA I and HLA II genes. The HLA profile of an individual consists in our case of six (homozygous in all genes) to 12 (heterozygous in all genes) different HLA alleles. Let H represent the set of all possible HLA I and HLA II alleles. A particular HLA profile h is encoded as a binary vector with zeros everywhere, apart from the positions corresponding to the HLA alleles of the HLA profile. Note, thereby homozygosity is not modeled. We model adaptation as the conditional probability that a sequence s occurs under pressure from the host HLA profile similarly to [35]: (1) Assuming independence among sites and relevance of only frequent sites, the conditional probability over the sequence s can be decomposed to the product over the conditional probabilities over all m frequent sites sj (per-site model): (2) Similarly, a hypothetical model estimating the likelihood of the sequence s without any HLA pressure is defined as: (3) The conditional probabilities P(sj|H = h) and P(sj|H = ⌀) for each site are referred to as the HLA model and the baseline model, respectively. Identification and encoding of frequent sites. Since all patients in the study are infected with the subtype C variant of HIV-1, we align all nucleotide sequences to the subtype C consensus sequence using the alignment tool MAFFT (version 7.407) [50]. The subtype C consensus sequence is retrieved using the HIV Sequence Alignments tool from the Los Alamos HIV sequence database (www.hiv.lanl.gov/content/sequence/NEWALIGN/align.html). We correct and translate the nucleotide alignment using the Codon Align Tool from the Los Alamos HIV sequence database (www.hiv.lanl.gov/content/sequence/CodonAlign/codonalign.html). The alignment positions are mapped to the corresponding HXB2 reference gene with Genbank accession ‘AAB50258.1’ (gag) using the alignment tool MAFFT (version 7.407) [50]. Ambiguous amino acids X in the env sequence are not considered and set to NA. Frameshifts and stop codons are disregarded and set to gaps. Each site in the sequence s with at least two frequent (1% prevalence) amino acid variants is selected as potential site sj under HLA pressure. For each frequent site and each hypothesis (HLA and baseline model), a multinomial Bayesian generalized linear mixed model is built, where each frequent amino acid is considered a class, and all non-frequent amino acids are grouped together to an ‘OTHER’ class. Bayesian generalized linear mixed models. We model the conditional probabilities for site adaptation (see Eqs 2 and 3) using separate multinomial Bayesian generalized linear mixed models (GLMMs). GLMMs are tailored for data with non-normal response distributions and dependency structures in the observations by combining the properties of generalized linear models (GLMs) [51, 52] and linear mixed models (LMMs). While GLMs model non-normal response distributions (such as binomial) via link functions of the means (e.g. logistic regression), LMMs enable to model not only population-level effects but also group-level effects assuming dependency structures in the samples. Mathematically, GLMMs have the following form excluding the residuals (ϵ) [53]: (4) where Y is the response variable, β and u the coefficients for the population and group-level effects, respectively, X and Z the corresponding design matrices and g(x) a link function relating the response Y to the linear predictor η. Thus, between-subject correlations, like the phylogenetic relatedness of some viruses, can be modeled as a group-level effect. While y, X and Z are given by the data, β and are unknown and have to be estimated. We use Markov chain Monte-Carlo (MCMC) based Bayesian GLMMs, since they are more robust and accurate in their parameter estimations of the group-level effects in contrast to classical maximum likelihood (ML) and restricted maximum likelihood (REML) methods [54]. In non-Bayesian frameworks, the group-level effect vector u is treated as part of the error term and thus likelihood computation requires the integration over the likelihood of all group-level effects, which might be analytically intractable for complex group-level structures [55]. In Bayesian settings where posterior distributions of the parameters are estimated by combining likelihood and prior distributions, both u and β are treated as parameters, allowing more accurate variance estimates for the group-level effects. We use the MCMC Bayesian GLMM implementation of the R [56] package brms [57] that provides an interface to the STAN software [58]. By implementing Hamiltonian Monte Carlo [59] and the No-U-Turn Sampler (NUTS) [60], Stan allows for faster convergence compared to conventional MCMC methods. Another advantage of Bayesian models is the possibility to include the prior information of the parameters into the model. The prior knowledge that only few HLA alleles have potential influence on a variant site [31, 35] is modeled using the horseshoe prior that has a global parameter τ shrinking most of the coefficients to zero and a local parameter λ, which is a heavy-tailed half-Cauchy (C+(0, 1)) prior, allowing some coefficients to escape the shrinkage [46]. Thus, the the horseshoe prior for the D population level coefficients β = (β1, …, βD) has the following form: (5) In addition, we regularize the horseshoe prior by setting the ratio of the expected number of non-zero coefficients to the expected number of zero coefficients to 10%. All other parameters of the horseshoe prior are set to default. For the remaining coefficients the default priors of the brm function are used (non or very weakly informative priors). Estimating per-site adaptation. For each frequent site, we model an HLA model (see Eq 2) and a baseline model (see Eq 3) using multinomial Bayesian generalized linear mixed models (GLMMs) as implemented by the brms package [57] in R [56]. Both models estimate the probability distribution of each site sj spanning over the space Y of all frequent amino acid variants (and ‘OTHER’ for the non-frequent variants’) conditioned on the potential confounders age, sex, and ethnicity. Age is defined as the interval between sample extraction date and birthday and scaled to mean 0 and variance 1. If missing, months and days are set to the first day and month, respectively. Due to the ambiguous recording of ethnicity groups, samples are assigned to either African, Caucasian, or ‘Other’ ethnicity. Sex is modeled as a binary feature. Though deep sequencing has been performed, we use for this study only the consensus sequences derived using a 10% prevalence cutoff, which is commonly used in the research community [61]. The NGS reads were mapped with a customized version of MinVar [62]. Predicting if a polymorphism is under HLA pressure or not is confounded by the phylogenetic relatedness of the viral sequences. As proposed by [63], the phylogeny of the viral sequences of the subjects is incorporated as group-level effect (1|subject) into the model using the option cov_ranef = list(subject = A). Here, A denotes the computed covariance-matrix of the phylogenetic tree calculated using the vcv.phylo function from the ape package. A phylogenetic tree is constructed based on the nucleotide sequences of the p24 protein from the chronic_lowCD4 data set using the RAxML software (version 8.2.12) [64] under the GTRGAMMA model. Thus, the formula to compute the HLA model taking all HLA alleles H as potential covariates into the model has the following form: (6) in contrast to the baseline model, which estimates the probability that the frequent site is not under HLA pressure: (7) The logistic function is used as a link function. As described in previous sections, the horseshoe prior is used on all population-level effects [46]. Alleles in H are converted to four digit resolution. Alleles with alternative expression (suffix ‘L’, ‘S’, ‘C’, ‘A’, or ‘Q’) are treated separately from the normally expressed allele. The complete call to compute the per-site models using the brms package is provided in the code repository. Calculation of adaptation score. We define the adaptation score, as proposed by [35], as: (8) where the per-site likelihood P(sj|H = h) and P(sj|H = ⌀) are defined by Eqs 2 and 3, respectively. For better interpretation, we also transform the estimated adaptation x = adapt(s, h) using a sigmoidal function g(x) to a range of -1 to 1 [35]: Thereby, a positive adaptation score denotes that the sequence has more likely occurred under HLA pressure than without, and vice versa. Logo computation. The adaptation score can be decomposed into the likelihood ratios per frequent variant sites (see Eq 8). Odds ratios above or below 1 indicate that either the polymorphism at the site sj is more likely to be under HLA pressure, or vice versa. We use this information to provide a visual logo depicting the amino acids that contributed most to the adaptation score. Therefore, only sites with odds ratios differing from 1 (and an offset of 0.01 to account for the variance) are considered. The contribution is scaled by the maximum contribution. In order to use the existing Weblogo 3.0 software to produce the logos [65], we create a pseudo-alignment with 100 sequences with length of the number of important sites. Each position in the alignment represents a polymorphism site. The sequences contain the polymorphism at this position with a frequency equal to the scaled contribution and a gap for the remaining sequences. Thus, the logo is a consensus logo for the pseudo-alignment. Notation. Let S be a random variable representing the set of all possible HIV-1 amino acid sequences of a particular protein of length L. A particular sequence s is a realization of S covering all sites l = 1, …, L of the protein. A particular site sl can be realized by any amino acid (aa). Since we do not have enough power to find an HLA-restricted polymorphism at a very conserved site, we restrict the sites to m frequent single amino acid sites sj with j = 1, …, m, which are defined by sites that vary over the set of all HIV-1 sequences in their amino acid realization. A site is defined as frequent, if the particular amino acid is observed in at least 1% of the sequences. The host immune system is represented by the HLA alleles of the HLA I and HLA II genes. The HLA profile of an individual consists in our case of six (homozygous in all genes) to 12 (heterozygous in all genes) different HLA alleles. Let H represent the set of all possible HLA I and HLA II alleles. A particular HLA profile h is encoded as a binary vector with zeros everywhere, apart from the positions corresponding to the HLA alleles of the HLA profile. Note, thereby homozygosity is not modeled. We model adaptation as the conditional probability that a sequence s occurs under pressure from the host HLA profile similarly to [35]: (1) Assuming independence among sites and relevance of only frequent sites, the conditional probability over the sequence s can be decomposed to the product over the conditional probabilities over all m frequent sites sj (per-site model): (2) Similarly, a hypothetical model estimating the likelihood of the sequence s without any HLA pressure is defined as: (3) The conditional probabilities P(sj|H = h) and P(sj|H = ⌀) for each site are referred to as the HLA model and the baseline model, respectively. Identification and encoding of frequent sites. Since all patients in the study are infected with the subtype C variant of HIV-1, we align all nucleotide sequences to the subtype C consensus sequence using the alignment tool MAFFT (version 7.407) [50]. The subtype C consensus sequence is retrieved using the HIV Sequence Alignments tool from the Los Alamos HIV sequence database (www.hiv.lanl.gov/content/sequence/NEWALIGN/align.html). We correct and translate the nucleotide alignment using the Codon Align Tool from the Los Alamos HIV sequence database (www.hiv.lanl.gov/content/sequence/CodonAlign/codonalign.html). The alignment positions are mapped to the corresponding HXB2 reference gene with Genbank accession ‘AAB50258.1’ (gag) using the alignment tool MAFFT (version 7.407) [50]. Ambiguous amino acids X in the env sequence are not considered and set to NA. Frameshifts and stop codons are disregarded and set to gaps. Each site in the sequence s with at least two frequent (1% prevalence) amino acid variants is selected as potential site sj under HLA pressure. For each frequent site and each hypothesis (HLA and baseline model), a multinomial Bayesian generalized linear mixed model is built, where each frequent amino acid is considered a class, and all non-frequent amino acids are grouped together to an ‘OTHER’ class. Bayesian generalized linear mixed models. We model the conditional probabilities for site adaptation (see Eqs 2 and 3) using separate multinomial Bayesian generalized linear mixed models (GLMMs). GLMMs are tailored for data with non-normal response distributions and dependency structures in the observations by combining the properties of generalized linear models (GLMs) [51, 52] and linear mixed models (LMMs). While GLMs model non-normal response distributions (such as binomial) via link functions of the means (e.g. logistic regression), LMMs enable to model not only population-level effects but also group-level effects assuming dependency structures in the samples. Mathematically, GLMMs have the following form excluding the residuals (ϵ) [53]: (4) where Y is the response variable, β and u the coefficients for the population and group-level effects, respectively, X and Z the corresponding design matrices and g(x) a link function relating the response Y to the linear predictor η. Thus, between-subject correlations, like the phylogenetic relatedness of some viruses, can be modeled as a group-level effect. While y, X and Z are given by the data, β and are unknown and have to be estimated. We use Markov chain Monte-Carlo (MCMC) based Bayesian GLMMs, since they are more robust and accurate in their parameter estimations of the group-level effects in contrast to classical maximum likelihood (ML) and restricted maximum likelihood (REML) methods [54]. In non-Bayesian frameworks, the group-level effect vector u is treated as part of the error term and thus likelihood computation requires the integration over the likelihood of all group-level effects, which might be analytically intractable for complex group-level structures [55]. In Bayesian settings where posterior distributions of the parameters are estimated by combining likelihood and prior distributions, both u and β are treated as parameters, allowing more accurate variance estimates for the group-level effects. We use the MCMC Bayesian GLMM implementation of the R [56] package brms [57] that provides an interface to the STAN software [58]. By implementing Hamiltonian Monte Carlo [59] and the No-U-Turn Sampler (NUTS) [60], Stan allows for faster convergence compared to conventional MCMC methods. Another advantage of Bayesian models is the possibility to include the prior information of the parameters into the model. The prior knowledge that only few HLA alleles have potential influence on a variant site [31, 35] is modeled using the horseshoe prior that has a global parameter τ shrinking most of the coefficients to zero and a local parameter λ, which is a heavy-tailed half-Cauchy (C+(0, 1)) prior, allowing some coefficients to escape the shrinkage [46]. Thus, the the horseshoe prior for the D population level coefficients β = (β1, …, βD) has the following form: (5) In addition, we regularize the horseshoe prior by setting the ratio of the expected number of non-zero coefficients to the expected number of zero coefficients to 10%. All other parameters of the horseshoe prior are set to default. For the remaining coefficients the default priors of the brm function are used (non or very weakly informative priors). Estimating per-site adaptation. For each frequent site, we model an HLA model (see Eq 2) and a baseline model (see Eq 3) using multinomial Bayesian generalized linear mixed models (GLMMs) as implemented by the brms package [57] in R [56]. Both models estimate the probability distribution of each site sj spanning over the space Y of all frequent amino acid variants (and ‘OTHER’ for the non-frequent variants’) conditioned on the potential confounders age, sex, and ethnicity. Age is defined as the interval between sample extraction date and birthday and scaled to mean 0 and variance 1. If missing, months and days are set to the first day and month, respectively. Due to the ambiguous recording of ethnicity groups, samples are assigned to either African, Caucasian, or ‘Other’ ethnicity. Sex is modeled as a binary feature. Though deep sequencing has been performed, we use for this study only the consensus sequences derived using a 10% prevalence cutoff, which is commonly used in the research community [61]. The NGS reads were mapped with a customized version of MinVar [62]. Predicting if a polymorphism is under HLA pressure or not is confounded by the phylogenetic relatedness of the viral sequences. As proposed by [63], the phylogeny of the viral sequences of the subjects is incorporated as group-level effect (1|subject) into the model using the option cov_ranef = list(subject = A). Here, A denotes the computed covariance-matrix of the phylogenetic tree calculated using the vcv.phylo function from the ape package. A phylogenetic tree is constructed based on the nucleotide sequences of the p24 protein from the chronic_lowCD4 data set using the RAxML software (version 8.2.12) [64] under the GTRGAMMA model. Thus, the formula to compute the HLA model taking all HLA alleles H as potential covariates into the model has the following form: (6) in contrast to the baseline model, which estimates the probability that the frequent site is not under HLA pressure: (7) The logistic function is used as a link function. As described in previous sections, the horseshoe prior is used on all population-level effects [46]. Alleles in H are converted to four digit resolution. Alleles with alternative expression (suffix ‘L’, ‘S’, ‘C’, ‘A’, or ‘Q’) are treated separately from the normally expressed allele. The complete call to compute the per-site models using the brms package is provided in the code repository. Calculation of adaptation score. We define the adaptation score, as proposed by [35], as: (8) where the per-site likelihood P(sj|H = h) and P(sj|H = ⌀) are defined by Eqs 2 and 3, respectively. For better interpretation, we also transform the estimated adaptation x = adapt(s, h) using a sigmoidal function g(x) to a range of -1 to 1 [35]: Thereby, a positive adaptation score denotes that the sequence has more likely occurred under HLA pressure than without, and vice versa. Logo computation. The adaptation score can be decomposed into the likelihood ratios per frequent variant sites (see Eq 8). Odds ratios above or below 1 indicate that either the polymorphism at the site sj is more likely to be under HLA pressure, or vice versa. We use this information to provide a visual logo depicting the amino acids that contributed most to the adaptation score. Therefore, only sites with odds ratios differing from 1 (and an offset of 0.01 to account for the variance) are considered. The contribution is scaled by the maximum contribution. In order to use the existing Weblogo 3.0 software to produce the logos [65], we create a pseudo-alignment with 100 sequences with length of the number of important sites. Each position in the alignment represents a polymorphism site. The sequences contain the polymorphism at this position with a frequency equal to the scaled contribution and a gap for the remaining sequences. Thus, the logo is a consensus logo for the pseudo-alignment. Statistical analyses We perform a one-sided Wilcoxon rank-sum test to compare the adaptation scores (i) between different data sets and (ii) with respect to different clinical characteristics. For settings, where the data is paired (random data set—same subjects, R5-FPR analysis—matched CD4 count, heterologous—autologous viruses), a one-sided Wilcoxon signed-rank test is performed. A significance threshold of 0.05 is set for all hypothesis tests. Results and discussion Validation of the adaptation score We trained our adaptation model on data from a cohort consisting of 274 chronically-infected, untreated, subtype C, HIV-1 patients, all having a CD4+ T cell count less than 500 cells/mm3 and on average a log viral load of 4.87 (’chronic_lowCD4’ data set). In addition, 38 samples from the same study cohort with a CD4+ T cell count above 500 cells/mm3 (’chronic_highCD4’ data set) were available. Apart from the CD4+ T cell count, the two data sets are comparable with regard to potential confounders and clinical variables (see Table 1). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Summary statistics for the variables of interest for both data sets. https://doi.org/10.1371/journal.pcbi.1010355.t001 Performing several runs of 10-fold cross-validation revealed that the predicted adaptation score is quite robust, changing with an average standard deviation around 0.1. Though there exists no ground truth for HLA adaptation, we set some requirements that a valid adaptation score should fulfill, which can be seen in the following. Study cohort contains HLA adapted sequences. We assume that by construction the study cohort should harbor some HLA adapted sequences. 62% of the samples from the chronic_lowCD4 data set (n = 274) are estimated to be adapted (adaptation score >0.1), compared to 47% of the chronic_highCD4 data set (n = 38). The adaptation scores of the chronic_lowCD4 data set are taken from a 10-fold cross-validation, while the adaptation scores of the chronic_highCD4 data set are predicted using the full chronic_lowCD4 data set for training. Fig 1A shows the distribution of the adaptation score in the chronic_lowCD4 data set and the chronic_highCD4 data set. Note, there is however no ground truth on the true HLA adaptation status of the virus in the cohort. In the following, we analyze the predicted adaptation score distribution between different cohorts. Statistically, HIV-1 isolates of patients with CD4+ T cell count below 500 (chronic_lowCD4 data set) are significantly more adapted than patients with higher CD4+ T cell count (one-sided, unpaired Wilcoxon rank-sum test, p-value = 1.97e-2). The comparison of the chronic_lowCD4 data set with the chronic_highCD4 data set is however not straightforward. On the one hand, the size of the chronic_highCD4 data set is quite small compared to the chronic_lowCD4 data set. On the other hand, while we exclude the patients with the higher CD4+ T cell count from the training process as a precaution because they might be less chronic, this assumption does not have to be true and the samples cannot be treated to test the hypothesis that chronically-infected patients have more adapted viruses compared to patients with shorter infection duration. Last but not least, HLA-1 adapted viruses are assumed to escape the CTL response, resulting in fewer infected CD4+ T cells being killed. As a consequence, it is not necessarily the case that patients with higher CD4+ T cell count have less adapted viruses compared to patients with a lower CD4+ T cell count. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Histograms of the predicted adaptation scores of the chronic lowCD4 data set compared to four other data sets. A: Histogram of the adaptation scores of the chronic_lowCD4 data set (red) and the chronic_highCD4 data set (turquoise). Dashed line represents the mean adaptation score per data set. The mean adaptation score is 0.19 for the chronic_lowCD4 data set and 0.05 for the chronic_highCD4 data set. B: Histogram of the adaptation scores of the chronic_lowCD4 data set (red) and the the averaged adaptation scores of the random data set (turquoise). Dashed line represents the mean adaptation score per data set. The mean adaptation score is 0.19 for the chronic_lowCD4 data set and -0.67 for the random data set. C: Histogram of estimated adapatation score for each HLA profile and autologous and heterologous viruses. Estimated adaptation scores for each HLA profile and its autologous virus (red) and heterologous viruses of the cohort (turquoise). The adaptation scores of the heterologous viruses are averaged. Dashed line represents the mean adaptation score per data set. The mean adaptation score for autologous viruses is 0.19 and -0.12 for heterologous viruses. D: Histogram of the estimated adaptation scores for the chronically-infected data set (turquoise) and the acutely infected data set (red). Dashed line represents the mean adaptation score per data set. Comparison of HLA adaptation in acutely- and chronically-infected HIV-1 patients. The mean adaptation score is 0.20 for the chronic_lowCD4 data set based only on HLA I alleles and -0.24 for the acute data set, respectively. https://doi.org/10.1371/journal.pcbi.1010355.g001 Random HLA profile leads to non-adaptedness. We expect that viruses in the study cohort are more adapted to the host’s HLA profile than to a random HLA profile. Therefore, we predicted the HLA adaptation of the viral sequences of the cohort to a random HLA profile (100 times). Adaptation scores in the random data set are averaged per patient over 100 draws. Only 10% of the random samples (n = 274) are predicted to be adapted. As expected, the adaptation of the same virus to a randomized HLA profile is significantly lower than to its host HLA profile (one-sided, paired Wilcoxon signed-rank test, p-value = 1.50e-44). Fig 1B shows the distribution of the estimated adaptation scores for the random data set compared to the chronic_lowCD4 data set. Autologous viruses more adapted than heterologous viruses. We observed that the adaptation score of the harbored virus to its host (autologous virus) is higher (p-value = 1.48e-31) in contrast to the adaptation of the other viruses in the cohort to the same HLA profile (heterologous virus). This meets our expectation, since we define the adaptation score to reflect how likely the virus acquired escape mutations specific to the host HLA profile. Fig 1C shows the adaptation scores of the autologous virus and the averaged heterologous viruses for each subject (HLA profile). Viruses in acute phase less adapted than in chronic phase. We expect that viruses from acutely-infected HIV-1 patients should be less adapted than from chronically-infected HIV-1 patients due to the shorter exposure to the immune system. Fig 1D shows a histogram of the estimated adaptation scores for the acute and the chronic data sets. Since only the HLA I profile was available for the acute data set, we built an adaptation model based only on the HLA I profile for this purpose. We observed that viral strains from acutely infected patients have significantly lower estimated adaptation scores compared to the chronically-infected HIV-1 patients from our cohort (one-sided, unpaired Wilcoxon rank-sum test, p-value = 4.17e-5). Note that viruses from acutely-infected patients might also carry HLA-related escape mutations due to transmission. Validation of the per-site models Non-informative per-site models have no influence on the adaptation score. In contrast to the overall adaptation score, it is possible to evaluate the performance of the per-site models. This is useful for the interpretation and validation of the model but irrelevant for the quality of the adaptation score. For each frequent site, we compute the likelihood ratio of a model that estimates the likelihood that the site is under HLA pressure (HLA model) and a hypothetical model that assumes no HLA pressure (baseline). Thereby, the estimated per-site adaptations are directly adjusted by a baseline model and calibrated among all sites. Thus, including sites which are not under HLA pressure will more likely contribute with a factor of 1 to the overall adaptation score and, consequently, have no influence. This allows to take all frequent sites into consideration without any preselection or apriori knowledge. Note, by definition of the adaptation score, the adaptation of each frequent site contributes with the same weight to the overall adaptation score. All per-site models reached the Gelman-Rubin convergence criteria by having an Rhat value less than or equal to 1. Informative models learn HLA footprints. While it is not the focus of the study, we can identify sites with a likelihood ratio over 1, indicating a potential association between the frequent site and the HLA profile. In the study cohort, we identified 68 frequent sites in the p24 protein. Out of the corresponding 68 per-site HLA models, 21 had an averaged AUC under the precision-recall ROC curve higher than the averaged precision-recall baseline, where the precision-recall baseline is computed as the ratio of positive samples in the data set. Precision-recall was computed for each possible amino acid at a frequent site via 10-fold cross-validation. If models are evaluated by the performance to predict each frequent single amino acid polymorphism (SAP) separately, 52 models out of 210 perform better than the precision-recall baseline. Table 2 shows the top 10 polymorphisms with precision-recall AUC exceeding the baseline. Further analyzing the learned coefficients of the per-site models with high performance revealed that the models learned known footprints for subtype C such as the association between the T242N mutation and the HLA alleles HLA-B*57:01/02/03 or HLA-B*58:01 as well as the T186S escape mutation associated with HLA-B*81:01 [66–68]. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Top ten potential HLA-restricted sites and single amino acid polymorphisms (SAPs) with respect to precision-recall baseline performance. The performance of the HLA model at a specific site and for a specific SAP is computed as the AUC under the precision-recall curve (PRROC). https://doi.org/10.1371/journal.pcbi.1010355.t002 Interpretable adaptation score by providing logos for each virus. For each frequent variant site, an odds ratio above or below 1 (with an offset of 0.1) indicates whether the amino acid at this site is more likely under HLA pressure or not. This information can be used to compute a logo revealing the amino acids that contributed the most to the adaptation score. This information helps the user to understand the results for different inputs. Fig 2 shows the logo for the patient with the highest adaptation score in the cohort. The known HLA escape mutation T186S [69] has the highest contribution to the predicted adaptation score. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Logo for the patient with the highest adaptation score. The logo shows the viral polymorphisms that have the highest contribution to the adaptation score of this patient. Blue capital letters indicate adapted amino acids, while orange lowercase letters reflect non-adapted amino acids. The height of the letters reflects the contribution to the adaptation score and is scaled by the maximum contribution. The x-axis denotes the corresponding sites in the HXB2 virus. https://doi.org/10.1371/journal.pcbi.1010355.g002 HLA adaptation associated with CD4+ T cell count but not viral load We analyzed the estimated adaptation score with respect to viral load, CD4+ T cell count and coreceptor usage. On the one hand, we tested whether patients with adapted and non-adapted viruses differ in these variables, where adapted is defined as an adaptation score > 0.1 and non-adapted as an adaptation score < -0.1, based on the expected variance of 0.1 (see Fig 3). On the other hand, we analyzed whether viruses of patients with different known levels of these variables differ in their adaptation (see Fig 4). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Difference in clinical variables based on HLA adaption. Measurement of CD4 + T cell (CD4), logarithmized viral load (VL), FPR, and the FPR of R5 viruses matched based on their CD4 count (R5-FPR) stratified among adapted (red) and non-adapted(turquoise) viruses. https://doi.org/10.1371/journal.pcbi.1010355.g003 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Adaptation score for different levels of CD4+ T cell count (CD4) and coreceptor usage (coreceptor). https://doi.org/10.1371/journal.pcbi.1010355.g004 Though HLA class-I restricted polymorphism are known to be predictive for viral load and CD4+ T cell count in general [26, 70], we observed only a correlation between the estimated adaptation scores (based on HLA I and HLA II alleles) and the CD4+ count (Pearson correlation coefficient -0.16, p-value = 0.02) but not with viral load (0.04, p-value = 0.88). Note, however, that the study cohort consists of rather chronically-infected patients at a later stage of infection, where other factors more likely affect fluctuations in the viral load than the HLA adaptation, and a difference between controllers and non-controllers, for example, is not expected to be seen as in the beginning of the infection. We also observed that adapted viruses do not have statistically significant higher viral loads than non-adapted viruses (one-sided, unpaired Wilcoxon rank-sum test, p-value = 1.86e-1), and that patients with low viral load have not less adapted viruses (one-sided, unpaired Wilcoxon rank-sum test, p-value = 8.54e-2). In addition to the significant correlation between the CD4+ T cell count and adaptation score, we observed that patients with AIDS (CD4+ T cell count < 200) have more adapted viruses than patients with higher CD4+ T cell counts (one-sided, unpaired, Wilcoxon rank-sum test, p-value = 3.20e-3). CD4+ T cell count was also lower in patients with adapted viruses compared to non-adapted (Wilcoxon rank-sum test, p-value = 1.27e-3). Adaptation associated with coreceptor usage Using our adaptation score, we investigated the relationship between HLA adaptation and coreceptor usage. More precisely, we analyzed the hypothesis that high HLA adaptation might trigger the coreceptor switch in a similar way as a weak immune system (measured by a low number of CD4+ T cell counts). Coreceptor usage was determined with the false positive rate (FPR) of the coreceptor prediction tool geno2pheno[coreceptor] [17]. The provided FPR corresponds to the confidence with which the sequence is classified as X4-capable. The higher the FPR, the more likely the sequence is not X4-capable, but R5-capable. Table 3 shows the average adaptation scores stratified for coreceptor usage. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Averaged and maximum adaptation score stratified on the coreceptor usage and two data sets. https://doi.org/10.1371/journal.pcbi.1010355.t003 We observed a negative correlation between estimated adaptation score and corresponding FPR (Pearson correlation coefficient of -0.15, p-value = 0.03). This means that the more adapted the virus, the higher the likelihood that the virus is classified as X4-capable. This was further confirmed by the observation that X4-capable viruses are more adapted compared to R5 viruses (Wilcoxon rank-sum test, p-value = 1.34e-2, see Fig 4) and that, in general, adapted viruses have a lower FPR (rather X4 variants) compared to non-adapted viruses (Wilcoxon rank-sum test, p-value = 6.76e-3, see Fig 3). Note, since the variants are already determined as X4-capable, it is impossible to show if the emergence of X4-variants is driven by HLA adaptation. This analysis would require longitudinal data where the emergence of the coreceptor switch is captured. To rule out the possibility that higher adaptation of the X4 variants occurs due to longer exposure to the host immune system in contrast to R5 variants, the exact duration of infection is required. However, we observed that even among all R5 viruses, higher adaptation is associated with lower FPR, indicating that more adapted R5 samples might be closer to the coreceptor switch compared to non-adapted samples (one-sided paired Wilcoxon signed-rank test, p-value = 2.21e-2). Since the CD4+ T cell count is a major confounder for the coreceptor usage, we have matched for this test adapted and non-adapted R5 samples with similar CD4+ count (± 50 cells/mm3). Note, high adaptation of an R5 variant in a chronically-infected patient can also occur due to the long exposure to the immune system, since a coreceptor switch is only observed in 50% of the patients. Validation of the adaptation score We trained our adaptation model on data from a cohort consisting of 274 chronically-infected, untreated, subtype C, HIV-1 patients, all having a CD4+ T cell count less than 500 cells/mm3 and on average a log viral load of 4.87 (’chronic_lowCD4’ data set). In addition, 38 samples from the same study cohort with a CD4+ T cell count above 500 cells/mm3 (’chronic_highCD4’ data set) were available. Apart from the CD4+ T cell count, the two data sets are comparable with regard to potential confounders and clinical variables (see Table 1). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Summary statistics for the variables of interest for both data sets. https://doi.org/10.1371/journal.pcbi.1010355.t001 Performing several runs of 10-fold cross-validation revealed that the predicted adaptation score is quite robust, changing with an average standard deviation around 0.1. Though there exists no ground truth for HLA adaptation, we set some requirements that a valid adaptation score should fulfill, which can be seen in the following. Study cohort contains HLA adapted sequences. We assume that by construction the study cohort should harbor some HLA adapted sequences. 62% of the samples from the chronic_lowCD4 data set (n = 274) are estimated to be adapted (adaptation score >0.1), compared to 47% of the chronic_highCD4 data set (n = 38). The adaptation scores of the chronic_lowCD4 data set are taken from a 10-fold cross-validation, while the adaptation scores of the chronic_highCD4 data set are predicted using the full chronic_lowCD4 data set for training. Fig 1A shows the distribution of the adaptation score in the chronic_lowCD4 data set and the chronic_highCD4 data set. Note, there is however no ground truth on the true HLA adaptation status of the virus in the cohort. In the following, we analyze the predicted adaptation score distribution between different cohorts. Statistically, HIV-1 isolates of patients with CD4+ T cell count below 500 (chronic_lowCD4 data set) are significantly more adapted than patients with higher CD4+ T cell count (one-sided, unpaired Wilcoxon rank-sum test, p-value = 1.97e-2). The comparison of the chronic_lowCD4 data set with the chronic_highCD4 data set is however not straightforward. On the one hand, the size of the chronic_highCD4 data set is quite small compared to the chronic_lowCD4 data set. On the other hand, while we exclude the patients with the higher CD4+ T cell count from the training process as a precaution because they might be less chronic, this assumption does not have to be true and the samples cannot be treated to test the hypothesis that chronically-infected patients have more adapted viruses compared to patients with shorter infection duration. Last but not least, HLA-1 adapted viruses are assumed to escape the CTL response, resulting in fewer infected CD4+ T cells being killed. As a consequence, it is not necessarily the case that patients with higher CD4+ T cell count have less adapted viruses compared to patients with a lower CD4+ T cell count. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Histograms of the predicted adaptation scores of the chronic lowCD4 data set compared to four other data sets. A: Histogram of the adaptation scores of the chronic_lowCD4 data set (red) and the chronic_highCD4 data set (turquoise). Dashed line represents the mean adaptation score per data set. The mean adaptation score is 0.19 for the chronic_lowCD4 data set and 0.05 for the chronic_highCD4 data set. B: Histogram of the adaptation scores of the chronic_lowCD4 data set (red) and the the averaged adaptation scores of the random data set (turquoise). Dashed line represents the mean adaptation score per data set. The mean adaptation score is 0.19 for the chronic_lowCD4 data set and -0.67 for the random data set. C: Histogram of estimated adapatation score for each HLA profile and autologous and heterologous viruses. Estimated adaptation scores for each HLA profile and its autologous virus (red) and heterologous viruses of the cohort (turquoise). The adaptation scores of the heterologous viruses are averaged. Dashed line represents the mean adaptation score per data set. The mean adaptation score for autologous viruses is 0.19 and -0.12 for heterologous viruses. D: Histogram of the estimated adaptation scores for the chronically-infected data set (turquoise) and the acutely infected data set (red). Dashed line represents the mean adaptation score per data set. Comparison of HLA adaptation in acutely- and chronically-infected HIV-1 patients. The mean adaptation score is 0.20 for the chronic_lowCD4 data set based only on HLA I alleles and -0.24 for the acute data set, respectively. https://doi.org/10.1371/journal.pcbi.1010355.g001 Random HLA profile leads to non-adaptedness. We expect that viruses in the study cohort are more adapted to the host’s HLA profile than to a random HLA profile. Therefore, we predicted the HLA adaptation of the viral sequences of the cohort to a random HLA profile (100 times). Adaptation scores in the random data set are averaged per patient over 100 draws. Only 10% of the random samples (n = 274) are predicted to be adapted. As expected, the adaptation of the same virus to a randomized HLA profile is significantly lower than to its host HLA profile (one-sided, paired Wilcoxon signed-rank test, p-value = 1.50e-44). Fig 1B shows the distribution of the estimated adaptation scores for the random data set compared to the chronic_lowCD4 data set. Autologous viruses more adapted than heterologous viruses. We observed that the adaptation score of the harbored virus to its host (autologous virus) is higher (p-value = 1.48e-31) in contrast to the adaptation of the other viruses in the cohort to the same HLA profile (heterologous virus). This meets our expectation, since we define the adaptation score to reflect how likely the virus acquired escape mutations specific to the host HLA profile. Fig 1C shows the adaptation scores of the autologous virus and the averaged heterologous viruses for each subject (HLA profile). Viruses in acute phase less adapted than in chronic phase. We expect that viruses from acutely-infected HIV-1 patients should be less adapted than from chronically-infected HIV-1 patients due to the shorter exposure to the immune system. Fig 1D shows a histogram of the estimated adaptation scores for the acute and the chronic data sets. Since only the HLA I profile was available for the acute data set, we built an adaptation model based only on the HLA I profile for this purpose. We observed that viral strains from acutely infected patients have significantly lower estimated adaptation scores compared to the chronically-infected HIV-1 patients from our cohort (one-sided, unpaired Wilcoxon rank-sum test, p-value = 4.17e-5). Note that viruses from acutely-infected patients might also carry HLA-related escape mutations due to transmission. Study cohort contains HLA adapted sequences. We assume that by construction the study cohort should harbor some HLA adapted sequences. 62% of the samples from the chronic_lowCD4 data set (n = 274) are estimated to be adapted (adaptation score >0.1), compared to 47% of the chronic_highCD4 data set (n = 38). The adaptation scores of the chronic_lowCD4 data set are taken from a 10-fold cross-validation, while the adaptation scores of the chronic_highCD4 data set are predicted using the full chronic_lowCD4 data set for training. Fig 1A shows the distribution of the adaptation score in the chronic_lowCD4 data set and the chronic_highCD4 data set. Note, there is however no ground truth on the true HLA adaptation status of the virus in the cohort. In the following, we analyze the predicted adaptation score distribution between different cohorts. Statistically, HIV-1 isolates of patients with CD4+ T cell count below 500 (chronic_lowCD4 data set) are significantly more adapted than patients with higher CD4+ T cell count (one-sided, unpaired Wilcoxon rank-sum test, p-value = 1.97e-2). The comparison of the chronic_lowCD4 data set with the chronic_highCD4 data set is however not straightforward. On the one hand, the size of the chronic_highCD4 data set is quite small compared to the chronic_lowCD4 data set. On the other hand, while we exclude the patients with the higher CD4+ T cell count from the training process as a precaution because they might be less chronic, this assumption does not have to be true and the samples cannot be treated to test the hypothesis that chronically-infected patients have more adapted viruses compared to patients with shorter infection duration. Last but not least, HLA-1 adapted viruses are assumed to escape the CTL response, resulting in fewer infected CD4+ T cells being killed. As a consequence, it is not necessarily the case that patients with higher CD4+ T cell count have less adapted viruses compared to patients with a lower CD4+ T cell count. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Histograms of the predicted adaptation scores of the chronic lowCD4 data set compared to four other data sets. A: Histogram of the adaptation scores of the chronic_lowCD4 data set (red) and the chronic_highCD4 data set (turquoise). Dashed line represents the mean adaptation score per data set. The mean adaptation score is 0.19 for the chronic_lowCD4 data set and 0.05 for the chronic_highCD4 data set. B: Histogram of the adaptation scores of the chronic_lowCD4 data set (red) and the the averaged adaptation scores of the random data set (turquoise). Dashed line represents the mean adaptation score per data set. The mean adaptation score is 0.19 for the chronic_lowCD4 data set and -0.67 for the random data set. C: Histogram of estimated adapatation score for each HLA profile and autologous and heterologous viruses. Estimated adaptation scores for each HLA profile and its autologous virus (red) and heterologous viruses of the cohort (turquoise). The adaptation scores of the heterologous viruses are averaged. Dashed line represents the mean adaptation score per data set. The mean adaptation score for autologous viruses is 0.19 and -0.12 for heterologous viruses. D: Histogram of the estimated adaptation scores for the chronically-infected data set (turquoise) and the acutely infected data set (red). Dashed line represents the mean adaptation score per data set. Comparison of HLA adaptation in acutely- and chronically-infected HIV-1 patients. The mean adaptation score is 0.20 for the chronic_lowCD4 data set based only on HLA I alleles and -0.24 for the acute data set, respectively. https://doi.org/10.1371/journal.pcbi.1010355.g001 Random HLA profile leads to non-adaptedness. We expect that viruses in the study cohort are more adapted to the host’s HLA profile than to a random HLA profile. Therefore, we predicted the HLA adaptation of the viral sequences of the cohort to a random HLA profile (100 times). Adaptation scores in the random data set are averaged per patient over 100 draws. Only 10% of the random samples (n = 274) are predicted to be adapted. As expected, the adaptation of the same virus to a randomized HLA profile is significantly lower than to its host HLA profile (one-sided, paired Wilcoxon signed-rank test, p-value = 1.50e-44). Fig 1B shows the distribution of the estimated adaptation scores for the random data set compared to the chronic_lowCD4 data set. Autologous viruses more adapted than heterologous viruses. We observed that the adaptation score of the harbored virus to its host (autologous virus) is higher (p-value = 1.48e-31) in contrast to the adaptation of the other viruses in the cohort to the same HLA profile (heterologous virus). This meets our expectation, since we define the adaptation score to reflect how likely the virus acquired escape mutations specific to the host HLA profile. Fig 1C shows the adaptation scores of the autologous virus and the averaged heterologous viruses for each subject (HLA profile). Viruses in acute phase less adapted than in chronic phase. We expect that viruses from acutely-infected HIV-1 patients should be less adapted than from chronically-infected HIV-1 patients due to the shorter exposure to the immune system. Fig 1D shows a histogram of the estimated adaptation scores for the acute and the chronic data sets. Since only the HLA I profile was available for the acute data set, we built an adaptation model based only on the HLA I profile for this purpose. We observed that viral strains from acutely infected patients have significantly lower estimated adaptation scores compared to the chronically-infected HIV-1 patients from our cohort (one-sided, unpaired Wilcoxon rank-sum test, p-value = 4.17e-5). Note that viruses from acutely-infected patients might also carry HLA-related escape mutations due to transmission. Validation of the per-site models Non-informative per-site models have no influence on the adaptation score. In contrast to the overall adaptation score, it is possible to evaluate the performance of the per-site models. This is useful for the interpretation and validation of the model but irrelevant for the quality of the adaptation score. For each frequent site, we compute the likelihood ratio of a model that estimates the likelihood that the site is under HLA pressure (HLA model) and a hypothetical model that assumes no HLA pressure (baseline). Thereby, the estimated per-site adaptations are directly adjusted by a baseline model and calibrated among all sites. Thus, including sites which are not under HLA pressure will more likely contribute with a factor of 1 to the overall adaptation score and, consequently, have no influence. This allows to take all frequent sites into consideration without any preselection or apriori knowledge. Note, by definition of the adaptation score, the adaptation of each frequent site contributes with the same weight to the overall adaptation score. All per-site models reached the Gelman-Rubin convergence criteria by having an Rhat value less than or equal to 1. Informative models learn HLA footprints. While it is not the focus of the study, we can identify sites with a likelihood ratio over 1, indicating a potential association between the frequent site and the HLA profile. In the study cohort, we identified 68 frequent sites in the p24 protein. Out of the corresponding 68 per-site HLA models, 21 had an averaged AUC under the precision-recall ROC curve higher than the averaged precision-recall baseline, where the precision-recall baseline is computed as the ratio of positive samples in the data set. Precision-recall was computed for each possible amino acid at a frequent site via 10-fold cross-validation. If models are evaluated by the performance to predict each frequent single amino acid polymorphism (SAP) separately, 52 models out of 210 perform better than the precision-recall baseline. Table 2 shows the top 10 polymorphisms with precision-recall AUC exceeding the baseline. Further analyzing the learned coefficients of the per-site models with high performance revealed that the models learned known footprints for subtype C such as the association between the T242N mutation and the HLA alleles HLA-B*57:01/02/03 or HLA-B*58:01 as well as the T186S escape mutation associated with HLA-B*81:01 [66–68]. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Top ten potential HLA-restricted sites and single amino acid polymorphisms (SAPs) with respect to precision-recall baseline performance. The performance of the HLA model at a specific site and for a specific SAP is computed as the AUC under the precision-recall curve (PRROC). https://doi.org/10.1371/journal.pcbi.1010355.t002 Interpretable adaptation score by providing logos for each virus. For each frequent variant site, an odds ratio above or below 1 (with an offset of 0.1) indicates whether the amino acid at this site is more likely under HLA pressure or not. This information can be used to compute a logo revealing the amino acids that contributed the most to the adaptation score. This information helps the user to understand the results for different inputs. Fig 2 shows the logo for the patient with the highest adaptation score in the cohort. The known HLA escape mutation T186S [69] has the highest contribution to the predicted adaptation score. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Logo for the patient with the highest adaptation score. The logo shows the viral polymorphisms that have the highest contribution to the adaptation score of this patient. Blue capital letters indicate adapted amino acids, while orange lowercase letters reflect non-adapted amino acids. The height of the letters reflects the contribution to the adaptation score and is scaled by the maximum contribution. The x-axis denotes the corresponding sites in the HXB2 virus. https://doi.org/10.1371/journal.pcbi.1010355.g002 Non-informative per-site models have no influence on the adaptation score. In contrast to the overall adaptation score, it is possible to evaluate the performance of the per-site models. This is useful for the interpretation and validation of the model but irrelevant for the quality of the adaptation score. For each frequent site, we compute the likelihood ratio of a model that estimates the likelihood that the site is under HLA pressure (HLA model) and a hypothetical model that assumes no HLA pressure (baseline). Thereby, the estimated per-site adaptations are directly adjusted by a baseline model and calibrated among all sites. Thus, including sites which are not under HLA pressure will more likely contribute with a factor of 1 to the overall adaptation score and, consequently, have no influence. This allows to take all frequent sites into consideration without any preselection or apriori knowledge. Note, by definition of the adaptation score, the adaptation of each frequent site contributes with the same weight to the overall adaptation score. All per-site models reached the Gelman-Rubin convergence criteria by having an Rhat value less than or equal to 1. Informative models learn HLA footprints. While it is not the focus of the study, we can identify sites with a likelihood ratio over 1, indicating a potential association between the frequent site and the HLA profile. In the study cohort, we identified 68 frequent sites in the p24 protein. Out of the corresponding 68 per-site HLA models, 21 had an averaged AUC under the precision-recall ROC curve higher than the averaged precision-recall baseline, where the precision-recall baseline is computed as the ratio of positive samples in the data set. Precision-recall was computed for each possible amino acid at a frequent site via 10-fold cross-validation. If models are evaluated by the performance to predict each frequent single amino acid polymorphism (SAP) separately, 52 models out of 210 perform better than the precision-recall baseline. Table 2 shows the top 10 polymorphisms with precision-recall AUC exceeding the baseline. Further analyzing the learned coefficients of the per-site models with high performance revealed that the models learned known footprints for subtype C such as the association between the T242N mutation and the HLA alleles HLA-B*57:01/02/03 or HLA-B*58:01 as well as the T186S escape mutation associated with HLA-B*81:01 [66–68]. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Top ten potential HLA-restricted sites and single amino acid polymorphisms (SAPs) with respect to precision-recall baseline performance. The performance of the HLA model at a specific site and for a specific SAP is computed as the AUC under the precision-recall curve (PRROC). https://doi.org/10.1371/journal.pcbi.1010355.t002 Interpretable adaptation score by providing logos for each virus. For each frequent variant site, an odds ratio above or below 1 (with an offset of 0.1) indicates whether the amino acid at this site is more likely under HLA pressure or not. This information can be used to compute a logo revealing the amino acids that contributed the most to the adaptation score. This information helps the user to understand the results for different inputs. Fig 2 shows the logo for the patient with the highest adaptation score in the cohort. The known HLA escape mutation T186S [69] has the highest contribution to the predicted adaptation score. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Logo for the patient with the highest adaptation score. The logo shows the viral polymorphisms that have the highest contribution to the adaptation score of this patient. Blue capital letters indicate adapted amino acids, while orange lowercase letters reflect non-adapted amino acids. The height of the letters reflects the contribution to the adaptation score and is scaled by the maximum contribution. The x-axis denotes the corresponding sites in the HXB2 virus. https://doi.org/10.1371/journal.pcbi.1010355.g002 HLA adaptation associated with CD4+ T cell count but not viral load We analyzed the estimated adaptation score with respect to viral load, CD4+ T cell count and coreceptor usage. On the one hand, we tested whether patients with adapted and non-adapted viruses differ in these variables, where adapted is defined as an adaptation score > 0.1 and non-adapted as an adaptation score < -0.1, based on the expected variance of 0.1 (see Fig 3). On the other hand, we analyzed whether viruses of patients with different known levels of these variables differ in their adaptation (see Fig 4). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Difference in clinical variables based on HLA adaption. Measurement of CD4 + T cell (CD4), logarithmized viral load (VL), FPR, and the FPR of R5 viruses matched based on their CD4 count (R5-FPR) stratified among adapted (red) and non-adapted(turquoise) viruses. https://doi.org/10.1371/journal.pcbi.1010355.g003 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Adaptation score for different levels of CD4+ T cell count (CD4) and coreceptor usage (coreceptor). https://doi.org/10.1371/journal.pcbi.1010355.g004 Though HLA class-I restricted polymorphism are known to be predictive for viral load and CD4+ T cell count in general [26, 70], we observed only a correlation between the estimated adaptation scores (based on HLA I and HLA II alleles) and the CD4+ count (Pearson correlation coefficient -0.16, p-value = 0.02) but not with viral load (0.04, p-value = 0.88). Note, however, that the study cohort consists of rather chronically-infected patients at a later stage of infection, where other factors more likely affect fluctuations in the viral load than the HLA adaptation, and a difference between controllers and non-controllers, for example, is not expected to be seen as in the beginning of the infection. We also observed that adapted viruses do not have statistically significant higher viral loads than non-adapted viruses (one-sided, unpaired Wilcoxon rank-sum test, p-value = 1.86e-1), and that patients with low viral load have not less adapted viruses (one-sided, unpaired Wilcoxon rank-sum test, p-value = 8.54e-2). In addition to the significant correlation between the CD4+ T cell count and adaptation score, we observed that patients with AIDS (CD4+ T cell count < 200) have more adapted viruses than patients with higher CD4+ T cell counts (one-sided, unpaired, Wilcoxon rank-sum test, p-value = 3.20e-3). CD4+ T cell count was also lower in patients with adapted viruses compared to non-adapted (Wilcoxon rank-sum test, p-value = 1.27e-3). Adaptation associated with coreceptor usage Using our adaptation score, we investigated the relationship between HLA adaptation and coreceptor usage. More precisely, we analyzed the hypothesis that high HLA adaptation might trigger the coreceptor switch in a similar way as a weak immune system (measured by a low number of CD4+ T cell counts). Coreceptor usage was determined with the false positive rate (FPR) of the coreceptor prediction tool geno2pheno[coreceptor] [17]. The provided FPR corresponds to the confidence with which the sequence is classified as X4-capable. The higher the FPR, the more likely the sequence is not X4-capable, but R5-capable. Table 3 shows the average adaptation scores stratified for coreceptor usage. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Averaged and maximum adaptation score stratified on the coreceptor usage and two data sets. https://doi.org/10.1371/journal.pcbi.1010355.t003 We observed a negative correlation between estimated adaptation score and corresponding FPR (Pearson correlation coefficient of -0.15, p-value = 0.03). This means that the more adapted the virus, the higher the likelihood that the virus is classified as X4-capable. This was further confirmed by the observation that X4-capable viruses are more adapted compared to R5 viruses (Wilcoxon rank-sum test, p-value = 1.34e-2, see Fig 4) and that, in general, adapted viruses have a lower FPR (rather X4 variants) compared to non-adapted viruses (Wilcoxon rank-sum test, p-value = 6.76e-3, see Fig 3). Note, since the variants are already determined as X4-capable, it is impossible to show if the emergence of X4-variants is driven by HLA adaptation. This analysis would require longitudinal data where the emergence of the coreceptor switch is captured. To rule out the possibility that higher adaptation of the X4 variants occurs due to longer exposure to the host immune system in contrast to R5 variants, the exact duration of infection is required. However, we observed that even among all R5 viruses, higher adaptation is associated with lower FPR, indicating that more adapted R5 samples might be closer to the coreceptor switch compared to non-adapted samples (one-sided paired Wilcoxon signed-rank test, p-value = 2.21e-2). Since the CD4+ T cell count is a major confounder for the coreceptor usage, we have matched for this test adapted and non-adapted R5 samples with similar CD4+ count (± 50 cells/mm3). Note, high adaptation of an R5 variant in a chronically-infected patient can also occur due to the long exposure to the immune system, since a coreceptor switch is only observed in 50% of the patients. Conclusion Here, we introduced a novel computational approach to jointly estimate HLA I and HLA II adaptation of HIV-1 using Bayesian generalized linear mixed models. In addition, we presented a new study cohort of 312 treatment-naive, subtype C, chronically-infected HIV-1 patients from South Africa, where we sequenced the viral gag (and env) protein with corresponding HLA class I and II alleles for the training of our models. Apart from validating that our adaptation score inherited appropriate characteristics, we showed that the models underlying the adaptation score are biologically meaningful by learning well-known HLA-restricted polymorphisms. Using our approach and the data, we investigated the relationship between HLA adaptation and coreceptor usage of HIV-1, which had been unexplored up to now. We observed that X4-capable viruses are more adapted compared to R5-capable viruses (Wilcoxon rank-sum test, p-value = 1.34e-2). Moreover, even among all R5-capable viruses, higher adaptation is associated with lower FPR, indicating that more adapted R5 variants might be closer to the coreceptor switch compared to non-adapted variants (Wilcoxon signed-rank test, p-value = 2.21e-2). Thus, HLA adaptation might be another factor that should be considered prior to the administration of CCR5 coreceptor antagonists. It might also be useful in predicting how imminent the coreceptor switch is. In general, the estimated adaptation score allows to measure and understand HIV-1’s adaptation to the immune system. The adaptation score can be used to guide the design of suitable immunogens as vaccine targets by selecting sites that are more likely to be non-adapted to the immune system. Since the approach itself is not HIV-1 specific, the presented method can be also applied to study any virus-host adaptation. We encourage the usage of Bayesian GLMMs for modeling virus-host adaptation due to their ability to adjust for phylogenetic dependencies in the data and to handle highly overparameterized settings within the model. In light of current and potential future viral threats to mankind, such as SARS-CoV-2 or Ebola or MERS-CoV, this flexible, data non-intensive method can be useful to reveal and analyze virus-host dynamics of new viruses where little data is available. Future studies of the study cohort are required to further evaluate how the adaptation score is coherent with CTL escape experiments. While the study cohort was appropriate to learn HLA adaptation, it only allows to study the coreceptor switch and the role of HLA adaptation on it from a retrospective angle. Moreover, the number of CXCR4-using variants with intermediate to high CD4+ T cell count was very low. Consequently, a study cohort with longitudinal data on coreceptor usage and intermediate to high CD4+ T cell count would give additional insights. This would not only allow to investigate if HLA II adaptation occurs prior to the coreceptor switch, but also if the degree of adaptation is associated with the time until the coreceptor switch occurs. Note that the presented adaptation score here is a simple approach that can be easily optimized and extended in different ways. Given the available data, we restricted the analysis to subtype C infected patients and the p24 protein. However, our approach can also be applied to other subtypes and/or combined over different viral proteins. Further, we used the viral consensus sequences instead of the NGS sequences, since we aimed at predicting the adaptation per sequence. Still, the within-subject relatedness of different reads per virus could be easily incorporated into the Bayesian models. A larger data set might improve the current adaptation score by better representing the population with respect to the HLA repertoire and potential frequent variant sites, resulting in more informative per-site models. We assumed independence of all variant sites for the computation of the adaptation score as used in related work [35, 36]. Considering potential dependencies between variant sites would have raised the complexity of the model, which would have not been supported well by the small sample size. Therefore, we also did not perform independence tests. However, the proposed framework is easily extendable to support more dependency structures such as HLA linkage, or by relaxing the assumption of independence among all sites to capture compensatory mutations by incorporating additional random components. Another assumption is that each frequent variant site has the same probability to be under HLA pressure. Prior knowledge about common HLA epitopes can be added to the model by weighting the per-site likelihood odds ratios accordingly. However, if a site is more likely to be under HLA pressure, given by the underlying data, by construction of the adaptation score, the likelihood odds ratio should contribute with a higher factor to the overall adaptation score. Decomposing the adaptation score based on the potential adaptation of each frequent variant site is very advantageous with respect to model explainability and to settings, where little prior information exists. However, it requires the computation of two models per frequent variant sites, leading to a high number of models. Currently, the per-site models are not optimized with regard to parameter and predictor selection. To avoid overfitting and p-value based selection, we forced each model to capture our prior beliefs that the model should be unbiased with regard to sex, age, ethnicity and not be hampered by phylogenetic relatedness. While we ensured that the models are all converging according to the Rubin-Gelman criterion, we do not perform visual checks of the Bayesian GLMMs, such as prior and posterior predictive checks. Though this is a standard procedure for Bayesian GLMMs, it was not feasible in our case due to the high number of models. In our case, this was also not mandatory. Setting the horseshoe prior for the beta coefficients was based on our apriori knowledge that only few HLA alleles and clinical variables should have influence on a site. Setting potential non-optimal parameters might lead to non-informative per-site models. While we might lose some potential information for these sites, the quality of the overall adaptation score remains guaranteed by calibrating the per-site models with a baseline model. For computational reasons, it might be also beneficial to reduce the computation of the adaptation score based on only the informative per-site models. Acknowledgments Sample collection was funded by the Bill & Melinda Gates Foundation (Grant 38580 and OPP38580_01) for the project “Global HIV Vaccine Research Cryorepository—GHRC”.

Rothschild, Jeremy;Ma, Tianyi;Milstein, Joshua N.;Zilman, Anton

doi: 10.1371/journal.pcbi.1010868pmid: 38039342

Introduction Ecological competition is a ubiquitous feature of multi-species communities. It often manifests itself through direct antagonistic interactions between species, such as bacterial toxins, metabolic waste products and parasitic infections [1–3]. Competition also commonly occurs indirectly through various exploitative scenarios that deplete communal resources. Computational models of the dynamics of populations, framed in the context of a competition for finite system resources (e.g., light, food, population density, etc.) [4–12], have defined various heuristic measures of this competition for resources, such as the niche overlap, competition strength, and carrying capacity. Although these measures are commonly used to describe the dynamics of population growth and co-existence, a deeper understanding of the processes that govern the structure of ecological communities is acquired by exploring the mechanisms of the resource competition that underlie these coarse-grained, aggregate parameters [13–18]. Among the various resources required for population maintenance and growth, physical space is essential for expansion and access to additional resources [19–21]. In fact, individuals inherently require physical space for both their own growth and those of their progeny. Spatial competition can result in complicated patterning, synchrony of population distributions, spatial segregation into different niches within the environment and hosts, as well as other non-trivial dynamics [20, 22–25]. In bacterial communities, a variety of spatially ordered configurations may emerge from similarly distributed initial populations. This spatial structure plays an important role in medicine, industrial fabrication, and food production [26–30]. In bacterial biofilms, for instance, microbial populations form complex structures wherein various species segregate [31–33]. Different layers of bacterial species within a biofilm may have different sensitivity and resistance to antibiotics that restrict our ability to treat associated infections [34, 35]. The biogeography of bacteria in the digestive tract, which form the human digestive microbiome, illustrates another spatially heterogeneous ecology [36, 37]. In particular, the intestinal tract hosts diverse microbiota whose complex physical structures, such as mucus densities and epithelial crypts, have direct implications on the long-term composition of the bacterial community [38]. It is, therefore, necessary to understand how spatial constraints, arising from a confining environment and crowding/exclusion by other bacteria, shape the dynamics of each species and the overall patterning of the populations. Spatial constraints may also have ramifications for the overall ecological diversity. For instance, the boundary between expanding fronts of different bacterial populations, grown on solid substrates, fluctuates superdiffusively. This encourages accelerated genetic drift that may limit diversity more rapidly than neutral mutation models void of any spatial dynamics [39]. Alternatively, diversity may be increased in ecosystems wherein species undertake differing strategies in relation to the space they occupy—sometimes referred to as distinct spatial ‘niches’. For instance, trade-offs between motility and competitive ability may allow for coexistence between competing species [40]. It is well known that diversity may be strongly influenced by the invasion of external species. As initially noted by Gause et al., extinctions are frequently observed in a closed competitive ecosystem within a laboratory setting even though the similar ecosystem persists indefinitely in nature [41–44]. This suggests that invasion events, which implicitly rely on a partitioning of the space between the local and meta-community, contribute crucially to the population dynamics by reintroducing individuals into the ecosystem [45–48]. For instance, persistent diversity is observed through the fragmentation of continuous ecosystems in studies of patch-models of ecology and theories of island biogeography [16, 49–52]. The contest for space is critical in environments with small total populations, which accentuates the individual composition of the colony, such as within intestinal crypts [53, 54]. In recent years, microfluidic devices have started to provide controlled platforms for exploring the population dynamics of small bacterial microcolonies. Experiments in microfluidic monolayer devices, of various geometries, have shown that small populations of asymmetric bacteria, like E. coli, can align into highly ordered arrangements [55]. These populations behave differently from their well-mixed counterparts given the densely packed nature of their confinement [56, 57]. Cell morphology and the confining geometry are observed to greatly affect the ordering and fixation probability of cell populations in these devices [57–62]. Certain models, like the classical Moran model, subsume spatial factors into the well-mixed assumption that the species abundances are uniformly distributed in a certain location. Other models of ecological dynamics, such as island models, have incorporated aspects of special heterogeneity [63–65]. However, well-mixed continuous models that describe the concentration of bacterial populations generally neglect explicit spatial exclusion. Although many factors may influence competition for space, fundamentally cells in these densely packed populations must exclude each other through mechanical interactions. Consequently, modeling the competition of bacteria in small, confined environments requires explicit consideration of the system’s spatial configuration to correctly describe the competitive dynamics. We find that in confined environments, the spatial variation in species location may generate profound spatial disparities in reproductive opportunities. The collective division of cells engenders a phenomenon of skewed collective growth of the species domains, displacing a significant proportion of cells towards the peripheral boundaries, see Fig 1A. In this scenario, cells positioned in central regions of the space enjoy the advantage of prolonged reproductive success over several generations, while those situated at the environment’s periphery face rapid extinction. This spatial variation in reproductive value carries significant consequences for the process of ecological dynamics. Over time, it promotes the emergence of long-lived lineages originating from the central cells, while lineages from cells at the ecosystem edges die out faster. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Illustration of minimal models of ecological competition used in this paper. (A) Schematic portrayal of the skewed collective growth. Left: dividing cells push on their neighbors which results in preferential movement of the cells toward the ends of the channel. Right: the skewed collective growth results in the asymmetric dynamics of the inter-species boundary. (B) The classical Moran model within a well-mixed population of two species of bacteria. Change in the abundance, n, of a species relies on selecting an individual of that species to give birth while simultaneously having another individual die. (C) A spatial exclusion model for 1-dimensional competition within a micro-channel. Instead of a well-mixed population, the cell populations are segregated, each to one side of the open channel; see text. https://doi.org/10.1371/journal.pcbi.1010868.g001 Previous studies have examined quantitative aspects of this spatial exclusion on population diversity. Inspired by the somatic evolution processes, [66] examined one-dimensional pushing dynamics of a linear a chain of cells growing out of a compartment with one exit, modelling a mother machine geometry. The study showed that spatial exclusion slowed down somatic evolution and delayed the onset of cancer within tissues of multicellular organisms. However, the linear pushing process studied was uni-directional, with cells leaving the system only from one end of the lane. As a result, the founder cells at the closed end of the pushing queue remained indefinitely in the process, and in the final fixated state, all cells were progeny of one founder cell. Removing this physical constraint, a recent study explored the lineage diversity of a microbial population in a channel with two open ends [57], combining experiments and mathematical modeling. Focusing on the time of complete loss of lineage diversity in a system where at the initial configuration each cell belongs to a different species, the study revealed that lineage diversity can rapidly diminish within channels. However, the dynamics of fixation, extinction and invasion in a population of two species with fitness differences competing due to spatial exclusion in the one-dimensional space remain incompletely understood. In this paper, we systematically explore how spatial constraints imposed by cellular interactions influence the population dynamics of competing cellular species, confirming and expanding previous work. Inspired by the experimental setting of a microfluidic chemostat, we investigate how two species of bacteria compete through physical exclusion in an open, single lane microchannel [57, 67]. The paper is structured as follows. We first review a general model of competing well-mixed populations—the Moran model—before describing an extension to this model that incorporates spatial exclusion. We calculate the probabilities and the mean first-passage times of fixation for this spatial exclusion model. We then explore how fitness differences between the two species—for instance, differences in the bacterial growth rate or doubling time—shape the competitive landscape by affecting the time to and probability of fixation. Finally, the model is used to investigate invasion events that perturb a local community and the characteristics of a successful invasion. Models and methods We characterize the state of the system by the abundances of the competing species (i.e. the numbers of individuals of each species found in the ecosystem) which reflect the dynamics and the evolution of the community. We focus on the processes where the competition amounts to a zero-sum game: different populations compete for dominance under a constraint of finite total population size, N, determined by limitations of the inhabited space. Thus, in a two species system with a finite total population size studied here, fixing the abundance of the population of the first species, n, determines the abundance of the population of the second species, N − n, and both species compete to maintain their non-zero abundances in the system. The constraint of a fixed population size means that the dynamics of this 2 species system can be mapped onto a one-dimensional process defined by the abundance n. Several important models have been used to study the effects of different competition mechanisms on the species abundance and the community structures. A classical, highly influential model—the Moran model (and its variants)—has served as the paradigm for understanding the effects of stochastic ecological drift and natural selection on the diversity of a well-mixed population [59, 68–71]. A closely related model, Hubbell’s neutral theory of biodiversity has been used to describe the emergence of the species abundance distribution in a neutral immigration-birth-death process [72]. Among others, Lotka-Voltera models further explore the role of species interactions and niche overlap on the interspecific competition; their frameworks can also be roughly mapped to the Moran model in their neutral regimes [16, 73, 74]. The fundamental stochastic nature of the ecological processes underlies all these models, where stochastic fluctuations of the abundances emerge from the demographic noise (i.e., the inherent randomness of birth and death events in a population). We model the population dynamics of two competing species as a discrete stochastic process denoting the probability of being in a state with one species abundance at n (and the other species abundance at N − n) at time t is p(n, t). The population abundance of a species can change either through births or deaths of the individual cells, with the probability of a birth or death in the population n in an interval of time Δt denoted as T+(n → n + 1, Δt) or T−(n → n − 1, Δt), respectively [75, 76]. The evolution of the probability p(n, t), is governed by a one-dimensional forward master equation (ME) (1) where are the transition rates for events of an increase or decrease in abundance [77]. We are interested in the process of fixation wherein the abundance of one species approaches N, effectively outcompeting the other species by removing it from the system. This fixation can be viewed as a first-passage process that occurs when the abundance of a species reaches either of the absorbing states, at n = 0 and n = N, at which point the system settles at steady-state with one species dominating indefinitely [16, 78]. In these processes, the first-passage probability and the mean first-passage time (MFPT) are characteristics of the system which elucidate the dynamics of the process [76]. The mean-first passage time to either fixation state, from a starting abundance n, τ(n), relates the average time the competition between the two species lasts before one takes over and is described by the backward equation (2) The master Eqs 1 and 2 can be solved numerically to obtain the probability and MFPT of fixation, (see Section A in S1 Appendix). Alternatively, the Fokker-Planck (FP) expansion to order of the backward equation is (3) where the drift and diffusion terms are defined as A(f) = r+(f) − r−(f) and B(f) = r+(f) + r−(f) with the transformation and f = n/N. A FP expansion in 1/N of the probability of the first passage to settle in either absorbing abundance states F ∈ {0, 1} results in a continuous description (see Section B in S1 Appendix) (4) where PF(f) is the probability of being absorbed at the state F starting from an abundance f. For instance, P1(f) is the probability that a species with fractional abundance f will fill the space to fixate at the absorbing abundance F = 1. An equation can also be derived for the MFPT conditioned on the fixation to an absorbing state. The discrete form of these equations all have exact numerical solutions [79] (see Section C in S1 Appendix). The solutions to these equations depend on the choice of the birth and death rates of the model. In the classical Moran model, which represents mixed populations without spatial structure, a random individual from a fixed and finite population of size N is selected to give birth at each time step while, simultaneously, a random individual is selected to be removed from the system to make room for the progeny and maintain a constant total population N (see Fig 1A and Section C in S1 Appendix). Building on this paradigmatic model, we consider a system where N individuals of two species are constrained to a one-dimensional space (a channel open at both ends) as shown in Fig 1B. We assume that the channel is always full and that the two species are segregated such that only one boundary separates the populations. Without loss of generality, we take n to describe the number of individuals belonging to the species on the left side of the lane (species 1). Contrary to the Moran model, the transition rates of the populations now depend on the spatial arrangement of the cells. A cell at any location can divide and produce a progeny, but death events only happen when a cell is pushed out of either end of the channel. In this system, the relative species abundance (frequency) delineates the location of the boundary that separates the species. When an individual cell grows, the boundary shifts right or left, as illustrated in Fig 1C. As n increases, the boundary between the two species moves to the right with probability T+(n → n + 1); conversely, decreases in n result in the boundary moving to the left with probability T−(n → n − 1). This competitive process continues until a sequence of jumps makes the boundary reach either end of the channel (i.e., n = 0 or n = N) with only one surviving species, which is said to fixate. Like the Moran model, fitness differences modify the selection probabilities. The probability of selecting a specific individual i from species 1 for birth with relative fitness difference w—related to the selection parameter s commonly used in evolutionary dynamics as w = 1 + s—is , whereas the probability for individual i of species 2 is [80, 81]. Subsequent to the birth, the offspring must create space for itself by pushing upon the adjacent cells situated on either side of its progenitor. This triggers a cascade of coordinated cellular displacements along the channel, culminating in the expulsion of a terminal cell from the microchannel. This skewed collective growth causes the population boundary to preferentially drift towards an edge of the channel, eventually expelling one of the two species and fixating the other. In a recent experiment in open-ended micro-channels, it was observed that cells are more likely to grow in the direction of the closer channel opening because fewer cells need to be pushed in that direction. As a first approximation, the probability for a cell to grow towards one of the openings can be expected to scale linearly with the number of individuals between the cell location and the other opening, as was indeed observed in [57]. The magnitude of this bias towards an end of the channel has been estimated in this particular experimental system [57]; however there are as of yet no systematic estimates for different experimental conditions. In this work, we make the minimal assumption that the probability that an individual cell at position i divides to the right is proportional to the number of individuals that are to the left of it pright = (i − 1)/(N − 1) and vice versa pleft = (N − i)/(N − 1), see also [57] and SI. This choice corresponds to a somewhat larger bias than estimated and modelled in [57] and the effects of changing this assumption will be investigated in the future. However, the effects of varying this bias on the investigated quantities (invasion probabilities) appear to be minimal [57] (See also Discussion). This spatial exclusion model is illustrated in Fig 1C. Thus, the rate at which the boundary moves to the right in an interval of time Δt is the sum of the rates for each individual of species 1 to grow to the right (5) whereas the rate of the boundary moving to the left is (6) Here, we have rescaled the basal rate to r = 1/(N − 1)Δt. The summation of individual cell growth rates contributes to the cumulative growth of the entire cell population, thereby influencing the skewed directional movement towards either extremity of the channel. These rates can be substituted in Eqs 3 and 4 and solved for the dynamics of the spatial exclusion model. Results Spatial exclusion gives rise to sharp sigmoidal fixation probabilities and exponentially fast MFPTs We first consider a neutral case where species are functionally equivalent without fitness differences between them (w = 1) (e.g., two populations of cells whose phenotypic differences offer no upper hand or two identical lineages with a common ancestor). In contrast to the neutral Moran model where the drift term is zero, the drift term in the spatial exclusion model is strictly positive: A(f) = N[(w − 1)f2 + 2f − 1]/2[1 + f(w − 1)]. Although an analytical solution to Eq 4 is not available for w ≠ 1, the equation can be easily numerically integrated (see also Section A in S1 Appendix). Fig 2 shows the results of a comparison between the neutral Moran model and the spatial exclusion model. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Competition outcomes for neutral dynamics. Fokker-Planck approximation for the fixation probabilities and mean-first passage times (MFPTs) for the Moran model (dotted lines) and the spatial exclusion model (solid lines) for various system sizes (N = 10, 100, 1000). (A) The fixation probability in the spatial model is sigmoidal around an inflection point at f = 1/2 and becomes increasingly steep, approaching a step-function at large N. The Moran model, in contrast, always predicts a linear fixation probability equal to the fractional abundance. (B) For the spatial exclusion model, the MFPTs at varying N collapse onto each other for most fractional abundances except around the inflection point. The dynamics away from f = 1/2 follow a deterministic path to fixation approximated by τdet (dashed red line) from Eq 7. Notably, the MFPTs in the spatial model are significantly faster than those predicted by the Moran model. (C) The maximal MFPT of the Moran model is linear in N whereas the maximal MFPT of the spatial model grows substantially slower and sub-linear in N; see Fig 3C. https://doi.org/10.1371/journal.pcbi.1010868.g002 We find that the probability for a species to fixate in the spatial exclusion model as a function of its initial fractional abundance is a sigmoidal function. This differs significantly from the linear dependence predicted by the Moran model (see Fig 2A). Any minority population (with a starting fraction f < 1/2), is much less likely to take over the population than in the Moran model. Conversely, any majority population (f > 1/2) is much more likely to succeed at fixating within the lane. The slope of this sigmoidal probability depends on the length of the microchannel (or the total population N) approaching a step function for large N. The inflection point is found at the equiprobable takeover abundance feq = 1/2, defined as the abundance at which both species are equally likely to take over the channel. For the Moran model, the mean time for either species to fixate (the unconditional fixation time) grows linearly with the system size. For the spatial exclusion model, the mean time to fixation is much shorter than that of the Moran model for all initial boundary positions as shown in Fig 2B) (for the MFPT conditioned on the success of one of the two species, see Section D in S1 Appendix). Interestingly, the MFPT curves for different N collapse onto each other away from the central peak at the equiprobable takeover abundance of 1/2. Thus, the time to fixation does not depend on system size unless the initial fractions of the two populations are closely balanced, in which case it is approximately logarithmic in N as discussed below. This independence of the dynamics on N can be heuristically explained by investigating the behaviour of Eq 3. For the spatial model, the first term on the right-hand side (the drift term) is small, A(f) ≈ 0, around the peak of the MFPT. The dynamics around the peak are dominated by the second (diffusion) term on the right-hand side of Eq 3, which scales like 1/N. Conversely, away from the peak, the drift term dominates the expression for large N. In this case, Eq 3 can be approximated by a first-order ODE (7) The solution to this equation is the time to fixation for a process with a deterministic velocity A(f), (8) where Θ is the Heaviside function. Given that the deterministic velocity is proportional to N, the fixation is independent of N. The displacement of the fractional abundance is exponential in time when the change in the fractional abundance is governed by this deterministic velocity. The mean time to fixation of the spatial exclusion model substantially differs from the prediction of the Moran model, even close to the peak, where the MFPT depends on N. This difference between the two models for the MFPTs to fixation is most apparent at a starting fraction of f = 1/2 where A(f) = 0. In Fig 2C, the MFPT to fixation in the neutral spatial exclusion model grows only sub-polynomially with N rather than linearly as in the classical Moran model. We return to the quantification of this sub-polynomial growth in N below. Fitness differences break the symmetry of the fixation probability and engender longer MFPTs Fig 2 presents results of unconditional fixation times and probabilities of fixation for neutral populations with different but functionally identical species. However, more generally, phenotypically dissimilar species exhibit differences in their dynamics such as growth and death rates, efficiency of resource consumption, etc., which may impact their overall fitness in the environment. In this section, we present the results for the population fixation in the presence of a relative fitness difference, w, between two species, shown in Fig 3 for w = 1, 1.5, 10, 100. Very small relative fitness differences (such as w = 1.1) do not significantly alter the probability and MFPT to fixation compared to the neutral model as shown in Section D in S1 Appendix. Although results are shown for w > 1, results for fitness differences w′ < 1 can be obtained by interchanging species 1 and 2 with w = 1/w′. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Competition outcomes for dynamics with fitness differences. Fokker-Planck approximation for the fixation probabilities and the mean-first passage times (MFPTs) for both the Moran model (dotted lines) and the spatial exclusion model (solid lines) for the values of relative fitness difference w = 1, 1.5, 10, 100. The fractional abundance f is the abundance of the species with higher fitness. (A) Fitness difference increases the fixation probability of the fitter species, shifting the fixation probabilities curves towards lower abundances compared to the neutral case; N = 100. Note that the probabilities of fixation for the Moran model with w = 10 and w = 100 are effectively indistinguishable. (B) In the spatial model the maximum MFPT slowly increases with fitness in contrast to the Moran model where the MFPTs tend to diminish with w. For the spatial model, the dynamics outside a narrow region around follow a quasi-deterministic path to fixation approximated by τdet, (shown in a dashed green line for w = 100) obtained from Eq 8. (C) The first passage time for the neutral (w = 1) Moran model exhibits linear growth in N which slows down for higher w. For the spatial model, the maximal fixation times (MFPTs) scale sublinearly with N, with minimal effect of the fitness difference. See Fig 4 for further discussion. https://doi.org/10.1371/journal.pcbi.1010868.g003 In the Moran model, the probability of fixation is strongly favored towards the species with a fitness advantage for even modest values of w, (Fig 3A), with the probability of fixation being p(f) = (1 − w−fN)/(1 − w−N) for the fitter species [82]. Selection quickly skews the fixation probabilities so that the species with the advantage almost always fixates regardless of its initial abundance. On the contrary, similar fitness differences do not influence population dynamics as markedly in the spatial exclusion model. Whereas the probability of fixation as a function of the initial fraction f changes drastically in the Moran model, from a linear function at w = 1 to a concave function without an inflection point for w > 1, within the spatial model, the shape of these curves remains relatively unchanged for all values of w (see Fig 3A). Rather, fitness differences shift the inflection point of the sigmoidal fixation probability curves towards lower initial fractional abundances , but a species is still significantly favoured to fixate when its fractional abundance is above the inflection point. The fact that the effect of fitness difference is blunted by the spatial competition is also reflected in the probability of fixation averaged over all initial abundances, see Section E in S1 Appendix: the average probability of fixation for the Moran model is higher than the average probability in the spatial exclusion model. Fitness advantages also impact the dynamics of fixation. As shown in Fig 3B, higher fitness differences in the Moran model reduce the mean fixation time (unconditioned on the success of a species). In the spatial exclusion model, although a greater fitness leads to faster fixation conditioned on the species success (see Section D in S1 Appendix), this is not always the case when considering the unconditional mean fixation time. The dependence of the unconditional mean fixation time on the relative fitness in the spatial model is non-monotonic and location dependent as shown in Fig 3B. In particular, the mean fixation time is maximal for populations initialized at abundances close to the equiprobable takeover abundance , τ(feq) for any fitness (see Fig 3B). The fixation time marginally increases with the fitness difference between the species scaling logarithmically with 1/(w − 1) as follows from Eq 8. For this reason, the times to fixation for various fitness differences almost collapse onto each other for all system sizes N, as shown in Fig 3C. Accordingly, in the spatial exclusion model, the longest timescale of the system is determined by the competition dynamics around feq where the probabilities of either species successfully taking over are roughly equivalent. The asymptotic scaling with the system size N of this maximal MFPT in the presence of fitness differences is similar to the asymptotic scaling in the neutral case (see Fig 3C)—much slower than the Moran time and sublinear with N, as further explored in discussion around Fig 4. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Heuristic approximation of the MFPT and scaling with N. (A) Graphs of the potential U(f) with the location of the maximum, , indicated at the peak of the curves. The shaded areas represent regions along the channel, f, where the stochastic behaviour heuristically dominates the dynamics (N = 100). (B) The maximal mean fixation time (solid lines) as a function of system size increases with fitness advantage and scales approximately logratihmically with system size N. The quasi-deterministic approximation τa(feq) (dashed lines) is very close to the exact dynamics. The maximal MFPT for w = 1.0 and w = 1.5 overlap at this scale. Colour coding for both panels is presented in the legend of panel A. https://doi.org/10.1371/journal.pcbi.1010868.g004 For initial conditions outside the close vicinity of the equiprobable abundance feq fixation follows a quasi-deterministic course where the fixation times are almost independent of the system size for any initial condition. The deterministic approximation to the fixation time calculated from Eq 8 is shown in green dashed line in Fig 3B and serves as an excellent approximation except in the close viscinity of feq. In the close viscinity of feq species abundance dynamics correspond to a “tug-of-war”; the two species abundances fluctuate around the equiprobable takeover abundance, with both species trying to take control of the channel by their abundance. Once the abundance diffuses away from the equiprobable point and one species becomes dominant, the dynamics quickly switches to a deterministic trajectory where skewed collective growth further accelerates the growth of the dominant species abundance, causing the rapid collapse in the abundance of the other species. Size dependence of the fixation times in different dynamical regimes in large systems competing in a tug-of-war The dependence of the fixation time of any species on the size of the system N is of crucial importance for the the speed of diversity loss, and therefore for the inference from experimental observations that commonly probe only the transient composition of evolving ecosystems. In Fig 3C we have numerically seen that the asymptotic behaviour of the MFPT to fixation grows sublinearly in N for the spatial exclusion model (see Section D in S1 Appendix). In this section, we investigate further the large N scaling of MFPTs—a regime relevant for many experimental systems that are often comprised of large numbers of individuals such as crevices in biological systems (e.g. gastrointestinal crypts [83–85]) or narrow channels in which cells navigate their environment (e.g. in Antarctic icesheets [86, 87]). We note that the results closely follow the asymptotic scaling already for N ∼ 100, on the order of the typical number of bacteria in microfluidic channels and mother machines. The one-dimensional Fokker-Planck equation for the MFPT to fixation, Eq 3, may be rewritten as (9) where (10) U(f), sometimes referred to as the Fokker-Planck potential, describes an effective potential landscape in which the boundary between the two species moves (see Section B in S1 Appendix). A general compact integral form of the solution to Eq 3, which is shown in Section G in S1 Appendix, can be evaluated for different potentials representing different population dynamics. For the model defined in Eqs 5 and 6, U(f) is a unimodal distribution with an unstable maximum found at for w ≥ 1 (see Fig 4A). Thus, unlike the more familiar problem of calculating the MFPT to cross over a potential barrier (Kramers Theory) [76], here we are interested in finding the MFPT to descend from a potential peak starting at an unstable point. This distinction means that the saddle-point approximation—which is commonly employed for the evaluation of asymptotic of the MFPTs shown in Eq 3—is inadequate in this case. To derive an approximation for the asymptotic behaviour of the fixation MFPT, we heuristically separate the space into regions of predominantly deterministic or stochastic dynamics. As shown in Section G in S1 Appendix, the boundaries between these two regions naturally emerge from the integral form of the MFPT as the solutions to the equation (11) and are depicted in Fig 4A. The interval is the region around feq where the dynamics are dominated by stochastic diffusion with the frequency-dependent diffusion coefficient B(f) arising from a “tug-of-war” dynamics between the two species. Conversely, outside of this region, the dynamics are well approximated as deterministic and are dominated by an ecological drift with a frequency-dependent velocity A(f) arising from the skewed collective growth towards the channel exits. Thus, if the initial fractional abundance of species 1 lies in the deterministic region x ∉ Xt, the time to fixation is well approximated by the deterministic time τdet(f) of Eq 8. On the other hand, if the initial fractional abundance is in the stochastic region x ∈ Xt, its motion is dominated at first by diffusion until it reaches one of the boundaries within a time , after which it typically continues along a deterministic trajectory. For illustrative purposes, we examine the fixation time starting at f = feq given that there is an equal probability to diffuse to either boundary from this position. Thus the maximal MFPT can be approximated as (12) which scales as for large N, as is shown in Fig 4B (see also Section G in S1 Appendix). As shown in Section G in S1 Appendix, numerically τdif ≈ 0.4 and is independent of N or the relative fitness difference; further analysis is needed to determine the constant analytically. Consequently, the longest timescale observed for species to fixate in the system increases slowly with the length of the channel so that rapid fixation may occur even in very long channels. Invasions in spatial models are less likely to succeed but succeed on shorter timescales on average than in well-mixed models Once one species has fixated, it remains dominant in the channel unless an external event, such as an invasion or mutation, perturbs the system by introducing a new species variant that could compete against the established strain [88–90]. In microchannels, a mutation event could introduce a new variant at any location, but immigration is normally possible only from the edges. However, bacterial populations that grow in wider microchannels have been shown to organize into parallel lanes aligning and growing along the axis of the channels. These aligned lanes permit rare immigration events from one lane into another previously fixated lane, which can be viewed as the invasion of a new species into a lane [57, 91]. We are interested in the probability and the mean time of a successful invasion, wherein an individual cell of the invading species is inserted anywhere in the channel. Success of an invasion is defined as the invading species taking over and fixating within the lane/channel. Whereas the MFPTs presented in previous sections were the fixation times for any species to fixate, the invasion fixation time refers to the MFPT conditioned on the success of the invader. In the well-mixed Moran model, when an individual from a new species invades a previously fixated system, the fractional abundance of the invading species is 1/N. As derived in earlier sections, the probabilities of a successful invasion fixation for the Moran model (P1(1/N) = (1 + w−1)/(1 + w−N)) are depicted in Figs 2 and 3. In contrast to earlier sections, the appearance of an invader in the one-lane model results in two rather than one inter-species boundaries as illustrated in Fig 5A. Nevertheless, the mathematical framework of competitive spatial exclusion described above allows us to model inter-species dynamics after such an invasion event, as shown in Fig 5A. The rules that determine the probability for a cell to be selected for birth and death are identical to those outlined previously. However, the system defined by the two inter-species boundaries is now two-dimensional and, accordingly, has added states with rates defining additional transitions. The expressions for these rates and the corresponding two variable master equation are outlined in Section F in S1 Appendix. Given that the state space for the spatial invasion model is now two-dimensional, we rely on numerical solutions for the discrete dynamics instead of solving the corresponding 2D Fokker-Planck equation. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Invasion into a fixated channel. (A) Illustration of the invasion/mutation model wherein an invader (blue) with relative fitness difference w infiltrates a native population (orange). The invasion is shown at the center for illustrative purposes. (B, C, D): Numerical solutions to the Master equation for the spatial model (solid lines with × symbols) and Moran model (dotted lines with + symbols). (B) The probability of a successful invasion as a function of the insertion location (N = 100). Invasion events at the center of the channel are the most likely to succeed, but a relative fitness difference by the invading species increases the range of likely successful invasions. (C) The average probability of a successful invasion obtained by averaging with uniform weight the probability of successful invasion at all locations along the micro-channel. For w = 1, both models show similar behaviour. (D) The average mean time of a successful invasion in the spatial model increases with the population size, but less rapidly than in the Moran model. As the relative fitness difference increases, the mean time of successful invasions averaged over all invasion events decreases as well. Logarithmic dependence on N of the complete lineage fixation time derived in [57] is shown as the green dashed line. https://doi.org/10.1371/journal.pcbi.1010868.g005 The outcome of an invasion in the spatial exclusion model is strongly dependent on the initial location of the invasion event, x. We find that invasions starting about the center of the channel are the most likely to succeed in pushing out the established species, see Fig 5B; this has previously been shown in the neutral case [57]. The probability that an invasion event succeeds increases with the fitness advantage of the invading species and successful invasions occur over a broader range of locations along the channel. To provide a global measure of invasion success, we calculate the average invasion probability over all initial conditions. Notably, the average invasion probability in the neutral spatial model(w = 1) is identical to the invasion probability of the neutral Moran model and is equal to 1/N as shown in Fig 5C. Intuitively, since all cells are identical and an invasion can arise at any location, one of the N cells will become the ancestor of the entire population. This implies that the average probability of a successful invasion is 1/N. As demonstrated in Fig 5C, the probability of a successful invasion increases with fitness more sharply for the Moran model than for the spatial model. However, the probability of successful invasion remains low even for an invasive species with a ten-fold increase in fitness advantage, which is still less likely to fixate than the native species. This suggests that the spatial exclusion dynamics modeled here limit the competitive edge of strains with fitness advantages. Although the average probability for the invasive species to fixate is identical under neutral conditions for both the Moran model and the spatial exclusion model, the dynamics of fixation differ between the two models. A global measure of the invasion dynamics is the average MFPT of a successful invasion, weighted by the probability of success at each initial invasion location, shown in Fig 5D (see Section B in S1 Appendix for details on conditional MFPTs). In the neutral case, this time is equivalent to the mean time to the complete loss of lineage diversity considered in [57]. The conditional MFPT of a successful invasion is larger for the Moran model than the spatial model in the neutral case. This is also true for an invader when fitness differences are incorporated: successful invasions fixate more rapidly in the presence of spatial exclusion than in the well-mixed model. We find that the average conditional MFPT of a successful invasion in the spatial model changes non-monotonically with fitness advantage in Fig 5D. At low fitness differences (w ≤ 10), the conditional MFPT decreases when fitness differences increase (see Section B in S1 Appendix); intuitively, an increase in fitness difference advantages the growth of the invader such that it takes over more rapidly. This is contrary to the behaviour of the conditional MFPT at very high fitness (w > 10): the average conditional MFPT of invasion increases with fitness difference increases, see Section F in S1 Appendix for details. Spatial exclusion gives rise to sharp sigmoidal fixation probabilities and exponentially fast MFPTs We first consider a neutral case where species are functionally equivalent without fitness differences between them (w = 1) (e.g., two populations of cells whose phenotypic differences offer no upper hand or two identical lineages with a common ancestor). In contrast to the neutral Moran model where the drift term is zero, the drift term in the spatial exclusion model is strictly positive: A(f) = N[(w − 1)f2 + 2f − 1]/2[1 + f(w − 1)]. Although an analytical solution to Eq 4 is not available for w ≠ 1, the equation can be easily numerically integrated (see also Section A in S1 Appendix). Fig 2 shows the results of a comparison between the neutral Moran model and the spatial exclusion model. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Competition outcomes for neutral dynamics. Fokker-Planck approximation for the fixation probabilities and mean-first passage times (MFPTs) for the Moran model (dotted lines) and the spatial exclusion model (solid lines) for various system sizes (N = 10, 100, 1000). (A) The fixation probability in the spatial model is sigmoidal around an inflection point at f = 1/2 and becomes increasingly steep, approaching a step-function at large N. The Moran model, in contrast, always predicts a linear fixation probability equal to the fractional abundance. (B) For the spatial exclusion model, the MFPTs at varying N collapse onto each other for most fractional abundances except around the inflection point. The dynamics away from f = 1/2 follow a deterministic path to fixation approximated by τdet (dashed red line) from Eq 7. Notably, the MFPTs in the spatial model are significantly faster than those predicted by the Moran model. (C) The maximal MFPT of the Moran model is linear in N whereas the maximal MFPT of the spatial model grows substantially slower and sub-linear in N; see Fig 3C. https://doi.org/10.1371/journal.pcbi.1010868.g002 We find that the probability for a species to fixate in the spatial exclusion model as a function of its initial fractional abundance is a sigmoidal function. This differs significantly from the linear dependence predicted by the Moran model (see Fig 2A). Any minority population (with a starting fraction f < 1/2), is much less likely to take over the population than in the Moran model. Conversely, any majority population (f > 1/2) is much more likely to succeed at fixating within the lane. The slope of this sigmoidal probability depends on the length of the microchannel (or the total population N) approaching a step function for large N. The inflection point is found at the equiprobable takeover abundance feq = 1/2, defined as the abundance at which both species are equally likely to take over the channel. For the Moran model, the mean time for either species to fixate (the unconditional fixation time) grows linearly with the system size. For the spatial exclusion model, the mean time to fixation is much shorter than that of the Moran model for all initial boundary positions as shown in Fig 2B) (for the MFPT conditioned on the success of one of the two species, see Section D in S1 Appendix). Interestingly, the MFPT curves for different N collapse onto each other away from the central peak at the equiprobable takeover abundance of 1/2. Thus, the time to fixation does not depend on system size unless the initial fractions of the two populations are closely balanced, in which case it is approximately logarithmic in N as discussed below. This independence of the dynamics on N can be heuristically explained by investigating the behaviour of Eq 3. For the spatial model, the first term on the right-hand side (the drift term) is small, A(f) ≈ 0, around the peak of the MFPT. The dynamics around the peak are dominated by the second (diffusion) term on the right-hand side of Eq 3, which scales like 1/N. Conversely, away from the peak, the drift term dominates the expression for large N. In this case, Eq 3 can be approximated by a first-order ODE (7) The solution to this equation is the time to fixation for a process with a deterministic velocity A(f), (8) where Θ is the Heaviside function. Given that the deterministic velocity is proportional to N, the fixation is independent of N. The displacement of the fractional abundance is exponential in time when the change in the fractional abundance is governed by this deterministic velocity. The mean time to fixation of the spatial exclusion model substantially differs from the prediction of the Moran model, even close to the peak, where the MFPT depends on N. This difference between the two models for the MFPTs to fixation is most apparent at a starting fraction of f = 1/2 where A(f) = 0. In Fig 2C, the MFPT to fixation in the neutral spatial exclusion model grows only sub-polynomially with N rather than linearly as in the classical Moran model. We return to the quantification of this sub-polynomial growth in N below. Fitness differences break the symmetry of the fixation probability and engender longer MFPTs Fig 2 presents results of unconditional fixation times and probabilities of fixation for neutral populations with different but functionally identical species. However, more generally, phenotypically dissimilar species exhibit differences in their dynamics such as growth and death rates, efficiency of resource consumption, etc., which may impact their overall fitness in the environment. In this section, we present the results for the population fixation in the presence of a relative fitness difference, w, between two species, shown in Fig 3 for w = 1, 1.5, 10, 100. Very small relative fitness differences (such as w = 1.1) do not significantly alter the probability and MFPT to fixation compared to the neutral model as shown in Section D in S1 Appendix. Although results are shown for w > 1, results for fitness differences w′ < 1 can be obtained by interchanging species 1 and 2 with w = 1/w′. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Competition outcomes for dynamics with fitness differences. Fokker-Planck approximation for the fixation probabilities and the mean-first passage times (MFPTs) for both the Moran model (dotted lines) and the spatial exclusion model (solid lines) for the values of relative fitness difference w = 1, 1.5, 10, 100. The fractional abundance f is the abundance of the species with higher fitness. (A) Fitness difference increases the fixation probability of the fitter species, shifting the fixation probabilities curves towards lower abundances compared to the neutral case; N = 100. Note that the probabilities of fixation for the Moran model with w = 10 and w = 100 are effectively indistinguishable. (B) In the spatial model the maximum MFPT slowly increases with fitness in contrast to the Moran model where the MFPTs tend to diminish with w. For the spatial model, the dynamics outside a narrow region around follow a quasi-deterministic path to fixation approximated by τdet, (shown in a dashed green line for w = 100) obtained from Eq 8. (C) The first passage time for the neutral (w = 1) Moran model exhibits linear growth in N which slows down for higher w. For the spatial model, the maximal fixation times (MFPTs) scale sublinearly with N, with minimal effect of the fitness difference. See Fig 4 for further discussion. https://doi.org/10.1371/journal.pcbi.1010868.g003 In the Moran model, the probability of fixation is strongly favored towards the species with a fitness advantage for even modest values of w, (Fig 3A), with the probability of fixation being p(f) = (1 − w−fN)/(1 − w−N) for the fitter species [82]. Selection quickly skews the fixation probabilities so that the species with the advantage almost always fixates regardless of its initial abundance. On the contrary, similar fitness differences do not influence population dynamics as markedly in the spatial exclusion model. Whereas the probability of fixation as a function of the initial fraction f changes drastically in the Moran model, from a linear function at w = 1 to a concave function without an inflection point for w > 1, within the spatial model, the shape of these curves remains relatively unchanged for all values of w (see Fig 3A). Rather, fitness differences shift the inflection point of the sigmoidal fixation probability curves towards lower initial fractional abundances , but a species is still significantly favoured to fixate when its fractional abundance is above the inflection point. The fact that the effect of fitness difference is blunted by the spatial competition is also reflected in the probability of fixation averaged over all initial abundances, see Section E in S1 Appendix: the average probability of fixation for the Moran model is higher than the average probability in the spatial exclusion model. Fitness advantages also impact the dynamics of fixation. As shown in Fig 3B, higher fitness differences in the Moran model reduce the mean fixation time (unconditioned on the success of a species). In the spatial exclusion model, although a greater fitness leads to faster fixation conditioned on the species success (see Section D in S1 Appendix), this is not always the case when considering the unconditional mean fixation time. The dependence of the unconditional mean fixation time on the relative fitness in the spatial model is non-monotonic and location dependent as shown in Fig 3B. In particular, the mean fixation time is maximal for populations initialized at abundances close to the equiprobable takeover abundance , τ(feq) for any fitness (see Fig 3B). The fixation time marginally increases with the fitness difference between the species scaling logarithmically with 1/(w − 1) as follows from Eq 8. For this reason, the times to fixation for various fitness differences almost collapse onto each other for all system sizes N, as shown in Fig 3C. Accordingly, in the spatial exclusion model, the longest timescale of the system is determined by the competition dynamics around feq where the probabilities of either species successfully taking over are roughly equivalent. The asymptotic scaling with the system size N of this maximal MFPT in the presence of fitness differences is similar to the asymptotic scaling in the neutral case (see Fig 3C)—much slower than the Moran time and sublinear with N, as further explored in discussion around Fig 4. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Heuristic approximation of the MFPT and scaling with N. (A) Graphs of the potential U(f) with the location of the maximum, , indicated at the peak of the curves. The shaded areas represent regions along the channel, f, where the stochastic behaviour heuristically dominates the dynamics (N = 100). (B) The maximal mean fixation time (solid lines) as a function of system size increases with fitness advantage and scales approximately logratihmically with system size N. The quasi-deterministic approximation τa(feq) (dashed lines) is very close to the exact dynamics. The maximal MFPT for w = 1.0 and w = 1.5 overlap at this scale. Colour coding for both panels is presented in the legend of panel A. https://doi.org/10.1371/journal.pcbi.1010868.g004 For initial conditions outside the close vicinity of the equiprobable abundance feq fixation follows a quasi-deterministic course where the fixation times are almost independent of the system size for any initial condition. The deterministic approximation to the fixation time calculated from Eq 8 is shown in green dashed line in Fig 3B and serves as an excellent approximation except in the close viscinity of feq. In the close viscinity of feq species abundance dynamics correspond to a “tug-of-war”; the two species abundances fluctuate around the equiprobable takeover abundance, with both species trying to take control of the channel by their abundance. Once the abundance diffuses away from the equiprobable point and one species becomes dominant, the dynamics quickly switches to a deterministic trajectory where skewed collective growth further accelerates the growth of the dominant species abundance, causing the rapid collapse in the abundance of the other species. Size dependence of the fixation times in different dynamical regimes in large systems competing in a tug-of-war The dependence of the fixation time of any species on the size of the system N is of crucial importance for the the speed of diversity loss, and therefore for the inference from experimental observations that commonly probe only the transient composition of evolving ecosystems. In Fig 3C we have numerically seen that the asymptotic behaviour of the MFPT to fixation grows sublinearly in N for the spatial exclusion model (see Section D in S1 Appendix). In this section, we investigate further the large N scaling of MFPTs—a regime relevant for many experimental systems that are often comprised of large numbers of individuals such as crevices in biological systems (e.g. gastrointestinal crypts [83–85]) or narrow channels in which cells navigate their environment (e.g. in Antarctic icesheets [86, 87]). We note that the results closely follow the asymptotic scaling already for N ∼ 100, on the order of the typical number of bacteria in microfluidic channels and mother machines. The one-dimensional Fokker-Planck equation for the MFPT to fixation, Eq 3, may be rewritten as (9) where (10) U(f), sometimes referred to as the Fokker-Planck potential, describes an effective potential landscape in which the boundary between the two species moves (see Section B in S1 Appendix). A general compact integral form of the solution to Eq 3, which is shown in Section G in S1 Appendix, can be evaluated for different potentials representing different population dynamics. For the model defined in Eqs 5 and 6, U(f) is a unimodal distribution with an unstable maximum found at for w ≥ 1 (see Fig 4A). Thus, unlike the more familiar problem of calculating the MFPT to cross over a potential barrier (Kramers Theory) [76], here we are interested in finding the MFPT to descend from a potential peak starting at an unstable point. This distinction means that the saddle-point approximation—which is commonly employed for the evaluation of asymptotic of the MFPTs shown in Eq 3—is inadequate in this case. To derive an approximation for the asymptotic behaviour of the fixation MFPT, we heuristically separate the space into regions of predominantly deterministic or stochastic dynamics. As shown in Section G in S1 Appendix, the boundaries between these two regions naturally emerge from the integral form of the MFPT as the solutions to the equation (11) and are depicted in Fig 4A. The interval is the region around feq where the dynamics are dominated by stochastic diffusion with the frequency-dependent diffusion coefficient B(f) arising from a “tug-of-war” dynamics between the two species. Conversely, outside of this region, the dynamics are well approximated as deterministic and are dominated by an ecological drift with a frequency-dependent velocity A(f) arising from the skewed collective growth towards the channel exits. Thus, if the initial fractional abundance of species 1 lies in the deterministic region x ∉ Xt, the time to fixation is well approximated by the deterministic time τdet(f) of Eq 8. On the other hand, if the initial fractional abundance is in the stochastic region x ∈ Xt, its motion is dominated at first by diffusion until it reaches one of the boundaries within a time , after which it typically continues along a deterministic trajectory. For illustrative purposes, we examine the fixation time starting at f = feq given that there is an equal probability to diffuse to either boundary from this position. Thus the maximal MFPT can be approximated as (12) which scales as for large N, as is shown in Fig 4B (see also Section G in S1 Appendix). As shown in Section G in S1 Appendix, numerically τdif ≈ 0.4 and is independent of N or the relative fitness difference; further analysis is needed to determine the constant analytically. Consequently, the longest timescale observed for species to fixate in the system increases slowly with the length of the channel so that rapid fixation may occur even in very long channels. Invasions in spatial models are less likely to succeed but succeed on shorter timescales on average than in well-mixed models Once one species has fixated, it remains dominant in the channel unless an external event, such as an invasion or mutation, perturbs the system by introducing a new species variant that could compete against the established strain [88–90]. In microchannels, a mutation event could introduce a new variant at any location, but immigration is normally possible only from the edges. However, bacterial populations that grow in wider microchannels have been shown to organize into parallel lanes aligning and growing along the axis of the channels. These aligned lanes permit rare immigration events from one lane into another previously fixated lane, which can be viewed as the invasion of a new species into a lane [57, 91]. We are interested in the probability and the mean time of a successful invasion, wherein an individual cell of the invading species is inserted anywhere in the channel. Success of an invasion is defined as the invading species taking over and fixating within the lane/channel. Whereas the MFPTs presented in previous sections were the fixation times for any species to fixate, the invasion fixation time refers to the MFPT conditioned on the success of the invader. In the well-mixed Moran model, when an individual from a new species invades a previously fixated system, the fractional abundance of the invading species is 1/N. As derived in earlier sections, the probabilities of a successful invasion fixation for the Moran model (P1(1/N) = (1 + w−1)/(1 + w−N)) are depicted in Figs 2 and 3. In contrast to earlier sections, the appearance of an invader in the one-lane model results in two rather than one inter-species boundaries as illustrated in Fig 5A. Nevertheless, the mathematical framework of competitive spatial exclusion described above allows us to model inter-species dynamics after such an invasion event, as shown in Fig 5A. The rules that determine the probability for a cell to be selected for birth and death are identical to those outlined previously. However, the system defined by the two inter-species boundaries is now two-dimensional and, accordingly, has added states with rates defining additional transitions. The expressions for these rates and the corresponding two variable master equation are outlined in Section F in S1 Appendix. Given that the state space for the spatial invasion model is now two-dimensional, we rely on numerical solutions for the discrete dynamics instead of solving the corresponding 2D Fokker-Planck equation. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Invasion into a fixated channel. (A) Illustration of the invasion/mutation model wherein an invader (blue) with relative fitness difference w infiltrates a native population (orange). The invasion is shown at the center for illustrative purposes. (B, C, D): Numerical solutions to the Master equation for the spatial model (solid lines with × symbols) and Moran model (dotted lines with + symbols). (B) The probability of a successful invasion as a function of the insertion location (N = 100). Invasion events at the center of the channel are the most likely to succeed, but a relative fitness difference by the invading species increases the range of likely successful invasions. (C) The average probability of a successful invasion obtained by averaging with uniform weight the probability of successful invasion at all locations along the micro-channel. For w = 1, both models show similar behaviour. (D) The average mean time of a successful invasion in the spatial model increases with the population size, but less rapidly than in the Moran model. As the relative fitness difference increases, the mean time of successful invasions averaged over all invasion events decreases as well. Logarithmic dependence on N of the complete lineage fixation time derived in [57] is shown as the green dashed line. https://doi.org/10.1371/journal.pcbi.1010868.g005 The outcome of an invasion in the spatial exclusion model is strongly dependent on the initial location of the invasion event, x. We find that invasions starting about the center of the channel are the most likely to succeed in pushing out the established species, see Fig 5B; this has previously been shown in the neutral case [57]. The probability that an invasion event succeeds increases with the fitness advantage of the invading species and successful invasions occur over a broader range of locations along the channel. To provide a global measure of invasion success, we calculate the average invasion probability over all initial conditions. Notably, the average invasion probability in the neutral spatial model(w = 1) is identical to the invasion probability of the neutral Moran model and is equal to 1/N as shown in Fig 5C. Intuitively, since all cells are identical and an invasion can arise at any location, one of the N cells will become the ancestor of the entire population. This implies that the average probability of a successful invasion is 1/N. As demonstrated in Fig 5C, the probability of a successful invasion increases with fitness more sharply for the Moran model than for the spatial model. However, the probability of successful invasion remains low even for an invasive species with a ten-fold increase in fitness advantage, which is still less likely to fixate than the native species. This suggests that the spatial exclusion dynamics modeled here limit the competitive edge of strains with fitness advantages. Although the average probability for the invasive species to fixate is identical under neutral conditions for both the Moran model and the spatial exclusion model, the dynamics of fixation differ between the two models. A global measure of the invasion dynamics is the average MFPT of a successful invasion, weighted by the probability of success at each initial invasion location, shown in Fig 5D (see Section B in S1 Appendix for details on conditional MFPTs). In the neutral case, this time is equivalent to the mean time to the complete loss of lineage diversity considered in [57]. The conditional MFPT of a successful invasion is larger for the Moran model than the spatial model in the neutral case. This is also true for an invader when fitness differences are incorporated: successful invasions fixate more rapidly in the presence of spatial exclusion than in the well-mixed model. We find that the average conditional MFPT of a successful invasion in the spatial model changes non-monotonically with fitness advantage in Fig 5D. At low fitness differences (w ≤ 10), the conditional MFPT decreases when fitness differences increase (see Section B in S1 Appendix); intuitively, an increase in fitness difference advantages the growth of the invader such that it takes over more rapidly. This is contrary to the behaviour of the conditional MFPT at very high fitness (w > 10): the average conditional MFPT of invasion increases with fitness difference increases, see Section F in S1 Appendix for details. Discussion Spatial exclusion can significantly alter the competitive dynamics between species in densely populated bacterial communities. To understand bacterial competition in confined geometries, we investigated the competition between two cellular species confined to an open 1D microchannel lane. To this end, we have developed a spatial exclusion model that explicitly accounts for the mechanical exclusion between cells, in contrast to non-spatial well-mixed models like the paradigmatic Moran model. In this study, we examine two competition scenarios of populations of two species in a one-dimensional channel. In the first case, each species population is segregated to separate sides of the channel, resulting in a single boundary between the two populations. The MFPT analyzed in this scenario corresponds to the time for the system to fixate; in other words, the fixation is not conditioned on the fixation of either species. In the one boundary scenario, we find that the probability of species fixation in the spatial exclusion model shows a much sharper sigmoidal dependence on the initial fractional abundance in contrast to the Moran model where the corresponding probability is equal to its initial fractional abundance. The inflection point of the sigmoidal curve—where the probabilities of fixation of either species are equal—is located at the initial abundance f = 1/2 for neutral populations without relative fitness difference (w = 1), but shifts to lower initial abundances of the fitter species () at higher values of the relative fitness difference. This result in the spatial model is different from the equivalent quantity in the Moran model: the equiprobable takeover abundance in well-mixed models depends on the relative fitness difference as well as the system size N, see Section C in S1 Appendix. With an increase in the population size, N, the sigmoidal curve approaches a step function, effectively setting a threshold in the initial abundance above (below) which the cells will always (never) fixate. The overall dynamics of competition can be characterized by the mean time of fixation (unconditioned on either species dominating the channel). The mean fixation times in the spatial exclusion model are sped up in comparison to the predictions of the Moran model. The fixation time is essentially independent of the total population size for most initial abundances compared to the well-mixed models (where the MFPT scales linearly with system size), and is dominated by a quasi-deterministic exponential escape towards fixation arising from skewed collective growth of the cells toward channel exits. The fixation time shows a weak dependence on population size that is well approximated by an asymptotic logarithmic scaling in population size only for the initial abundances near the inflection point of the fixation probability curve, where the maximum timescales are observed (see Figs 2C and 3C). Although the species with a relative fitness difference maintains a competitive advantage within our spatial exclusion model, this competitive advantage does not affect the fixation dynamics to the same extent as it does within the Moran model. Indeed, within the spatial exclusion model, the equiprobable takeover abundance occurs at lower initial fractional abundances of the fitter species than in the neutral model, see Fig 3A. However, the competitive advantage a bacterial species gains by having higher relative fitness than its competitor is significantly less in this constrained environment than in a well-mixed system devoid of spatial limitations. Although births of the more fit species happen more frequently, the spatial organization of the cells make their deaths more frequent as well (as they fall out of the channel) resulting in a reduction in the competitive advantage compared to the well-mixed model. For the 1D channel with fitness differences, we showed that one species is deterministically favoured to out-compete the other when the initial fractional abundances are not close to the equiprobable takeover abundance. By contrast, for initial abundances in a region close to the equiprobable takeover abundance, the dynamics are dominated by stochastic diffusion. Either species is roughly equal likely to take control of the channel within this region with the two opposing species competing in a “tug-of-war”. We find that the “tug-of-war” concludes with one species outcompeting its rival in times that scale approximately as when starting in the diffusive regime. This is significantly different from the paradigmatic Moran model which predicts a fixation. More generally, we have developed a heuristic framework to approximate the asympotic dependence (at large N) of the mean first-passage times in a “tug-of-war” process with concave potentials. More precise approaches to this framework will be considered in the future. The heuristic framework for calculating the asymptotic MFPTs is of general interest and has applications well beyond bacterial population dynamics. The results of the competition between the species are due to these “tug-of-war” dynamics engendered by the skewed collective growth, which may be applied to other systems with drift and diffusion terms leading to a concave effective potential as in Fig 4A. For instance, the transport of organelles and other cellular cargo has been described by a “tug-of-war” wherein competing sets of molecular motors pull in opposite directions with the drift depending on the number of motors on either side of the cargo, much like our model [92]. Moreover, competition between populations of cancerous and healthy cells display a “tug-of-war” effective potential that recovers a probability of cancer development that is sinusoidal as a function of the initial fractional abundance of cancerous cells, as in the spatial exclusion model [93]. Our methodology can predict the probabilities of and mean times to a clinical outcome determining the rapidity of the disease. As expected, an increase in the relative fitness increases the probability of fixation and shortens the fixation time of the species with the higher fitness. However, mechanisms that convey selective advantage in division rates can be difficult to maintain due to their strain on metabolism and increased resource costs [94, 95]. As shown in Section D in S1 Appendix, the decrease in the fixation times of a higher fitness species conditioned on successful fixation can be quite small compared to the neutral case. In the long term, this introduces an evolutionary tradeoff between the increased fitness and the cost of maintaining it. In this paper, we used a minimalistic representation of the skewed collective growth reflecting crowding within the channel. Different models with higher or lower degrees of bias are possible, including models where the basal division rate directly depends on the local conditions such as adhesion to the walls, and the position of the cell in the channel especially in long channels [67, 96]. These models will be studied in the future. In the second scenario studied in this paper, an invading individual is inserted into a homogeneous inhabitant population in the channel at any location, creating two inter-species boundaries at either side of the invader. This scenario corresponds to the dynamics of the two-boundary invasion model wherein we explore the probability and time for the invader to successfully fixate in the channel. If the invasion is on the edge of the channel, this is identical to a special case of the single boundary case. For these invasions, we calculated the probability and the MFPT of a successful invasion for different invasion locations, assuming that the invasion is likely to occur at every point in the channel with equal probability. Overall, an invader with a relative fitness difference increases the likelihood that an invasion at any location is successful, with the invasions most likely to succeed occurring in the middle of the channel. Similar to the Moran model, the probability of a successful invasion (averaged over all initial conditions in the channel) decreases as the total population size increases. Koldaeva et al. investigated the loss of lineage diversity in a channel for a neutral model. In a single lane, their results describe the mean first-passage time in which the progeny of a bacterium takes over the whole lane. This is closely related to the invasion scenario studied in our paper, where we calculate the first-passage time for an invading bacterium to fixate starting from different initial locations in the channel. Notably, for the neutral case, the mean time of successful neutral invasion when averaged across all possible initial locations in the channel weighted with the invasion probability from that location is numerically very close to the mean time in which diversity is lost in Koldaeva et al.’s study. Both their findings and our results in Fig 5D agree that this aggregate diversity loss time grows approximately logarithmically with system size. As discussed above, the time for a species to fixate in the one-dimensional boundary (two domain) case also exhibits logarithmic scaling with system size at the equiprobable takeover abundance, as shown in Fig 2C. There is a connection between these two timescales, accounting for their similar scaling behavior. Given that most successful invasions occur when initiated from the center of the lane, as depicted in Fig 5B, the average invasion time (Fig 5D) is dominated by the invaders originating near the center. Alongside this, the fixation time of an invader starting near the center is the same (up to a factor) as the fixation time in the one-boundary case starting at the location of equiprobable abundance in the neutral case. Thus, the logarithmic scaling of the average invasion time is closely linked to the logarithmic scaling of the fixation time in the one-boundary case. In recent years, microfluidic monolayer devices (MMD)—such as mother machines, chemostats, etc.—have been designed to study single cell bacterial growth and generational dynamics [67, 96, 97]. Our results model the behaviour of competing populations in a single-lane, open chemostat and could be directly tested within such a device. However, our spatial model can also be extended to more complex 2D MMDs that support multi-lane channels. Pill-shaped bacteria, such as E. coli, are observed to grow constrained to 1D lanes within wide, open-ended microchannels [57, 58, 98]. As a first approximation, the larger channels can be viewed as many 1D lanes that interact through rare immigration events. In this simplified view, the dynamics of channel fixation can be decoupled from the lane fixation if the time between lane invasions is longer than the fixation time within a lane. An interesting direction for future work would be to combine the probability of a successful invasion found in Fig 5 with a rate of invasion to model 2D competition and check if the logarithmic scaling previously reported is recovered as we extend the model to higher dimensions. In summary, we have shown that explicitly incorporating spatial interactions arising from cell growth and division within dense bacterial populations can have important consequences for both the overall composition and the rate of species exclusion from the system. In densely packed microenvironments, bacteria will align into lanes to maintain their growth in the confined geometries of the space they inhabit. In the one-dimensional geometry of a channel, the skewed collective growth of cells fosters the “tug-of-war” dynamics between two populations, showcasing a distinct form of competition compared to well-mixed populations. Our results describe the outcome of the competition for space between species, predicting the timescale in which the competition terminates and the likelihood that a species survives in the ecosystem. Our results provide insights into the processes involved in the formation and maintenance of complex bacterial ecosystems such as biofilms, intestinal flora, or various persistent infections. Likewise, the mathematical techniques developed here may more broadly be applied to a range of competitive dynamical systems, from cellular transport to cancer. Supporting information S1 Appendix. Section A. Calculation of the discrete mean first-passage time to fixation. We review the calculation of the probability and exact mean first-passage time for the discrete models discussed in this main text. Section B. The Fokker-Planck approximation. The Fokker-Planck approximations corresponding to the continuous limit of the exact discrete dynamics are derived. Section C. The Moran model. Details of the classical Moran model with fitness differences are summarized. The differences between results of the Fokker-Planck equation and the master equation are explored. Section D. The spatial exclusion model. Further details of the spatial exclusion model are clarified, including the MFPT of fixation conditioned on the success of one of the two species. Intermediate fitness differences are plotted. A comparison of the Fokker-Planck and master equation results is shown. Section E. Average probabilities and MFPT. The MFPT and probability of fixation are averaged over all initial fractional abundances for the one-boundary problem. Section F. Invasion into a fixated channel. The mathematical details of the inva.sion dynamics discussed above are outlined. Additional results on a minimal invasion from the side of the channel are also shown. Section G. The heuristic asymptotic MFPT approximation. The asymptotic in N behaviour of the maximal MFPT at the equiprobable takeover abundance is explored. A mathematical derivation involving an approximation to the MFPT of fixation is investigated. https://doi.org/10.1371/journal.pcbi.1010868.s001 (PDF) Acknowledgments The authors are indebted to the members of the Milstein, Zilman and Goyal labs for numerous discussions.

Zhang, Xing;Yin, Hongmei;Ling, Fei;Zhan, Jian;Zhou, Yaoqi

doi: 10.1371/journal.pcbi.1011330pmid: 38060617

1. Introduction De novo protein design is considered as an inverse problem of de novo protein structure prediction, that is, to find a sequence that would fold into a given structure, instead of predicting its structure for a given sequence. Both problems have been long dominated by energy-based approaches: energy-guided fragment reassembly in the case of protein structure prediction [1] and energy-guided sequence design in the case of protein design [2]. The progress for solving both problems (poor accuracy for de novo structure prediction and low success rate for designed sequences, respectively), however, were hampered by the lack of an accurate energy function to describe the solvent-mediated interactions between amino acid residues of proteins [2,3]. The energy functions for protein design were typically modified from protein folding studies that can be categorized as molecular mechanics force fields (e.g. EGAD [4]), statistical energy functions (e.g. ABACUS [5,6]), and mixed statistical, empirical, and physical force fields (e.g. RosettaDesign [7]). More recently, we developed a new protein design technique called OSCAR-design [8] that is based on a purely mathematical scoring function. This scoring function employed series expansion in distance and orientation dependence with mixing coefficients optimized for sequence recovery, sidechain modeling, and loop selection. The optimization effort leads to an average recovery of wild-type sequences at ~40%, similar to that achieved by the state-of-the-art technique RosettaDesign 3.12 with mixed physical and statistical energy terms, suggesting the bottleneck of an energy-based method. To avoid an energy function, we developed the first direct prediction of sequences from structures by a simple artificial neural network called SPIN [9] (Sequence Profiles by Integrated Neural networks). By combining local-fragment-derived sequence profiles and nonlocal-energy functions, SPIN achieved a sequence recovery of 30% among 50 test proteins (denoted as TS50). SPIN2 [10] employed a deep three-layer neural network and additional structural features to improve SPIN, achieving a higher recovery of 34% on TS50. At the meantime, Qi et al. [11] employed a combination of three neural networks to predict amino-acid type of a center residue from structure fragments constructed with k-nearest neighbors (KNN) residues. With a preset k of 15, this method achieved 34% recovery in the 5-fold cross validation on a dataset constructed on PDB with 30% sequence identity cutoff. These early methods based on MLP (Multi-Layer Perceptrons), which is a basic architecture of artificial neural networks, provided a proof of concept for deep learning-based protein design given a fixed backbone structure. Rapid advances in deep learning techniques enable the breakthrough of AlphaFold2 [12] and RoseTTAFold [13] in protein structure prediction by avoiding the need for an energy function through the end-to-end learning [14,15]. During the same period, there is a rapid employment of deep learning in protein design by convolutional neural networks (CNN) such as SPROF [16], DenseCPD [17], and ProDCoNN [18], graph neural networks (GNN) such as GraphTrans [19], GCA [20], GVP [21], AlphaDesign [22], ESM-IF [23], ProteinMPNN [24], PiFold [25], and LM-DESIGN [26], and transformer such as ABACUS-R [27] and ProDesign-LE [28]. These AI-driven backbone-based design substantially improved sequence recovery from 30–34% in MLP based techniques to ~50–55% by PiFold [25] or LM-DESIGN [26]. Moreover, the methods have developed from fixed backbone design, flexible design based on energy-based structure generation (SCUBA [29]) and sequence-based structure prediction [30,31] to structure and sequence generators [32–34]. This work focuses on the use of GNN for further improving AI-based fixed-backbone design as it appears to improve over MLP and CNN-based models in the latest development [35]. GraphTrans [19] represented protein backbone structures as a graph, in which residues were represented by node features containing distance and orientation between sequential adjacent Cα atoms and backbone dihedral angles, while inter-residue distances and orientations were encoded as edge features. With an encoder-decoder model constructed with graph attention layers, node features were updated to predict the sequences by sequential autoregressive decoding. GraphTrans achieved 35.82% recovery after training and testing on a dataset set constructed on CATH 4.2 database (denoted as CATH4.2). GCA [20] (Global Context Aware generative protein design) improved GraphTrans by appending a global module after the local module (the graph attention layer). It improved the recovery on CATH4.2 to 37.64%. GVP [21] (Geometric Vector Perceptron) decoupled vector and scale information in graph features and proposed a network module to update geometrically sensitive representations. By simply replacing MLP layers employed in GraphTrans with GVP layers, the recovery on CATH4.2 increased to 39.47%. ProteinMPNN [24] improved GraphTrans with three modifications: replacing edge features with interatomic distances between all five atoms (including a virtual Cβ atom) on backbones, updating edge features in GNN, and replacing the sequential decoding order with random decoding order in autoregressive decoding. These modifications further improved sequence recovery on CATH4.2 to 45.96%. PiFold [25] introduced virtual atoms determined by backbone position and learnable parameters. Besides, the autoregressive decoding was replaced by one-shot decoding. PiFold achieved 51.66% recovery on CATH4.2 with orders of magnitude efficiency improvement. The above methods can be further improved by using additional training data, scaling model sizes, and integration with large pretrained models. For example, ESM-IF [23] stack a scaled GVP model with a large transformer model and trained with over 1.2 million of structures predicted by AlphaFold2. The model containing over 142 million parameters significantly improved the recovery of the GVP model in CATH 4.2 dataset from 42.2% to 51.3%. LM-DESIGN [34] used a large protein language model, which was pretrained with over 50 million protein sequences, as a decoder to sampled protein sequences with an encoder from GVP, ProteinMPNN, and PiFold. With about 650 million of additional pretrained parameters, this method brought over 5% improvement to these methods. For the PiFold model reimplemented in this work, we found an increase from 50.22% to 55.65% for the sequence recovery. These GNN-based methods [19–26], however, utilized k-nearest neighbors (KNN) to construct a graph for feature initialization and local information passing. The KNN graph construction significantly reduces the computational cost from passing information between all node pairs in a graph, since k is much smaller than the length of a sequence in most cases. Moreover, a proper setting of k is expected to prevent the modules for local information passing from overfitting the non-local information. Typically, k is set as 30 because many studies [19,24] suggested it sufficient for local information passing in GNN-based, fixed-backbone protein design. However, the local structures defined by a fixed number of neighbors might not be capable of handling dense local structures and sparse local structures at the same time. In this study, we proposed SPIN-CGNN, a deep graph neural network-based method for the fixed backbone design, in which a protein structure graph is constructed with a distance-based contact map. This contact map-based graph construction (CGraph) enables GNN to handle a varied number of neighbors within a preset distance cutoff. In addition, we introduced information of symmetric and second order edges to update edge features. The symmetric edge information enabled information sharing inside an edge pair that connects two nodes. The information on second-order edges is expected to capture high-order interactions between two nodes from their shared neighbors. We found that this SPIN-CGNN achieved 54.81% for sequence recovery. This was achieved by employing a small model of 5.58 million parameters in the absence of pretrained models. Moreover, we further evaluated the method according to amino-acid substitution matrix, sequence complexity, and the deviation of the query structure to the structure predicted by AlphaFold2. These performance measures further support the improvement of SPIN-CGNN over or comparable to existing state-of-the-art techniques. 2. Methods Fig 1 shows the overall workflow of SPIN-CGNN. The backbone structure in SPIN-CGNN is first represented as a graph with a distance-based contact graph representation. It takes in the coordinates of backbone atoms including N, Cα, C, O, and transforms them as node features for residues and edge features for inter-residues relationship. Moreover, the connections in the graph including edges and second-order edges were recorded for the computation of neural networks. Then, the represented graph will be inputted into the SPIN-CGNN blocks to iteratively extract structural features. An MLP will be applied on the updated node features to predict the probability of 20 amino acids for each position. Finally, protein sequences can be sampled based on the predicted probabilities. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. The overall workflow of SPIN-CGNN, which employed a contact-based graph with improved edge and node updates. https://doi.org/10.1371/journal.pcbi.1011330.g001 2.1. Datasets Many GNN-based methods have employed the CATH 4.2 dataset from Ingraham’s [19] to assess their performance. This dataset was constructed by: a) collecting all chains with no more than 500 residues from the CATH 4.2 at 40% sequence identity cutoff; b) randomly split all collected chains into 80%/10%/10% subsets for training, validation, and test; c) removed the entries from these subsets to ensure that there was no overlap in CATH topology (also known as fold) classification between the subsets. Overall, the dataset consisted of a total of 18024, 608, and 1120 structure-sequence pairs in the training, validation, and test subsets from 950, 100, and 150 structural folds, respectively. Thus, there were overlaps in structural folds within each subset. To avoid performance bias toward specific structural folds within the CATH4.2 test set, we calculated the TM-scores [36] between all test structures and found some structure pairs with high levels of similarity, as illustrated in S1 Fig. Thus, we created a non-structural redundant version of the test set by iteratively removing entries that had the most structural similar entries (TM-score > 0.4 [36]) in the test set until no structurally similar entry pairs remained. Ultimately, the resulting CATH4.2-StructNR193 test set contained 193 structure-sequence pairs, after removing 927 out of the original 1120 entries. To generate another fully independent test set, we performed the following: a) gathering all PDB structures released after CATH4.2 (April 9th, 2019); b) extracting chains with no less than 30 residues; c) calculating the TM-score between each chain and all existing entries from the new test set, the CATH4.2 test set, the CATH4.2 validation set, and the CATH4.2 training set; and d) retaining the chain if the maximum TM-score with all existing entries is no more than 0.4. Finally, we obtained a structural non-redundant test set with 156 entries, which we named as PDB-StructNR156. Recently, several methods enabled the generation of near-native structures for protein design, including hallucination and diffusion models. We aimed to further evaluate the performance of fixed backbone protein design methods using generated structures. For hallucination methods, we utilized the ‘Hallucination129’ test set that includes 129 hallucinated structures [31]. For diffusion models, we constructed a new ‘Diffusion100’ test set by: a) generated 1000 structures with SE(3) diffusion model [37]; b) test the refoldability with the process from the paper of SE(3)-diffusion model. Specifically, we designed 8 sequences for each generated structure with ProteinMPNN, predicted structures for each sequence with ESMFold [38], calculated the median refold RMSD of 8 designed sequences; c) picked the 100 generated structures with the smallest median refold RMSD. 2.2. Graph representation Contact Map-based Graph Construction: We defined the neighbors in a graph by contacts between the virtual Cβ atoms with a distance cutoff. The coordinate of virtual Cβ atom of each residue is calculated by where ui is the directional vector from atom to atom vi is the directional vector from atom Ci to atom . Notably, we removed all self-loops in the contact graph (a residue is not a neighbor of itself). In the contact graph, a residue can receive information from different number of neighbors, depending on the local density around the residue. The edges in a contact graph are symmetric: whenever residue i is a neighbor of residue j, residue j would be a neighbor of residue i. We constructed the node features and edge features as in PiFold [25]. In addition to conventional features widely employed such as distance and direction between backbone atoms, the PiFold features further incorporated bond lengths, bond angles, and learnable virtual atoms. A series of ablation studies conducted to discern the individual contributions of these components revealed a slight enhancement in the recovery performance of PiFold in fixed backbone design. Node Features: For the node features of residue i, we defined a local coordinate system to construct rotation-invariant features, where , and . The unit directional vectors from to Ni, Ci, Oi, and in the local coordinate system were collected as node directional features. Bond angles and torsion angles of continuous Ni, , and Ci were encoded as sine and cosine values as node angle features. Specifically, the bond angles included , and , where a−b−c denotes the angle between and . The torsion angles included rotational angles around (i.e., ω, ϕ, ψ, respectively). The distances from to Ni, Ci, Oi, and the virtual were encoded with the Gaussian radial basis function (RBF) from ProteinMPNN [24] as node distance features. Finally, the initial node features of residue i were constructed by concatenating all unit vector, angle, and distance features. Edge Features: The edge features of residue j to residue i also contain unit vector, angle, and distance features. All interatomic unit directional vectors from five atoms (Cα, C, N, O, and the virtual Cβ) of residue j to those of residue i were calculated and rotated with the local coordinate system Qi, in total of 25 edge unit vector features. The interatomic distances between atoms of residue i to atoms of residue j, including five main chain atoms (Cα, C, N, and O plus the virtual Cβ) and three virtual atoms determined by learnable parameters, were collected and encoded with an RBF as 64 edge distance features in total. The coordinate of three virtual atoms of residue i can be calculated as where , and (xn, yn, zn) is a set of learnable parameters of the nth virtual atoms. These virtual atoms were employed for capturing complementary information with real atoms. The number of virtual atoms was set as 3 in this study, according to PiFold [25]. The rotation from Qj to Qi was encoded with the quaternion function as the edge angle features. Additionally, the sequential relative distance (the difference in sequence positions, i−j) from residue j to residue i was encoded, with a positional encoding function [39], and append into the edge features. Here, we set the dimension of positional encoding function and RBF as 16. Thus, the total dimensions of node and edge features are 96 and 1119, respectively. Note that any residues with any missing coordinates of atoms Cα, C, N, and O will be masked. 2.3. Network architecture The graph neural network in SPIN-CGNN was built by stacking 10 Contact Graph Neural Network (CGNN) blocks to fit the message passing in contact map-based graphs. In a CGNN block, edge features were updated first (S2 Fig) and these updated edge features were then utilized to update node features (S3 Fig). 2.3.1. Edge update. Updating edge features has been proved to be useful for improving model performance in many related works including ProteinMPNN [24] and PiFold [25]. If we denote the node feature of node i in layer l as and the edge feature from to as eij, the edge update is performed by simply aggregating the information from adjacent nodes and edge itself: where is the edge update by a MLP module, and ‖ represents a concatenate operation. We further enriched edge updates by edge symmetry. We defined eij as the symmetric edge of eji in a graph. To ensure that the information passing inside symmetric edge pairs is symmetric, each edge is concatenated with its symmetric edge to produce the symmetric edge information by: In addition to edge symmetry, we introduced the second-order edges (SOE) by considering the edges associated with their shared neighbors. More specifically, given representing the neighbor nodes for node i, the shared neighbors of node i and node j can be represented as . For a shared neighbor node , we defined the combination of ein and enj as the SOE from node i to node j though node n. The edge update information from the SOE can be captured with a MLP module from the concatenated feature of , and . By taking the average of all second-order edges of eij, we can calculate the SOE information as: where is the number of nodes in . A second MLP module is applied to to specifically extract adjacent information from the basic edge update information : The above updates were merged by a selective kernel module to produce the final edge update (S4A Fig), similar to the selective kernel convolution from SK-Net [40]: where SK represents selective kernel, and is the merged edge update information from the graph. To prevent overfitting, the edge information would be updated with dropout, residual connection, and layer normalization: Adding a position-wise feedforward module after attention module has been found to significantly improve the network performance [39]. Therefore, we performed the final update with such typical operation: where FFN represent the position-wise feedforward module. 2.3.2. Node update. Node updates in CGNN blocks were performed by integrating local information from neighboring nodes and global information from the whole graph. More specifically, we extract local information by aggregating information from neighboring nodes of the center node with a typical graph-attention module, in which the attention scores were calculated by: where is the query features of node i, is the key features of , and is the attention score of for the local information of node i. We further calculated the value features which were concatenated from edge feature , and its adjacent node features and . The summation of neighboring attention-scaled value features yields the local information : In addition to the local updates, we also used global updates to account for nonlocal interactions. If we define as all nodes in a protein structure graph, we can calculate the global context Gl by summing scaled value features from all nodes, in which both value feature and attention were calculated from the node feature itself as below: The global update was then calculated from the concatenated features of node feature and global context, with an MLP: Similarly, a selective kernel module (S4B Fig) was used to merge global with local update information: Same as edge update, the node features were updated with the graph update information: followed by a FFN update: to reduce the possibility of overfitting. 2.3.3. Selective kernel. The selective kernel from SKNet [40] (Selective Kernel Networks) was designed for and has been widely used to merge multi-scale features captured by convolutional kernels with different kernel sizes. In SPIN-CGNN, it was employed to adaptively merge a set of features from different update modules. Given a feature set ℱ containing n features, a selective kernel simply summarizes all features and squeeze the dimension with an MLP: A set of MLPs were employed for the excitation from fsqueeze to the dimension-wise weight of each feature, and normalized by Softmax function: Finally, the merged feature fmerge is calculated by summarizing all features that are dimension-wisely scaled by the attention: 2.4. Training The dimensions of edge and node features were set as 128 for all layers. All models were trained by minimizing the cross entropy between output logits and native sequences for 100 epochs with AdamW optimizer [41]. The learning rate was adjusted according to OneCycle learning rate schedule [42] with a max learning rate of 0.004. We set the drop probability as 0.1 for all dropout operators. Training data were randomly grouped with a maximum batch size of 4096 residues. We employed mixed precision to accelerate the training speed and reduce the GPU memory occupation in all experiments. All other settings followed the default setting of PyTorch.1.13 [43]. 2.5. Performance measure The most widely used criteria for evaluating the methods for fixed backbone design are perplexity and recovery. The perplexity on test set is calculated by exponentiated categorical cross-entropy loss per residue: where SN is a sequence with N residues from the test set is the i-th native residue and is the corresponding predicted probability from the model. Perplexity is a measure that accounts for the certainty around the native amino acid residues. Lower perplexity values indicate smaller deviation from native residue types. Recovery, measuring the ability of the model to reconstruct the native sequence of a protein, is calculated by the percentage of the identity of designed sequences to native sequences: This measure, however, only compared the top ranked prediction and does not reflect fluctuation around the top ranked prediction. We further examined the frequencies of each amino-acid-residue type given by native sequences and designed sequences (amino acid compositions). The similarity between native frequencies and predicted frequencies can be measured by the relative deviations: where X is native frequencies and X′ is the predicted frequencies. It is known that surface residues are more difficult to recover, we calculated the fraction of surface residues for each target protein. The residue-wise SASA (Solvent Accessible Surface Area) was obtained by BioPython [44]. The SASA for each residue is divided by the maximum allowed solvent accessibility (MaxASA) [45] of the residue type to yield the relative accessible surface area (RSA). Finally, we classified the residues with RSA smaller than 0.2 as core residues, and the others as surface residues as in OSCAR-design [8]. However, the above criteria are based on native sequences. Many sequences can fold into the same structure. Some sequences can fail to fold into a target structure despite high sequence identity to the native sequence because a few mutations may well destabilize the structure. Thus, we also examined low complexity regions, which is the subsequences that normally lead to intrinsically disordered regions and the inability to fold into the target structure. We detected these subsequences using the SEG algorithm [46]. SEG identifies approximate segments of low complexity using a sliding window in the first pass and optimized these segments in the second pass. The optimized segments with the information measure lower than a given threshold will be marked as an LCR. Specifically, we run the SEG algorithm with the default setting from NCBI C++ toolkit [47], which is 12 for sliding window size and 2.2 for low-information cutoff. The fractions of low complexity regions were calculated as a criterion for analysis, denoted as LCR. Additionally, we measured the frequencies of amino acid substitutions in the designed sequences from the native sequences by calculation the BLOSUM score and the Pearson correlation coefficient of the confusion matrix with BLOSUM62[48] as the reference of native amino acids substitution. The BLOSUM score is calculated as the summation of BLOSUM62 values of the native amino acid, weighted by the predicted probability. It should be mentioned here that we did not calculate BLOSUM score for OSCAR-design and RosettaFixBB because it does not yield the probability of amino acid residues in one design as in AI-based methods. Although in principle one could perform 100 designs for each protein to obtain the probability, computational requirement is prohibitive for our available computing resource when applying to the large dataset we are employing here for test. The calculation of confusion matrix followed ESM-IF [23], in which the substitution scores between native sequences and designed sequences were calculated by using the same log odds ratio formula as: in the BLOSUM62 substitution matrix. For two amino acid types x and y, the substitution score is: where p(x, y) is the jointly likelihood that native amino acid x is substituted by predicted amino acid y, q(x) is the frequencies of amino acid x in the native distribution, and q(y) is the frequencies of amino acid y in the predicted distribution. Finally, we performed a test to evaluated the performance of methods by measuring the structure deviations between the target structures and the AlphaFold2-predicted structures for designed sequences. Notably, we run AlphaFold2 with default setting while excluding PDB templates to avoid the influence of the template searched by high sequence recovery. Three criteria were employed including TM-score, RMSD (Root-Mean-Square Deviation), GDT-TS (Global Distance Test–Total Score) [49] on the superposition Cα coordinate. Specifically, the distance cutoff used in GDT-TS is 1, 2, 4, and 8 Å, same as that in CASP [49]. 2.6. Method comparison We employed four methods for comparison including two energy-based methods RosettaFixBB and OSCAR-design, and two GNN-based method ProteinMPNN and PiFold. For the energy-based method RosettaFixBB and OSCAR-design, we designed sequences with the default setting. For deep learning-based methods ProteinMPNN and PiFold, we reimplemented them with the source code and the training setting from their paper. Specifically, we reimplemented ProteinMPNN model with its source code and training setting: negative-log likelihood loss, transformer learning rate schedule, batch size 6000 residues, training epochs 100, Adam optimizer, 30 residue neighbors, no coordinate noise. The median recovery of the reimplemented model tested on CATH4.2 test set is 46.15%, which is consistent to the reported recovery of ProteinMPNN reimplemented model (45.96%) [25]. We also reimplemented PiFold with its source code and training setting: negative-log likelihood loss, OneCycle learning rate schedule, a batch size of 4096 residues, training epochs of 100, Adam optimizer, 30 residue neighbors, and 3 virtual atoms. The median recovery of the reimplemented model tested on CATH4.2 test set is 51.55%, which is also consistent to the reported recovery (51.66%) [25]. 2.1. Datasets Many GNN-based methods have employed the CATH 4.2 dataset from Ingraham’s [19] to assess their performance. This dataset was constructed by: a) collecting all chains with no more than 500 residues from the CATH 4.2 at 40% sequence identity cutoff; b) randomly split all collected chains into 80%/10%/10% subsets for training, validation, and test; c) removed the entries from these subsets to ensure that there was no overlap in CATH topology (also known as fold) classification between the subsets. Overall, the dataset consisted of a total of 18024, 608, and 1120 structure-sequence pairs in the training, validation, and test subsets from 950, 100, and 150 structural folds, respectively. Thus, there were overlaps in structural folds within each subset. To avoid performance bias toward specific structural folds within the CATH4.2 test set, we calculated the TM-scores [36] between all test structures and found some structure pairs with high levels of similarity, as illustrated in S1 Fig. Thus, we created a non-structural redundant version of the test set by iteratively removing entries that had the most structural similar entries (TM-score > 0.4 [36]) in the test set until no structurally similar entry pairs remained. Ultimately, the resulting CATH4.2-StructNR193 test set contained 193 structure-sequence pairs, after removing 927 out of the original 1120 entries. To generate another fully independent test set, we performed the following: a) gathering all PDB structures released after CATH4.2 (April 9th, 2019); b) extracting chains with no less than 30 residues; c) calculating the TM-score between each chain and all existing entries from the new test set, the CATH4.2 test set, the CATH4.2 validation set, and the CATH4.2 training set; and d) retaining the chain if the maximum TM-score with all existing entries is no more than 0.4. Finally, we obtained a structural non-redundant test set with 156 entries, which we named as PDB-StructNR156. Recently, several methods enabled the generation of near-native structures for protein design, including hallucination and diffusion models. We aimed to further evaluate the performance of fixed backbone protein design methods using generated structures. For hallucination methods, we utilized the ‘Hallucination129’ test set that includes 129 hallucinated structures [31]. For diffusion models, we constructed a new ‘Diffusion100’ test set by: a) generated 1000 structures with SE(3) diffusion model [37]; b) test the refoldability with the process from the paper of SE(3)-diffusion model. Specifically, we designed 8 sequences for each generated structure with ProteinMPNN, predicted structures for each sequence with ESMFold [38], calculated the median refold RMSD of 8 designed sequences; c) picked the 100 generated structures with the smallest median refold RMSD. 2.2. Graph representation Contact Map-based Graph Construction: We defined the neighbors in a graph by contacts between the virtual Cβ atoms with a distance cutoff. The coordinate of virtual Cβ atom of each residue is calculated by where ui is the directional vector from atom to atom vi is the directional vector from atom Ci to atom . Notably, we removed all self-loops in the contact graph (a residue is not a neighbor of itself). In the contact graph, a residue can receive information from different number of neighbors, depending on the local density around the residue. The edges in a contact graph are symmetric: whenever residue i is a neighbor of residue j, residue j would be a neighbor of residue i. We constructed the node features and edge features as in PiFold [25]. In addition to conventional features widely employed such as distance and direction between backbone atoms, the PiFold features further incorporated bond lengths, bond angles, and learnable virtual atoms. A series of ablation studies conducted to discern the individual contributions of these components revealed a slight enhancement in the recovery performance of PiFold in fixed backbone design. Node Features: For the node features of residue i, we defined a local coordinate system to construct rotation-invariant features, where , and . The unit directional vectors from to Ni, Ci, Oi, and in the local coordinate system were collected as node directional features. Bond angles and torsion angles of continuous Ni, , and Ci were encoded as sine and cosine values as node angle features. Specifically, the bond angles included , and , where a−b−c denotes the angle between and . The torsion angles included rotational angles around (i.e., ω, ϕ, ψ, respectively). The distances from to Ni, Ci, Oi, and the virtual were encoded with the Gaussian radial basis function (RBF) from ProteinMPNN [24] as node distance features. Finally, the initial node features of residue i were constructed by concatenating all unit vector, angle, and distance features. Edge Features: The edge features of residue j to residue i also contain unit vector, angle, and distance features. All interatomic unit directional vectors from five atoms (Cα, C, N, O, and the virtual Cβ) of residue j to those of residue i were calculated and rotated with the local coordinate system Qi, in total of 25 edge unit vector features. The interatomic distances between atoms of residue i to atoms of residue j, including five main chain atoms (Cα, C, N, and O plus the virtual Cβ) and three virtual atoms determined by learnable parameters, were collected and encoded with an RBF as 64 edge distance features in total. The coordinate of three virtual atoms of residue i can be calculated as where , and (xn, yn, zn) is a set of learnable parameters of the nth virtual atoms. These virtual atoms were employed for capturing complementary information with real atoms. The number of virtual atoms was set as 3 in this study, according to PiFold [25]. The rotation from Qj to Qi was encoded with the quaternion function as the edge angle features. Additionally, the sequential relative distance (the difference in sequence positions, i−j) from residue j to residue i was encoded, with a positional encoding function [39], and append into the edge features. Here, we set the dimension of positional encoding function and RBF as 16. Thus, the total dimensions of node and edge features are 96 and 1119, respectively. Note that any residues with any missing coordinates of atoms Cα, C, N, and O will be masked. 2.3. Network architecture The graph neural network in SPIN-CGNN was built by stacking 10 Contact Graph Neural Network (CGNN) blocks to fit the message passing in contact map-based graphs. In a CGNN block, edge features were updated first (S2 Fig) and these updated edge features were then utilized to update node features (S3 Fig). 2.3.1. Edge update. Updating edge features has been proved to be useful for improving model performance in many related works including ProteinMPNN [24] and PiFold [25]. If we denote the node feature of node i in layer l as and the edge feature from to as eij, the edge update is performed by simply aggregating the information from adjacent nodes and edge itself: where is the edge update by a MLP module, and ‖ represents a concatenate operation. We further enriched edge updates by edge symmetry. We defined eij as the symmetric edge of eji in a graph. To ensure that the information passing inside symmetric edge pairs is symmetric, each edge is concatenated with its symmetric edge to produce the symmetric edge information by: In addition to edge symmetry, we introduced the second-order edges (SOE) by considering the edges associated with their shared neighbors. More specifically, given representing the neighbor nodes for node i, the shared neighbors of node i and node j can be represented as . For a shared neighbor node , we defined the combination of ein and enj as the SOE from node i to node j though node n. The edge update information from the SOE can be captured with a MLP module from the concatenated feature of , and . By taking the average of all second-order edges of eij, we can calculate the SOE information as: where is the number of nodes in . A second MLP module is applied to to specifically extract adjacent information from the basic edge update information : The above updates were merged by a selective kernel module to produce the final edge update (S4A Fig), similar to the selective kernel convolution from SK-Net [40]: where SK represents selective kernel, and is the merged edge update information from the graph. To prevent overfitting, the edge information would be updated with dropout, residual connection, and layer normalization: Adding a position-wise feedforward module after attention module has been found to significantly improve the network performance [39]. Therefore, we performed the final update with such typical operation: where FFN represent the position-wise feedforward module. 2.3.2. Node update. Node updates in CGNN blocks were performed by integrating local information from neighboring nodes and global information from the whole graph. More specifically, we extract local information by aggregating information from neighboring nodes of the center node with a typical graph-attention module, in which the attention scores were calculated by: where is the query features of node i, is the key features of , and is the attention score of for the local information of node i. We further calculated the value features which were concatenated from edge feature , and its adjacent node features and . The summation of neighboring attention-scaled value features yields the local information : In addition to the local updates, we also used global updates to account for nonlocal interactions. If we define as all nodes in a protein structure graph, we can calculate the global context Gl by summing scaled value features from all nodes, in which both value feature and attention were calculated from the node feature itself as below: The global update was then calculated from the concatenated features of node feature and global context, with an MLP: Similarly, a selective kernel module (S4B Fig) was used to merge global with local update information: Same as edge update, the node features were updated with the graph update information: followed by a FFN update: to reduce the possibility of overfitting. 2.3.3. Selective kernel. The selective kernel from SKNet [40] (Selective Kernel Networks) was designed for and has been widely used to merge multi-scale features captured by convolutional kernels with different kernel sizes. In SPIN-CGNN, it was employed to adaptively merge a set of features from different update modules. Given a feature set ℱ containing n features, a selective kernel simply summarizes all features and squeeze the dimension with an MLP: A set of MLPs were employed for the excitation from fsqueeze to the dimension-wise weight of each feature, and normalized by Softmax function: Finally, the merged feature fmerge is calculated by summarizing all features that are dimension-wisely scaled by the attention: 2.3.1. Edge update. Updating edge features has been proved to be useful for improving model performance in many related works including ProteinMPNN [24] and PiFold [25]. If we denote the node feature of node i in layer l as and the edge feature from to as eij, the edge update is performed by simply aggregating the information from adjacent nodes and edge itself: where is the edge update by a MLP module, and ‖ represents a concatenate operation. We further enriched edge updates by edge symmetry. We defined eij as the symmetric edge of eji in a graph. To ensure that the information passing inside symmetric edge pairs is symmetric, each edge is concatenated with its symmetric edge to produce the symmetric edge information by: In addition to edge symmetry, we introduced the second-order edges (SOE) by considering the edges associated with their shared neighbors. More specifically, given representing the neighbor nodes for node i, the shared neighbors of node i and node j can be represented as . For a shared neighbor node , we defined the combination of ein and enj as the SOE from node i to node j though node n. The edge update information from the SOE can be captured with a MLP module from the concatenated feature of , and . By taking the average of all second-order edges of eij, we can calculate the SOE information as: where is the number of nodes in . A second MLP module is applied to to specifically extract adjacent information from the basic edge update information : The above updates were merged by a selective kernel module to produce the final edge update (S4A Fig), similar to the selective kernel convolution from SK-Net [40]: where SK represents selective kernel, and is the merged edge update information from the graph. To prevent overfitting, the edge information would be updated with dropout, residual connection, and layer normalization: Adding a position-wise feedforward module after attention module has been found to significantly improve the network performance [39]. Therefore, we performed the final update with such typical operation: where FFN represent the position-wise feedforward module. 2.3.2. Node update. Node updates in CGNN blocks were performed by integrating local information from neighboring nodes and global information from the whole graph. More specifically, we extract local information by aggregating information from neighboring nodes of the center node with a typical graph-attention module, in which the attention scores were calculated by: where is the query features of node i, is the key features of , and is the attention score of for the local information of node i. We further calculated the value features which were concatenated from edge feature , and its adjacent node features and . The summation of neighboring attention-scaled value features yields the local information : In addition to the local updates, we also used global updates to account for nonlocal interactions. If we define as all nodes in a protein structure graph, we can calculate the global context Gl by summing scaled value features from all nodes, in which both value feature and attention were calculated from the node feature itself as below: The global update was then calculated from the concatenated features of node feature and global context, with an MLP: Similarly, a selective kernel module (S4B Fig) was used to merge global with local update information: Same as edge update, the node features were updated with the graph update information: followed by a FFN update: to reduce the possibility of overfitting. 2.3.3. Selective kernel. The selective kernel from SKNet [40] (Selective Kernel Networks) was designed for and has been widely used to merge multi-scale features captured by convolutional kernels with different kernel sizes. In SPIN-CGNN, it was employed to adaptively merge a set of features from different update modules. Given a feature set ℱ containing n features, a selective kernel simply summarizes all features and squeeze the dimension with an MLP: A set of MLPs were employed for the excitation from fsqueeze to the dimension-wise weight of each feature, and normalized by Softmax function: Finally, the merged feature fmerge is calculated by summarizing all features that are dimension-wisely scaled by the attention: 2.4. Training The dimensions of edge and node features were set as 128 for all layers. All models were trained by minimizing the cross entropy between output logits and native sequences for 100 epochs with AdamW optimizer [41]. The learning rate was adjusted according to OneCycle learning rate schedule [42] with a max learning rate of 0.004. We set the drop probability as 0.1 for all dropout operators. Training data were randomly grouped with a maximum batch size of 4096 residues. We employed mixed precision to accelerate the training speed and reduce the GPU memory occupation in all experiments. All other settings followed the default setting of PyTorch.1.13 [43]. 2.5. Performance measure The most widely used criteria for evaluating the methods for fixed backbone design are perplexity and recovery. The perplexity on test set is calculated by exponentiated categorical cross-entropy loss per residue: where SN is a sequence with N residues from the test set is the i-th native residue and is the corresponding predicted probability from the model. Perplexity is a measure that accounts for the certainty around the native amino acid residues. Lower perplexity values indicate smaller deviation from native residue types. Recovery, measuring the ability of the model to reconstruct the native sequence of a protein, is calculated by the percentage of the identity of designed sequences to native sequences: This measure, however, only compared the top ranked prediction and does not reflect fluctuation around the top ranked prediction. We further examined the frequencies of each amino-acid-residue type given by native sequences and designed sequences (amino acid compositions). The similarity between native frequencies and predicted frequencies can be measured by the relative deviations: where X is native frequencies and X′ is the predicted frequencies. It is known that surface residues are more difficult to recover, we calculated the fraction of surface residues for each target protein. The residue-wise SASA (Solvent Accessible Surface Area) was obtained by BioPython [44]. The SASA for each residue is divided by the maximum allowed solvent accessibility (MaxASA) [45] of the residue type to yield the relative accessible surface area (RSA). Finally, we classified the residues with RSA smaller than 0.2 as core residues, and the others as surface residues as in OSCAR-design [8]. However, the above criteria are based on native sequences. Many sequences can fold into the same structure. Some sequences can fail to fold into a target structure despite high sequence identity to the native sequence because a few mutations may well destabilize the structure. Thus, we also examined low complexity regions, which is the subsequences that normally lead to intrinsically disordered regions and the inability to fold into the target structure. We detected these subsequences using the SEG algorithm [46]. SEG identifies approximate segments of low complexity using a sliding window in the first pass and optimized these segments in the second pass. The optimized segments with the information measure lower than a given threshold will be marked as an LCR. Specifically, we run the SEG algorithm with the default setting from NCBI C++ toolkit [47], which is 12 for sliding window size and 2.2 for low-information cutoff. The fractions of low complexity regions were calculated as a criterion for analysis, denoted as LCR. Additionally, we measured the frequencies of amino acid substitutions in the designed sequences from the native sequences by calculation the BLOSUM score and the Pearson correlation coefficient of the confusion matrix with BLOSUM62[48] as the reference of native amino acids substitution. The BLOSUM score is calculated as the summation of BLOSUM62 values of the native amino acid, weighted by the predicted probability. It should be mentioned here that we did not calculate BLOSUM score for OSCAR-design and RosettaFixBB because it does not yield the probability of amino acid residues in one design as in AI-based methods. Although in principle one could perform 100 designs for each protein to obtain the probability, computational requirement is prohibitive for our available computing resource when applying to the large dataset we are employing here for test. The calculation of confusion matrix followed ESM-IF [23], in which the substitution scores between native sequences and designed sequences were calculated by using the same log odds ratio formula as: in the BLOSUM62 substitution matrix. For two amino acid types x and y, the substitution score is: where p(x, y) is the jointly likelihood that native amino acid x is substituted by predicted amino acid y, q(x) is the frequencies of amino acid x in the native distribution, and q(y) is the frequencies of amino acid y in the predicted distribution. Finally, we performed a test to evaluated the performance of methods by measuring the structure deviations between the target structures and the AlphaFold2-predicted structures for designed sequences. Notably, we run AlphaFold2 with default setting while excluding PDB templates to avoid the influence of the template searched by high sequence recovery. Three criteria were employed including TM-score, RMSD (Root-Mean-Square Deviation), GDT-TS (Global Distance Test–Total Score) [49] on the superposition Cα coordinate. Specifically, the distance cutoff used in GDT-TS is 1, 2, 4, and 8 Å, same as that in CASP [49]. 2.6. Method comparison We employed four methods for comparison including two energy-based methods RosettaFixBB and OSCAR-design, and two GNN-based method ProteinMPNN and PiFold. For the energy-based method RosettaFixBB and OSCAR-design, we designed sequences with the default setting. For deep learning-based methods ProteinMPNN and PiFold, we reimplemented them with the source code and the training setting from their paper. Specifically, we reimplemented ProteinMPNN model with its source code and training setting: negative-log likelihood loss, transformer learning rate schedule, batch size 6000 residues, training epochs 100, Adam optimizer, 30 residue neighbors, no coordinate noise. The median recovery of the reimplemented model tested on CATH4.2 test set is 46.15%, which is consistent to the reported recovery of ProteinMPNN reimplemented model (45.96%) [25]. We also reimplemented PiFold with its source code and training setting: negative-log likelihood loss, OneCycle learning rate schedule, a batch size of 4096 residues, training epochs of 100, Adam optimizer, 30 residue neighbors, and 3 virtual atoms. The median recovery of the reimplemented model tested on CATH4.2 test set is 51.55%, which is also consistent to the reported recovery (51.66%) [25]. 3. Results 3.1. Impact of graph constructions We examined the impact of using the contact maps at different cutoff distances and compared them against the K-nearest-neighbor graph (k = 30, KNN-30) employing the CATH4.2-StructNR193 and the PDB-StructNR156 test sets. Performance was evaluated using perplexity and median recovery. To make a fair comparison, Table 1 compares KNN-30 to CGNN all at without employing CGNN edge information (named as Model 1). The results indicated that KNN-30 has a better performance than CGraph-8 (Contact graph at 8Å distance cutoff) for both test sets. However, increasing distance cutoff (from 8Å to 10Å, and then 12 Å) improves over KNN-30 in perplexity and sequence recovery. At 12 Å cutoff, there is ~1% increase in median sequence recovery, and 2–3% reduction of perplexity from KNN-30 to CGraph-12. We note that even at 12Å cutoff, the average number of neighbors (25 or 29) is still smaller than 30 employed in KNN-30. We fixed the cutoff at 12Å for all subsequent analysis because further increasing the cutoff will only lead to minor improvement at the expense of higher computational requirement. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Contact-based versus K-nearest neighbors in the absence of edge information for all methods (Model 1 for SPIN-CGNN) according to perplexity and median sequence recovery for two test datasets (CATH4.2-StructNR193 and PDB-StructNR156). https://doi.org/10.1371/journal.pcbi.1011330.t001 3.2. Ablation test for CGNN edge updates Table 2 examines the effect of second-order edge (SOE) and symmetric edge updates by constructing Model 2 and Model 3, respectively, as well as the SPIN-CGNN with both SOE and symmetric edge updates. Table 2 shows that without both edge updates Model 1 leads to the worst performance in perplexity and median sequence recovery. Removing SOE also led to statistically significant increase of perplexity and reduction of median recovery from SPIN-CGNN. Although symmetric edge updates do improve the CGNN model when SOE update is absent (from Model 1 to Model 2), it contributes little when the SOE update is performed, indicating that the information captured by symmetric edge updates may have been covered by SOE updates. The overall effect of introducing edge updates is ~1% increase in sequence recovery and 4–6% reduction in perplexity. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Impact of CGNN edge updates (symmetric information and second-order edge (SOE) information) according to perplexity and median sequence recovery for two test datasets (CATH4.2-StructNR193 and PDB-StructNR156). https://doi.org/10.1371/journal.pcbi.1011330.t002 3.3. Ablation test for selective kernel (SK) Table 3 examines the effect of the feature integrating module, selective kernels (SKs), compared to average pooling on features to be integrated. Specifically, we constructed three additional models: Model 4, where all SKs in both edge and node updates were replaced by average pooling, Model 5, where only SKs in edge updates were replaced, and Model 6, where SKs in only node updates were replaced. Table 3 indicates that Model 4 had the worst or second-to-the-worst perplexity and median sequence recovery. The cumulative improvement due to the use of SK is 3% reduction in perplexity and 1% increase in sequence recovery. We also evaluated the effect of SK with comparison to MLP modules (S1 Table) and observed similar improvement of using SK. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Impact of the use of selective kernels in node update and edge update according to perplexity and median sequence recovery for two test datasets (CATH4.2-StructNR193 and PDB-StructNR156). https://doi.org/10.1371/journal.pcbi.1011330.t003 We noted that there is a large gap in the performance between CATH4.2-StructNR193 and PDB-StructNR156 test sets. We found that this is mainly because surface residues are less conserved and, thus, harder to recover in computational design and the PDB-StructNR156 test set has 6.5% less surface residues (54.5% versus 61.0%), and thus, with about 5% higher sequence recovery for designed sequences than the CATH4.2-StructNR193 test set. As shown in S5 Fig, there is an overall similarity in the dependence of recovery on the fraction of surface residues, indicating the robustness of SPIN-CGNN on unseen structures. Moreover, Deep-learning-based methods consistently outperform energy-based techniques (OSCAR-design and RosettaFixBB) in both core and surface regions. 3.4. Method comparison on the whole CATH4.2 test set Table 4 compared SPIN-CGNN to a number of other methods for fixed-backbone protein design that employed the same CATH4.2 training, validation and test sets. This is based on the whole test set (rather than structurally non-redundant set) as we do not have the performance for all individual proteins for most methods. As we can see, SPIN-CGNN achieved the best performance in terms of both perplexity and recovery for the whole test set, as well as for two subsets of small and single-chain proteins with 3–4% improvement of recovery and 10–20% improvement in perplexity over the next best PiFold for those methods without a pretrained language model. Compared to LM-DESIGN, which employed the language model for enhancing the method PiFold, our method continues to improve over perplexity by 10% with a slightly lower sequence recovery (1.0%). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 4. Method comparison on the whole CATH4.2 test set according to perplexity and median native sequence recovery. https://doi.org/10.1371/journal.pcbi.1011330.t004 3.5. Method comparison on structural non-redundant test sets To confirm that the above improvement by SPIN-CGNN over other methods was not due to biased structural redundancy, Table 5 compared the performance of SPIN-CGNN, OSCAR-design, ProteinMPNN, and PiFold on CATH4.2-StructNR193 and PDB-StructNR156 test sets. Here, we employed RosettaFixBB and OSCAR-design as examples of the energy-based techniques, PiFold and ProteinMPNN as examples of modern deep learning models. The results confirmed that SPIN-CGNN has the lowest perplexity (10–15% reduction from the second-best method PiFold, highest sequence recovery (3–4% increase from PiFold). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 5. Comparison of sequences designed by SPIN-CGNN, RosettaFixBB, OSCAR-design, ProteinMPNN, and PiFold on CATH4.2-StructNR193 and PDB-StructNR156 test sets according to perplexity, median sequence recovery, median relative deviation of the frequency of amino-acid residue types, the median relative BLOSUM score, the fraction of low complexity regions, conservation of hydrophobic and hydrophilic sequence positions, the mean steric clash count of refolded structures, and the difference between refolded and target structures in term of RMSD, GDT-TS and TM-score. https://doi.org/10.1371/journal.pcbi.1011330.t005 3.6. Method comparison on sequence compositions of amino acid residues Obviously, fluctuation around native sequences (perplexity) and the recovery of native sequences are only one aspect to measure the quality of predicted sequences. The diversity of amino acid residues employed is another measure for designed sequences. A well-designed sequence should take the advantage of the diversity of amino acid residues. Fig 2 compares the frequency of each amino acid residue types employed in native sequences and in the sequences designed by SPIN-CGNN, RosettaFixBB, OSCAR-design, ProteinMPNN, and PiFold, for CATH4.2-StructNR193 (A) and PDB-StructNR156 (B) test sets. There is a large deviation of ProteinMPNN from native frequencies due to its over-employment of A, E, L, and V and under-employment of H, M, Q, R, and W, as shown in Fig 2. The imbalance of residue usages by ProteinMPNN led to the highest median relative deviation of 0.357 (0.306), compared to 0.141 (0.169) by RosettaFixBB, 0.145 (0.110) by PiFold, 0.099 (0.063) by SPIN-CGNN, and 0.078 (0.067) by OSCAR-design for the CATH4.2-StructNR193 (PDB-StructNR156) dataset (Table 5). Thus, SPIN-CGNN (and OSCAR-design) has much more natural sequence compositions than PiFold and ProteinMPNN (SPIN-CGNN is 32 or 43% better than PiFold, depending on the dataset). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Deviation of the frequency of an amino acid in designed sequences from that in the native sequences by RosettaFixBB, OSCAR-design, ProteinMPNN, PiFold and SPIN-CGNN. (A) CATH4.2-StructNR193 and (B) PDB-StructNR156 test set. https://doi.org/10.1371/journal.pcbi.1011330.g002 3.7. Substitutions between amino acids The phenomenon of amino acid substitutions offers the possibility of different sequences to attain the same target protein structure. This is due to the fact that some positions in the protein structure permit the interchange of amino acids without affecting structural stability. We obtained amino acid substitutions in fixed backbone protein design methods by computing the position-wise confusion matrix between the predicted and native amino acids. Such confusion matrix can be used to compare to BLOSUM62 matrix, that describes the likelihood of amino acid replacements in native sequences. As shown in Fig 3, we can see the confusion matrix of SPIN-CGNN presented a similar pattern to the reference BLOSUM62 matrix: most positive substitutions in BLOSUM62 matrix are also positives values in the confusion matrix of SPIN-CGNN. The Pearson correlation coefficient of SPIN-CGNN on CATH4.2-StructNR193 test set was calculated to be 0.899, compared to 0.841 by RosettaFixBB, 0.884 by OSCAR-Design (S6 Fig), 0.839 by ProteinMPNN (S7 Fig), and 0.890 by PiFold (S8 Fig). We also obtained the correlation coefficients of these methods on PDB-StructNR156 test set. Similarly, the correlation coefficient given by SPIN-CGNN (0.869) is higher than that of RosettaFixBB (0.850), ProteinMPNN (0.853) and PiFold (0.863). Notably, the energy-based method OSCAR-design outperformed all deep learning-based methods with the highest coefficients of 0.888 for this test set. We also calculated the BLOSUM score, a summation of BLOSUM62 values weighted by the predicted probability, as a composite metric of perplexity and amino acid substitution. The BLOSUM score of the methods was further normalized by dividing it with the BLOSUM score of the native sequences, where the probabilities of residues were substituted with the one-hot encoded native sequence. As presented in Table 5, SPIN-CGNN outperformed ProteinMPNN and PiFold on both CATH4.2-StructNR193 and PDB-StructNR156 test sets with respect to the median relative BLOSUM score (0.442 / 0.517 for SPIN-CGNN, 0.362 / 0.433 for ProteinMPNN, and 0.394 / 0.459 for PiFold). These results highlight the stronger overall ability of SPIN-CGNN to capture evolution information than other deep learning techniques. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Confusion matrix of SPIN-CGNN in comparison to the reference matrix BLOSUM62 on the CATH4.2-StructNR193 test set. Positive values (colored) indicate substitutions between amino acids. ρ denotes the Pearson correlation coefficient between confusion matrix of SPIN-CGNN and BLOSUM62. https://doi.org/10.1371/journal.pcbi.1011330.g003 3.8. Sequence complexity The presence and distribution of low-complexity regions (LCR) within protein sequences plays a crucial role in both their structural and functional properties, making it a vital aspect of protein design. Higher LCR fractions in designed sequences as compared to native sequences may result in protein’s structural instability. The fractions of LCR for native sequences are at 4.12% and 4.27% for the CATH4.2-StructNR193 and the PDB-StructNR156 test sets, respectively. All designed sequences had more LCRs as shown in Table 5. SPIN-CGNN has the lowest (5.5% for the PDB-StructNR156 test set) or the third lowest (11.2% for the CATH4.2-StructNR193, behind OSCAR-design and RosettaFixBB) fractions of LCRs. Compared to other deep learning techniques, SPIN-CGNN is 1%-2% improvement over PiFold. ProteinMPNN has the worst performance as expected because it over-employed small hydrophobic residues such as A, L, and V (Fig 2). 3.9. Hydrophobicity conservation One important requirement for soluble proteins is that hydrophobic residues should be mostly buried inside the core whereas surface residues are dominated by hydrophilic residues to ensure solubility and prevent hydrophobic aggregation. We examined the conservation of hydrophobic and hydrophilic sequence positions of design sequences by defining hydrophobic (Ile, Leu, Met, Phe, Cys, Trp, Pro, Val, Ala and Gly) and hydrophilic (Ser, Thr, Asn, Gln, Asp, Glu, His, Arg, Lys and Tyr) residue positions according to the native sequence. As Table 5 shows that SPIN-CGNN has the highest conservation in hydrophobicity positions (~3% over the next best PiFold) for both non-redundant test sets. 3.10. Deviation of target structures from the structures predicted by AlphaFold2 based on designed sequences To further evaluate whether the designed sequences would fold into target structures as expected, we employed AlphaFold2 [12] without using templates as a part of input to predict the structures of designed sequences and measured the root-mean-square deviation (RMSD), global distance test-total score (GDT-TS), and TM-score between predicted structures and target structures. As shown in Table 5, the predicted structures of sequences designed by SPIN-CGNN achieved the smallest median RMSD of 2.09 Å, the greater median GDT-TS of 79.13, and the highest median TM-score of 0.857 on the CATH4.2-StructNR193 test set, comparing to that of RosettaFixBB (5.01 Å, 52.91, and 0.759, respectively), OSCAR-design (3.17 Å, 63.95, and 0.819, respectively), ProteinMPNN (2.75 Å, 67.97, and 0.812, respectively), and PiFold (2.41 Å, 75.88, and 0.847, respectively). As a reference, we also run AlphaFold2 on native sequences and measured the structure deviation (1.85 Å, 83.82, and 0.863, respectively). We displayed the refoldability of different methods as a distribution in Fig 4. PiFold and SPIN-CGNN have comparable performance (no statistically significant difference) in term of refoldability by AlphaFold2. SPIN-CGNN also have comparable performance to native sequences, in term of RMSD on CATH4.2-StructNR193. We noted that including templates in AlphaFold2 prediction only leads to small improvement in refolded structures as shown in S2 Table and S9 Fig. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Deviations of the structures of designed sequences predicted by AlphaFold2 from their respective target structures on three separate test sets from left to right panels (CATH4.2-StructNR193, PDB-StructNR156, Hallucination129, and Diffusion100 test sets) evaluated according to RMSD (Å), GDT-TS, and TM-score (from top to bottom panels). The statistical significance of the difference of a given method to SPIN-CGNN was marked with ‘**’ for highly statistically significant (p-value<0.01), ‘*’ for statistically significant (0.01<p-value<0.05), and ‘-’ for not statistically significant (p-value>0.05). Specific p-values are presented in S4 Table. https://doi.org/10.1371/journal.pcbi.1011330.g004 Additionally, we count the steric clash within each AlphaFold2-predicted structures and compare SPIN-CGNN to other methods according to the mean clash count, as shown in Table 5. The mean clash count of SPIN-CGNN on CATH4.2-StructNR193 is 0.078, which is the second lowest comparing to OSCAR-design (0.088), ProteinMPNN (0.130) and PiFold (0.321), behind RosettaFixBB (0.053). Notably, the mean clash count of RosettaFixBB and ProteinMPNN could be under-estimated due to their over-employment of small residues such as Ala. 3.11. Performance on test sets of generated structures To further evaluate SPIN-CGNN and compare with other methods, we expand our experiments with two generated structure test sets. The Hallucination129 test set is made of the structures generated by hallucination and, thus, each structure does not have a native sequence to calculate sequence recovery. The same is true for the Diffusion100 set. As shown in Table 6, OSCAR-design has the lowest fraction of LCR (11%), compared to 15% by SPIN-CGNN, 17% by PiFold, 20% by RosettaFixBB and 33% by ProteinMPNN. For the difference between target structures and AlphaFold2-predicted structures (Fig 4), only RosettaFixBB and ProteinMPNN has highly significant worse performance on GDT-TS and TM-score, and RMSD from PiFold, SPIN-CGNN, and OSCAR-design. The deviations of refolded from native structures given by PiFold, SPIN-CGNN, and OSCAR-design are statistically similar to each other. Adding Gaussian noise of 0.02 Å standard deviation to coordinates did not lead to further improvement of deep learning techniques on artificially generated structures, unlike a previous report [24] (S3 Table). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 6. Comparison of sequences designed by SPIN-CGNN, ProDesign-LE, RosettaFixBB, OSCAR-design, ProDesign-LE, ProteinMPNN, and PiFold on the Hallucination129 and Diffusion100 test sets according to the fraction of Low-Complexity Regions (LCR), the mean steric clash count of refolded structures, and the difference between refolded and target structures in term of RMSD, GDT-TS and TM-score. https://doi.org/10.1371/journal.pcbi.1011330.t006 On the Diffusion100 test set, which is made of structures generated by a diffusion model, the refoldabilities of all methods are worse than that on the Hallucination129 test set, which may be possibly due to the worse designability of this diffusion model. Compared to the other methods, SPIN-CGNN have better refoldability than RosettaFixBB and ProteinMPNN, while comparable to OSCAR-design and PiFold (with statistically insignificant difference). It’s worth mentioning that LCRs of deep learning methods are even higher (23.16% for SPIN-CGNN, 32.73% for PiFold, and 56.32% for ProteinMPNN) than that on the Hallucination129 test set. By comparison, the LCRs of sequences designed by OSCAR-design on the Diffusion100 test set (5.01%) is significantly lower than all the other methods, demonstrating its robustness on sequence complexity. Additionally, we calculated AA composition deviations for methods on both generated structures test sets, with the average native AA composition from the CATH4.2-StructNR193 and PDB-StructNR156 test set as a reference (S10 Fig). The AA composition deviations of all methods are significantly higher than that on native structures. There are two possible reasons: a) design methods are not robust for unseen structures; b) the generated structures are biased toward most popular amino acid residues due to fixed backbone conformations (See more in the discussion section). 3.12. Case study To further understand how SPIN-CGNN improved fixed backbone protein design with contact graph (CGraph), we presented a case in Fig 5, where the structure of NADP-reducing hydrogenase subunit HndA (PDB 2AUV Chain A) was employed for protein design. SPIN-CGNN outperformed PiFold with a higher recovery (30.59% vs. 28.24%) and a much smaller refold RMSD (5.66 Å vs. 14.02 Å). The CGraph12 graph construction method captured more information from the compact core, as shown in Fig 5A with 47 neighboring residues for residue 10, compared to a fixed number of 30, when KNN-30 were employed. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. An illustrative example to highlight the dense local contacts accounted by SPIN-CGNN for improving fixed backbone design. (A) The neighbors for residue 10 (yellow) in PDB 2AUV chain A, which has the greatest number of neighbors determined by CGraph12 (magenta). (B) The AlphaFold2-predicted structure of sequence designed by SPIN-CGNN (magenta), aligned on the native PDB structure (green). (C) The neighbors determined by KNN-30 (cyan) for the same protein. (D) The corresponding AlphaFold2-predicted structure of sequence designed by PiFold (cyan). https://doi.org/10.1371/journal.pcbi.1011330.g005 3.1. Impact of graph constructions We examined the impact of using the contact maps at different cutoff distances and compared them against the K-nearest-neighbor graph (k = 30, KNN-30) employing the CATH4.2-StructNR193 and the PDB-StructNR156 test sets. Performance was evaluated using perplexity and median recovery. To make a fair comparison, Table 1 compares KNN-30 to CGNN all at without employing CGNN edge information (named as Model 1). The results indicated that KNN-30 has a better performance than CGraph-8 (Contact graph at 8Å distance cutoff) for both test sets. However, increasing distance cutoff (from 8Å to 10Å, and then 12 Å) improves over KNN-30 in perplexity and sequence recovery. At 12 Å cutoff, there is ~1% increase in median sequence recovery, and 2–3% reduction of perplexity from KNN-30 to CGraph-12. We note that even at 12Å cutoff, the average number of neighbors (25 or 29) is still smaller than 30 employed in KNN-30. We fixed the cutoff at 12Å for all subsequent analysis because further increasing the cutoff will only lead to minor improvement at the expense of higher computational requirement. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Contact-based versus K-nearest neighbors in the absence of edge information for all methods (Model 1 for SPIN-CGNN) according to perplexity and median sequence recovery for two test datasets (CATH4.2-StructNR193 and PDB-StructNR156). https://doi.org/10.1371/journal.pcbi.1011330.t001 3.2. Ablation test for CGNN edge updates Table 2 examines the effect of second-order edge (SOE) and symmetric edge updates by constructing Model 2 and Model 3, respectively, as well as the SPIN-CGNN with both SOE and symmetric edge updates. Table 2 shows that without both edge updates Model 1 leads to the worst performance in perplexity and median sequence recovery. Removing SOE also led to statistically significant increase of perplexity and reduction of median recovery from SPIN-CGNN. Although symmetric edge updates do improve the CGNN model when SOE update is absent (from Model 1 to Model 2), it contributes little when the SOE update is performed, indicating that the information captured by symmetric edge updates may have been covered by SOE updates. The overall effect of introducing edge updates is ~1% increase in sequence recovery and 4–6% reduction in perplexity. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Impact of CGNN edge updates (symmetric information and second-order edge (SOE) information) according to perplexity and median sequence recovery for two test datasets (CATH4.2-StructNR193 and PDB-StructNR156). https://doi.org/10.1371/journal.pcbi.1011330.t002 3.3. Ablation test for selective kernel (SK) Table 3 examines the effect of the feature integrating module, selective kernels (SKs), compared to average pooling on features to be integrated. Specifically, we constructed three additional models: Model 4, where all SKs in both edge and node updates were replaced by average pooling, Model 5, where only SKs in edge updates were replaced, and Model 6, where SKs in only node updates were replaced. Table 3 indicates that Model 4 had the worst or second-to-the-worst perplexity and median sequence recovery. The cumulative improvement due to the use of SK is 3% reduction in perplexity and 1% increase in sequence recovery. We also evaluated the effect of SK with comparison to MLP modules (S1 Table) and observed similar improvement of using SK. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Impact of the use of selective kernels in node update and edge update according to perplexity and median sequence recovery for two test datasets (CATH4.2-StructNR193 and PDB-StructNR156). https://doi.org/10.1371/journal.pcbi.1011330.t003 We noted that there is a large gap in the performance between CATH4.2-StructNR193 and PDB-StructNR156 test sets. We found that this is mainly because surface residues are less conserved and, thus, harder to recover in computational design and the PDB-StructNR156 test set has 6.5% less surface residues (54.5% versus 61.0%), and thus, with about 5% higher sequence recovery for designed sequences than the CATH4.2-StructNR193 test set. As shown in S5 Fig, there is an overall similarity in the dependence of recovery on the fraction of surface residues, indicating the robustness of SPIN-CGNN on unseen structures. Moreover, Deep-learning-based methods consistently outperform energy-based techniques (OSCAR-design and RosettaFixBB) in both core and surface regions. 3.4. Method comparison on the whole CATH4.2 test set Table 4 compared SPIN-CGNN to a number of other methods for fixed-backbone protein design that employed the same CATH4.2 training, validation and test sets. This is based on the whole test set (rather than structurally non-redundant set) as we do not have the performance for all individual proteins for most methods. As we can see, SPIN-CGNN achieved the best performance in terms of both perplexity and recovery for the whole test set, as well as for two subsets of small and single-chain proteins with 3–4% improvement of recovery and 10–20% improvement in perplexity over the next best PiFold for those methods without a pretrained language model. Compared to LM-DESIGN, which employed the language model for enhancing the method PiFold, our method continues to improve over perplexity by 10% with a slightly lower sequence recovery (1.0%). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 4. Method comparison on the whole CATH4.2 test set according to perplexity and median native sequence recovery. https://doi.org/10.1371/journal.pcbi.1011330.t004 3.5. Method comparison on structural non-redundant test sets To confirm that the above improvement by SPIN-CGNN over other methods was not due to biased structural redundancy, Table 5 compared the performance of SPIN-CGNN, OSCAR-design, ProteinMPNN, and PiFold on CATH4.2-StructNR193 and PDB-StructNR156 test sets. Here, we employed RosettaFixBB and OSCAR-design as examples of the energy-based techniques, PiFold and ProteinMPNN as examples of modern deep learning models. The results confirmed that SPIN-CGNN has the lowest perplexity (10–15% reduction from the second-best method PiFold, highest sequence recovery (3–4% increase from PiFold). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 5. Comparison of sequences designed by SPIN-CGNN, RosettaFixBB, OSCAR-design, ProteinMPNN, and PiFold on CATH4.2-StructNR193 and PDB-StructNR156 test sets according to perplexity, median sequence recovery, median relative deviation of the frequency of amino-acid residue types, the median relative BLOSUM score, the fraction of low complexity regions, conservation of hydrophobic and hydrophilic sequence positions, the mean steric clash count of refolded structures, and the difference between refolded and target structures in term of RMSD, GDT-TS and TM-score. https://doi.org/10.1371/journal.pcbi.1011330.t005 3.6. Method comparison on sequence compositions of amino acid residues Obviously, fluctuation around native sequences (perplexity) and the recovery of native sequences are only one aspect to measure the quality of predicted sequences. The diversity of amino acid residues employed is another measure for designed sequences. A well-designed sequence should take the advantage of the diversity of amino acid residues. Fig 2 compares the frequency of each amino acid residue types employed in native sequences and in the sequences designed by SPIN-CGNN, RosettaFixBB, OSCAR-design, ProteinMPNN, and PiFold, for CATH4.2-StructNR193 (A) and PDB-StructNR156 (B) test sets. There is a large deviation of ProteinMPNN from native frequencies due to its over-employment of A, E, L, and V and under-employment of H, M, Q, R, and W, as shown in Fig 2. The imbalance of residue usages by ProteinMPNN led to the highest median relative deviation of 0.357 (0.306), compared to 0.141 (0.169) by RosettaFixBB, 0.145 (0.110) by PiFold, 0.099 (0.063) by SPIN-CGNN, and 0.078 (0.067) by OSCAR-design for the CATH4.2-StructNR193 (PDB-StructNR156) dataset (Table 5). Thus, SPIN-CGNN (and OSCAR-design) has much more natural sequence compositions than PiFold and ProteinMPNN (SPIN-CGNN is 32 or 43% better than PiFold, depending on the dataset). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Deviation of the frequency of an amino acid in designed sequences from that in the native sequences by RosettaFixBB, OSCAR-design, ProteinMPNN, PiFold and SPIN-CGNN. (A) CATH4.2-StructNR193 and (B) PDB-StructNR156 test set. https://doi.org/10.1371/journal.pcbi.1011330.g002 3.7. Substitutions between amino acids The phenomenon of amino acid substitutions offers the possibility of different sequences to attain the same target protein structure. This is due to the fact that some positions in the protein structure permit the interchange of amino acids without affecting structural stability. We obtained amino acid substitutions in fixed backbone protein design methods by computing the position-wise confusion matrix between the predicted and native amino acids. Such confusion matrix can be used to compare to BLOSUM62 matrix, that describes the likelihood of amino acid replacements in native sequences. As shown in Fig 3, we can see the confusion matrix of SPIN-CGNN presented a similar pattern to the reference BLOSUM62 matrix: most positive substitutions in BLOSUM62 matrix are also positives values in the confusion matrix of SPIN-CGNN. The Pearson correlation coefficient of SPIN-CGNN on CATH4.2-StructNR193 test set was calculated to be 0.899, compared to 0.841 by RosettaFixBB, 0.884 by OSCAR-Design (S6 Fig), 0.839 by ProteinMPNN (S7 Fig), and 0.890 by PiFold (S8 Fig). We also obtained the correlation coefficients of these methods on PDB-StructNR156 test set. Similarly, the correlation coefficient given by SPIN-CGNN (0.869) is higher than that of RosettaFixBB (0.850), ProteinMPNN (0.853) and PiFold (0.863). Notably, the energy-based method OSCAR-design outperformed all deep learning-based methods with the highest coefficients of 0.888 for this test set. We also calculated the BLOSUM score, a summation of BLOSUM62 values weighted by the predicted probability, as a composite metric of perplexity and amino acid substitution. The BLOSUM score of the methods was further normalized by dividing it with the BLOSUM score of the native sequences, where the probabilities of residues were substituted with the one-hot encoded native sequence. As presented in Table 5, SPIN-CGNN outperformed ProteinMPNN and PiFold on both CATH4.2-StructNR193 and PDB-StructNR156 test sets with respect to the median relative BLOSUM score (0.442 / 0.517 for SPIN-CGNN, 0.362 / 0.433 for ProteinMPNN, and 0.394 / 0.459 for PiFold). These results highlight the stronger overall ability of SPIN-CGNN to capture evolution information than other deep learning techniques. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Confusion matrix of SPIN-CGNN in comparison to the reference matrix BLOSUM62 on the CATH4.2-StructNR193 test set. Positive values (colored) indicate substitutions between amino acids. ρ denotes the Pearson correlation coefficient between confusion matrix of SPIN-CGNN and BLOSUM62. https://doi.org/10.1371/journal.pcbi.1011330.g003 3.8. Sequence complexity The presence and distribution of low-complexity regions (LCR) within protein sequences plays a crucial role in both their structural and functional properties, making it a vital aspect of protein design. Higher LCR fractions in designed sequences as compared to native sequences may result in protein’s structural instability. The fractions of LCR for native sequences are at 4.12% and 4.27% for the CATH4.2-StructNR193 and the PDB-StructNR156 test sets, respectively. All designed sequences had more LCRs as shown in Table 5. SPIN-CGNN has the lowest (5.5% for the PDB-StructNR156 test set) or the third lowest (11.2% for the CATH4.2-StructNR193, behind OSCAR-design and RosettaFixBB) fractions of LCRs. Compared to other deep learning techniques, SPIN-CGNN is 1%-2% improvement over PiFold. ProteinMPNN has the worst performance as expected because it over-employed small hydrophobic residues such as A, L, and V (Fig 2). 3.9. Hydrophobicity conservation One important requirement for soluble proteins is that hydrophobic residues should be mostly buried inside the core whereas surface residues are dominated by hydrophilic residues to ensure solubility and prevent hydrophobic aggregation. We examined the conservation of hydrophobic and hydrophilic sequence positions of design sequences by defining hydrophobic (Ile, Leu, Met, Phe, Cys, Trp, Pro, Val, Ala and Gly) and hydrophilic (Ser, Thr, Asn, Gln, Asp, Glu, His, Arg, Lys and Tyr) residue positions according to the native sequence. As Table 5 shows that SPIN-CGNN has the highest conservation in hydrophobicity positions (~3% over the next best PiFold) for both non-redundant test sets. 3.10. Deviation of target structures from the structures predicted by AlphaFold2 based on designed sequences To further evaluate whether the designed sequences would fold into target structures as expected, we employed AlphaFold2 [12] without using templates as a part of input to predict the structures of designed sequences and measured the root-mean-square deviation (RMSD), global distance test-total score (GDT-TS), and TM-score between predicted structures and target structures. As shown in Table 5, the predicted structures of sequences designed by SPIN-CGNN achieved the smallest median RMSD of 2.09 Å, the greater median GDT-TS of 79.13, and the highest median TM-score of 0.857 on the CATH4.2-StructNR193 test set, comparing to that of RosettaFixBB (5.01 Å, 52.91, and 0.759, respectively), OSCAR-design (3.17 Å, 63.95, and 0.819, respectively), ProteinMPNN (2.75 Å, 67.97, and 0.812, respectively), and PiFold (2.41 Å, 75.88, and 0.847, respectively). As a reference, we also run AlphaFold2 on native sequences and measured the structure deviation (1.85 Å, 83.82, and 0.863, respectively). We displayed the refoldability of different methods as a distribution in Fig 4. PiFold and SPIN-CGNN have comparable performance (no statistically significant difference) in term of refoldability by AlphaFold2. SPIN-CGNN also have comparable performance to native sequences, in term of RMSD on CATH4.2-StructNR193. We noted that including templates in AlphaFold2 prediction only leads to small improvement in refolded structures as shown in S2 Table and S9 Fig. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Deviations of the structures of designed sequences predicted by AlphaFold2 from their respective target structures on three separate test sets from left to right panels (CATH4.2-StructNR193, PDB-StructNR156, Hallucination129, and Diffusion100 test sets) evaluated according to RMSD (Å), GDT-TS, and TM-score (from top to bottom panels). The statistical significance of the difference of a given method to SPIN-CGNN was marked with ‘**’ for highly statistically significant (p-value<0.01), ‘*’ for statistically significant (0.01<p-value<0.05), and ‘-’ for not statistically significant (p-value>0.05). Specific p-values are presented in S4 Table. https://doi.org/10.1371/journal.pcbi.1011330.g004 Additionally, we count the steric clash within each AlphaFold2-predicted structures and compare SPIN-CGNN to other methods according to the mean clash count, as shown in Table 5. The mean clash count of SPIN-CGNN on CATH4.2-StructNR193 is 0.078, which is the second lowest comparing to OSCAR-design (0.088), ProteinMPNN (0.130) and PiFold (0.321), behind RosettaFixBB (0.053). Notably, the mean clash count of RosettaFixBB and ProteinMPNN could be under-estimated due to their over-employment of small residues such as Ala. 3.11. Performance on test sets of generated structures To further evaluate SPIN-CGNN and compare with other methods, we expand our experiments with two generated structure test sets. The Hallucination129 test set is made of the structures generated by hallucination and, thus, each structure does not have a native sequence to calculate sequence recovery. The same is true for the Diffusion100 set. As shown in Table 6, OSCAR-design has the lowest fraction of LCR (11%), compared to 15% by SPIN-CGNN, 17% by PiFold, 20% by RosettaFixBB and 33% by ProteinMPNN. For the difference between target structures and AlphaFold2-predicted structures (Fig 4), only RosettaFixBB and ProteinMPNN has highly significant worse performance on GDT-TS and TM-score, and RMSD from PiFold, SPIN-CGNN, and OSCAR-design. The deviations of refolded from native structures given by PiFold, SPIN-CGNN, and OSCAR-design are statistically similar to each other. Adding Gaussian noise of 0.02 Å standard deviation to coordinates did not lead to further improvement of deep learning techniques on artificially generated structures, unlike a previous report [24] (S3 Table). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 6. Comparison of sequences designed by SPIN-CGNN, ProDesign-LE, RosettaFixBB, OSCAR-design, ProDesign-LE, ProteinMPNN, and PiFold on the Hallucination129 and Diffusion100 test sets according to the fraction of Low-Complexity Regions (LCR), the mean steric clash count of refolded structures, and the difference between refolded and target structures in term of RMSD, GDT-TS and TM-score. https://doi.org/10.1371/journal.pcbi.1011330.t006 On the Diffusion100 test set, which is made of structures generated by a diffusion model, the refoldabilities of all methods are worse than that on the Hallucination129 test set, which may be possibly due to the worse designability of this diffusion model. Compared to the other methods, SPIN-CGNN have better refoldability than RosettaFixBB and ProteinMPNN, while comparable to OSCAR-design and PiFold (with statistically insignificant difference). It’s worth mentioning that LCRs of deep learning methods are even higher (23.16% for SPIN-CGNN, 32.73% for PiFold, and 56.32% for ProteinMPNN) than that on the Hallucination129 test set. By comparison, the LCRs of sequences designed by OSCAR-design on the Diffusion100 test set (5.01%) is significantly lower than all the other methods, demonstrating its robustness on sequence complexity. Additionally, we calculated AA composition deviations for methods on both generated structures test sets, with the average native AA composition from the CATH4.2-StructNR193 and PDB-StructNR156 test set as a reference (S10 Fig). The AA composition deviations of all methods are significantly higher than that on native structures. There are two possible reasons: a) design methods are not robust for unseen structures; b) the generated structures are biased toward most popular amino acid residues due to fixed backbone conformations (See more in the discussion section). 3.12. Case study To further understand how SPIN-CGNN improved fixed backbone protein design with contact graph (CGraph), we presented a case in Fig 5, where the structure of NADP-reducing hydrogenase subunit HndA (PDB 2AUV Chain A) was employed for protein design. SPIN-CGNN outperformed PiFold with a higher recovery (30.59% vs. 28.24%) and a much smaller refold RMSD (5.66 Å vs. 14.02 Å). The CGraph12 graph construction method captured more information from the compact core, as shown in Fig 5A with 47 neighboring residues for residue 10, compared to a fixed number of 30, when KNN-30 were employed. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. An illustrative example to highlight the dense local contacts accounted by SPIN-CGNN for improving fixed backbone design. (A) The neighbors for residue 10 (yellow) in PDB 2AUV chain A, which has the greatest number of neighbors determined by CGraph12 (magenta). (B) The AlphaFold2-predicted structure of sequence designed by SPIN-CGNN (magenta), aligned on the native PDB structure (green). (C) The neighbors determined by KNN-30 (cyan) for the same protein. (D) The corresponding AlphaFold2-predicted structure of sequence designed by PiFold (cyan). https://doi.org/10.1371/journal.pcbi.1011330.g005 4. Discussion We proposed SPIN-CGNN, a deep learning-based method for fixed backbone protein design. Our approach incorporates several key elements, including contact-map graph construction, second-order edge updates, and selective kernels. Ablation studies reveal that the power of SPIN-CGNN lies in the cumulative improvement resulting from these multiple changes. Comparing our method with the most recently developed deep learning-based approaches, ProteinMPNN and PiFold, we found that SPIN-CGNN consistently outperformed them across various metrics including perplexity, sequence recovery, amino-acid composition, amino-acid substitution, low complexity regions, and conservations of hydrophobic/hydrophilic positions. Refolding of designed sequences by AlphaFold2 indicates that the structures produced by SPIN-CGNN are comparable to those by PiFold but significantly closer to target structures than ProteinMPNN. Interestingly, a recently developed energy-based technique, OSCAR-design, produced comparable performance to PiFold and SPIN-CGNN for structures refolded by AlphaFold2 for Hallucination129 and Diffusion test sets. Depending on the datasets, OSCAR-design can have the closest amino-acid composition and low complexity region to the native composition compared to ProteinMPNN, PiFold and SPIN-CGNN, indicating that there is something that deep learning techniques can be improved further. When we detected the structural redundancy within the CATH4.2 test set, we also found structure pairs with high structure similarity (TM-score>0.4) between all three CATH4.2 subsets (training, validation, and test). To reduce the potential of overfitting, we removed all structures in the training set that have TM-score >0.4 with any structures in the validation and test sets. This led to a much smaller training set of 9311 structures, compared to 18024 proteins in the original set. This new training set, however, led to a poorer performance for those unseen structures. For example, the median TM-score refolded by AlphaFold2 was reduced from 0.924 trained by the whole training set to 0.823 trained by the new training set for SPIN-CGNN for the PDB StructNR156. The similar behavior was observed for PiFold and ProteinMPNN. The worse performance by using TM-score <0.4 is not only because it is too strict for removing many nonredundant folds but also because TM-score is just a global metric and the larger set can keep more abundant local structural information which are very useful for training. In this case, it will be difficult to separate the contributions from these two different effects. Thus, we employed the whole training set as it improves the generalizability over the smaller training set. We would like to emphasize the importance of using different measures to computationally assess the designed sequences. This is because native sequence recovery and the deviation from the native sequence (perplexity) only reflect one aspect of designed sequences. High sequence identity does not assure foldability as a few mutations are often found sufficient to disrupt structural stability. Moreover, too many hydrophobic residues on the protein surface will lead to insoluble and aggregated proteins and low complexity in sequences often leads to intrinsically disordered regions. In addition, the refoldability by AlphaFold2 is not that sensitive to the sequences given by different methods because designed sequences are now highly similar to the wild type sequences and, thus, naturally occurring homologous sequences were employed as a part of the input for AlphaFold2 (Tables 5 and 6). Here we showed that the fraction of low complexity regions remains much higher than native sequences and energy-based techniques, indicating the room for further improvement. It is well known that backbone bond angles and torsion angles may contain information biased toward certain amino acid residues. For example, the N−Cα−C angle is within [121, 126] for histidine and within [117, 122] for leucine. Unlike other residues, Gly does not any forbidden regions in the ϕ−ψ torsion angle space. This leads to the question if removing these angles will lead to a significant change in method performance. To address this question, we trained a SPIN-CGNN model with inter-atomic distances as the edge features only. We found that the recovery of the distance-only model on native structures drops only slightly from 52.5% to 51.21% on the CATH4.2-StructNR193 test set and from 58.8% to 57.36% on the PDB-StructNR156 test set. Thus, the angles only have a small impact on residue-type recovery. Two sets of generated model protein structures were also employed for testing various protein design techniques. They are Hallucination and Diffusion sets. Table 6 shows that although most methods can refold designed sequences to the corresponding generated structures well, both physical and AI-based methods have elevated factions of low-complexity regions and larger deviation of amino acid compositions from natural proteins. This may be because “generated” structures, particularly from diffusion models, do not explicitly consider specific sidechains. As a result, the geometrical characters of main chains and the physical space between mainchain atoms in these designed structures maybe more favorable for most popular residue types (as shown in Fig 2). These structures would be prone to produce sequences with low complexity, regardless of physical or deep-learning based methods. It is noted that the number of parameters employed by SPIN-CGNN is 5.58 million, comparable to 4.13 million by PiFold. It has a similar inference time as PiFold. For a 500-residue protein, the inference time is 0.09 second by SPIN-CGNN, compared to 0.03 second by PiFold, 0.83 by ProteinMPNN. Supporting information S1 Fig. High structural similarity pairs (TM score > 0.98) within the CATH4.2 test set. The alignments of structure pairs were presented between two test structures in green and cyan. https://doi.org/10.1371/journal.pcbi.1011330.s001 (TIF) S2 Fig. Edge updating in the CGNN block. https://doi.org/10.1371/journal.pcbi.1011330.s002 (TIF) S3 Fig. Node updating in the CGNN block. https://doi.org/10.1371/journal.pcbi.1011330.s003 (TIF) S4 Fig. Selective kernels for edge update (A) and node update (B) in the CGNN block. https://doi.org/10.1371/journal.pcbi.1011330.s004 (TIF) S5 Fig. The median sequence recovery of protein targets as a function of the fraction of surface residues on CATH4.2-StructNR193 (A) and PDB-Struct156 (B) test sets given by SPIN-CGNN, in comparison with a number of other methods as labeled. Nearly identical dependence on fraction of surface residues by SPIN-CGNN for two different test sets indicates the robustness of the methods for different datasets. https://doi.org/10.1371/journal.pcbi.1011330.s005 (TIF) S6 Fig. Confusion matrix given by OSCAR-design. https://doi.org/10.1371/journal.pcbi.1011330.s006 (TIF) S7 Fig. Confusion matrix given by ProteinMPNN. https://doi.org/10.1371/journal.pcbi.1011330.s007 (TIF) S8 Fig. Confusion matrix given by PiFold. https://doi.org/10.1371/journal.pcbi.1011330.s008 (TIF) S9 Fig. Deviations of the structures of designed sequences predicted by AlphaFold2 with PDB templates from their respective target structures on four separate test sets from left to right panels (CATH4.2-StructNR193, PDB-StructNR156, Hallucination129, and Diffusion100 test set) evaluated according to RMSD (Å), GDT-TS, and TM-score (from top to bottom panels). The statistical significance of the difference of a given method to SPIN-CGNN was marked with ‘**’ for highly statistically significant (p-value<0.01), ‘*’ for statistically significant (0.01<p-value<0.05), and ‘-’ for not statistically significant (p-value>0.05). https://doi.org/10.1371/journal.pcbi.1011330.s009 (TIF) S10 Fig. Deviation of the frequency of an amino acid in designed sequences from that in the native sequences by RosettaFixBB, OSCAR-design, ProteinMPNN, PiFold and SPIN-CGNN. (A) Hallucination129 and (B) Diffusion100 test set. https://doi.org/10.1371/journal.pcbi.1011330.s010 (TIF) S1 Table. Impact of the use of selective kernels in node update and edge update according to perplexity and median sequence recovery for two test datasets (CATH4.2-StructNR193 and PDB-StructNR156). Two-layer MLP modules were employed to substitute SK modules in models without selective kernels (no-SK models). https://doi.org/10.1371/journal.pcbi.1011330.s011 (DOCX) S2 Table. Refoldability tested by AlphaFold2 with PDB templates. https://doi.org/10.1371/journal.pcbi.1011330.s012 (DOCX) S3 Table. Adding Gaussian noises to the structural coordinates slightly improved the performance of deep-learning methods (SPIN-CGNN, ProteinMPNN, and PiFold) on the Hallucination129 test set, according to the fraction of Low-Complexity Regions (LCR) and the difference between refolded and target structures in term of RMSD, GDT-TS and TM-score, except an increase of low complexity regions for SPIN-CGNN and PiFold but a reduction of LCR for ProteinMPNN. https://doi.org/10.1371/journal.pcbi.1011330.s013 (DOCX) S4 Table. Statistical significance between a given method to SPIN-CGNN for the structural difference between target structures and AlphaFold2-predicted structures for designed sequences according to Root Mean Square Deviation (RMSD), Global Distance Test-Total Score (GDT-TS) and TM-Score for three test sets. ‘**’ denoted p-value < 0.01, ‘*’ denoted 0.01 < p-value < 0.05, and ‘-’ denoted P-value > 0.05. https://doi.org/10.1371/journal.pcbi.1011330.s014 (DOCX) Acknowledgments We thank Professor Dongbo Bu for his suggestion of the method for the BLOSUM score calculation. The work was done by using the supercomputing facility of the Shenzhen Bay Laboratory.

Belliard, Frédérique;Maineri, Angelica Maria;Plomp, Esther;Padilla, Andrés Felipe Ramos;Sun, Junzi;Jeddi, Maryam Zare

doi: 10.1371/journal.pcbi.1011668pmid: 38096152

Introduction The FAIR data principles promote good data stewardship by leveraging the Findability, Accessibility, Interoperability, and Reusability (FAIR) of research data and software [1–4]. These principles aim to facilitate the discovery, access, integration, and reuse of research data and software by both humans and machines, with the ultimate goal of enhancing the transparency, reproducibility, interoperability, and impact of research. While the FAIR principles are not a single standard [5], they do emphasise the need for standardisation in the way research objects are described, stored, and shared. For this reason, the implementation of the FAIR principles often involves discussions over which practice, resource, or technology should be adopted as standard by a (research) community. By promoting consistent and well-defined data structures, controlled vocabularies, and metadata, the FAIR principles can help make research objects more easily comparable and reusable across different disciplinary and spatial contexts. Despite the benefits of the FAIR principles and their widespread endorsement on behalf of research institutes, publishers, and funders, these principles have not been evenly adopted in all disciplines [6]. There is still a lack of data and code sharing (with estimates between 1% and 20% [7–11]—although there are higher sharing rates in, for example, genomic research [12]). Furthermore, not every discipline has access to metadata standards or discipline specific repositories. One of the main challenges to the wider implementation of the FAIR principles is linked to the social dynamics underlying standardisation processes. Standardisation is a complex process that involves the creation of agreed-upon rules across time and space ([13]; p. 71). This process is difficult to facilitate without sufficient leadership, resources, and time. Standardisation processes may also create frictions linked to imposing one solution on previously varied practices and to authority and governance issues (who decides on which standard to adopt?). These social dynamics are key to the successful implementation of the FAIR principles. In this context, we shift our focus away from the specific research objects involved in standardisation processes and instead focus on the community aspect of standardisation. Specifically, we consider the strategies and approaches that can be employed to engage research communities in fruitful discussions about standardisation in the context of implementing the FAIR principles. In our view, the successful implementation of the FAIR principles relies on the buy-in and participation of the research community that will have to actually implement the principles. To assist in the facilitation of standardisation discussions within individual research communities, we have developed the 10 rules as a reference point (see Fig 1 for an overview). Input on these rules has been initially provided by experts (including researchers, data supporters, students, and service providers) at the Netherlands Open Science Festival on September 1, 2022 (see S1 Text for more details) and a call for contributions via FAIR connect [14]. It is important to note that not all research communities will be at the same stage of adoption of the FAIR principles, and some of these steps may be deemed unnecessary or irrelevant depending on the specific needs and circumstances of a given community. Our perspective will be biased towards the Dutch context as the Open Science Festival was hosted primarily for researchers in the Netherlands, and all authors are based at Dutch institutes. Nevertheless, we hope that these rules serve as a useful resource for researchers, Research Data Management (RDM) support staff, and research data infrastructure providers, looking to effectively promote the adoption of the FAIR principles within their own communities. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Overview of the 10 rules to starting FAIR discussions. It is important to define the community that needs to be involved (Rule 1) and gain partnerships and support (Rule 2). In these discussions, the social aspects of standardisation should be considered (Rule 3). It is therefore important to establish the benefits of standardisation processes (Rule 4) and to address the existing barriers (Rule 5). Keeping this in mind, it will become possible to set up minimum metadata requirements (Rule 6), documentation standards (Rule 7), and to identify the infrastructure that the community can make use of or should establish (Rule 8). In these efforts, the long-term sustainability should be considered (Rule 9). For each of these steps, it is important to share experiences (Rule 10). https://doi.org/10.1371/journal.pcbi.1011668.g001 Rules Rule 1: Define the community you want to approach Discussions surrounding the adjustment of workflows to facilitate FAIR practices should occur within a research community, defined as a group of stakeholders (such as individual researchers, research support staff, and data infrastructure providers) that have a shared interest in streamlining their efforts to implement the FAIR principles. As explained by Timmermans and Epstein [13], standardisation is inherently a social process that requires the commitment and endorsement of multiple actors to be effective. The community aspects of FAIR implementation are embedded in the original FAIR principles [2] and made explicit in principle R1.3 (“(meta)data meet domain-relevant community standards”). For instance, in the framework of FAIR Implementation Profiles (FIPs, a methodology that has been introduced to document FAIR implementation choices), the community aspect is captured by the concept of FAIR Implementation Community [15,16]. How to adequately define and engage a community remains, however, an open challenge. Rule 1 recognises that identifying the appropriate research community is a crucial step in facilitating discussions on standardisation. The stakeholder who wishes to initiate a FAIR discussion may be already part of the community or not; regardless, it is important to provide a clear definition of what the community to approach is. Research communities can be based on various factors, such as the type of data being generated or used, a shared institutional affiliation, or a specific research project. A community can constitute a formal entity, or it can be an informal group, and it can exist for a determined time span, or be long-lasting [15]. It is important that a community self-identifies as such, as this can increase the level of commitment and engagement among members. Research communities typically involve individuals in a variety of different roles, such as researchers, RDM support staff, lab technicians, and students. Once the community has been identified, the levels of understanding of FAIR implementation and FAIR standards should be gauged, using resources such as the FAIR-Aware tool developed by DANS [17] or the How to FAIR quiz from the Danish National Forum for Research Data Management [18]. Disparities in understanding among different stakeholders may present challenges to the standardisation process—though a diversity in perspectives can be beneficial, as elaborated in Rule 3. Depending on the features of the community, there may be different ways to get in touch with the community members: In the case of an informal community, for instance, it may be necessary to proceed via “snowballing,” with one identified member suggesting other ones and so on. In the case of formalised communities, instead, there may be people with specific roles (such as community managers) who already have open communication channels with the community. In both cases, reaching out to RDM experts, research infrastructures, or scientific associations may be beneficial (see Rule 2), as they may be aware of existing or similar initiatives, or be able to suggest people to contact. Rule 2: Identify sources of support and partnership Attempting data standardisation is a complex process that should not be done alone. Support and partnerships are most likely to be found from the RDM support team at your institute (usually based at the library), the scientific association of your discipline (see Table 1 for some examples), or other enthusiastic individuals already involved more closely with the adoption of the FAIR principles. Failing to engage with these stakeholders may result in a lack of awareness and recognition of the need for promoting the FAIR principles in your community within your institute or association. We recommend prioritising seeking out this type of support or partnerships, as it could prove to be beneficial in the long run, even if funding or resources may not be immediately available. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Examples of organisations that can help finding RDM experts grouped by spatial focus (primarily in Europe) and domain specificity. The examples are meant to give an indication, not an exhaustive overview. This overview is available at https://github.com/AngelicaMaineri/awesome-RDM-support/blob/main/README.md under a CC0 licence to allow reuse and extension. https://doi.org/10.1371/journal.pcbi.1011668.t001 If you are a researcher, start by checking if an RDM team is available at your institution. The RDM team will be able to point to existing resources, tools, and information that can save time. The RDM team can also provide support in raising awareness, as they should already be involved with promoting the adoption of FAIR principles within the research community. This RDM team will likely have experience with providing workshops, training programmes, setting up policies and recommendations, and hosting events. The RDM team may have already set up materials or resources that can be tailored to specific needs. Vice versa, if you consider yourself as an RDM specialist or an infrastructure provider, make sure to seek partnership with researchers in the community, as they can contribute their domain-specific knowledge as well as their perspectives of potential data (re)users. Regardless of your role, the scientific association or individual researchers already involved with the FAIR principles may have already set up data standardisation processes that you can join, will provide connections to your community (see Rule 1), and can provide further support. Ideally, there is funding involved in standardisation efforts. Both RDM support and scientific associations may also have access to funding, or they may be able to connect you to other funding opportunities. Funding agencies in the Netherlands allow for funding data management activities in their projects (NWO and ZonMW). Open Science Funds may be available through Open Science Communities or at a (inter)national level (see NWO or FAIR impact). Once you have defined your community and found further support, the next step is to consider the social aspects of standardisation processes (Rule 3). Rule 3: Take into account the social aspects of standardisation As discussed in Rule 1, adopting research workflows is a social process that requires the participation and commitment of multiple actors within a research community [13]. Standardisation efforts can stop or become obsolete when a community loses interest [19]. Therefore, Rule 3 highlights the importance of considering the social aspects that may influence the consensus-building process when facilitating discussions on the adoption of FAIR practices and offers practical examples. The rule is quite broad in its formulation because the way in which these social dynamics manifest themselves may vary depending on the community and the stakeholders involved. We aim to illustrate some potential situations and actionable recommendations. First of all, the creation and adoption of standards may involve trust and authority issues. For example, the introduction of new standards may generate opposition and resistance if community members do not perceive them as legitimate or do not trust the authority of those proposing the standards. To mitigate the risk of frictions, involving the community is key. In particular, highlighting best practices via real use cases from the community can be effective in showing the value of standardised FAIR practices and promoting the use of existing standards. Generating trust in the process can be facilitated by having regular meetings, including occasional in-person meetings if possible, and by planning clear feedback processes so that concerns from the community can be addressed. When there are no existing best practices yet within your own community, examples and recommendations can be adopted from other communities who are further along in adopting the FAIR principles (see Rule 2: RDM experts can help reaching out to more mature communities; but also see Rule 10: Sharing experiences is pivotal!). Building on existing practices may save time and resources that can be used more efficiently in the standardisation process. Additionally, standardisation can sometimes result in a reduction of heterogeneity within a community, as it involves the creation of agreed-upon rules that may limit the range of previously enacted practices and even supersede previously adopted, perhaps outdated, standards. Therefore, it is important to carefully consider the potential impacts of standardisation on the variety of research practices within a community. The community-based approach to the FAIR principles, alongside the FIPs, which support the documentation of FAIR implementation choices [16], makes it possible to create multiple standards, as long as cross-standard interoperability is kept into account. For instance, the CEDAR Metadata Tools allow the creation of community-specific metadata templates while reusing existing ontologies and value sets, therefore enabling diverse solutions within a shared framework (see Rule 6 for more recommendations on creating metadata models). There are also solutions to make existing standards more easily findable and reusable. One such solution is FAIRsharing, which serves as a repository of FAIR-enabling standards and other FAIR resources. This platform was created to address the issue of excessive fragmentation in the developments of standards [20]. Standardisation often requires researchers to invest time and resources into changing their existing workflows, which can be a challenging task. The disruption of existing practices has also been reported to be an important barrier to adoption of standards in the industry context (see [21] and Rule 5). Therefore, it is important to clearly communicate the benefits and incentives of adopting the FAIR principles to the research community. By clearly communicating the value of the FAIR principles and engaging in meaningful dialogue with the research community, it is possible to facilitate a more effective and efficient standardisation process (see also Rule 4). Rule 4: Establish the benefits of standardisation for the community It can be helpful to establish benefits of standardisation in order to convince others to get involved and motivate them to change their workflows. Some examples of benefits are listed below and may not always be directly applicable to your community. Personal benefits by being involved in this process include: Direct impact on the eventual results, ensuring that the standardisation processes are applicable to your research. Extension of professional network and positive effects on your professional reputation. Long-term standardisation can be more cost efficient by increasing data reuse, preventing duplication of research/trials, improving quality of the data (reduction of data errors and increase of reproducibility), and facilitating integration of datasets. FAIR standardisation processes also reflect positively on institutions (see Rule 2 on where to find help in your institute). Benefits for institutions include: Streamlining data management processes and being cost efficient. Increasing the reputation and trust in research findings from the related research groups [22]. Rewarding and recognising data as a valuable research output in academia. Facilitating collaboration and innovation [22]. Increasing the value of existing data by facilitating reuse and, thereby, increasing the return on the initial investment on data collection. After the benefits have been established, the next step is to identify the barriers (Rule 5). Rule 5: Identify community-experienced barriers It is important to recognise that the barriers and challenges to implementing the FAIR principles and standardising data management practices may vary widely across different research domains and communities. This may involve performing a gap analysis to identify areas where additional support or resources are needed for the community (identified in Rule 1), or identifying case studies of successful FAIR implementation in similar disciplines that can serve as best practice for the community (see also Rule 3). Below are some of the barriers that we have encountered, either in the literature or in our own professional experience: Data requirements: The types of data more commonly used by a community as well as their size and heterogenous nature may pose challenges in terms of storage, management, and accessibility [23–25]. Ethical and legal barriers: The handling and sharing of sensitive political or personal data may be subject to strict regulations (such as the General Data Protection Regulation (GDPR)) and require additional considerations to ensure compliance [9,23–29]. Data sharing can also result in economic damage when disease data are shared and impact tourism and trade [29]. Different research environments: When there are critical resource shortages (such as the absence of research networks and lack of infrastructural support), there may be more immediate concerns that should be addressed [30]. Intellectual property and licensing: Intellectual property (IP) issues (such as data transfer and processing agreements) may arise when data are shared or reused [9,24,25], particularly when multiple stakeholders are involved. Not everyone may have access to proprietary software used in data analyses. Lack of incentives: Some researchers may not see the value in making their data FAIR or may not perceive a need to share their data with others (see Rule 3) [23,24,26–29,31–33]. Cultural barriers: for example, considering data sharing to hamper future publications if there is no reciprocity in the form of appropriate credit for data sharing [25,26,29,31], or a lack of trust in data being correctly interpreted and used [23,24,27–29,32]. Lack of institutional data policy, support, and training [34]. Lack of infrastructure (see Rule 8) to share data [23,25,29,31,32]. Lack of compliance monitoring by institutes, funders, or journals with policies regarding the FAIR principles, decreasing the need for researchers to comply with these policies [9,23,32,35]. Limited awareness about best practices, FAIR principles, and standards [28]. Emphasis on novel research may result in data generation rather than reuse, integration, and maintenance [33]. Possible criticism and fear that results will be invalidated [23,29]. Limited time and/or lack of resources [9,23,25–29,32,34]. To identify and address these barriers, you should discuss them with your community (Rules 1 and 3). Pending on the identified barriers, there will be different solutions to address them. Some of these barriers (limited awareness, lack of expertise and best practices) can be addressed by data standardisation and defining more explicitly what information is most valuable in the data management workflows. In Rule 6, this is addressed by going deeper into how information requirements can be established. Rule 6: Set up minimum metadata requirements Metadata is information about the data that provides context and allows for proper interpretation and reuse. A metadata standard is a structured form of documenting and describing this information. Several metadata standards are already in use, such as Dublin Core and DataCite ([36]; see the Digital Curation Center for an overview of metadata standards, or use FAIRsharing to browse metadata standards). Dublin Core consists of 15 general elements that make this standard easy to use across disciplines. Nevertheless, most disciplines and research communities may require more detailed metadata than those provided by Dublin Core in order to manage and document their research data effectively. This will eventually result in data shared in research repositories that are better aligned with the FAIR principles. It is therefore helpful to look for discipline-specific metadata standards or guidelines (see FAIRsharing.org [20] or the Metadata Standards Catalog). When there are no metadata standards or minimum metadata requirements available for your community, a more advanced step is to start creating these minimal metadata requirements. This can be a complex task, particularly when your community spans over different fields that use distinctive terminology to describe data [37]. To start with setting up minimum metadata requirements, it is important to first establish who needs to be involved, as community engagement will be crucial [37] (Rule 1). You may also need to consider who is most suitable to lead this process, as this will require some degree of authority, expertise, and trust within your community (see Rule 3). Other communities have already been successful in developing minimum metadata requirements, such as the Earth Sciences [38,39], Bionano Sciences [40], Biomedical Sciences [41], and -omics Sciences [42–45]. The lessons learned from these communities can be taken into account, although their approaches may not always fit with your research community that may have different requirements and challenges (Rule 5). Minimum metadata requirements can be about data/sample preprocessing, experimental analysis, quality control, preregistration—any aspect related to the research process. The metadata requirements should provide guidelines for essential information requirements while at the same time be flexible to meet each researcher’s objectives [41,46]. There is a need to “strike the right balance between minimising the barriers to data submission and maximising opportunities for data reuse” [41]. After you identified the relevant stakeholders (Rules 1 and 2), you can follow the recommendations below to start setting up minimum metadata requirements, or a Minimum Information Standard, in a research community: Review existing practices such as metadata standards, guidelines, and use cases [38,39,42,43,46]. If there are no existing efforts, you can start with a call for guidelines [44], set up a working group/project team [39,45], or a network [47]. You can get a team together by organising a workshop or conference session [41,42,48,49]. Ideally, funding is available for standardisation efforts (for example, NIH funding [19]) or should be applied for [47,48] (see also Rule 1). Adhering to the minimum metadata requirements should be as effortless as possible to enable widespread adoption [46,48] (see also Rules 3 and 4). Minimum metadata requirements are the first step towards standardisation. Additional developments will be needed for standardisation [42,46], involving the research community at each step. To establish community consensus, the research community should be asked for input and feedback, through community discussions, workshops, and surveys [38,41–44,48]. Only through active community involvement will a functional solution be achieved [45] (see also Rule 3). To ensure practical and effortless implementation of the standards by journal editors, reviewers, and data repositories, it is important to gather their feedback [39,41,45,49]. Once progression has been made, it is important to communicate this to the research community, via public documentation, reports, or publications (see also Rule 10 on sharing experiences). To support community uptake, it can be helpful to provide training, support, or to have champions involved that can promote the standards [38]. The benefits (Rule 4) of adjusting existing workflows should be clear. A great way to get started is to review the work by ESS-DIVE in establishing a community-centric metadata reporting format [38]. Crystal-Ornelas and colleagues [38] share guidelines (Box 1) and details about their process. A next step could be to develop an ontology (see Courtot and colleagues [50]’s 10 simple rules on this topic). Rule 7: Set up documentation standards In addition to the minimum metadata requirements described earlier (Rule 6), documentation is the next step that supports reuse of research outputs. The type of documentation needed depends on the purpose, expertise, and context of your community. If the primary purpose of the documentation is to be published in a research repository (see Rule 8) or to comply with the funders’ policies, the documentation must, at a minimum, be designed to achieve that purpose. Two general levels of documentation can be considered when documenting research data: project level and data level (also referred to as study-level and object-level documentation, respectively). The project-level documentation provides context for the collection, methodology, structure, and validation of data, while the data-level documentation consists of the variable names, descriptions, classifications, file formats, and software details. In other words, project-level documentation is about what is around data, and data-level documentation is about data itself [51]. Examples of project-level documentation are Data Management Plans (DMPs), Software Management Plans (SMPs), and the use of Preregistration and Registered Reports. The first two provide project-level documentation by describing the context of data and software; in DMPs, describing how data were collected and the methods used to validate it [52]; in SMPs, describing how software works, the purpose, the outputs, and its (continuous) development. Research communities may select standard templates for DMPs, taking into account the requirements of their organisation or funder (see for examples the list of public templates on DMPTool), while for SMPs, such standards are under development [53]. Preregistration, on the other hand, involves the public disclosure of research plans before data collection, analysis, and reporting are completed, with the goal of increasing transparency in the knowledge creation process from its inception to the results [54,55]. In an effort to standardise the information requested for preregistering a study and ideally simplify the process for researchers, templates have been developed (see the list of templates on the Open Science Framework website). At the data level, once the minimum metadata requirements are established (Rule 6), it will be easier to describe variables or file formats that the community will use and then expand to documentation guidelines. Documentation can be used to describe how to organise data, such as spreadsheets [56], workflows (see the Data Curation Network Primers), and to provide information on standards for dates and times (such as ISO 8601 or RFC3339). In addition, it may be possible to use documentation from other communities. In particular, code programming communities use standard style guides that are also widely used within research communities (such as PEP 8 for Python, Tidyverse for R; see also guidance by the Code Refinery [57]). A recommended solution for documentation could be codebooks, which describe the variables with their units, summarising choices made during the research process, and outlining the experimental study design [58]. Ideally, codebooks should be in a structured/standard format (for example, Data Documentation Initiative Codebook). Recently, tools have been developed that can automatically generate standardised metadata, reducing the (time) barriers to writing comprehensive codebooks (codebook R Package [59]). Ultimately, the choice of documentation standardisation should facilitate communication and collaboration between researchers and those who reuse their data. Rule 8 outlines the process of choosing the infrastructure to share the data and the accompanying documentation. Rule 8: Identify infrastructure to share data In order to get a clear idea of the infrastructure that can be used to share data, first, it is important to follow the requirements and guidelines of your institution, funders, and/or collaborators. Generally, data repositories are considered to be the ideal infrastructure to share data in a reliable manner [60]. Generic repositories, such as Zenodo, OSF, or Figshare, are widely used for preserving and sharing research data (see Table 2 for some examples). Your institution may also already have a repository that you can promote within your community. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. A list of common repositories outlined in more detail in [61]. https://doi.org/10.1371/journal.pcbi.1011668.t002 Generic and institutional repositories are generally not designed with the needs of a specific community in mind, which is where discipline-specific infrastructure may play an important role. Discipline-specific infrastructure is especially beneficial if standard data formats are used and enforced, ideally via user-friendly interfaces and with training provided where needed [62]. There are many domain-specific repositories (see NIH list of domain-specific repositories). It is therefore important to determine which type of data repository will better serve the needs of your community. Communicating a preferred infrastructure to share data may result in data that are more findable (as data are shared using the same infrastructure) and may reduce the cognitive load for individual researchers within your community as they do not have to look for suitable data repositories themselves. Before sharing data via a data repository, or promoting your repository to the community, you will need to verify that the repository follows the minimum requirements to be considered useful and adheres to the FAIR principles. Think that a repository needs to: Have a clear policy on how data will be managed, as well as a privacy policy and terms of use. Provide sufficient data storage size for the dataset. The geographic location where the data are saved (for restricted access dataset that contains personal data). Assign a persistent identifier (such as digital object identifier (DOI)) to be able to cite the data. Allow you to include a licence to your data (such as a Creative Commons licence). Make sure data are available/accessible and discoverable. Repositories can enhance their discoverability by being included in databases such as re3data (https://www.re3data.org [63], FAIRsharing (https://fairsharing.org [20]), and the EOSC portal (https://eosc-portal.eu). Allow revisions to be made to the dataset in the future. In some cases, institutional or generic repositories do not fulfil the requirements of your community (see Rule 5), and there may not be a discipline-specific data repository available. Setting up a specific repository may be a good option when there are sufficient resources and plans for the long-term sustainability of the infrastructure (see Rule 9). The main advantage of arranging your own repository infrastructure is that you have greater control over how data are documented and presented to the public and/or researchers. Specific data repository infrastructure may also improve the data quality of the datasets [64]. However, creating and using this infrastructure leads to additional costs (especially when dealing with large quantities of data). In addition, clear documentation and training materials (see Rule 9) are required to engage researchers to use the repository. Rule 9: Plan for the long-term actions As mentioned in Rule 8, discipline-specific infrastructures require resources and should be sustainable for the long term. When these infrastructures are set up by a small group, maintenance and sustainability is challenging as many researchers move across institutes and countries. While sustainability can be achieved by charging for repository services, it is also important to consider that not all researchers have access to these resources. Researchers are generally working on projects that eventually run out of funding, especially at the stage of data sharing. It is therefore important to consider who pays for long-term data sharing and maintenance. Maintenance plans and governance of infrastructure and standards should be transparently communicated (see also Rule 3). To plan for the long term and to establish robust policies for the repository, repositories could aim for certification (via CoreTrustSeal, ISO 16363, or Nestor), although this is a resource-intensive process. Resources are also needed for the maintenance of any created metadata standards (Rule 6). Standardisation is a continuous process and will require evaluation on their practical applicability (for example, metadata standards may become obsolete or deprecated when they are no longer applicable [19]). Standardisation is also a continuous learning process. Researchers may not be familiar with the standardisation efforts and will need a place to start, support, or training resources. It can also be important to monitor whether standardisations are followed appropriately—some form of manual curation may always be needed to avoid errors or incomplete entries. All of these processes take up resources in the long term. To facilitate long-term sustainability, it is better to use open formats and infrastructure built using open source software. This prevents lock-in to certain services and allows community members to continuously contribute. It is also important to consider how the infrastructure will scale when future use is increasing and user input may become more heterogeneous [65]. Individual researchers can improve the longevity of research data by starting to make use of data repositories and make their data available in open formats, following the repository guidelines. When researchers are already using data repositories, they can promote the use of data repositories to their colleagues and share any available training materials—especially if they are in leading roles or responsible for the training of other researchers. Individual researchers can also financially support data repositories by including a budget for data curation in their research proposals or by requesting funding from their institutions for data curation. Individual researchers, or research data support staff, can also provide other types of resources to data repositories by performing data curation tasks, data peer review, or taking on communication tasks. To foster a culture of data standardisation and sharing, it is needed to recognise the efforts of researchers who adopt minimum metadata requirements (Rule 4). Ideally, this happens at the institutional level. Research communities can also recognise practices during annual meetings and conferences or by awarding prizes (for example, the Open Science Community Amsterdam Awards that took place on January 26, OSCA 2023). Rule 10: Share experiences Sharing experiences about the standardisation process facilitates learning from existing efforts and identifying best practices that can facilitate the standardisation journey. By reaching out to local RDM support or other community stakeholders (see Rules 1 and 2), you have hopefully benefitted of the experiences of others as well. It is therefore important to share experiences gained from each of the rules listed here and as illustrated in Fig 1. Experiences and insights can be shared via case studies, best practices, and lessons learned from standardisation efforts. Venues to share these experiences may include journals (such as PLOS), preprint servers, publishing forums (such as FAIR connect, where an earlier version of this article was shared [14]), data repositories (such as Zenodo), blogs (for example, [66]), social media, or conferences and meetings. Rule 1: Define the community you want to approach Discussions surrounding the adjustment of workflows to facilitate FAIR practices should occur within a research community, defined as a group of stakeholders (such as individual researchers, research support staff, and data infrastructure providers) that have a shared interest in streamlining their efforts to implement the FAIR principles. As explained by Timmermans and Epstein [13], standardisation is inherently a social process that requires the commitment and endorsement of multiple actors to be effective. The community aspects of FAIR implementation are embedded in the original FAIR principles [2] and made explicit in principle R1.3 (“(meta)data meet domain-relevant community standards”). For instance, in the framework of FAIR Implementation Profiles (FIPs, a methodology that has been introduced to document FAIR implementation choices), the community aspect is captured by the concept of FAIR Implementation Community [15,16]. How to adequately define and engage a community remains, however, an open challenge. Rule 1 recognises that identifying the appropriate research community is a crucial step in facilitating discussions on standardisation. The stakeholder who wishes to initiate a FAIR discussion may be already part of the community or not; regardless, it is important to provide a clear definition of what the community to approach is. Research communities can be based on various factors, such as the type of data being generated or used, a shared institutional affiliation, or a specific research project. A community can constitute a formal entity, or it can be an informal group, and it can exist for a determined time span, or be long-lasting [15]. It is important that a community self-identifies as such, as this can increase the level of commitment and engagement among members. Research communities typically involve individuals in a variety of different roles, such as researchers, RDM support staff, lab technicians, and students. Once the community has been identified, the levels of understanding of FAIR implementation and FAIR standards should be gauged, using resources such as the FAIR-Aware tool developed by DANS [17] or the How to FAIR quiz from the Danish National Forum for Research Data Management [18]. Disparities in understanding among different stakeholders may present challenges to the standardisation process—though a diversity in perspectives can be beneficial, as elaborated in Rule 3. Depending on the features of the community, there may be different ways to get in touch with the community members: In the case of an informal community, for instance, it may be necessary to proceed via “snowballing,” with one identified member suggesting other ones and so on. In the case of formalised communities, instead, there may be people with specific roles (such as community managers) who already have open communication channels with the community. In both cases, reaching out to RDM experts, research infrastructures, or scientific associations may be beneficial (see Rule 2), as they may be aware of existing or similar initiatives, or be able to suggest people to contact. Rule 2: Identify sources of support and partnership Attempting data standardisation is a complex process that should not be done alone. Support and partnerships are most likely to be found from the RDM support team at your institute (usually based at the library), the scientific association of your discipline (see Table 1 for some examples), or other enthusiastic individuals already involved more closely with the adoption of the FAIR principles. Failing to engage with these stakeholders may result in a lack of awareness and recognition of the need for promoting the FAIR principles in your community within your institute or association. We recommend prioritising seeking out this type of support or partnerships, as it could prove to be beneficial in the long run, even if funding or resources may not be immediately available. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Examples of organisations that can help finding RDM experts grouped by spatial focus (primarily in Europe) and domain specificity. The examples are meant to give an indication, not an exhaustive overview. This overview is available at https://github.com/AngelicaMaineri/awesome-RDM-support/blob/main/README.md under a CC0 licence to allow reuse and extension. https://doi.org/10.1371/journal.pcbi.1011668.t001 If you are a researcher, start by checking if an RDM team is available at your institution. The RDM team will be able to point to existing resources, tools, and information that can save time. The RDM team can also provide support in raising awareness, as they should already be involved with promoting the adoption of FAIR principles within the research community. This RDM team will likely have experience with providing workshops, training programmes, setting up policies and recommendations, and hosting events. The RDM team may have already set up materials or resources that can be tailored to specific needs. Vice versa, if you consider yourself as an RDM specialist or an infrastructure provider, make sure to seek partnership with researchers in the community, as they can contribute their domain-specific knowledge as well as their perspectives of potential data (re)users. Regardless of your role, the scientific association or individual researchers already involved with the FAIR principles may have already set up data standardisation processes that you can join, will provide connections to your community (see Rule 1), and can provide further support. Ideally, there is funding involved in standardisation efforts. Both RDM support and scientific associations may also have access to funding, or they may be able to connect you to other funding opportunities. Funding agencies in the Netherlands allow for funding data management activities in their projects (NWO and ZonMW). Open Science Funds may be available through Open Science Communities or at a (inter)national level (see NWO or FAIR impact). Once you have defined your community and found further support, the next step is to consider the social aspects of standardisation processes (Rule 3). Rule 3: Take into account the social aspects of standardisation As discussed in Rule 1, adopting research workflows is a social process that requires the participation and commitment of multiple actors within a research community [13]. Standardisation efforts can stop or become obsolete when a community loses interest [19]. Therefore, Rule 3 highlights the importance of considering the social aspects that may influence the consensus-building process when facilitating discussions on the adoption of FAIR practices and offers practical examples. The rule is quite broad in its formulation because the way in which these social dynamics manifest themselves may vary depending on the community and the stakeholders involved. We aim to illustrate some potential situations and actionable recommendations. First of all, the creation and adoption of standards may involve trust and authority issues. For example, the introduction of new standards may generate opposition and resistance if community members do not perceive them as legitimate or do not trust the authority of those proposing the standards. To mitigate the risk of frictions, involving the community is key. In particular, highlighting best practices via real use cases from the community can be effective in showing the value of standardised FAIR practices and promoting the use of existing standards. Generating trust in the process can be facilitated by having regular meetings, including occasional in-person meetings if possible, and by planning clear feedback processes so that concerns from the community can be addressed. When there are no existing best practices yet within your own community, examples and recommendations can be adopted from other communities who are further along in adopting the FAIR principles (see Rule 2: RDM experts can help reaching out to more mature communities; but also see Rule 10: Sharing experiences is pivotal!). Building on existing practices may save time and resources that can be used more efficiently in the standardisation process. Additionally, standardisation can sometimes result in a reduction of heterogeneity within a community, as it involves the creation of agreed-upon rules that may limit the range of previously enacted practices and even supersede previously adopted, perhaps outdated, standards. Therefore, it is important to carefully consider the potential impacts of standardisation on the variety of research practices within a community. The community-based approach to the FAIR principles, alongside the FIPs, which support the documentation of FAIR implementation choices [16], makes it possible to create multiple standards, as long as cross-standard interoperability is kept into account. For instance, the CEDAR Metadata Tools allow the creation of community-specific metadata templates while reusing existing ontologies and value sets, therefore enabling diverse solutions within a shared framework (see Rule 6 for more recommendations on creating metadata models). There are also solutions to make existing standards more easily findable and reusable. One such solution is FAIRsharing, which serves as a repository of FAIR-enabling standards and other FAIR resources. This platform was created to address the issue of excessive fragmentation in the developments of standards [20]. Standardisation often requires researchers to invest time and resources into changing their existing workflows, which can be a challenging task. The disruption of existing practices has also been reported to be an important barrier to adoption of standards in the industry context (see [21] and Rule 5). Therefore, it is important to clearly communicate the benefits and incentives of adopting the FAIR principles to the research community. By clearly communicating the value of the FAIR principles and engaging in meaningful dialogue with the research community, it is possible to facilitate a more effective and efficient standardisation process (see also Rule 4). Rule 4: Establish the benefits of standardisation for the community It can be helpful to establish benefits of standardisation in order to convince others to get involved and motivate them to change their workflows. Some examples of benefits are listed below and may not always be directly applicable to your community. Personal benefits by being involved in this process include: Direct impact on the eventual results, ensuring that the standardisation processes are applicable to your research. Extension of professional network and positive effects on your professional reputation. Long-term standardisation can be more cost efficient by increasing data reuse, preventing duplication of research/trials, improving quality of the data (reduction of data errors and increase of reproducibility), and facilitating integration of datasets. FAIR standardisation processes also reflect positively on institutions (see Rule 2 on where to find help in your institute). Benefits for institutions include: Streamlining data management processes and being cost efficient. Increasing the reputation and trust in research findings from the related research groups [22]. Rewarding and recognising data as a valuable research output in academia. Facilitating collaboration and innovation [22]. Increasing the value of existing data by facilitating reuse and, thereby, increasing the return on the initial investment on data collection. After the benefits have been established, the next step is to identify the barriers (Rule 5). Rule 5: Identify community-experienced barriers It is important to recognise that the barriers and challenges to implementing the FAIR principles and standardising data management practices may vary widely across different research domains and communities. This may involve performing a gap analysis to identify areas where additional support or resources are needed for the community (identified in Rule 1), or identifying case studies of successful FAIR implementation in similar disciplines that can serve as best practice for the community (see also Rule 3). Below are some of the barriers that we have encountered, either in the literature or in our own professional experience: Data requirements: The types of data more commonly used by a community as well as their size and heterogenous nature may pose challenges in terms of storage, management, and accessibility [23–25]. Ethical and legal barriers: The handling and sharing of sensitive political or personal data may be subject to strict regulations (such as the General Data Protection Regulation (GDPR)) and require additional considerations to ensure compliance [9,23–29]. Data sharing can also result in economic damage when disease data are shared and impact tourism and trade [29]. Different research environments: When there are critical resource shortages (such as the absence of research networks and lack of infrastructural support), there may be more immediate concerns that should be addressed [30]. Intellectual property and licensing: Intellectual property (IP) issues (such as data transfer and processing agreements) may arise when data are shared or reused [9,24,25], particularly when multiple stakeholders are involved. Not everyone may have access to proprietary software used in data analyses. Lack of incentives: Some researchers may not see the value in making their data FAIR or may not perceive a need to share their data with others (see Rule 3) [23,24,26–29,31–33]. Cultural barriers: for example, considering data sharing to hamper future publications if there is no reciprocity in the form of appropriate credit for data sharing [25,26,29,31], or a lack of trust in data being correctly interpreted and used [23,24,27–29,32]. Lack of institutional data policy, support, and training [34]. Lack of infrastructure (see Rule 8) to share data [23,25,29,31,32]. Lack of compliance monitoring by institutes, funders, or journals with policies regarding the FAIR principles, decreasing the need for researchers to comply with these policies [9,23,32,35]. Limited awareness about best practices, FAIR principles, and standards [28]. Emphasis on novel research may result in data generation rather than reuse, integration, and maintenance [33]. Possible criticism and fear that results will be invalidated [23,29]. Limited time and/or lack of resources [9,23,25–29,32,34]. To identify and address these barriers, you should discuss them with your community (Rules 1 and 3). Pending on the identified barriers, there will be different solutions to address them. Some of these barriers (limited awareness, lack of expertise and best practices) can be addressed by data standardisation and defining more explicitly what information is most valuable in the data management workflows. In Rule 6, this is addressed by going deeper into how information requirements can be established. Rule 6: Set up minimum metadata requirements Metadata is information about the data that provides context and allows for proper interpretation and reuse. A metadata standard is a structured form of documenting and describing this information. Several metadata standards are already in use, such as Dublin Core and DataCite ([36]; see the Digital Curation Center for an overview of metadata standards, or use FAIRsharing to browse metadata standards). Dublin Core consists of 15 general elements that make this standard easy to use across disciplines. Nevertheless, most disciplines and research communities may require more detailed metadata than those provided by Dublin Core in order to manage and document their research data effectively. This will eventually result in data shared in research repositories that are better aligned with the FAIR principles. It is therefore helpful to look for discipline-specific metadata standards or guidelines (see FAIRsharing.org [20] or the Metadata Standards Catalog). When there are no metadata standards or minimum metadata requirements available for your community, a more advanced step is to start creating these minimal metadata requirements. This can be a complex task, particularly when your community spans over different fields that use distinctive terminology to describe data [37]. To start with setting up minimum metadata requirements, it is important to first establish who needs to be involved, as community engagement will be crucial [37] (Rule 1). You may also need to consider who is most suitable to lead this process, as this will require some degree of authority, expertise, and trust within your community (see Rule 3). Other communities have already been successful in developing minimum metadata requirements, such as the Earth Sciences [38,39], Bionano Sciences [40], Biomedical Sciences [41], and -omics Sciences [42–45]. The lessons learned from these communities can be taken into account, although their approaches may not always fit with your research community that may have different requirements and challenges (Rule 5). Minimum metadata requirements can be about data/sample preprocessing, experimental analysis, quality control, preregistration—any aspect related to the research process. The metadata requirements should provide guidelines for essential information requirements while at the same time be flexible to meet each researcher’s objectives [41,46]. There is a need to “strike the right balance between minimising the barriers to data submission and maximising opportunities for data reuse” [41]. After you identified the relevant stakeholders (Rules 1 and 2), you can follow the recommendations below to start setting up minimum metadata requirements, or a Minimum Information Standard, in a research community: Review existing practices such as metadata standards, guidelines, and use cases [38,39,42,43,46]. If there are no existing efforts, you can start with a call for guidelines [44], set up a working group/project team [39,45], or a network [47]. You can get a team together by organising a workshop or conference session [41,42,48,49]. Ideally, funding is available for standardisation efforts (for example, NIH funding [19]) or should be applied for [47,48] (see also Rule 1). Adhering to the minimum metadata requirements should be as effortless as possible to enable widespread adoption [46,48] (see also Rules 3 and 4). Minimum metadata requirements are the first step towards standardisation. Additional developments will be needed for standardisation [42,46], involving the research community at each step. To establish community consensus, the research community should be asked for input and feedback, through community discussions, workshops, and surveys [38,41–44,48]. Only through active community involvement will a functional solution be achieved [45] (see also Rule 3). To ensure practical and effortless implementation of the standards by journal editors, reviewers, and data repositories, it is important to gather their feedback [39,41,45,49]. Once progression has been made, it is important to communicate this to the research community, via public documentation, reports, or publications (see also Rule 10 on sharing experiences). To support community uptake, it can be helpful to provide training, support, or to have champions involved that can promote the standards [38]. The benefits (Rule 4) of adjusting existing workflows should be clear. A great way to get started is to review the work by ESS-DIVE in establishing a community-centric metadata reporting format [38]. Crystal-Ornelas and colleagues [38] share guidelines (Box 1) and details about their process. A next step could be to develop an ontology (see Courtot and colleagues [50]’s 10 simple rules on this topic). Rule 7: Set up documentation standards In addition to the minimum metadata requirements described earlier (Rule 6), documentation is the next step that supports reuse of research outputs. The type of documentation needed depends on the purpose, expertise, and context of your community. If the primary purpose of the documentation is to be published in a research repository (see Rule 8) or to comply with the funders’ policies, the documentation must, at a minimum, be designed to achieve that purpose. Two general levels of documentation can be considered when documenting research data: project level and data level (also referred to as study-level and object-level documentation, respectively). The project-level documentation provides context for the collection, methodology, structure, and validation of data, while the data-level documentation consists of the variable names, descriptions, classifications, file formats, and software details. In other words, project-level documentation is about what is around data, and data-level documentation is about data itself [51]. Examples of project-level documentation are Data Management Plans (DMPs), Software Management Plans (SMPs), and the use of Preregistration and Registered Reports. The first two provide project-level documentation by describing the context of data and software; in DMPs, describing how data were collected and the methods used to validate it [52]; in SMPs, describing how software works, the purpose, the outputs, and its (continuous) development. Research communities may select standard templates for DMPs, taking into account the requirements of their organisation or funder (see for examples the list of public templates on DMPTool), while for SMPs, such standards are under development [53]. Preregistration, on the other hand, involves the public disclosure of research plans before data collection, analysis, and reporting are completed, with the goal of increasing transparency in the knowledge creation process from its inception to the results [54,55]. In an effort to standardise the information requested for preregistering a study and ideally simplify the process for researchers, templates have been developed (see the list of templates on the Open Science Framework website). At the data level, once the minimum metadata requirements are established (Rule 6), it will be easier to describe variables or file formats that the community will use and then expand to documentation guidelines. Documentation can be used to describe how to organise data, such as spreadsheets [56], workflows (see the Data Curation Network Primers), and to provide information on standards for dates and times (such as ISO 8601 or RFC3339). In addition, it may be possible to use documentation from other communities. In particular, code programming communities use standard style guides that are also widely used within research communities (such as PEP 8 for Python, Tidyverse for R; see also guidance by the Code Refinery [57]). A recommended solution for documentation could be codebooks, which describe the variables with their units, summarising choices made during the research process, and outlining the experimental study design [58]. Ideally, codebooks should be in a structured/standard format (for example, Data Documentation Initiative Codebook). Recently, tools have been developed that can automatically generate standardised metadata, reducing the (time) barriers to writing comprehensive codebooks (codebook R Package [59]). Ultimately, the choice of documentation standardisation should facilitate communication and collaboration between researchers and those who reuse their data. Rule 8 outlines the process of choosing the infrastructure to share the data and the accompanying documentation. Rule 8: Identify infrastructure to share data In order to get a clear idea of the infrastructure that can be used to share data, first, it is important to follow the requirements and guidelines of your institution, funders, and/or collaborators. Generally, data repositories are considered to be the ideal infrastructure to share data in a reliable manner [60]. Generic repositories, such as Zenodo, OSF, or Figshare, are widely used for preserving and sharing research data (see Table 2 for some examples). Your institution may also already have a repository that you can promote within your community. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. A list of common repositories outlined in more detail in [61]. https://doi.org/10.1371/journal.pcbi.1011668.t002 Generic and institutional repositories are generally not designed with the needs of a specific community in mind, which is where discipline-specific infrastructure may play an important role. Discipline-specific infrastructure is especially beneficial if standard data formats are used and enforced, ideally via user-friendly interfaces and with training provided where needed [62]. There are many domain-specific repositories (see NIH list of domain-specific repositories). It is therefore important to determine which type of data repository will better serve the needs of your community. Communicating a preferred infrastructure to share data may result in data that are more findable (as data are shared using the same infrastructure) and may reduce the cognitive load for individual researchers within your community as they do not have to look for suitable data repositories themselves. Before sharing data via a data repository, or promoting your repository to the community, you will need to verify that the repository follows the minimum requirements to be considered useful and adheres to the FAIR principles. Think that a repository needs to: Have a clear policy on how data will be managed, as well as a privacy policy and terms of use. Provide sufficient data storage size for the dataset. The geographic location where the data are saved (for restricted access dataset that contains personal data). Assign a persistent identifier (such as digital object identifier (DOI)) to be able to cite the data. Allow you to include a licence to your data (such as a Creative Commons licence). Make sure data are available/accessible and discoverable. Repositories can enhance their discoverability by being included in databases such as re3data (https://www.re3data.org [63], FAIRsharing (https://fairsharing.org [20]), and the EOSC portal (https://eosc-portal.eu). Allow revisions to be made to the dataset in the future. In some cases, institutional or generic repositories do not fulfil the requirements of your community (see Rule 5), and there may not be a discipline-specific data repository available. Setting up a specific repository may be a good option when there are sufficient resources and plans for the long-term sustainability of the infrastructure (see Rule 9). The main advantage of arranging your own repository infrastructure is that you have greater control over how data are documented and presented to the public and/or researchers. Specific data repository infrastructure may also improve the data quality of the datasets [64]. However, creating and using this infrastructure leads to additional costs (especially when dealing with large quantities of data). In addition, clear documentation and training materials (see Rule 9) are required to engage researchers to use the repository. Rule 9: Plan for the long-term actions As mentioned in Rule 8, discipline-specific infrastructures require resources and should be sustainable for the long term. When these infrastructures are set up by a small group, maintenance and sustainability is challenging as many researchers move across institutes and countries. While sustainability can be achieved by charging for repository services, it is also important to consider that not all researchers have access to these resources. Researchers are generally working on projects that eventually run out of funding, especially at the stage of data sharing. It is therefore important to consider who pays for long-term data sharing and maintenance. Maintenance plans and governance of infrastructure and standards should be transparently communicated (see also Rule 3). To plan for the long term and to establish robust policies for the repository, repositories could aim for certification (via CoreTrustSeal, ISO 16363, or Nestor), although this is a resource-intensive process. Resources are also needed for the maintenance of any created metadata standards (Rule 6). Standardisation is a continuous process and will require evaluation on their practical applicability (for example, metadata standards may become obsolete or deprecated when they are no longer applicable [19]). Standardisation is also a continuous learning process. Researchers may not be familiar with the standardisation efforts and will need a place to start, support, or training resources. It can also be important to monitor whether standardisations are followed appropriately—some form of manual curation may always be needed to avoid errors or incomplete entries. All of these processes take up resources in the long term. To facilitate long-term sustainability, it is better to use open formats and infrastructure built using open source software. This prevents lock-in to certain services and allows community members to continuously contribute. It is also important to consider how the infrastructure will scale when future use is increasing and user input may become more heterogeneous [65]. Individual researchers can improve the longevity of research data by starting to make use of data repositories and make their data available in open formats, following the repository guidelines. When researchers are already using data repositories, they can promote the use of data repositories to their colleagues and share any available training materials—especially if they are in leading roles or responsible for the training of other researchers. Individual researchers can also financially support data repositories by including a budget for data curation in their research proposals or by requesting funding from their institutions for data curation. Individual researchers, or research data support staff, can also provide other types of resources to data repositories by performing data curation tasks, data peer review, or taking on communication tasks. To foster a culture of data standardisation and sharing, it is needed to recognise the efforts of researchers who adopt minimum metadata requirements (Rule 4). Ideally, this happens at the institutional level. Research communities can also recognise practices during annual meetings and conferences or by awarding prizes (for example, the Open Science Community Amsterdam Awards that took place on January 26, OSCA 2023). Rule 10: Share experiences Sharing experiences about the standardisation process facilitates learning from existing efforts and identifying best practices that can facilitate the standardisation journey. By reaching out to local RDM support or other community stakeholders (see Rules 1 and 2), you have hopefully benefitted of the experiences of others as well. It is therefore important to share experiences gained from each of the rules listed here and as illustrated in Fig 1. Experiences and insights can be shared via case studies, best practices, and lessons learned from standardisation efforts. Venues to share these experiences may include journals (such as PLOS), preprint servers, publishing forums (such as FAIR connect, where an earlier version of this article was shared [14]), data repositories (such as Zenodo), blogs (for example, [66]), social media, or conferences and meetings. Conclusions Adopting the FAIR principles and adjusting research workflows is a complex and time-consuming process. The recommendations that we share emphasise the need to identify the community that needs to be involved (Rule 1) and find support as well as the relevant stakeholders that need to be involved (Rule 2). As adjusting existing workflows is primarily a social issue (Rule 3), it is important to identify the benefits (Rule 4) and to address the existing barriers (Rule 5). Keeping this in mind, it will become possible to set up minimum metadata requirements (Rule 6), documentation standards (Rule 7), and identify the infrastructure that the community can make use of or should establish (Rule 8). It is important for infrastructure, and also for metadata standards, to consider the long-term sustainability of the efforts (Rule 9). Crucial to each of these steps is the sharing of the lessons learned and the materials created so that others do not have to start from scratch (Rule 10). By following these recommendations, you should be able to more successfully engage your community in discussions that will result in successful implementations of the FAIR principles. Supporting information S1 Text. Process to getting to “Ten simple rules for starting FAIR discussions in your community.” https://doi.org/10.1371/journal.pcbi.1011668.s001 (DOCX) Acknowledgments We are grateful to the over 40 participants at the Session “Starting FAIR discussions increasing standardisation in your research community” at the Open Science Festival in Amsterdam on September 1, 2022, for their initial input on the checklist (see S1 Text for more details). Thanks to FAIRconnect for helpful comments by Erik Schultes and Barbara Magagna. Author contributions were set up via Tenzing [67].

Hussaini, Sayedeh;Lädke, Sarah L.;Schröder-Schetelig, Johannes;Venkatesan, Vishalini;Uribe, Raúl A. Quiñonez;Richter, Claudia;Majumder, Rupamanjari;Luther, Stefan

doi: 10.1371/journal.pcbi.1011660pmid: 38060618

Introduction Excitable media are complex dynamical systems that have the ability to support the formation and sustenance of non-linear excitation patterns, such as the spiral wave (in a 2 dimensional- (2D) system) and scroll wave (in a 3 dimensional- (3D) system) [1, 2]. In the heart, the occurrence of these waves is associated with life-threatening ventricular tachyarrhythmia, rhythm disorders, which are considered to be major precursors of sudden cardiac death [3–6]. Clinically, arrhythmias are treated by removing all abnormal electrical activity from the heart, to allow the system to restore its regular function. This is most effectively achieved by electrical defibrillation. Electrical defibrillation applies a global electrical high-energy shock to the heart. Unfortunately, despite its high success rate, electrical defibrillation has considerable negative side effects such as intense pain, trauma, and tissue damage [7–12]. These disadvantages motivate the search for low-energy alternatives to conventional defibrillation [13–16]. For this to work, a deeper understanding of arrhythmia dynamics during successful termination is required. In principle, the applied electric field causes the intrinsic heterogeneities of the heart muscle to form virtual electrodes. These electrodes generate new excitation waves [17–22] that propagate intramurally and interact with the preexisting electrical abnormality. On the one hand, these interactions force synchronization of these abnormal waves at high stimulation amplitude. This leads to a high termination rate of arrhythmias. On the other hand, a decrease in stimulation amplitude leads to a lower termination rate, which could be due to the fact that the generation rate of the abnormal waves is higher than the termination rate during the wave-wave interaction. Nevertheless, it is of great interest to understand the underlying mechanism of low amplitude cardiac arrhythmia termination, as the common side effects mentioned above may be reduced. Depending on the spatiotemporal dynamics of the abnormal electrical activity, namely the spiral wave and scroll wave within cardiac tissue, the approach to control the arrhythmia must be adapted. During conventional defibrillation, the mechanism is based on ensuring a complete phase reset of the heart’s electrical activity and is therefore almost immediate. However, this may not be the only mechanism underlying successful defibrillation, as termination of arrhythmias with transient time has also been observed, numerically and experimentally, in studies of [23–25]. Therefore, understanding spiral wave dynamics and the mechanisms underlying successful defibrillation are of great interest to the nonlinear dynamics and complex excitable systems community. It is important to emphasize that such in-depth experimental investigations require powerful research tools. In addition to probes that can overcome the major challenges of tracking and visualizing scroll waves in cardiac tissue [26, 27], tools are needed that allow controlled and reversible manipulation of cellular processes. Optogenetics is a technology that enables such control at the cell, tissue, and organ level. This technique is utilized in a vast range of study purposes, from basic [28–32] to application studies [23, 33–37]. In these works, arrhythmia in genetically modified cardiac tissue is controlled by different global and structured illumination patterns. At low light intensities, it was observed that the arrhythmia is not terminated abruptly but only after a few rotations of spiral waves. These observations prompted us to investigate in more detail the progression of arrhythmia dynamics during termination. Here, we use optogenetics to investigate the mechanisms that lead to the termination of spiral waves at different energy levels, both in in silico and in ex vivo systems. Defibrillation energy is given by regulating the intensity and duration of the applied light pulse. We find that the main mechanism for successful termination of spiral waves at high light intensities is the abrupt excitation of the entire medium containing the wave, which prevents its further propagation and results in a spontaneous termination. However, at low light intensities with a long pulse length, we observe slow termination of a spiral wave in both systems of in silico and ex vivo. In the latter case, the termination of a spiral wave rotating in an transgenic mouse heart, is proceeded by prolongation of the last action potential. However, in the former case, a progressive dissolution of the spiral wave core, in a 2D domain of ventricular mouse heart, was followed by a termination with a transient time. Materials and methods Ethics statement All experiments in the intact mouse heart were done in accordance with the guidelines from Directive 2010/63/EU of the European Parliament on the protection of animals used for scientific purposes and the current version of the German animal welfare law and were reported to our animal welfare representatives. The experimental protocol was approved by the responsible animal welfare authority (Lower Saxony State Office for Consumer protection and Food Safety). Humane welfare-oriented procedures were carried out in accordance with the Guide for the Care and Use of Laboratory Animals and done after recommendations of the Federation of Laboratory Animal Science Associations (FELASA). Numerical study For our numerical studies, we use a 2D continuum model of the membrane voltage V spread across an optogenetically modified monolayer of cardiac cells. Spatiotemporal evolution of V is described using a reaction-diffusion-type partial differential equation: (1) Here, Cm = 1.0 μF/cm2 is the specific capacitance of a single cell membrane, is the diffusion coefficient, which accounts for intercellular coupling, and is set to obtain conduction velocity of a propagating plane wave to 43.9 cm/s. We describe the total ionic current (Iion) produced by a cell according to the formulation of [38, 39], who developed this model for adult mouse ventricular cardiomyocytes. Our simulation domain contains 100 × 100 grid points, which translates to a physical size of 25 mm × 25 mm. In order to incorporate light-sensitivity to the cardiac cells we combine a 4-state model of Channelrhodopsin-2 (ChR2, a light-gated protein) [40] to the Bondarenko model [38, 39, 41]. The ChR2 current (IChR2) is described in Eqs (2) and (3). (2) (3) Here, gChR2 is the maximum conductance of the ChR2 ion channel, G(V) is the voltage-dependent rectification function. O1 and O2 are open state gating variables, as opposed to two closed state gating variables C1 and C2, that together comprise the 4-state ion channel. C1 and C2 are described in Ref. [40]. γ is the ratio O2/O1, and EChR2 = 0mV is the reversal potential of this non-selective cation channel. In our numerical studies we considered 25 different initial conditions. The source code for our numerical simulations is provided in S1 Code. Calculation of core radius. In our simulations, to calculate the core radius of the spiral wave under global illumination, we first use the Canny filter method to detect the edge, and then a circle is identified and fitted to the core area using the Hough transform method (see S2 Fig). Experimental study Experimental measurements on control of ventricular arrhythmia in the intact mouse heart. In experiments, we report results of our ex vivo studies using Langendorff-perfused hearts obtained from α-MHC-ChR2 transgenic mice (source: Dr. S. Sonntag, PolyGene AG, Switzerland). To induce arrhythmias, we apply 30 electrical pulses with an amplitude of 2.3–2.5V, width of 3–7ms, and a frequency of 30–50Hz using a needle electrode. To stabilize the arrhythmia, (i) the concentration of KCl in tyrode solution was reduced from 4mM to 2mM, and (ii), 100 μM Pinacidil, (a KATP channel activator) was added to the tyrode. In all our experiments we define arrhythmias to be persistent, if they last at least 5s. For global illumination, we placed three equidistant LEDs (wavelength of 460 nm) each with an angular separation of 120° around the bath and simultaneously illuminate the heart from all three directions with a single blue light pulse generated by the three LEDs [23]. In our experimental studies we used 5 different mouse hearts with 10 trials each. Optical mapping experiments on ventricular arrhythmia and its termination in the intact mouse heart. In this study, we use optical mapping using potentiometric fluorescent dye to visualize the dynamics of ventricular arrhythmias in an intact optogenetic mouse heart [42]. To this end, we first introduce the contraction upcoupler blebbistatin (c = 5 μM) into the heart via perfusion. This leads to a decrease in the mechanical contraction of the heart (after about 20 minutes), which results in a significant reduction in the distortion of the fluorescence signals [43]. To record the changes of the membrane voltage of cardiomyocytes, we stain the heart with a bolus injection of a voltage-sensitive dye: the red-shifted dye Di-4-ANBDQPQ (c = 50 μM, Thermo Fisher Scientific). During optical mapping imaging, excitation light from a 625nm mounted LED (M625L3, Thorlabs) was first filtered through a bandpass filter (FF02–628/40-25, Semrock) and then reflected through a dichroic long-pass mirror (FF685-Di02–25x36, Semrock). Then, the emitted light was first collected by a 775 ± 70nm bandpass filter (FF01–775/140-25, Semrock) and finally by the camera (EMCCD, Cascade 128+, Photometrics) with a spatial resolution of 64 × 64 pixels (133 μm per pixel) at 1 kHz. Optical Mapping recordings were acquired with software MultiRecorder and analysed with software PythonAnalyser (both developed by Research Group Biomedical Physics). Electrical signals were recorded with 16-channel data acquisition system MP160 and software AcqKnowledge (BIOPAC Systems, Inc.). We observed the crosstalk between the blue stimulation light and the fluorescent dye leads to an increase in the fluorescence baseline. Therefore, to counteract this signal artifact, the optical signal is re-normalized by division to the recorded stimulation light signal. Ethics statement All experiments in the intact mouse heart were done in accordance with the guidelines from Directive 2010/63/EU of the European Parliament on the protection of animals used for scientific purposes and the current version of the German animal welfare law and were reported to our animal welfare representatives. The experimental protocol was approved by the responsible animal welfare authority (Lower Saxony State Office for Consumer protection and Food Safety). Humane welfare-oriented procedures were carried out in accordance with the Guide for the Care and Use of Laboratory Animals and done after recommendations of the Federation of Laboratory Animal Science Associations (FELASA). Numerical study For our numerical studies, we use a 2D continuum model of the membrane voltage V spread across an optogenetically modified monolayer of cardiac cells. Spatiotemporal evolution of V is described using a reaction-diffusion-type partial differential equation: (1) Here, Cm = 1.0 μF/cm2 is the specific capacitance of a single cell membrane, is the diffusion coefficient, which accounts for intercellular coupling, and is set to obtain conduction velocity of a propagating plane wave to 43.9 cm/s. We describe the total ionic current (Iion) produced by a cell according to the formulation of [38, 39], who developed this model for adult mouse ventricular cardiomyocytes. Our simulation domain contains 100 × 100 grid points, which translates to a physical size of 25 mm × 25 mm. In order to incorporate light-sensitivity to the cardiac cells we combine a 4-state model of Channelrhodopsin-2 (ChR2, a light-gated protein) [40] to the Bondarenko model [38, 39, 41]. The ChR2 current (IChR2) is described in Eqs (2) and (3). (2) (3) Here, gChR2 is the maximum conductance of the ChR2 ion channel, G(V) is the voltage-dependent rectification function. O1 and O2 are open state gating variables, as opposed to two closed state gating variables C1 and C2, that together comprise the 4-state ion channel. C1 and C2 are described in Ref. [40]. γ is the ratio O2/O1, and EChR2 = 0mV is the reversal potential of this non-selective cation channel. In our numerical studies we considered 25 different initial conditions. The source code for our numerical simulations is provided in S1 Code. Calculation of core radius. In our simulations, to calculate the core radius of the spiral wave under global illumination, we first use the Canny filter method to detect the edge, and then a circle is identified and fitted to the core area using the Hough transform method (see S2 Fig). Calculation of core radius. In our simulations, to calculate the core radius of the spiral wave under global illumination, we first use the Canny filter method to detect the edge, and then a circle is identified and fitted to the core area using the Hough transform method (see S2 Fig). Experimental study Experimental measurements on control of ventricular arrhythmia in the intact mouse heart. In experiments, we report results of our ex vivo studies using Langendorff-perfused hearts obtained from α-MHC-ChR2 transgenic mice (source: Dr. S. Sonntag, PolyGene AG, Switzerland). To induce arrhythmias, we apply 30 electrical pulses with an amplitude of 2.3–2.5V, width of 3–7ms, and a frequency of 30–50Hz using a needle electrode. To stabilize the arrhythmia, (i) the concentration of KCl in tyrode solution was reduced from 4mM to 2mM, and (ii), 100 μM Pinacidil, (a KATP channel activator) was added to the tyrode. In all our experiments we define arrhythmias to be persistent, if they last at least 5s. For global illumination, we placed three equidistant LEDs (wavelength of 460 nm) each with an angular separation of 120° around the bath and simultaneously illuminate the heart from all three directions with a single blue light pulse generated by the three LEDs [23]. In our experimental studies we used 5 different mouse hearts with 10 trials each. Optical mapping experiments on ventricular arrhythmia and its termination in the intact mouse heart. In this study, we use optical mapping using potentiometric fluorescent dye to visualize the dynamics of ventricular arrhythmias in an intact optogenetic mouse heart [42]. To this end, we first introduce the contraction upcoupler blebbistatin (c = 5 μM) into the heart via perfusion. This leads to a decrease in the mechanical contraction of the heart (after about 20 minutes), which results in a significant reduction in the distortion of the fluorescence signals [43]. To record the changes of the membrane voltage of cardiomyocytes, we stain the heart with a bolus injection of a voltage-sensitive dye: the red-shifted dye Di-4-ANBDQPQ (c = 50 μM, Thermo Fisher Scientific). During optical mapping imaging, excitation light from a 625nm mounted LED (M625L3, Thorlabs) was first filtered through a bandpass filter (FF02–628/40-25, Semrock) and then reflected through a dichroic long-pass mirror (FF685-Di02–25x36, Semrock). Then, the emitted light was first collected by a 775 ± 70nm bandpass filter (FF01–775/140-25, Semrock) and finally by the camera (EMCCD, Cascade 128+, Photometrics) with a spatial resolution of 64 × 64 pixels (133 μm per pixel) at 1 kHz. Optical Mapping recordings were acquired with software MultiRecorder and analysed with software PythonAnalyser (both developed by Research Group Biomedical Physics). Electrical signals were recorded with 16-channel data acquisition system MP160 and software AcqKnowledge (BIOPAC Systems, Inc.). We observed the crosstalk between the blue stimulation light and the fluorescent dye leads to an increase in the fluorescence baseline. Therefore, to counteract this signal artifact, the optical signal is re-normalized by division to the recorded stimulation light signal. Experimental measurements on control of ventricular arrhythmia in the intact mouse heart. In experiments, we report results of our ex vivo studies using Langendorff-perfused hearts obtained from α-MHC-ChR2 transgenic mice (source: Dr. S. Sonntag, PolyGene AG, Switzerland). To induce arrhythmias, we apply 30 electrical pulses with an amplitude of 2.3–2.5V, width of 3–7ms, and a frequency of 30–50Hz using a needle electrode. To stabilize the arrhythmia, (i) the concentration of KCl in tyrode solution was reduced from 4mM to 2mM, and (ii), 100 μM Pinacidil, (a KATP channel activator) was added to the tyrode. In all our experiments we define arrhythmias to be persistent, if they last at least 5s. For global illumination, we placed three equidistant LEDs (wavelength of 460 nm) each with an angular separation of 120° around the bath and simultaneously illuminate the heart from all three directions with a single blue light pulse generated by the three LEDs [23]. In our experimental studies we used 5 different mouse hearts with 10 trials each. Optical mapping experiments on ventricular arrhythmia and its termination in the intact mouse heart. In this study, we use optical mapping using potentiometric fluorescent dye to visualize the dynamics of ventricular arrhythmias in an intact optogenetic mouse heart [42]. To this end, we first introduce the contraction upcoupler blebbistatin (c = 5 μM) into the heart via perfusion. This leads to a decrease in the mechanical contraction of the heart (after about 20 minutes), which results in a significant reduction in the distortion of the fluorescence signals [43]. To record the changes of the membrane voltage of cardiomyocytes, we stain the heart with a bolus injection of a voltage-sensitive dye: the red-shifted dye Di-4-ANBDQPQ (c = 50 μM, Thermo Fisher Scientific). During optical mapping imaging, excitation light from a 625nm mounted LED (M625L3, Thorlabs) was first filtered through a bandpass filter (FF02–628/40-25, Semrock) and then reflected through a dichroic long-pass mirror (FF685-Di02–25x36, Semrock). Then, the emitted light was first collected by a 775 ± 70nm bandpass filter (FF01–775/140-25, Semrock) and finally by the camera (EMCCD, Cascade 128+, Photometrics) with a spatial resolution of 64 × 64 pixels (133 μm per pixel) at 1 kHz. Optical Mapping recordings were acquired with software MultiRecorder and analysed with software PythonAnalyser (both developed by Research Group Biomedical Physics). Electrical signals were recorded with 16-channel data acquisition system MP160 and software AcqKnowledge (BIOPAC Systems, Inc.). We observed the crosstalk between the blue stimulation light and the fluorescent dye leads to an increase in the fluorescence baseline. Therefore, to counteract this signal artifact, the optical signal is re-normalized by division to the recorded stimulation light signal. Results Fig 1A illustrates a single spiral wave with a circular core trajectory, indicated by a dashed box, in silico study. To investigate the electrical activity within the domain inside the spiral core and in its neighborhood, we measure the membrane voltage along a line parallel to the x-axis passing through the circular core (y = 12.5 mm). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Dynamics of a spiral wave in silico and ex vivo. A) Circular tip trajectory of a spiral wave, enclosing a core of diameter 0.3 mm, in a 25 mm × 25 mm simulation domain containing mouse ventricular cardiomyocytes. B) Space-time plot of the membrane voltage along the x-axis at y = 12.5 mm, for 500 mm of spiral wave rotation. The blurred region indicated with a rectangular dashed box demonstrates the core region of the spiral wave. C) Voltage distribution during 500 ms of simulation, along the x-axis with y = 12.5 mm. The gray shaded region illustrates the electrical activities inside the core. D) Fluorescence image of a single spiral wave in an optogenetic intact mouse heart in an arrhythmic state. E) Space-time plot of the membrane voltage along the dashed line in (D), for 350 ms of spiral wave rotation. https://doi.org/10.1371/journal.pcbi.1011660.g001 Temporal evolution of V measured along this line is shown in the space-time plot of the membrane voltage in Fig 1B. The core region of the spiral wave shows up as a blurred area on this plot. We have outlined this region using a dashed rectangle. The information presented in Fig 1B is shown slightly differently in Fig 1C, where the voltage along the line y = 12.5 mm is reported at different time points (indicated by the color bar). We observe that during 500 ms of spiral wave rotation, there is a local minimum in the value of V at the spiral core. The spatial extent of the core (0.3 mm in width) is indicated using the shaded grey rectangle in Fig 1C. Note that, within the core region the value of V increases with time, but always remains below the excitation threshold. Thus, no activity originates from the core around which the tip of the spiral continues to rotate. Fig 1D shows a single snapshot of a fluorescence image of an optogenetic intact mouse heart in an arrhythmic state. The depolarized region shows the arm of a single spiral wave, of which the core is located in the upper right of the image and is represented by a white cross. The space-time plot of the membrane voltage during 350 ms of recording along the dashed line is shown in Fig 1E. Since the entire dynamics of the spiral wave core is not in view in this recording, the ripples in the space-time diagram show only the time evolution of V of the arm of the spiral wave crossing the dashed line. To study the dynamics of arrhythmia termination during global illumination, we apply a single optical pulse with low light intensity (LI) and long pulse length (PL) to both systems of in silico and ex vivo during arrhythmic state. Fig 2A shows termination of the spiral wave in the 2D domain during global illumination with LI of 30 μW/mm2 and PL of 300 ms. We observe that the annihilation of the spiral wave occurs via gradual expansion of its core (see S1 Video). When light is applied globally to the excitable medium containing the spiral wave, the core region, which remains unexcited at all times by the spiral itself, becomes most susceptible to the applied excitation. Thus, the core depolarizes before any other part of the domain. By depolarizing the core we effectively convert functional reentry (free spiral wave activity) into the equivalent of an anatomical reentry, characterized by anchoring of the spiral wave around the light-induced heterogeneity at the core. At this point, the phase singularity, the unexcitable organizing center of the tip of the spiral wave, is moved into the boundary. Regions that respond next to the applied light stimulation are those that lie within the excitable gap. Thus, close to the core, subsequent excitation of the excitable gap effectively adds to the core size of the spiral, thereby contributing to its gradual expansion. Eventually, as the core expands all the way to the boundary, the rotational activity around it disappears and termination occurs (see Fig 2A for a full demonstration of the process). We refer to this mode of termination of the spiral wave, as core dissolution. The corresponding voltage time series of the recording electrode shows a very short termination transient time (the red shaded box), less than 100 ms. Thereafter, the system enters the constant elevated phase, which then decays to the resting state once the light is switched off (see Fig 2B). For ex vivo study we used a light stimulus with LI = 3.44 μW/mm2 and PL = 5 s. Fig 2C shows a monophasic action potential (MAP) signal (in black) before, during, and after illumination. It shows the sinus rhythm returns after 3190 ms of illumination. Fig 2D presents a closer look at the MAP signal of the heart during the last few rotations of arrhythmia and the first few action potentials of sinus rhythm. Fig 2E illustrates a sequence of fluorescence frames during the rotation of the spiral wave at the time between 4334 ms-4361 ms (indicated in Fig 2D with a green shaded box on the left). It shows the wave spreads first at the apex and then continues rotating clockwise within approximately 40 ms. The next three fluorescence imaging frames show the second last rotation of arrhythmia during 4427 ms to 4458 ms indicated in Fig 2D with a blue shaded box. The fluorescence frames from 4474 ms to 4526 ms illustrate the dynamics of the arrhythmia during the last rotation marked by a grey shaded box in Fig 2D. During this last rotation of the spiral wave the increase of the wavelength of the wave can be seen. This leads to a gradual propagation of the excitation wave (which is no longer a spiral wave at the time ≥ 4483 ms) over the entire area. Between 4483 ms and 4486 ms, the entire area is almost in the excited state and then falls back into the resting state at the time of ≈ 4526 ms (see S2 Video). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Control of arrhythmia by applying a single global optical pulse with low light intensity in in silico and ex vivo. A) Termination of a spiral wave in a two dimensional domain at light intensity (LI) of 30 μW/mm2 and Pulse length (PL) of 300 ms. B) Voltage-time series of a point (shown with a cross marker in (A)) before, during, and after illumination. C) A monophasic action potential (MAP) signal (in black) of an optogenetic intact mouse heart before, during, and after illumination (in blue) at LI of 3.44 μW/mm2 and PL of 5 s. D) shows a section of the MAP signal marked with a gray dashed box in (C). E) shows a series of fluorescence images of arrhythmia dynamics during illumination of a ventricular tachycardia in the mouse heart. https://doi.org/10.1371/journal.pcbi.1011660.g002 Comparing the transient termination time in the 2D numerical simulation and in the ex vivo experiment, shown as a red shaded area in Fig 2B and 2C, it can be seen that the transient time is relatively long in the case of the experiment. This could be due to the very different systems: a 2D cardiac tissue in which a single spiral wave rotates in a homogeneous single layer of cardiac tissue versus an intact mouse heart in which a spiral wave propagates in an inhomogeneous cardiac tissue with a more complex geometry. Moreover, light attenuation through the cardiac tissue in an intact heart enhances the complex spatiotemporal dynamics of the wave in the system. Our numerical results indicate that spiral wave termination via progressive dissolution of the core can be observed that at low LIs, whereas, at high LIs, termination occurs abruptly, similar to conventional electrical defibrillation. To develop a deeper understanding of the process of core dissolution, we apply a single global pulse to the spiral wave in a wide range of LIs from sub-threshold illumination (no excitation wave is triggered in this range of LIs) to supra-threshold illumination (an excitation wave is triggered in this range of LIs) with the PL of 500 ms. Fig 3A shows a space-time plot of the membrane voltage along a line in the range (shown in Fig 1A) from sub-threshold LIs, ⩽ 20 μW/mm2 to supra-threshold LIs, ⩾ 25 μW/mm2. We observe that for the sub-threshold illumination, increasing the LI increases the period of the spiral wave (Ts), which leads to a decrease in its conduction velocity [30]. In the supra-threshold régime, at low LIs (between 25–50 μW/mm2) an excitation wave is created at the core region which propagates through the whole domain and excites it to an elevated constant potential. Increasing LI leads to faster initiation of the excitation wave, reaching the elevated constant potential at shorter transient times. At very high LIs, (⩾ 80 μW/mm2) the system immediately enters the elevated constant potential when the light is switched on. A space-time plot of the membrane voltage of an intact mouse heart during 8500 ms sections of optical mapping recording is shown in Fig 3B. It shows that the arrhythmia consists of a single spiral wave with a stationary core dynamic and a rotational frequency of ≈ 20 Hz before illumination. During a single global optical pulse with a PL of 5000 ms and an LI of 3.21 μW/mm2, the frequency of the arrhythmia decreases to ≈ 18.75 Hz. It is important to note that the frequency reverses its original value of ≈ 20 Hz when the light is turned off. Fig 3C demonstrates a space-time plot of the membrane voltage of the same optical mapping recording as shown in Fig 2E. Here a single global optical illumination with LI of 3.44 μW/mm2 and PL of ≈ 5 s is applied to the mouse heart. It illustrates that during the first ≈ 4000 ms the arrhythmia dynamics consists of a single spiral wave with a non-stationary meandering core dynamics. Then, during the last nine rotations of the spiral wave, the wave propagates with a stationary core dynamic. Finally, the arrhythmia is terminated at ≈ 4500 ms due to the increase in action potential duration, and the first planar excitation wave of sinus rhythm propagates at ≈ 4900 ms. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Non-spontaneous termination of a spiral wave in silico and ex vivo. A) Space-time plots of the membrane voltage along the x-axis at y = 12.5 mm, in a two dimensional domain with an existing spiral wave during 500 ms of a single pulse global illumination. B) Space-time plot of membrane voltage of an intact mouse heart in an arrhythmic state before, during, and after illumination. C) Space-time plot of the membrane voltage of an intact mouse heart showing arrhythmia is terminated during illumination. https://doi.org/10.1371/journal.pcbi.1011660.g003 To investigate the transition dynamics into and out of the core dissolution process, we apply a shorter light stimulus PL = 150 ms, with simulation time of 300 ms, to the same range of LIs. We observe that at subthreshold LI, removal of the light pulse restores the system to its original state (see Fig 4A). At LI = 25 μW/mm2 and 30 μW/mm2 core dissolution stops when the light is turned off. At LI = 25 μW/mm2, the spiral wave continues to rotate at the same location within the domain. However, at LI = 30 μW/mm2, the new core position is shifted relative to the original state. At LI ⩾ 35 μW/mm2, the core dissolution continues and leads to the termination of the spiral wave. Fig 4B shows radius growth rate of the spiral wave’s core when global uniform illumination is applied at three different LI of 25, 30, and 35 μW/mm2. The growth rate was measured with the time interval of 10 ms. The slope of each fitted line shows an increase with increasing LI. The details of the slopes and intercept of each fitted line for the different LI of 25, 30, 35, 40, and 50 μW/mm2 are shown in Table 1. We also measured the minimum illumination time (PLcritical) to terminate the spiral wave at different LIs of 25–50 μW/mm2. Fig 4C shows the decrease in PLcritical with increase in LI, in which the PLcritical of 498, 190, 100, 100, and 80 ms corresponds to LIs of 25, 30, 35, 40, 45, and 50 μW/mm2, respectively. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Dynamics of a single spiral wave in silico study. A) Shows space-time plot of the membrane voltage along the x-axis at y = 12.5 mm, in a two dimensional domain with an existing spiral wave. A global pulse illumination with pulse length (PL) of 150 ms (shown with a blue trace on the left) was applied at different light intensities (LIs) during 350 ms simulation time. B) Shows the core size growth of the spiral wave with the time interval of 10 ms during global illumination at different LIs. C) Minimum PL (PLcritical) required to terminate the single spiral wave at various LIs. https://doi.org/10.1371/journal.pcbi.1011660.g004 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Comparison between the slopes and intercepts of the radius growth of the core of the spiral wave during global illumination at different LIs. https://doi.org/10.1371/journal.pcbi.1011660.t001 Next, to control the dynamics of the spiral wave in favor of termination, we apply a single global optical pulse to the pattern in Fig 1A and 1D at different LIs and PLs. Then, we compare the dose-response curves for arrhythmia termination in experiments and simulations. Dose-response diagram for optical stimulation displayed in Fig 5A and 5C shows increasing LI and PL, both independently and together, lead to an increase in arrhythmia termination efficiency. To calculate the termination time (both in the simulation and in the ex vivo experiment), the time between the switching on of the light and the last peak of the arrhythmia before termination is considered. The time required for the successful termination of arrhythmias with PL = 500 ms demonstrated in Fig 5B and 5D shows that the transient time is longer than 100 ms at low LIs (50 μW/mm2) in both numerical simulations and experiments. At high LIs (⩾ 80 μW/mm2), due to the abrupt annihilation of the spiral, the termination time is much shorter and varies in a narrow range of 10 to 20 ms. These results are presented in Fig 5D. The transient termination time of the experimental data shown in Fig 5B is in qualitative agreement with the numerical study (Fig 5A). Although the two systems (2D domain and intact mouse heart) are very different from each other, the similar trends in termination transient time suggest that the two systems must share a common dynamical feature. The kinetics of Channelrhodopsin-2 (ChR2) could possibly explain the similarity between numerics and experimental results. The activation of this protein leads to three different phases: 1) an initial peak of the incoming cation current (Ipeak), 2) decay of this peak to a constant phase with the width of the optical pulse duration (Icnst), and 3) decay of the current to the baseline when the light is switched off. Fig 1, supplementary, illustrates the kinetics of ChR2 current and corresponding membrane voltage activity at different LIs. It shows the application of sub-threshold LIs leads to a small inward current of IChR2 with low amplitude of Ipeak and Icnst. This leads to an increase in membrane voltage below the excitation threshold (indicated by the dashed line). However, the application of supra-threshold illumination (⩾ 25 μW/mm2) causes the large inward current of IChR2 to exceed the excitation threshold and an excitation activity called action potential occurs. The application of low-LI supra-threshold stimuli (e.g. 25 and 30 μW/mm2 in S1 Fig) results in slow activation of this channel with low Ipeak and Icnst, so that the arrhythmia can be terminated with a longer optical PL. This leads to a transient termination time long enough to include a few turns of the spiral wave. On the other hand, high-LI suprathreshold stimuli (e.g. 100 and 1000 μW/mm2 in S1 Fig) cause immediate activation of the ion channel, which results in the flow of a large current Ipeak in a very short time. This leads to immediate excitation of the heart tissue, which can lead to an abrupt termination without a long transient time. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Termination of arrhythmia applying a single pulse global illumination in silico and ex vivo. A) Termination rate of a single spiral wave in a two dimensional domain with size of 25 mm × 25 mm simulation (N = 25) at different light intensities (LIs) and pulse lengths (PLs) (percentage of successful attempts reported as mean ± SEM). B) Termination time at different LIs with PL of 300 ms. C) Arrhythmia termination rate vs. LI in Langendorff-perfused intact mouse hearts (N = 5) for different PLs. D) Termination time at different LIs and PL of 500 ms. https://doi.org/10.1371/journal.pcbi.1011660.g005 Discussion In this work, we investigate the mechanisms of arrhythmia termination during global illumination at low light intensities with a long pulse length. Our in silico studies on a 2D domain of ventricle mouse heart show arrhythmia termination with a transient time. During this time the core of the spiral wave is excited upon global illumination and leads to the dissolution of the core. During this process the excited region grows over time and pushes the arm of the spiral wave to the boundary, leading to its termination. In ex vivo studies on the intact optogenetic mouse heart, we have visualised arrhythmia termination with optical mapping technique. For a case of arrhythmia termination with a transient time, we observe an increase in the action potential during the last rotation of the spiral wave. This leads to a larger excitation of the heart surface so that the spiral wave cannot propagate further, resulting in its termination. In cardiac optogenetics, numerous studies have been carried out to control cardiac arrhythmias in murine hearts with different lighting patterns, such as structured [33, 34, 37] and global [23]. Uribe et al. applied global illumination to control cardiac arrhythmias in intact transgenic mouse hearts at different PLs and LIs [23]. They showed for some successful termination cases that there is a non-spontaneous termination at low LIs with a long PL of 1000 ms. Our numerical studies suggest that core dissolution may be the underlying mechanism leading to the slow termination that they have observed for the long illumination pulse with low-intensity. In another numerical study, we demonstrate that drift, a spatial translation of the spiral wave core along the illumination gradient, may underlie optogenetics defibrillation [30]. Dillon et al. provided optical mapping data showing the effect of electrical stimulation on the increase in action potential duration (APD) with simultaneous extension of the wavelength of the travelling wave [44]. With increase of APD, the excitable region for the spiral wave to propagate is reduced [24, 25] which may lead to a termination with transient time. Biasci et al. showed an incremental effect on APD by applying global sub-threshold illumination to a propagating planar wave in the domain [45]. All these studies show that the combination of different mechanisms can lead to the arrhythmia extinction at low-amplitude stimulation, or that occasionally one plays a dominant role over the others in causing termination of arrhythmias. In this work, the experimental studies were performed on intact mouse hearts, the geometry and structural features were completely different from the homogeneous 2D numerical system. Unlike 2D, in 3D ex vivo (or anatomical geometries for that matter), wave dynamics is determined not just by the area of excitable surface on the heart, but also by the depth of excitable tissue beneath the surface, which allows scroll filaments (a line of phase singularities in 3D) to resist removal by surface synchronization. Thus, arrhythmia termination occurs slowly in 3D, compared to 2D. This may explain why in experiments we see a smoother distribution for arrhythmia termination, than in 2D, where the dependence is rather abrupt. From a clinical perspective, it is important to understand the possible mechanisms of arrhythmia termination in thick cardiac tissue (e.g, from the left ventricular wall), which is 3D. Previous studies have shown that depending on the thickness of the tissue, a scroll filament can bend, twist, break up, grow and/or shrink inside the bulk of the tissue [46]. Presence of inhomogeneities inside the tissue further add to the complexity of the wave dynamics in that, they allow attachment/detachment of the filaments to/from their locations, as well as promote the formation of ‘seed waves’ which have the potential to regenerate a full scroll [47]. These dynamical behaviours can also occur under the influence of illumination on the surface of cardiac tissue during an arrhythmia. Depending on the true depth of penetration of the applied light, one can expect the tissue substrate to behave inhomogeneously. Thus, it would be interesting to observe how the proposed mechanism of slow termination of a scroll wave would take effect in such a scenario. We conjecture that the proposed mechanism of slow termination can then be seen to occur when the fibre orientation is mostly homogeneous, does not show sharp changes and light penetration is sufficiently deep. Conclusion Various studies have investigated and postulated different mechanisms underlying optogenetic defibrillation [23, 30, 48]. In this work, we investigate the mechanisms of arrhythmia termination during global illumination at very low supra-threshold light intensities with a long pulse length. Our in silico studies on two-dimensional domain of ventricle mouse heart show a slow termination of arrhythmia in which the core of the spiral wave is depolarized during global illumination, leading to its dissolution. During this process, which we refer to as core dissolution, the depolarized region grows over time and pushes the arm of the spiral wave to the boundary, leading to its termination. In ex vivo studies on the intact mouse heart, we have visualised arrhythmia termination with optical mapping technique. When arrhythmia termination is not instantaneous, we observe an increase in the action potential during the last rotation of the spiral wave. This leads to a larger depolarisation of the heart surface so that the spiral wave cannot propagate further, resulting in its termination. This work provides fundamental findings which could have implications for the improvement and development of new cardiac defibrillation techniques. Supporting information S1 Code. The source code used to obtain the numerical simulations reported in this manuscript. https://doi.org/10.1371/journal.pcbi.1011660.s001 (ZIP) S1 Fig. Kinetics of ChR2 and action potential of a single cell during sub- and supra-threshold illumination. The application of sub-threshold illumination (LIs of 10 and 20 μW/mm2) leads to the production of a small amount of inward IChR2 and elevation of the membrane voltage below the excitation threshold. The application of supra-threshold illumination at low LIs (25 and 30 μW/mm2) leads to the slow activation of the channel and in turn depolarization of the cell membrane. The application supra-threshold illumination with high LIs (100 and 1000 μW/mm2) results in the fast activation of the channel and depolarization of the membrane voltage to more positive values. https://doi.org/10.1371/journal.pcbi.1011660.s002 (EPS) S2 Fig. Calculation of core size. A) Shows a detected edge (white) using the Canny filter technique and a fitted circle (red) using the Hough transform method. B) illustrates the fitted circle on the snapshot of the spiral wave at a time step of 350 ms during illumination with a light intensity of 30 μW/mm2. https://doi.org/10.1371/journal.pcbi.1011660.s003 (EPS) S1 Video. In silico study of controlling a single spiral wave in a 2D domain by applying global illumination with LI of 30 μW/mm2. https://doi.org/10.1371/journal.pcbi.1011660.s004 (AVI) S2 Video. Ex vivo study of controlling a single spiral wave in an intact optogenetic mouse heart by applying global illumination with LI of 3.44 μW/mm2. https://doi.org/10.1371/journal.pcbi.1011660.s005 (MP4) Acknowledgments We thank all the members of biomedical physics group of Max Planck Institute for Dynamics and Self-Organization for their fruitful input. Our special thanks go to Marion Kunze, Tina Althaus, Andreas Barthel, and Laura Diaz for their technical support.

Zhao, Jichao;Sharma, Roshan;Kalyanasundaram, Anuradha;Kennelly, James;Bai, Jieyun;Li, Ning;Panfilov, Alexander;Fedorov, Vadim V.

doi: 10.1371/journal.pcbi.1011708pmid: 38109436

Introduction The sinoatrial node (SAN) is the primary pacemaker of the human heart, responsible for generating and efficiently regulating cardiac rhythm under various physiological conditions [1,2]. The human SAN is a single, “banana-shaped” 3D heterogeneous multicellular structure, composed of specialized pacemaker cells, adipose cells, immune cells, nerve fibers and importantly, ~35–50% dense connective tissue [3–7]. This structure is further compartmentalized into head, central and tail intranodal pacemakers characterized by heterogeneous ion channels and proteins that maintain pacemaking and conduction [8,9]. The distinctive fibrotic tissue in the human SAN together with fatty tissue, and low electrical coupling between pacemaker cells and atrial myocardium along the SAN border create electrical insulations of the intranodal pacemakers from the surrounding right atrial (RA) myocardium. This insulation may facilitate intranodal pacemaking and conduction as well as overcome the sink-source mismatch between the large RA myocardium (sink) and relatively small SAN pacemakers (source) as shown in the pioneer modelling study by Joyner and van Capelle [10]. Additionally, cardiac diseases including heart failure (HF) may lead to pathological molecular and structural (e.g., increased fibrosis) remodeling within the SAN pacemaker complex, resulting in SAN pacemaker and conduction dysfunction (SND) and reentrant arrhythmias [1,2,9,11]. The mechanisms of SAN function and dysfunction have been extensively studied in animal models [12,13]. These animal studies established a foundation of theories on heart rate regulation and possible mechanisms of SND or sick sinus syndrome but also highlighted large interspecies variations [14]. Recent ex-vivo studies reveal that the human SAN complex may be unique in both structural and electrophysiological aspects that limit the translational applications of animal studies to clinics [1,2,5]. Importantly, these studies revealed that the multiple SAN conduction pathways (SACPs) and intranodal pacemakers within the unique 3D SAN fibrotic structure may provide fail-safe mechanisms to ensure robust, uninterrupted SAN pacemaking and conduction [1,2,5,6]. Yet, due to the complexities of the human 3D structure and limited functional data on SAN conduction, the specific role of the SAN insulation/border, distinct SACPs and intranodal pacemaker molecular heterogeneity in the regulation of sinus rhythm in health and diseased hearts remains debatable. Thus, despite over a century of research on the SAN, limited knowledge of the relationship between human SAN function and the 3D structural-molecular microarchitecture of the human SAN pacemaker-conduction complex remains a critical barrier to properly understanding SAN dysfunction and arrhythmia mechanisms and developing new therapeutic approaches, e.g., biological pacemakers [15]. Computer models of cardiac electrical activation provide a powerful framework for understanding the structural and functional mechanisms of underlying variable physiological and pathological conditions, such as HF. Computer simulations also provide unique opportunities to test the role of each individual factor (e.g. region-specific fibrosis or pacemaker ion channel expression) in SAN pacemaking and conduction functions. These would be impossible to achieve in experimental or clinical studies [9,16]. However, currently existing computer models of the human SAN are either single pacemaker cell models [17] or 2D and 3D models, which do not incorporate human SAN-specific structural/functional/molecular data [18–20]. For these reasons, heart-specific computer models of the human SAN, incorporating molecular, structural and functional data from the same cohort of ex-vivo human hearts [1,5,6,8,9,21] may provide a powerful means to test novel hypotheses. The goal of this study was to define the key factors influencing human SAN pacemaking function and SAN dysfunction by developing and utilizing computer models of the human SAN complex. This novel SAN in silico model was based on data from our recent high-resolution near-infrared optical mapping, molecular, and detailed 3D histological imaging studies directly in the human heart ex-vivo [1,9]. Biophysics-based computer models of the human SAN were designed to simulate electrical pacemaking and conduction between SAN, SACPs, and RA based on realistic geometric loading and compartment-specific heterogeneity of molecular and ion channel expressions, as well as the impact of autonomic stimulation with adenosine, HF-induced remodeling and atrial pacing on SAN function. Methods Human SAN optical mapping and 3D reconstruction Near-infrared optical mapping data and histological imaging and reconstruction of human SAN used for the current SAN model was published previously in Li et al. 2017 and 2020[5,9] and described in S1 Text. Briefly, ex-vivo optical mapped donor human SAN preparations were histologically dissected for 3D structural reconstruction and analysis. 400 histology sections were imaged at a spatial resolution of 0.5×0.5 μm2 using a 20X digital slide scanner (Aperio ScanScope XT, Leica). The high-resolution histology images of the human SAN pacemaker complex were sequentially stacked, and artificial deformation across the z-axis was minimized using a novel 3D image alignment approach [5]. Subsequently, segmentation was performed on the stacks of Masson’s trichrome to separate the SAN from the neighbouring RA based on functional and structural data. Myocardial tissue was delineated from fat, blood vessels and fibrosis based on the colour intensity within the 3D SAN complex (Fig A in S1 Text). In addition, five SACPs (yellow color) were identified as 1–3 mm of myofibers with transitional cells in the SAN border that merged with RA myofibers [5]. High-resolution fiber fields were obtained using eigenanalysis of the structure tensor [16]. Human SAN computer model Based on the 3D reconstruction of the optically mapped human SAN complex, we developed a SAN-SACP-RA model to conduct computer simulations (Fig 1). The reconstructed 3D SAN-SACP-RA anatomical model had a size of 19.5x4.0x2.6 mm3 at an isotropic resolution of 40 μm3. The SAN computer model was obtained using a shadow of the 3D SAN model to the XY plane (parallel to epicardium) as shown in Fig B in S1 Text. As a result, the 2D representation of the entire 3D human SAN structure included all SACPs and the complete SAN head/center/tail, which is crucial for the aims of this study. In addition, the computer model used the myofiber field from histology data. Such model reproduced the geometry of electrical connections between the SAN and the neighbouring RA and was much more efficient to run than a computer model of the SAN directly based on the 3D histological data. We did not incorporate the SAN’s internal blood vessels into the model (as physical barriers) as they do not affect SAN and RA interaction. The insulating wall (at a uniform thickness of 3 pixels) between the SAN and RA was given a constant potential of -62.5 mV, which is the mean of the resting potentials of the RA and SAN cells, and a 0.001% diffusivity of the RA diffusivity [22–24]. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. The reconstruction of the human SAN anatomical and computer activation model. A representative human SAN computer model at an isotropic resolution of 40 μm3 was constructed using a shadow of the 3D immunehistological segmentation of the human SAN structure including SACPs [1] into a 2D plane. For this computer model, three different cellular kinetics models were developed for SAN center (1), head and tail (2), and neighboring RA (3). SAN–sinoatrial node, RA–right atrium, SACP–sino-atrial conduction pathway, CL–cycle length. https://doi.org/10.1371/journal.pcbi.1011708.g001 The cellular activation models for the human SAN center/head/tail and SACPs were adapted from the Fabbri et al. 2017 model [17], which is the most widely used human SAN model based on recent experimental data. The following modifications were made by considering SAN regional heterogeneity data from recent studies [1,2,9] (Fig 1). The ratios of INa, If and IK1 currents among the four SAN regions were listed in Table A in S1 Text, respectively [1]. The simulated SCLs for isolated SAN pacemakers in the SAN center and head/tail were 813 ms and 798 ms, respectively. The SACP cell models were not able to pace themselves. The baseline condition was considered to be without adenosine. In our computer model of the human SAN complex, the maximum concentration of acetylcholine (ACh), 60 nM, led to SAN arrest, which we referred to as 100% adenosine and a utilized dose of adenosine was represented as a percentage relative to this maximum value throughout this modeling study. To incorporate the effects of adenosine/ACh into the SAN cellular models, we utilized the same approach as described in the study by Fabbri et al.[17] for modeling the effects of ACh by modulating its concentration. The administration of ACh in the SAN activated the ACh-activated K+ current (IKACh), influencing If, ICaL and sarcoplasmic reticulum Ca2+ uptake. In addition, expression of the IKACh channel or A1 adenosine receptor (A1R) was modeled higher in the SAN center than in the SAN head/tail compartments based on data from human experimental studies [1]. Modelling of the relative expression of A1R in the SAN head/tail was achieved by changing the density (max conductance) of the IKACh current at the head and the tail of SAN. The RA cells were modeled by using the recently adapted human atrial Courtemanche et al. cell model [25]. In addition, the impact of adenosine/ACh on RA cells was modeled using the same formula by Grandi et al. [26] To simulate the electrical remodeling in the SAN complex under HF, we introduced If and INa current block by 20% in the SAN and SACPs and 5% INa block in the RA [9]. To simulate the impact of fibrosis in HF, we used 20% fibrosis in both SAN and SACP regions as we have done previously [9] (Table B in S1 Text). Electrical conduction among SAN pacemakers and RA cells was modeled using a mono-domain equation and solved using a parallelized finite difference approach. We used a spatial step of 0.04 mm and a temporal step of 0.0025 ms in our solver. A forward Euler method was used to solve the ordinary differential equations of cellular models. The electrical conduction attributable to intercellular electric coupling via gap junctions was simulated through the diffusion coefficient. In the model, we considered the regional differences in gap junctional coupling between the SAN center, SAN head/tail, SACPs and RA tissues by setting the diffusion coefficients at a ratio of 7:10:20:50 in these regions (Table C in S1 Text) [9]. In the models, an anisotropic diffusivity ratio of 1:10 was used as conducted in the past [9,16]. Human SAN optical mapping and 3D reconstruction Near-infrared optical mapping data and histological imaging and reconstruction of human SAN used for the current SAN model was published previously in Li et al. 2017 and 2020[5,9] and described in S1 Text. Briefly, ex-vivo optical mapped donor human SAN preparations were histologically dissected for 3D structural reconstruction and analysis. 400 histology sections were imaged at a spatial resolution of 0.5×0.5 μm2 using a 20X digital slide scanner (Aperio ScanScope XT, Leica). The high-resolution histology images of the human SAN pacemaker complex were sequentially stacked, and artificial deformation across the z-axis was minimized using a novel 3D image alignment approach [5]. Subsequently, segmentation was performed on the stacks of Masson’s trichrome to separate the SAN from the neighbouring RA based on functional and structural data. Myocardial tissue was delineated from fat, blood vessels and fibrosis based on the colour intensity within the 3D SAN complex (Fig A in S1 Text). In addition, five SACPs (yellow color) were identified as 1–3 mm of myofibers with transitional cells in the SAN border that merged with RA myofibers [5]. High-resolution fiber fields were obtained using eigenanalysis of the structure tensor [16]. Human SAN computer model Based on the 3D reconstruction of the optically mapped human SAN complex, we developed a SAN-SACP-RA model to conduct computer simulations (Fig 1). The reconstructed 3D SAN-SACP-RA anatomical model had a size of 19.5x4.0x2.6 mm3 at an isotropic resolution of 40 μm3. The SAN computer model was obtained using a shadow of the 3D SAN model to the XY plane (parallel to epicardium) as shown in Fig B in S1 Text. As a result, the 2D representation of the entire 3D human SAN structure included all SACPs and the complete SAN head/center/tail, which is crucial for the aims of this study. In addition, the computer model used the myofiber field from histology data. Such model reproduced the geometry of electrical connections between the SAN and the neighbouring RA and was much more efficient to run than a computer model of the SAN directly based on the 3D histological data. We did not incorporate the SAN’s internal blood vessels into the model (as physical barriers) as they do not affect SAN and RA interaction. The insulating wall (at a uniform thickness of 3 pixels) between the SAN and RA was given a constant potential of -62.5 mV, which is the mean of the resting potentials of the RA and SAN cells, and a 0.001% diffusivity of the RA diffusivity [22–24]. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. The reconstruction of the human SAN anatomical and computer activation model. A representative human SAN computer model at an isotropic resolution of 40 μm3 was constructed using a shadow of the 3D immunehistological segmentation of the human SAN structure including SACPs [1] into a 2D plane. For this computer model, three different cellular kinetics models were developed for SAN center (1), head and tail (2), and neighboring RA (3). SAN–sinoatrial node, RA–right atrium, SACP–sino-atrial conduction pathway, CL–cycle length. https://doi.org/10.1371/journal.pcbi.1011708.g001 The cellular activation models for the human SAN center/head/tail and SACPs were adapted from the Fabbri et al. 2017 model [17], which is the most widely used human SAN model based on recent experimental data. The following modifications were made by considering SAN regional heterogeneity data from recent studies [1,2,9] (Fig 1). The ratios of INa, If and IK1 currents among the four SAN regions were listed in Table A in S1 Text, respectively [1]. The simulated SCLs for isolated SAN pacemakers in the SAN center and head/tail were 813 ms and 798 ms, respectively. The SACP cell models were not able to pace themselves. The baseline condition was considered to be without adenosine. In our computer model of the human SAN complex, the maximum concentration of acetylcholine (ACh), 60 nM, led to SAN arrest, which we referred to as 100% adenosine and a utilized dose of adenosine was represented as a percentage relative to this maximum value throughout this modeling study. To incorporate the effects of adenosine/ACh into the SAN cellular models, we utilized the same approach as described in the study by Fabbri et al.[17] for modeling the effects of ACh by modulating its concentration. The administration of ACh in the SAN activated the ACh-activated K+ current (IKACh), influencing If, ICaL and sarcoplasmic reticulum Ca2+ uptake. In addition, expression of the IKACh channel or A1 adenosine receptor (A1R) was modeled higher in the SAN center than in the SAN head/tail compartments based on data from human experimental studies [1]. Modelling of the relative expression of A1R in the SAN head/tail was achieved by changing the density (max conductance) of the IKACh current at the head and the tail of SAN. The RA cells were modeled by using the recently adapted human atrial Courtemanche et al. cell model [25]. In addition, the impact of adenosine/ACh on RA cells was modeled using the same formula by Grandi et al. [26] To simulate the electrical remodeling in the SAN complex under HF, we introduced If and INa current block by 20% in the SAN and SACPs and 5% INa block in the RA [9]. To simulate the impact of fibrosis in HF, we used 20% fibrosis in both SAN and SACP regions as we have done previously [9] (Table B in S1 Text). Electrical conduction among SAN pacemakers and RA cells was modeled using a mono-domain equation and solved using a parallelized finite difference approach. We used a spatial step of 0.04 mm and a temporal step of 0.0025 ms in our solver. A forward Euler method was used to solve the ordinary differential equations of cellular models. The electrical conduction attributable to intercellular electric coupling via gap junctions was simulated through the diffusion coefficient. In the model, we considered the regional differences in gap junctional coupling between the SAN center, SAN head/tail, SACPs and RA tissues by setting the diffusion coefficients at a ratio of 7:10:20:50 in these regions (Table C in S1 Text) [9]. In the models, an anisotropic diffusivity ratio of 1:10 was used as conducted in the past [9,16]. Results SAN activation at baseline and with application of adenosine The developed control human SAN computer model reliably reproduced SAN rhythm or sinus cycle length (SCL), leading pacemaker location and electrical propagation pattern including preferential SACPs within the SAN complex at baseline and during adenosine administration, which were identical to these parameters recorded during our ex-vivo mapping experiment in the same ex-vivo human heart (Fig 2). Under baseline conditions in the computer model (Fig 2A left), the leading pacemaker was located in the SAN center (circle), and the earliest RA activation site was through the middle lateral SACP (magenta asterisk). The activation time within the SAN before exiting through SACP was ~75 ms (SAN conduction time—SACT). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. The computer activation model of the human SAN complex was validated using the optical mapping of the same human heart under both baseline and adenosine (Ado). A, The activation maps of the human SAN-RA in the computer model where the earliest pacemaker originated from the SAN center and propagated into the RA via the middle lateral SACP under baseline conditions. The introduction of Ado shifted the leading pacemaker to the SAN tail and changed the exit site to the superior lateral SACP as in ex-vivo optical mapping experiments. B, Optical mapping of the same human heart ex-vivo had qualitatively similar leading pacemaker sites and propagation patterns [1]. SAN–sino-atrial node, RA–right atrium, SACP–SAN conduction pathway, SACTsr–SAN conduction time during sinus rhythm, Ado–Adenosine, CT–crista terminals, IAS–interatrial septum, SVC–superior vena cava. https://doi.org/10.1371/journal.pcbi.1011708.g002 Qualitatively similar results were observed in the same human heart optically mapped ex-vivo at baseline conditions (Fig 2B left). In the presence of a higher concentration of adenosine (84% of the maximum dose) in the computer model, the leading pacemaker shifted to the SAN tail (Fig 2 right). In addition, the earliest RA activation site was activated through the superior lateral SACP. The SACT within the SAN was prolonged to 318 ms. An example of qualitatively similar results observed in the experiment in the presence of a high dose of adenosine is shown in Fig 2B right. In that case, under application of 10 μM of adenosine, we also observed the shift of the leading pacemaker to the tail, RA activation through the superior lateral SACP and SACT of 370 ms, which is comparable to the model (318 ms). The activation path of the electrical wave was the same in the model and in the experiment. Importantly, increasing the concentration of adenosine from 0% to 100% in the computer models (Figs 2A and 3A) led to progressive slowing of the SCL and SACT in parallel with a shift in the leading pacemaker and earliest atrial activation sites, until complete atrial arrest at 100% adenosine concentration. The simulation results show that the leading pacemaker shifted inferiorly from the SAN center to the SAN tail, while the earliest RA activation site first shifted to the inferior lateral SACP for adenosine concentration 50% (Fig 3A) and then to superior lateral SACP at adenosine concentration 84% (Fig 2A). Both SCL and SACTsr were increased in the computer model with an increasing dose of adenosine. Experimental results in the optically mapped human hearts ex-vivo (n = 11) at baseline and in the presence of low (10 μM) and high (100 μM) concentrations of adenosine are shown in Fig 3B. In functionally mapped explanted human SANs (n = 11), the increasing dose of adenosine led to a heart-specific pacemaker shift toward the head or tail of the SAN complex. An increase in concentrations of adenosine led to a higher chance of conducting via superior or inferior lateral SACPs as in the computer model. Also, 100 μM Ado led to cardiac or SAN arrest in five out of the 11 hearts similar to what we observed in our simulations. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. A shift in the leading pacemaker and earliest atrial activation sites in the human SAN model with varying Ado concentration. A, Increasing the presence of Ado (from 0% to 100%) in the computer model of the human SAN complex led to a shift in the leading pacemaker and earliest atrial activation sites, eventually exit block and complete atrial arrest (100% of Ado). B, A similar shift in the leading pacemaker and earliest atrial activation sites was observed in 11 optically mapped human hearts ex-vivo in the absence and presence of Ado [1]. The increasing dose of Ado led to a heart-specific pacemaker shift toward the head or tail of the SAN complex, and a higher chance of conducting via superior or inferior lateral SACPs as in computer model. Also, 100 μM Ado led to cardiac or SAN arrest in five out of the 11 hearts. SAN–sino-atrial node, SCL–sinus cycle length, SACP–sino-atrial conduction pathway, SACTsr–sinoatrial conduction time during sinus rhythm, Ado–Adenosine. https://doi.org/10.1371/journal.pcbi.1011708.g003 The heterogeneity in expression of A1 adenosine receptors or IKACh channel may explain pacemaker shifts The heterogeneity in expressions of A1R or IKACh channel in the human SAN ex-vivo was shown in our previous experimental studies [1] (Fig 4A). To understand the effect of this heterogeneity on SAN function, we implemented this heterogeneity in A1R[1] and hyperpolarization-activated cyclic nucleotide-gated channel subunits (HCN)[8] to our model with higher expression levels in the SAN center than its head/tail (Fig 4B). We performed a series of simulations of SAN activation patterns in which we increased the ratios of expressions of A1R (SAN head/tail to center) from 0.1 to 0.9 in the presence of 20% adenosine. We found that increasing heterogeneity resulted in gradual shift of the leading pacemaker from the SAN center to the tail (Figs 4C and C in S1 Text). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. The heterogeneity in expressions of A1R and regional source-sink ratios within the SAN complex explains leading pacemaker shifting. The higher expressions of A1R in the SAN center than that in the SAN head/tail was suggested in the human SAN ex-vivo [1] (A) and used in the computer model (B). C, Increasing expressions of A1R/IKACh channel in the SAN head/tail relative to the center (from 0.1 to 0.9) led to a shift in the leading pacemaker. D, Increasing the A1R expressions in the SAN tail (from 0.1 to 0.9) led to the leading pacemaker in the SAN head. E, Blocking two right SACPs (see black arrows) in the SAN head reduced electric sink/loading, shifting the leading pacemaker in the SAN head with 20% Ado. A1R –A1 Ado receptors. See other abbreviations in Fig 2. https://doi.org/10.1371/journal.pcbi.1011708.g004 We also modelled the superior-inferior gradient by changing the A1R expressions in SAN head vs tail (Fig 4D and 4E). We performed simulations in the presence of 20% adenosine where the expression of A1R in the tail was change from 0.1 to 0.9, and expression in the head and center were constant at 0.1 and 1.0, respectively (Fig D in S1 Text). Fig 4D shows two different scenarios of A1R expressions in SAN head vs tail: when A1R expressions is higher in the SAN tail vs head (0.9:0.1), the application of adenosine slowed SAN automaticity and shifted the leading pacemaker from SAN center (baseline condition) to head (superior). The pacemaker shift had opposite directionality (from center to tail) when SAN head and tail have the same A1R expression (0.1:0.1), despite of similar automaticity slowing (SCL from 930ms to 1229 ms vs 1134ms). We documented both these scenarios in ex-vivo human donor hearts studied with near-infrared transmural optical mapping as it shown in Fig 3B [1]. There is no leading pacemaker shift with homogenous A1R expressions at the SAN center, head and tail (Fig E in S1 Text). Thus in human hearts, the heterogeneity in the expression of A1R within the SAN pacemaker compartments (center, head and tail) could explain the shift of the leading pacemaker and earliest atrial activation sites under adenosine conditions. Interestingly, under similar conditions of adenosine, almost all leading pacemakers were always in the SAN tail/center and never in the SAN head. We hypothesized that it was due to the regional source-sink relationship within the SAN complex. The SAN head in this specific human heart had fewer SAN cells (electrical sources) and more SACPs (electrical loading) than the tail (three SACPs versus two SACPs), which made excitation more difficult. To test this hypothesis, we performed additional simulations in which we artificially blocked two lateral SACPs in the SAN head (see the two black arrows in Fig 4E right), thereby reducing the electrotonic load. As a result, the leading pacemaker shifted from the tail to the head in the presence of 20% adenosine. The characteristics of SACPs dictate the earliest atrial activation sites One of the results of pathological remodelling of cardiac tissue is a change in INa in the atrial myocardium [9]. We studied how the change in INa affects the functioning of the SAN complex. We demonstrated that INa in SACPs is one of the main determinants of the earliest atrial activation sites (Fig 5A). Varying the density of the INa current in the SACPs alone from its original value to 85% or 115% led to the slowing or acceleration of SACT, and the earliest atrial activation shift from the middle lateral SACP to the superior and inferior lateral SACPs or the middle lateral and the left inferior SACPs, respectively. However, the SCL and leading pacemaker remained unchanged (Fig 5A). Both INa and If currents within the SAN complex were inversely associated with SCL. However, it appeared that INa and If currents in the SAN did not influence the earliest atrial activation or the leading pacemaker sites (Fig 5B). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Modulation of INa current in SACPs indicates the earliest atrial activation sites, while If current only influences SCL. A, Varying the INa current in SACPs alone heavily influenced the earliest atrial activation sites. B, Varying the If current in SAN and SACPs did not change the earliest atrial activation sites. Both INa and If currents were positively associated with the heart rate. See abbreviations in Fig 2. https://doi.org/10.1371/journal.pcbi.1011708.g005 The role of the insulation boundary between the SAN and RA The existence of an insulation boundary between the SAN and the RA septum is widely observed and accepted. However, the existence of the insulation boundary between the SAN and the lateral RA is controversial. To illustrate the necessity and potential role of the insulation boundary between the SAN and the lateral RA, we performed computer simulations with and without the insulation boundary (Fig 6A). In the case of no insulation boundary, we considered situations with different diffusivities within the SAN complex (the diffusion coefficient is 100%, 50% and 25% of its normal value). Reducing diffusivities within the SAN complex led to increased heart rate and prolonged SACT within the SAN. We found that computer models with 100% and 50% diffusivities produced realistic activation time between 150–200 msec. Using computer models without the insulation boundary between the SAN and the lateral RA, we observed only a mild shift of the leading pacemaker and the earliest atrial activation sites with increasing adenosine (Fig 6B and 6C), which is not consistent with that commonly seen during experimental studies of animal and human hearts [1,2,9]. Furthermore, even 10% Ado was sufficient to induce SAN arrest (Fig 6B), which is 10 times lower than that required to induce SAN arrest with an insulation border (Fig 3A). Thus our simulations suggest that the insulation boundary between the SAN and the lateral RA is necessary for normal functioning of SAN and support findings from experimental studies with adenosine. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. The human SAN model without the insulation boundary between the SAN and the lateral RA (right) had difficulty in replicating dramatic shift in the leading pacemaker and earliest atrial activation sites. A, Activation patterns and earliest atrial activation sites in the computer model of the human SAN complex, and in the computer models of the SAN without the crista terminalis (CT) side insulation boundary at different conduction diffusivities. B and C, Activation maps in the computer models of the SAN without the insulation boundary at 100% conduction diffusivity and at 50% conduction diffusivity at the baseline and with increasing the concentrations of Ado up to 10% (B) and 50% (C) which led to SAN arrests, respectively. See other abbreviations in Fig 2. https://doi.org/10.1371/journal.pcbi.1011708.g006 The impact of HF-induced remodeling on SAN pacemaking and conduction We have also studied the effects of ion channel remodeling and fibrosis due to HF on SAN complex function. We implemented these changes to our model as shown in (Fig 7A). This figure also shows that electrical remodeling in the SAN cellular kinetics models led to depression of SAN pacemaking and increased SCL. In our computer simulations, SAN with HF conditions led to exit block even without the presence of adenosine (Fig 7B). When we reversed the fibrotic remodeling only in the SACPs (Fig 7C), we observed electrical activation and propagation in the RA at the baseline and in the presence of up to 50% of adenosine. We also found a similar trend in shifting leading pacemakers and earliest atrial activation sites. At 85% of adenosine, electrical remodeling without fibrosis in SACP led to exit block (Fig 7C right). We also found that the fibrotic remodeling in the SACPs alone (20% of fibrosis) without electrical remodeling, produced exit block similar to that under HF remodeling. On the other hand, in the computer model with HF ion channel remodeling only (Fig 7D), we again observed electrical activation and propagation in the RA from the baseline to the presence of 50% adenosine. At 85% adenosine, it led to SAN arrest. Our simulation results indicated that HF ionic channel remodeling influenced the SCL, earliest atrial activation sites and increased chances of SAN arrest, while fibrotic remodeling in SACPs increased the chance of SAN exit block. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. Modeling of diseased induced fibrotic and ion channel remodeling in SAN and SACPs at the baseline and with Ado. A, SAN and RA cellular models were further adapted by varying If and INa currents and 20% fibrosis were added to the model to simulate HF conditions. HF-induced electrical remodeling slowed the heart rate. B, In our computer simulations, SAN with HF conditions led to exit block even without the presence of Ado. C, When we reversed the fibrotic remodeling in the SACPs only, we observed electrical activation and propagation in the RA from the baseline and in the presence of up to 50% of Ado. D, With HF ionic channel remodeling only, electrical activation and propagation in the RA were observed from the baseline to the presence of 50% Ado. While 85% Ado led to SAN arrest. HF–heart failure, SCL–sinus cycle length, see other abbreviations in Fig 2. https://doi.org/10.1371/journal.pcbi.1011708.g007 SAN is prone to arrhythmia and exit block under INa channel block, adenosine and HF Finally, we evaluated the human SAN function after RA pacing (Figs 8 and F in S1 Text) and without any RA pacing (Fig G in S1 Text). In the computer model of the human SAN complex, a train of stimuli at a pacing cycle length of 500 ms was delivered from the right superior RA, and a typical activation pattern is shown in Fig 8A. Under baseline conditions, once the RA stimuli were terminated, the SAN recovered to its normal automaticity and function immediately. We performed systematic simulations in which we varied the degree of the INa channel block and concentration of adenosine. We found that depending on these parameters we can observe the following situations: a shift in the leading pacemaker and the earliest atrial activation sites, SAN-RA reentry, SAN exit block and arrest (Fig 8B). Typical propagation patterns for each of the cases are shown in Figs 8B and F in S1 Text. RA pacing led to both localized and macro-reentries. In contrast, we only observed macro reentries in the control SAN model with increasing INa block or adenosine without extrastimuli (Fig G in S1 Text). The SAN HF model produced SAN exit block post-RA pacing regardless of the presence or absence of INa block or adenosine. Interestingly, the SAN HF model without fibrotic remodeling in the SACPs had a higher tendency of exit block with increasing INa block or adenosine (Fig 8D). Thus we see that the application of adenosine in combination with INa channel block leads to cardiac arrhythmias and other serious pathologies. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 8. Atrial pacing induces SAN nodal reentrant arrhythmia during mild INa suppression and Adenosine (Ado), and SAN conduction block, particularly in HF models. A, Atrial pacing protocol at a frequency of 2 Hz for SAN function from the RA. The pacing site and SAN activation maps during RA pacing are shown. B, Top: Summary of propagation patterns post RA pacing in control SAN model with increasing INa block or Ado concentration. Here ’Normal’ (blue squares) refers to the leading pacemaker in the SAN center and preferential SAN exit through the middle lateral SACP. Bottom: Examples of post-RA pacing activation patterns in the SAN complex are shown. C, Summary of propagation patterns in the HF SAN model without fibrotic remodeling in the SACPs in the presence of increasing INa block or Ado. See abbreviations in Fig 2. https://doi.org/10.1371/journal.pcbi.1011708.g008 SAN activation at baseline and with application of adenosine The developed control human SAN computer model reliably reproduced SAN rhythm or sinus cycle length (SCL), leading pacemaker location and electrical propagation pattern including preferential SACPs within the SAN complex at baseline and during adenosine administration, which were identical to these parameters recorded during our ex-vivo mapping experiment in the same ex-vivo human heart (Fig 2). Under baseline conditions in the computer model (Fig 2A left), the leading pacemaker was located in the SAN center (circle), and the earliest RA activation site was through the middle lateral SACP (magenta asterisk). The activation time within the SAN before exiting through SACP was ~75 ms (SAN conduction time—SACT). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. The computer activation model of the human SAN complex was validated using the optical mapping of the same human heart under both baseline and adenosine (Ado). A, The activation maps of the human SAN-RA in the computer model where the earliest pacemaker originated from the SAN center and propagated into the RA via the middle lateral SACP under baseline conditions. The introduction of Ado shifted the leading pacemaker to the SAN tail and changed the exit site to the superior lateral SACP as in ex-vivo optical mapping experiments. B, Optical mapping of the same human heart ex-vivo had qualitatively similar leading pacemaker sites and propagation patterns [1]. SAN–sino-atrial node, RA–right atrium, SACP–SAN conduction pathway, SACTsr–SAN conduction time during sinus rhythm, Ado–Adenosine, CT–crista terminals, IAS–interatrial septum, SVC–superior vena cava. https://doi.org/10.1371/journal.pcbi.1011708.g002 Qualitatively similar results were observed in the same human heart optically mapped ex-vivo at baseline conditions (Fig 2B left). In the presence of a higher concentration of adenosine (84% of the maximum dose) in the computer model, the leading pacemaker shifted to the SAN tail (Fig 2 right). In addition, the earliest RA activation site was activated through the superior lateral SACP. The SACT within the SAN was prolonged to 318 ms. An example of qualitatively similar results observed in the experiment in the presence of a high dose of adenosine is shown in Fig 2B right. In that case, under application of 10 μM of adenosine, we also observed the shift of the leading pacemaker to the tail, RA activation through the superior lateral SACP and SACT of 370 ms, which is comparable to the model (318 ms). The activation path of the electrical wave was the same in the model and in the experiment. Importantly, increasing the concentration of adenosine from 0% to 100% in the computer models (Figs 2A and 3A) led to progressive slowing of the SCL and SACT in parallel with a shift in the leading pacemaker and earliest atrial activation sites, until complete atrial arrest at 100% adenosine concentration. The simulation results show that the leading pacemaker shifted inferiorly from the SAN center to the SAN tail, while the earliest RA activation site first shifted to the inferior lateral SACP for adenosine concentration 50% (Fig 3A) and then to superior lateral SACP at adenosine concentration 84% (Fig 2A). Both SCL and SACTsr were increased in the computer model with an increasing dose of adenosine. Experimental results in the optically mapped human hearts ex-vivo (n = 11) at baseline and in the presence of low (10 μM) and high (100 μM) concentrations of adenosine are shown in Fig 3B. In functionally mapped explanted human SANs (n = 11), the increasing dose of adenosine led to a heart-specific pacemaker shift toward the head or tail of the SAN complex. An increase in concentrations of adenosine led to a higher chance of conducting via superior or inferior lateral SACPs as in the computer model. Also, 100 μM Ado led to cardiac or SAN arrest in five out of the 11 hearts similar to what we observed in our simulations. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. A shift in the leading pacemaker and earliest atrial activation sites in the human SAN model with varying Ado concentration. A, Increasing the presence of Ado (from 0% to 100%) in the computer model of the human SAN complex led to a shift in the leading pacemaker and earliest atrial activation sites, eventually exit block and complete atrial arrest (100% of Ado). B, A similar shift in the leading pacemaker and earliest atrial activation sites was observed in 11 optically mapped human hearts ex-vivo in the absence and presence of Ado [1]. The increasing dose of Ado led to a heart-specific pacemaker shift toward the head or tail of the SAN complex, and a higher chance of conducting via superior or inferior lateral SACPs as in computer model. Also, 100 μM Ado led to cardiac or SAN arrest in five out of the 11 hearts. SAN–sino-atrial node, SCL–sinus cycle length, SACP–sino-atrial conduction pathway, SACTsr–sinoatrial conduction time during sinus rhythm, Ado–Adenosine. https://doi.org/10.1371/journal.pcbi.1011708.g003 The heterogeneity in expression of A1 adenosine receptors or IKACh channel may explain pacemaker shifts The heterogeneity in expressions of A1R or IKACh channel in the human SAN ex-vivo was shown in our previous experimental studies [1] (Fig 4A). To understand the effect of this heterogeneity on SAN function, we implemented this heterogeneity in A1R[1] and hyperpolarization-activated cyclic nucleotide-gated channel subunits (HCN)[8] to our model with higher expression levels in the SAN center than its head/tail (Fig 4B). We performed a series of simulations of SAN activation patterns in which we increased the ratios of expressions of A1R (SAN head/tail to center) from 0.1 to 0.9 in the presence of 20% adenosine. We found that increasing heterogeneity resulted in gradual shift of the leading pacemaker from the SAN center to the tail (Figs 4C and C in S1 Text). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. The heterogeneity in expressions of A1R and regional source-sink ratios within the SAN complex explains leading pacemaker shifting. The higher expressions of A1R in the SAN center than that in the SAN head/tail was suggested in the human SAN ex-vivo [1] (A) and used in the computer model (B). C, Increasing expressions of A1R/IKACh channel in the SAN head/tail relative to the center (from 0.1 to 0.9) led to a shift in the leading pacemaker. D, Increasing the A1R expressions in the SAN tail (from 0.1 to 0.9) led to the leading pacemaker in the SAN head. E, Blocking two right SACPs (see black arrows) in the SAN head reduced electric sink/loading, shifting the leading pacemaker in the SAN head with 20% Ado. A1R –A1 Ado receptors. See other abbreviations in Fig 2. https://doi.org/10.1371/journal.pcbi.1011708.g004 We also modelled the superior-inferior gradient by changing the A1R expressions in SAN head vs tail (Fig 4D and 4E). We performed simulations in the presence of 20% adenosine where the expression of A1R in the tail was change from 0.1 to 0.9, and expression in the head and center were constant at 0.1 and 1.0, respectively (Fig D in S1 Text). Fig 4D shows two different scenarios of A1R expressions in SAN head vs tail: when A1R expressions is higher in the SAN tail vs head (0.9:0.1), the application of adenosine slowed SAN automaticity and shifted the leading pacemaker from SAN center (baseline condition) to head (superior). The pacemaker shift had opposite directionality (from center to tail) when SAN head and tail have the same A1R expression (0.1:0.1), despite of similar automaticity slowing (SCL from 930ms to 1229 ms vs 1134ms). We documented both these scenarios in ex-vivo human donor hearts studied with near-infrared transmural optical mapping as it shown in Fig 3B [1]. There is no leading pacemaker shift with homogenous A1R expressions at the SAN center, head and tail (Fig E in S1 Text). Thus in human hearts, the heterogeneity in the expression of A1R within the SAN pacemaker compartments (center, head and tail) could explain the shift of the leading pacemaker and earliest atrial activation sites under adenosine conditions. Interestingly, under similar conditions of adenosine, almost all leading pacemakers were always in the SAN tail/center and never in the SAN head. We hypothesized that it was due to the regional source-sink relationship within the SAN complex. The SAN head in this specific human heart had fewer SAN cells (electrical sources) and more SACPs (electrical loading) than the tail (three SACPs versus two SACPs), which made excitation more difficult. To test this hypothesis, we performed additional simulations in which we artificially blocked two lateral SACPs in the SAN head (see the two black arrows in Fig 4E right), thereby reducing the electrotonic load. As a result, the leading pacemaker shifted from the tail to the head in the presence of 20% adenosine. The characteristics of SACPs dictate the earliest atrial activation sites One of the results of pathological remodelling of cardiac tissue is a change in INa in the atrial myocardium [9]. We studied how the change in INa affects the functioning of the SAN complex. We demonstrated that INa in SACPs is one of the main determinants of the earliest atrial activation sites (Fig 5A). Varying the density of the INa current in the SACPs alone from its original value to 85% or 115% led to the slowing or acceleration of SACT, and the earliest atrial activation shift from the middle lateral SACP to the superior and inferior lateral SACPs or the middle lateral and the left inferior SACPs, respectively. However, the SCL and leading pacemaker remained unchanged (Fig 5A). Both INa and If currents within the SAN complex were inversely associated with SCL. However, it appeared that INa and If currents in the SAN did not influence the earliest atrial activation or the leading pacemaker sites (Fig 5B). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Modulation of INa current in SACPs indicates the earliest atrial activation sites, while If current only influences SCL. A, Varying the INa current in SACPs alone heavily influenced the earliest atrial activation sites. B, Varying the If current in SAN and SACPs did not change the earliest atrial activation sites. Both INa and If currents were positively associated with the heart rate. See abbreviations in Fig 2. https://doi.org/10.1371/journal.pcbi.1011708.g005 The role of the insulation boundary between the SAN and RA The existence of an insulation boundary between the SAN and the RA septum is widely observed and accepted. However, the existence of the insulation boundary between the SAN and the lateral RA is controversial. To illustrate the necessity and potential role of the insulation boundary between the SAN and the lateral RA, we performed computer simulations with and without the insulation boundary (Fig 6A). In the case of no insulation boundary, we considered situations with different diffusivities within the SAN complex (the diffusion coefficient is 100%, 50% and 25% of its normal value). Reducing diffusivities within the SAN complex led to increased heart rate and prolonged SACT within the SAN. We found that computer models with 100% and 50% diffusivities produced realistic activation time between 150–200 msec. Using computer models without the insulation boundary between the SAN and the lateral RA, we observed only a mild shift of the leading pacemaker and the earliest atrial activation sites with increasing adenosine (Fig 6B and 6C), which is not consistent with that commonly seen during experimental studies of animal and human hearts [1,2,9]. Furthermore, even 10% Ado was sufficient to induce SAN arrest (Fig 6B), which is 10 times lower than that required to induce SAN arrest with an insulation border (Fig 3A). Thus our simulations suggest that the insulation boundary between the SAN and the lateral RA is necessary for normal functioning of SAN and support findings from experimental studies with adenosine. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. The human SAN model without the insulation boundary between the SAN and the lateral RA (right) had difficulty in replicating dramatic shift in the leading pacemaker and earliest atrial activation sites. A, Activation patterns and earliest atrial activation sites in the computer model of the human SAN complex, and in the computer models of the SAN without the crista terminalis (CT) side insulation boundary at different conduction diffusivities. B and C, Activation maps in the computer models of the SAN without the insulation boundary at 100% conduction diffusivity and at 50% conduction diffusivity at the baseline and with increasing the concentrations of Ado up to 10% (B) and 50% (C) which led to SAN arrests, respectively. See other abbreviations in Fig 2. https://doi.org/10.1371/journal.pcbi.1011708.g006 The impact of HF-induced remodeling on SAN pacemaking and conduction We have also studied the effects of ion channel remodeling and fibrosis due to HF on SAN complex function. We implemented these changes to our model as shown in (Fig 7A). This figure also shows that electrical remodeling in the SAN cellular kinetics models led to depression of SAN pacemaking and increased SCL. In our computer simulations, SAN with HF conditions led to exit block even without the presence of adenosine (Fig 7B). When we reversed the fibrotic remodeling only in the SACPs (Fig 7C), we observed electrical activation and propagation in the RA at the baseline and in the presence of up to 50% of adenosine. We also found a similar trend in shifting leading pacemakers and earliest atrial activation sites. At 85% of adenosine, electrical remodeling without fibrosis in SACP led to exit block (Fig 7C right). We also found that the fibrotic remodeling in the SACPs alone (20% of fibrosis) without electrical remodeling, produced exit block similar to that under HF remodeling. On the other hand, in the computer model with HF ion channel remodeling only (Fig 7D), we again observed electrical activation and propagation in the RA from the baseline to the presence of 50% adenosine. At 85% adenosine, it led to SAN arrest. Our simulation results indicated that HF ionic channel remodeling influenced the SCL, earliest atrial activation sites and increased chances of SAN arrest, while fibrotic remodeling in SACPs increased the chance of SAN exit block. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. Modeling of diseased induced fibrotic and ion channel remodeling in SAN and SACPs at the baseline and with Ado. A, SAN and RA cellular models were further adapted by varying If and INa currents and 20% fibrosis were added to the model to simulate HF conditions. HF-induced electrical remodeling slowed the heart rate. B, In our computer simulations, SAN with HF conditions led to exit block even without the presence of Ado. C, When we reversed the fibrotic remodeling in the SACPs only, we observed electrical activation and propagation in the RA from the baseline and in the presence of up to 50% of Ado. D, With HF ionic channel remodeling only, electrical activation and propagation in the RA were observed from the baseline to the presence of 50% Ado. While 85% Ado led to SAN arrest. HF–heart failure, SCL–sinus cycle length, see other abbreviations in Fig 2. https://doi.org/10.1371/journal.pcbi.1011708.g007 SAN is prone to arrhythmia and exit block under INa channel block, adenosine and HF Finally, we evaluated the human SAN function after RA pacing (Figs 8 and F in S1 Text) and without any RA pacing (Fig G in S1 Text). In the computer model of the human SAN complex, a train of stimuli at a pacing cycle length of 500 ms was delivered from the right superior RA, and a typical activation pattern is shown in Fig 8A. Under baseline conditions, once the RA stimuli were terminated, the SAN recovered to its normal automaticity and function immediately. We performed systematic simulations in which we varied the degree of the INa channel block and concentration of adenosine. We found that depending on these parameters we can observe the following situations: a shift in the leading pacemaker and the earliest atrial activation sites, SAN-RA reentry, SAN exit block and arrest (Fig 8B). Typical propagation patterns for each of the cases are shown in Figs 8B and F in S1 Text. RA pacing led to both localized and macro-reentries. In contrast, we only observed macro reentries in the control SAN model with increasing INa block or adenosine without extrastimuli (Fig G in S1 Text). The SAN HF model produced SAN exit block post-RA pacing regardless of the presence or absence of INa block or adenosine. Interestingly, the SAN HF model without fibrotic remodeling in the SACPs had a higher tendency of exit block with increasing INa block or adenosine (Fig 8D). Thus we see that the application of adenosine in combination with INa channel block leads to cardiac arrhythmias and other serious pathologies. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 8. Atrial pacing induces SAN nodal reentrant arrhythmia during mild INa suppression and Adenosine (Ado), and SAN conduction block, particularly in HF models. A, Atrial pacing protocol at a frequency of 2 Hz for SAN function from the RA. The pacing site and SAN activation maps during RA pacing are shown. B, Top: Summary of propagation patterns post RA pacing in control SAN model with increasing INa block or Ado concentration. Here ’Normal’ (blue squares) refers to the leading pacemaker in the SAN center and preferential SAN exit through the middle lateral SACP. Bottom: Examples of post-RA pacing activation patterns in the SAN complex are shown. C, Summary of propagation patterns in the HF SAN model without fibrotic remodeling in the SACPs in the presence of increasing INa block or Ado. See abbreviations in Fig 2. https://doi.org/10.1371/journal.pcbi.1011708.g008 Discussion This study presents the first comprehensive biophysical computer model of the human SAN complex based on direct molecular, structural and functional studies in the ex-vivo human heart. Our data show that the model can closely replicate pacemaking, SAN activation patterns and exit sites /earliest atrial activation through preferential SACP, as well as physiological SACT and SCL changes including the shift of the leading pacemaker in the presence of adenosine reported in the human heart ex-vivo [1,9]. Importantly, the novel modeling simulation results provide mechanistic insight into the crucial role of the structural and electrical heterogeneity of the human SAN in the pacemaking and conduction function. More specifically: The heterogeneity in the expression of adenosine A1 receptors (A1R) or the IKACh channels within the human SAN pacemaker compartments explains leading pacemaker and preferential SACP shifts in the presence of adenosine. The electrical insulation boundary between the SAN and RA except the SACP is required for normal SAN pacemaker and conduction function and to reproduce the leading pacemaker and the earliest atrial activation sites, observed in experimental and clinical studies. Importantly, the insulation prevents the high sensitivity of SAN pacemaking to adenosine, including complete SAN arrest seen at lower doses of adenosine in the models without the insulation boundary. The INa current density and fibrotic remodelling (e.g. in heart failure HF) in SACPs modulate the SAN conduction (e.g. exit block) and the preferential SACP/exits to the atria (e.g. earliest atrial activation). Intranodal INa current suppression or low-dose adenosine intervention leads to shifts in the leading pacemaker and the earliest atrial activation sites and renders the human SAN pacemaker-conduction complex vulnerable to SAN-RA reentry, SAN exit block and arrest. The SAN HF model had a higher incidence of exit block regardless of the presence or absence of INa block or adenosine. Structural and electrical heterogeneity of the SAN Since the discovery of the SAN by Keith and Flack [3] more than one century ago, significant strides in our understanding of the SAN complex have been made [5,6,12,27]. It is widely accepted that the heterogeneous distribution of specialized ion channels, intracellular Na+/Ca2+ handling proteins, gap junction channels and receptors within the human SAN complex are some of the few critical players in SAN pacemaking [5]. In particular, expressions of ion channels (e.g. Na+, Ca2+ and HCN1/4), and gap junction proteins are heterogeneous within the SAN pacemaker complex [27,28]. Furthermore, in contrast to other cardiac regions, the human SAN has more extensive fibrosis, which is further upregulated in SAN dysfunction or HF [6]. It is also well established that the existence of a septal area of conduction block, known as the block zone, prevents the spread of the electric impulse to the interatrial septum directly from the SAN. The precise mechanism of atrial activation by the leading pacemaker remains a controversy. In the past, the gradient model, in which there is a gradual change in the intrinsic properties of pacemaker cells from the center to the periphery of the SAN, was proposed to explain how SAN could effectively pace the atria [12] The intrinsic pacemaker activity is greater in cells from the periphery than from the centre of the SAN [29–33]. However, at the tissue level, the periphery of the SAN is connected to a large mass of atrial muscle in the crista terminalis (CT) through gap junctions resulting in the inhibition of peripheral pacemaker activity by the electrotonic influence of the highly hyperpolarized atrial muscle. On the other hand, central pacemaker cells, which are more distal from the atria, are less affected by atrial electrotonic effects. Therefore, leading pacemaker activity at baseline conditions always originates in the central SAN cells, although they are intrinsically slower than the peripheral pacemaker cells. These results were primarily based on small animals studies, including rabbit models, which show both lateral (towards CT) and superior-inferior gradient of intrinsic SAN pacemaker properties. However, in large animal models including canine and human SAN, the superior-inferior gradient is more prominent while lateral gradient is evident only across the SACPs, due to the larger CT myocardium requiring more lateral insulation [1,5,6,9]. Earlier Crick et al. [34] reported twice higher density of parasympathetic and sympathetic nerves fibers in the SAN center vs periphery (tail). The innervation gradient may explain why sympathetic activation can shift atrial exits to superior SACP and parasympathetic activation can slow sinus rhythm and shift atrial exits inferiorly [35]. However, in both human and canine SAN, the direction of intranodal pacemaker shift from center to head or tail does not always correlate with the closest superior or inferior SACP [1,36]. SAN activation can exit via the superior SACPs even though the leading pacemaker shifted inferiorly to the tail (Figs 2B and 3B). These studies suggest that SAN automaticity gradient and superior or inferior intranodal pacemaker shift (from center to head or center to tail) depend on the heart-specific SAN compartment molecular profiles. However, no one yet measure and compare intrinsic frequency of pacemaker cells isolated from different SAN pacemaker compartments (head, center and tail). Instead, we included the electrophysiological and molecular difference between the SAN center and head/tail (periphery), as well as transitional cells in preferential SACP for electrical coupling from SAN to RA. In keeping with the classical gradient model, our modelling results showed that isolated SAN pacemakers in the SAN center were slower than the SAN head/tail, which is consistent with most experimental results [29–33]. The most significant controversy surrounds the location and nature of the SAN in the human heart. Dobrzynski and her colleagues found an intermediate region with an expression of many ion channels between the SAN and RA in the human heart, which is similar to that seen in the periphery of the rabbit SAN [27]. They speculated that this region, termed the paranodal area, might contribute to pacemaker shift though they did not conduct any electrophysiological or optical mapping studies to support the claim. Using high-resolution optical mapping and histological studies in the human heart ex-vivo, we have shown that the human SAN complex is a 3D, specialized multi-compartment structure (head, center and tail) with higher expression of HCN and A1R proteins, lower gap junctional coupling and Nav1.5 in the SAN center than in the SAN head and tail [1,9]. Our comprehensive biophysical computer model of the human SAN pacemaker conduction complex was developed based on the functional and structural mapping in the human heart to investigate the role of human SAN structure and ion channels heterogeneity in SAN pacemaking and conduction functions. The novelty of our computer model of the human SAN Computer models provide a powerful tool for the quantitative examination of structural and electrical substrates and their individual contributions to cardiac arrhythmia mechanisms [9,16]. However, much less literature exists for multi-scale human SAN modeling, mainly due to the structural and functional complexity of the human SAN complex. The first computer modeling of the SAN by Joyner and Capelle [10] demonstrated that the electrical uncoupling of the pacemaker cells might be an essential design feature of a healthy SAN complex using a two-dimensional sheet model. Similar approaches were expanded to study the importance of gradients in membrane properties (e.g., the ionic current density of the INa) and electrical coupling in 2D [18] and 3D [19] models, and the mechanisms by which the SCN5A mutations (Na+ channel) impair cardiac pacemaking [37]. Earlier 3D computer models of the human SAN complex were developed primarily for studying the electrical conduction within the RA [4,28]. The more recent 3D computer model by Kharche et al. [20] was the first to study the role of the insulating border between the SAN and RA septum, and the paranodal area in the SAN function. However, they only used the simple three-current Fenton-Karma cellular models to simulate the human SAN and RA cell kinetics. In addition, SACPs in their model were not anatomically based, which led to extremely short un-physiological SAN conduction. To date, there is no comprehensive SAN model that integrated realistic anatomical structures and ion channel expressions from direct studies of the human SAN complex. As such, no computer model could successfully reproduce all functional observations at different conditions in normal and diseased human SAN, including the location of leading pacemakers, preferential SACP exits, physiological SAN conduction time (70–80 ms at baseline) and SCL (700–900 ms). Our study takes the next step in defining the key factors influencing human SAN pacemaking function and SAN dysfunction by developing and utilizing computer models of the human SAN based on the current knowledge of electrical and structural heterogeneity. The unique strength of our SAN computer model is that it was based on published data from high-resolution near-infrared optical mapping, molecular mapping and detailed 3D histological analyses of the human SAN complex ex-vivo [1,5,8,9]. More importantly, it included anatomically based SACPs directly identified from the 3D immunohistological analyses which provide realistic SAN to RA electric loading. In addition, we have adapted the most widely used human SAN and atrial cellular activation models, i.e., the Fabbri et al. [17] and Courtemanche et al. cell model [25]. Critical functional insights from the human SAN model Firstly, we have illustrated the role of structural and electrical heterogeneity in the shift of the leading pacemaker and the earliest atrial activation sites. Under baseline conditions, the leading pacemaker is more likely located in the SAN center due to less electrical loading (the relatively larger SAN center region with only one SACP, in contrast with the SAN head and tail), despite the fact that single isolated SAN head or tail cells have a shorter SCL than that in the center (798 ms versus 814ms). The difference in cellular SCL is caused by the electrical heterogeneity within the SAN complex, particularly, the higher If and lower INa currents in the SAN center, as widely reported. That is to say that the effect of the electrical heterogeneity is suppressed by the SAN structure (SAN-RA isolation layer and SACPs). In the presence of adenosine, the shift of the leading pacemaker from the SAN center to the head or tail is made possible by the higher A1R expression in the center, in addition to the heterogeneity of ion channels INa and If currents between the center and head/tail. Our modeling study also shows that the difference in the source-sink ratio between the SAN head and tail can also influence the new leading pacemaker site. Along with the change in the leading pacemaker, the earliest atrial activation sites shift in locations as well, though it does not always exit through the nearest SACP to the leading pacemaker. Secondly, our study sheds new light on the role of the insulation boundary between the SAN and neighbouring RA. It is the first time that a computer model of the 3D human SAN complex with the insulation boundary can replicate the dramatic shift in the leading pacemaker and the earliest atrial activation sites as commonly observed in the high-resolution optical mapping of the human heart ex-vivo and clinical studies [9]. On the other hand, modeling simulations suggest that computer models with reduced diffusivities within the SAN can only produce a modest shift. In addition, the leading pacemaker tends to localize close to the insulation boundary between the SAN and the septum which is not consistent with the reported results in the literature [9]. Therefore, our modeling study lends further support to the existence of the insulation boundary between the SAN and neighbouring RA, except SACPs for connecting RA electrically. Finally, our computer simulations indicate that the INa remodeling and fibrosis upregulation in the SACPs play a key role in the shift of the earliest atrial activation sites and in the SAN exit block. The reduction in INa current density and fibrotic remodeling slow down conduction is widely accepted and are well-known arrhythmogenic factors [38–41]. However, the novelty of this study is that the INa current density and fibrotic remodeling in SACPs are more important than in other SAN regions. Therefore, the SACP may be a potential therapeutic target in clinics for patients with SAN dysfunction. For instance, reversing fibrotic remodeling in the SACPs will improve the electrical conduction of the SAN to the heart and alleviate the need for electronic pacemaker implantation. Further development of our modeling analysis for clinical intervention may provide a powerful, safe approach to test novel treatments to treat SND. Study limitations Our computer model of the human SAN complex was based on a shadow of high-resolution histology images by projecting the SAN model to the imaging plane. It had realistic geometric regions proportional to the neighboring RA, and different SAN compartments, so it is not a complete 3D representation of the 3D human SAN complex. Due to the current paucity of human SAN compartment-specific electrophysiological data, we used the same cellular model for SAN periphery compartments (head and tail). A computationally efficient, biophysics-based computer model of the entire 3D human SAN pacemaker-conduction complex and RA directly based on the 3D imaging data is yet to develop and validate the insights learned from this study. However, we suggest that taking into account 3D effects may not conceptually affect the main conclusions drawn from our study as the effects of the SAN structural and molecular features (e.g. A1R and ionic channels) on the superior/inferior shift of the leading pacemaker and preferential SACP/earliest atrial activation sites are confirmed by human SAN experiments. Structural and electrical heterogeneity of the SAN Since the discovery of the SAN by Keith and Flack [3] more than one century ago, significant strides in our understanding of the SAN complex have been made [5,6,12,27]. It is widely accepted that the heterogeneous distribution of specialized ion channels, intracellular Na+/Ca2+ handling proteins, gap junction channels and receptors within the human SAN complex are some of the few critical players in SAN pacemaking [5]. In particular, expressions of ion channels (e.g. Na+, Ca2+ and HCN1/4), and gap junction proteins are heterogeneous within the SAN pacemaker complex [27,28]. Furthermore, in contrast to other cardiac regions, the human SAN has more extensive fibrosis, which is further upregulated in SAN dysfunction or HF [6]. It is also well established that the existence of a septal area of conduction block, known as the block zone, prevents the spread of the electric impulse to the interatrial septum directly from the SAN. The precise mechanism of atrial activation by the leading pacemaker remains a controversy. In the past, the gradient model, in which there is a gradual change in the intrinsic properties of pacemaker cells from the center to the periphery of the SAN, was proposed to explain how SAN could effectively pace the atria [12] The intrinsic pacemaker activity is greater in cells from the periphery than from the centre of the SAN [29–33]. However, at the tissue level, the periphery of the SAN is connected to a large mass of atrial muscle in the crista terminalis (CT) through gap junctions resulting in the inhibition of peripheral pacemaker activity by the electrotonic influence of the highly hyperpolarized atrial muscle. On the other hand, central pacemaker cells, which are more distal from the atria, are less affected by atrial electrotonic effects. Therefore, leading pacemaker activity at baseline conditions always originates in the central SAN cells, although they are intrinsically slower than the peripheral pacemaker cells. These results were primarily based on small animals studies, including rabbit models, which show both lateral (towards CT) and superior-inferior gradient of intrinsic SAN pacemaker properties. However, in large animal models including canine and human SAN, the superior-inferior gradient is more prominent while lateral gradient is evident only across the SACPs, due to the larger CT myocardium requiring more lateral insulation [1,5,6,9]. Earlier Crick et al. [34] reported twice higher density of parasympathetic and sympathetic nerves fibers in the SAN center vs periphery (tail). The innervation gradient may explain why sympathetic activation can shift atrial exits to superior SACP and parasympathetic activation can slow sinus rhythm and shift atrial exits inferiorly [35]. However, in both human and canine SAN, the direction of intranodal pacemaker shift from center to head or tail does not always correlate with the closest superior or inferior SACP [1,36]. SAN activation can exit via the superior SACPs even though the leading pacemaker shifted inferiorly to the tail (Figs 2B and 3B). These studies suggest that SAN automaticity gradient and superior or inferior intranodal pacemaker shift (from center to head or center to tail) depend on the heart-specific SAN compartment molecular profiles. However, no one yet measure and compare intrinsic frequency of pacemaker cells isolated from different SAN pacemaker compartments (head, center and tail). Instead, we included the electrophysiological and molecular difference between the SAN center and head/tail (periphery), as well as transitional cells in preferential SACP for electrical coupling from SAN to RA. In keeping with the classical gradient model, our modelling results showed that isolated SAN pacemakers in the SAN center were slower than the SAN head/tail, which is consistent with most experimental results [29–33]. The most significant controversy surrounds the location and nature of the SAN in the human heart. Dobrzynski and her colleagues found an intermediate region with an expression of many ion channels between the SAN and RA in the human heart, which is similar to that seen in the periphery of the rabbit SAN [27]. They speculated that this region, termed the paranodal area, might contribute to pacemaker shift though they did not conduct any electrophysiological or optical mapping studies to support the claim. Using high-resolution optical mapping and histological studies in the human heart ex-vivo, we have shown that the human SAN complex is a 3D, specialized multi-compartment structure (head, center and tail) with higher expression of HCN and A1R proteins, lower gap junctional coupling and Nav1.5 in the SAN center than in the SAN head and tail [1,9]. Our comprehensive biophysical computer model of the human SAN pacemaker conduction complex was developed based on the functional and structural mapping in the human heart to investigate the role of human SAN structure and ion channels heterogeneity in SAN pacemaking and conduction functions. The novelty of our computer model of the human SAN Computer models provide a powerful tool for the quantitative examination of structural and electrical substrates and their individual contributions to cardiac arrhythmia mechanisms [9,16]. However, much less literature exists for multi-scale human SAN modeling, mainly due to the structural and functional complexity of the human SAN complex. The first computer modeling of the SAN by Joyner and Capelle [10] demonstrated that the electrical uncoupling of the pacemaker cells might be an essential design feature of a healthy SAN complex using a two-dimensional sheet model. Similar approaches were expanded to study the importance of gradients in membrane properties (e.g., the ionic current density of the INa) and electrical coupling in 2D [18] and 3D [19] models, and the mechanisms by which the SCN5A mutations (Na+ channel) impair cardiac pacemaking [37]. Earlier 3D computer models of the human SAN complex were developed primarily for studying the electrical conduction within the RA [4,28]. The more recent 3D computer model by Kharche et al. [20] was the first to study the role of the insulating border between the SAN and RA septum, and the paranodal area in the SAN function. However, they only used the simple three-current Fenton-Karma cellular models to simulate the human SAN and RA cell kinetics. In addition, SACPs in their model were not anatomically based, which led to extremely short un-physiological SAN conduction. To date, there is no comprehensive SAN model that integrated realistic anatomical structures and ion channel expressions from direct studies of the human SAN complex. As such, no computer model could successfully reproduce all functional observations at different conditions in normal and diseased human SAN, including the location of leading pacemakers, preferential SACP exits, physiological SAN conduction time (70–80 ms at baseline) and SCL (700–900 ms). Our study takes the next step in defining the key factors influencing human SAN pacemaking function and SAN dysfunction by developing and utilizing computer models of the human SAN based on the current knowledge of electrical and structural heterogeneity. The unique strength of our SAN computer model is that it was based on published data from high-resolution near-infrared optical mapping, molecular mapping and detailed 3D histological analyses of the human SAN complex ex-vivo [1,5,8,9]. More importantly, it included anatomically based SACPs directly identified from the 3D immunohistological analyses which provide realistic SAN to RA electric loading. In addition, we have adapted the most widely used human SAN and atrial cellular activation models, i.e., the Fabbri et al. [17] and Courtemanche et al. cell model [25]. Critical functional insights from the human SAN model Firstly, we have illustrated the role of structural and electrical heterogeneity in the shift of the leading pacemaker and the earliest atrial activation sites. Under baseline conditions, the leading pacemaker is more likely located in the SAN center due to less electrical loading (the relatively larger SAN center region with only one SACP, in contrast with the SAN head and tail), despite the fact that single isolated SAN head or tail cells have a shorter SCL than that in the center (798 ms versus 814ms). The difference in cellular SCL is caused by the electrical heterogeneity within the SAN complex, particularly, the higher If and lower INa currents in the SAN center, as widely reported. That is to say that the effect of the electrical heterogeneity is suppressed by the SAN structure (SAN-RA isolation layer and SACPs). In the presence of adenosine, the shift of the leading pacemaker from the SAN center to the head or tail is made possible by the higher A1R expression in the center, in addition to the heterogeneity of ion channels INa and If currents between the center and head/tail. Our modeling study also shows that the difference in the source-sink ratio between the SAN head and tail can also influence the new leading pacemaker site. Along with the change in the leading pacemaker, the earliest atrial activation sites shift in locations as well, though it does not always exit through the nearest SACP to the leading pacemaker. Secondly, our study sheds new light on the role of the insulation boundary between the SAN and neighbouring RA. It is the first time that a computer model of the 3D human SAN complex with the insulation boundary can replicate the dramatic shift in the leading pacemaker and the earliest atrial activation sites as commonly observed in the high-resolution optical mapping of the human heart ex-vivo and clinical studies [9]. On the other hand, modeling simulations suggest that computer models with reduced diffusivities within the SAN can only produce a modest shift. In addition, the leading pacemaker tends to localize close to the insulation boundary between the SAN and the septum which is not consistent with the reported results in the literature [9]. Therefore, our modeling study lends further support to the existence of the insulation boundary between the SAN and neighbouring RA, except SACPs for connecting RA electrically. Finally, our computer simulations indicate that the INa remodeling and fibrosis upregulation in the SACPs play a key role in the shift of the earliest atrial activation sites and in the SAN exit block. The reduction in INa current density and fibrotic remodeling slow down conduction is widely accepted and are well-known arrhythmogenic factors [38–41]. However, the novelty of this study is that the INa current density and fibrotic remodeling in SACPs are more important than in other SAN regions. Therefore, the SACP may be a potential therapeutic target in clinics for patients with SAN dysfunction. For instance, reversing fibrotic remodeling in the SACPs will improve the electrical conduction of the SAN to the heart and alleviate the need for electronic pacemaker implantation. Further development of our modeling analysis for clinical intervention may provide a powerful, safe approach to test novel treatments to treat SND. Study limitations Our computer model of the human SAN complex was based on a shadow of high-resolution histology images by projecting the SAN model to the imaging plane. It had realistic geometric regions proportional to the neighboring RA, and different SAN compartments, so it is not a complete 3D representation of the 3D human SAN complex. Due to the current paucity of human SAN compartment-specific electrophysiological data, we used the same cellular model for SAN periphery compartments (head and tail). A computationally efficient, biophysics-based computer model of the entire 3D human SAN pacemaker-conduction complex and RA directly based on the 3D imaging data is yet to develop and validate the insights learned from this study. However, we suggest that taking into account 3D effects may not conceptually affect the main conclusions drawn from our study as the effects of the SAN structural and molecular features (e.g. A1R and ionic channels) on the superior/inferior shift of the leading pacemaker and preferential SACP/earliest atrial activation sites are confirmed by human SAN experiments. Conclusions Our novel biophysical computer model of a human SAN conduction complex combining ex-vivo functional and 3D structural imaging at the highest resolution to date illustrates for the first time the crucial role of the structural and electrical heterogeneity of the human SAN in the pacemaking function. Particularly, our results lend support to the necessity of the insulation boundary between the SAN and neighbouring RA for robust SAN pacemaker and conduction function. The study also suggests that the cardiac disease or drug modulations of the INa current and fibrosis in intranodal pacemaker compartments and/or SACPs may promote SAN reentrant arrhythmias. The further development of 3D computer models based on the human heart ex-vivo may provide a powerful, safe approach for preclinical testing of novel treatment for patients with SAN dysfunction worldwide. Supporting information S1 Text. Supplemental Materials. Table A. Relative ratios of the densities of the four key ionic channels (If, INa, IK1 and IKACh currents) among the human SAN head, center and tail, and SACPs used in the computer modeling of the control SAN. SACP- sinoatrial pathways, SAN–sinoatrial node, SCL–sinus cycle length. Table B. Relative ratios of the densities of the four key ionic channels: If, INa, IK1 and IKACh (A1R expression), and fibrosis among the human SAN head, center and tail, and SACPs used in the computer modeling of HF. SACP- sinoatrial pathways, HF–heart failure, A1R –A1 adenosine receptor, SAN–sinoatrial node, SCL–sinus cycle length. Table C. The regional differences in gap junctional coupling between the SAN center, SAN head/tail, SACPs and RA tissues by setting different diffusion coefficients in these regions. SACP- sinoatrial pathways, SAN–sinoatrial node, RA–right atrium. Fig A. 3D microstructural composition of the human SAN complex (an ex-vivo human donor heart,). A, 3D microstructure of all tissue types including myofibers, fibrosis, and fat in the SAN complex (red) and surrounding atrial tissue (green). B, The myofibers of the SAN complex and surrounding atrial tissue. C, The fibrotic fibers of the SAN complex and surrounding atrial tissue. D, The fat texture of the SAN complex and surrounding atrial tissue. SAN–sino-atrial node, CT–crista terminals, IAS–interatrial septum, SVC–superior vena cava, RAA–right atrial appendage. Fig B. The structure of human SAN computer model. The SAN model structure was obtained using a shadow of the 3D SAN reconstruction to the XY plane (parallel to epicardium) by project all 2Ds into one plane. As a result, the 2D representation of the entire 3D human SAN structure included all SACPs and the complete SAN head/center/tail. SAN–sinoatrial node, SACP–sinoatrial pathways. Fig C. Changes in leading pacemaker locations and SACPs in the presence of 20% adenosine due to increasing A1R expression level in the SAN head and tail from 0.1 to 0.9 while keeping A1R in the SAN center constant as 1. SACP- sinoatrial pathways, SAN–sinoatrial node, RA–right atrium. Fig D. Changes in leading pacemaker locations and SACPs in the presence of 20% adenosine due to increasing A1R expression level in the SAN tail from 0.1 to 0.9 while keeping A1R constant in the SAN head and in the center. A1R was set at 1 to the SAN center and 0.1 to the SAN head. SACP- sinoatrial pathways, SAN–sinoatrial node, RA–right atrium. Fig E. The heterogeneity of adenosine A1 receptors (A1R) expressions or the IKACh channel within the intranodal pacemaker compartments (head, center and tail) of the SAN complex is required for the shift in the leading pacemaker and the earliest atrial activation sites during adenosine. A, For the heterogeneous model, the A1R/IKACh expression in the SAN center is 10 times higher than in head/tail (10:1) was used in the heterogeneous A1R computer model. B, In the heterogeneous A1R model, administration of 20% Adenosine led to both the leading pacemaker shift (from center to tail) and the shift of the earliest atrial activation site/ preferential SACP from the lateral to inferior SACP. C, In contrast, in the SAN model with homogeneous A1R/IKACh expression, the same 20% Adenosine didn’t lead to the leading pacemaker and preferential SACP shifts but more severely suppressed SAN automaticity and conduction. The activation maps were almost identical for homogenous A1R with 20% Ado and baseline. SAN–sinoatrial node, SACTsr–SAN conduction time during sinus rhythm, SCL–sinus cycle length. Fig F. Four activation patterns were observed after the cession of right atrial (RA) pacing with a CL of 500 ms in control SAN model (see Fig 8B in the main manuscript). A, Normal SAN activation pattern: the first post-pacing SAN beat had the same activation pattern as SAN beats before pacing with the leading pacemaker in the center and preferential conduction exit through the middle lateral SACP. B, Mild changes in pacemaker/SACP post the RA pacing: the first post-pacing SAN beat had different activation compared with pre-pacing SAN activation with both leading pacemaker and SACP exit site shifts. C, Severe abnormal conduction: SAN macro reentries with slower intranodal conduction path between inferior and superior SACP and a CL of 451 ms spontaneously occurred after RA pacing. D, Severe abnormal conduction–Localized SAN reentry between two superior SACPs induced by RA pacing. Two action potential (AP) tracings are from RA, near the pacing site, and the other is located in the center of the SAN. SAN–sino-atrial node, SACP–SAN conduction pathway, Ado–Adenosine, AP–Action potential, CL–cycle length. Fig G. Summary of propagation patterns in control SAN model with increasing INa block or Ado (without extrastimuli). Here control refers to the leading pacemaker in the SAN center and consistent exit through the lateral middle SACP. SAN–sino-atrial node, SACP–SAN conduction pathway, Ado–Adenosine. https://doi.org/10.1371/journal.pcbi.1011708.s001 (DOCX) Acknowledgments We thank the Lifeline of Ohio Organ Procurement Organization and cardiac transplant surgery department of the Ohio State University Wexner Medical Center for providing the explanted donor hearts, as well as the generous donors and their families whose selfless donations make this lifesaving research possible.

Harkos, Constantinos;Stylianopoulos, Triantafyllos;Jain, Rakesh K.

doi: 10.1371/journal.pcbi.1011740pmid: 38113269

Introduction Immune checkpoint inhibitors (ICIs) have transformed the treatment of cancer. To date 8 different ICIs have been approved alone or in combination with other therapies for ~80 indications [1]. However, less than 20% of patients currently benefit from these treatments [2]. Moreover, many patients develop immune-related adverse events, some of which can be fatal [3]. The abnormal and immunosuppressive tumor microenvironment (TME) not only hinders the delivery of ICIs, but also renders them ineffective once they accrue in tumors [4]. One approach to overcome these challenges is to normalize the tumor vasculature and microenvironment using anti-angiogenic agents [5]. Indeed 7 different combinations of ICIs and anti-angiogenic agents have been approved by the US FDA recently and multiple trials are currently testing this approach [6,7]. Another approach to improve the outcome of immunotherapies is the direct injection of immunostimulatory agents, tethered to a polymer or another substrate, into the tumor [8–10]. Agents being evaluated for this purpose include pro-inflammatory cytokines, such as, interleukin 2 (IL2) and interleukin 12 (IL12) [11–13]. The goal of intratumoral injection of pro-inflammatory cytokines is to maximize their activity within the tumor while minimizing systemic exposure. After the cytokines are administered intratumorally, they can be cleared via the tumor-associated vasculature and the lymphatic system as well as escape from the tumor margin into the surrounding host tissue, resulting in potentially toxic levels in the circulation and the host organ [14–16]. A promising approach to increase tumor exposure and reduce adverse effects to normal tissues, is controlled release of cytokines from a polymer-conjugate injected into a tumor. One example of this approach is to fuse cytokines to collagen binding proteins, so that they are bound to collagen fibers within the tumor and do not clear rapidly from the tumor margin or by the vasculature. This strategy has been successful in enhancing treatment efficacy in preclinical studies [17–19]. Apart from the binding properties of the cytokine agent, its local exposure depends also on properties of the TME [20–24]. Specifically, the uniformly elevated interstitial fluid pressure (IFP) within the tumors (resulting from vascular hyperpermeability and dysfunctional lymphatics) decreases to normal values at the tumor-host tissue margin, causing steep pressure gradients at the tumor periphery and resulting in fluid flux from the tumor into the surrounding tissue. This can wash out the injected cytokines and reduce their concentration in the tumor region (Fig 1A) [20–24]. In addition, the hyper-permeability of tumor vessels might lead to the intravasation of the conjugated-cytokines into the vessels and their clearance via the circulation, a process that depends not only on the pore size of the vessel walls but also on the size, charge, configuration, and binding characteristics of the conjugated-cytokines [25–27]. Despite their importance in the effectiveness of intratumoral injection of cytokines, the role of these tumor parameters (i.e., vessel permeability, hydraulic conductivity, vessel density) and properties of the conjugated-cytokines (i.e., binding affinity, size) and their effects on the efficacy of intratumor injection of cytokines remain unexplored. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Model methodology. (a) Shcematic of various transport mechanisms considered in the model. The conjugated-cytokines are injected in the tumor region and can be transported via convection and diffusion to the host tissue and across the tumor vessel walls. Hyperpermeability of the tumor blood vessels and the lack of functional lymphatic vessels elevates interstitial fluid pressuse, inducing pressure gradients at the tumor periphery that drive transport of the conjugated-cytokines from the tumor to the host tissue via convection. The injected conjugated-cytokines can bind and unbind to the target (e.g., collagen fibers) in both tumor and host tissue. Cytokines produced by the immune cells can disperse via convection and difusion as well. Also immune cells can migrate (i.e., diffuse) from the tumor tissue to the host tissue and the reverse depending on the concentration gradients. (b) Model components of the immune system: IN represents the innate immune cells that induce cytolysis and produce antigen, e.g., Natural Killer cells. Immature APCs are the immature antigen presenting cells that can become antigen presenting cells (APCs). CD4 and CD8 represent effector CD4+ and CD8+ T cells. Production and activation of immune cells are affected by cytokines. The immune cells also produce cytokines. Oxygen supply increases cancer cells’ proliferation and tumor growth and decreases the apoptosis rate of the immune cells. Created with BioRender.com. https://doi.org/10.1371/journal.pcbi.1011740.g001 We have previously developed mathematical models of fluid and solute transport in tumors to investigate the role of vascular permeability, diffusion coefficient and hydraulic conductivity, binding and metabolism, interstitial fluid pressure, solid stresses as well as lymphatics [23,28–32]. Other in silico studies have examined the interactions of immune cells with cancer [33,34]. In addition, a recent intratumoral injection model examined the optimal cytokine design that increases intratumoral activity [18]. Although this model incorporated the binding-affinity of the conjugated-cytokines to their target, their transport into the blood circulation accounting for the conjugated-cytokines size and affinity, as well as temporal changes in model variables, they did not account for pathophysiological features and the spatial heterogeneity of the TME and the surrounding host tissue. To this end, building on our previous work, here we developed a mathematical model for intratumoral injection of conjugated-cytokines that accounts for i) spatiotemporal variations in model parameters, ii) the vascular and lymphatic function, iii) the hydraulic conductivity of the tumor and host tissue, iv) the interstitial fluid pressure, v) convection and diffusion within the tumor, from the tumor interstitial space to the blood vessels and the surrounding tissue, accounting explicitly for the size of the conjugated-cytokines, their binding affinity and vascular permeability, and vi) immune cells and cancer cells interactions (Table A in S1 Text). Two conjugated-cytokines cases were modeled: i) cytokines fused with mouse serum albumin (MSA) conjugated to the collagen binding protein, lumican [17], and ii) cytokines bound to aluminium hydroxide (alum) via ligand exchange between hydroxyls on the surface of alum and phosphoserine residues tagged to the cytokine by an alum-binding peptide [35]. We first assessed the robustness of our model by comparing model predictions with tumor growth data from these two independent studies [17,35]. Subsequently, we used the model to investigate the effect of the conjugated-cytokines size and binding affinity in conjunction with properties of the TME, on the efficacy of intratumorally injected conjugated-cytokines in reducing tumor growth. We further analyzed spatiotemporal changes in the concentration of the conjugated-cytokines and immune cells for a better understanding of the underlying mechanisms. Materials and methods A brief description of the basic components of the mathematical framework is presented here. A detailed description of the equations that form the mathematical model is found in S1 Text. The modeling framework consists of two steps. We first model the short time period immediately after the injection of the conjugated-cytokines from the needle into the tumor. Then, after the removal of the injection needle, we model the transport of the conjugated-cytokines into the tumor, its clearance through the blood vessels and tumor margin, as well as the growth of the tumor over a long time period. The first model simulates the injection of conjugated-cytokines inside a spherical tumor surrounded by host tissue (Fig 1 and Fig A in S1 Text). The conjugated-cytokine concentration profiles developed after the injection from the needle are used as initial conditions for the second model. The second model simulates cancer cell proliferation, the immune response and tumor growth (Fig 1 and Fig B in S1 Text). The model also accounts for transport of fluid and cytokines within the tumor, between the tumor and the host tissue as well as across the tumor vessel walls (Fig 1A). The model was developed and solved in COMSOL Multiphysics (COMSOL, Inc., Burlington, MA, USA) using the finite element method. Cytokine transport The conjugated-cytokines can be in a free state or bound to the target (bound state). Both convection and diffusion are considered for the transport of the free conjugated-cytokines within the tumor and the host region. The diffusion coefficient of the conjugated-cytokines are determined by experimental data based on the conjugate size [36]. Also, the conjugated-cytokines that are not bound can intravasate into the vessels through diffusion and convection based on Starling’s approximation for mass transfer [28,30,37]. The transport properties of the conjugated-cytokines across the tumor vessel wall (i.e., vascular permeability and reflection coefficient) are determined explicitly by the relative ratio of the conjugate size to the size of the pores of the vessel wall, so that we can account for tumors with low, moderately, and highly permeable vessels as well as for conjugated-cytokines of varying size. For each conjugated-cytokine case, cytokines fused with mouse serum albumin conjugated to lumican and cytokines bound to aluminium hydroxide, the molecular weight were taken from the respective study [17,35] to determine their diffusion coefficient and transport properties across the tumor vessel walls. The rate of clearance from blood was also determined by the conjugate size based on previous work [38]. Furthermore, due to the different conjugate design and the different nature of target (collagen vs alum) for each conjugated-cytokine case, the respective binding affinity was used. In addition to cancer cells, the model includes innate and adaptive immune cells. These cells produce pro-inflammatory cytokines in addition to the injected cytokine, so that the total population of pro-inflammatory cytokines includes the cytokines produced by the immune system, the injected conjugated-cytokines that are free to move and the injected conjugated-cytokines that are bound. The total pro-inflammatory cytokines can enhance the immune system’s response to kill cancer cells and reduce tumor growth. The types of immune cells and immune cell–cancer cell interactions considered in our model are shown in Fig 1B and described below. Immune response The simulation starts with a highly immunosuppressed TME by assuming initially no antigen presenting cells (APCs) or activated effector cells (e.g., effector CD4+ and CD8+ T cells), and predicts how the function of immune cells with positive effect on killing cancer cells impacts tumor growth. Due to the complex nature of the immune system and the immune cells—cancer cells interactions, we considered the immune cells in certain categories for simplicity. These include innate and adaptive immune cells. The innate cells are divided into two categories: cells that can induce cytolysis, such as Natural Killer (NK) cells, this category of cells can kill cancer cells and produce antigen, and the immature antigen presenting cells (IAPCs) that includes the dendritic cells and a sub-set of macrophages. When IAPCs interact with cancer cells or antigens they become APCs. The higher the number of APCs the more CD4+ and CD8+ T cells will reach the tumor and host tissue. In addition, and for simplicity, we did not include an explicit model of lymph nodes for the activation of T cells. Instead, we assumed that T cell activation takes place external to the tumor in lymph nodes where the T cells encounter APCs, but the activated T cells return to the same location in the tumor from which the APCs depart. Effector CD8+ T cells kill cancer cells and further increase the concentration of antigens in the region. Both CD4+ and CD8+ T cells produce pro-inflammatory cytokines to further increase the immune response (Fig 1B). Interstitial fluid flow Fluid flow within the tumor and host tissue is governed by Darcy’s law, taking into account the displacement of both the tumor and the surrounding normal tissue due to the growth of the tumor. Continuity of fluid velocity and fluid flux is applied at the tumor/host tissue interface [30]. The model also accounts for fluid flux across the tumor vessel walls based on Starling’s approximation [22,30,39,40]. The hydraulic conductivity of the tumor vessel wall is calculated based on the vessel walls pore size, following our previous research [28,39]. Oxygen transport The model considers oxygen transport from the vessels into the tumor and host tissue and transport within the tissue. Overall tumor growth depends on cancer cell number (concentration), which is determined by cancer cell proliferation (as a function of tissue oxygenation) and cancer cell killing by immune cells [41,42]. Details about the model variables as well as the baseline and initial values of the model parameters are given in Tables B and C in S1 Text. Cytokine transport The conjugated-cytokines can be in a free state or bound to the target (bound state). Both convection and diffusion are considered for the transport of the free conjugated-cytokines within the tumor and the host region. The diffusion coefficient of the conjugated-cytokines are determined by experimental data based on the conjugate size [36]. Also, the conjugated-cytokines that are not bound can intravasate into the vessels through diffusion and convection based on Starling’s approximation for mass transfer [28,30,37]. The transport properties of the conjugated-cytokines across the tumor vessel wall (i.e., vascular permeability and reflection coefficient) are determined explicitly by the relative ratio of the conjugate size to the size of the pores of the vessel wall, so that we can account for tumors with low, moderately, and highly permeable vessels as well as for conjugated-cytokines of varying size. For each conjugated-cytokine case, cytokines fused with mouse serum albumin conjugated to lumican and cytokines bound to aluminium hydroxide, the molecular weight were taken from the respective study [17,35] to determine their diffusion coefficient and transport properties across the tumor vessel walls. The rate of clearance from blood was also determined by the conjugate size based on previous work [38]. Furthermore, due to the different conjugate design and the different nature of target (collagen vs alum) for each conjugated-cytokine case, the respective binding affinity was used. In addition to cancer cells, the model includes innate and adaptive immune cells. These cells produce pro-inflammatory cytokines in addition to the injected cytokine, so that the total population of pro-inflammatory cytokines includes the cytokines produced by the immune system, the injected conjugated-cytokines that are free to move and the injected conjugated-cytokines that are bound. The total pro-inflammatory cytokines can enhance the immune system’s response to kill cancer cells and reduce tumor growth. The types of immune cells and immune cell–cancer cell interactions considered in our model are shown in Fig 1B and described below. Immune response The simulation starts with a highly immunosuppressed TME by assuming initially no antigen presenting cells (APCs) or activated effector cells (e.g., effector CD4+ and CD8+ T cells), and predicts how the function of immune cells with positive effect on killing cancer cells impacts tumor growth. Due to the complex nature of the immune system and the immune cells—cancer cells interactions, we considered the immune cells in certain categories for simplicity. These include innate and adaptive immune cells. The innate cells are divided into two categories: cells that can induce cytolysis, such as Natural Killer (NK) cells, this category of cells can kill cancer cells and produce antigen, and the immature antigen presenting cells (IAPCs) that includes the dendritic cells and a sub-set of macrophages. When IAPCs interact with cancer cells or antigens they become APCs. The higher the number of APCs the more CD4+ and CD8+ T cells will reach the tumor and host tissue. In addition, and for simplicity, we did not include an explicit model of lymph nodes for the activation of T cells. Instead, we assumed that T cell activation takes place external to the tumor in lymph nodes where the T cells encounter APCs, but the activated T cells return to the same location in the tumor from which the APCs depart. Effector CD8+ T cells kill cancer cells and further increase the concentration of antigens in the region. Both CD4+ and CD8+ T cells produce pro-inflammatory cytokines to further increase the immune response (Fig 1B). Interstitial fluid flow Fluid flow within the tumor and host tissue is governed by Darcy’s law, taking into account the displacement of both the tumor and the surrounding normal tissue due to the growth of the tumor. Continuity of fluid velocity and fluid flux is applied at the tumor/host tissue interface [30]. The model also accounts for fluid flux across the tumor vessel walls based on Starling’s approximation [22,30,39,40]. The hydraulic conductivity of the tumor vessel wall is calculated based on the vessel walls pore size, following our previous research [28,39]. Oxygen transport The model considers oxygen transport from the vessels into the tumor and host tissue and transport within the tissue. Overall tumor growth depends on cancer cell number (concentration), which is determined by cancer cell proliferation (as a function of tissue oxygenation) and cancer cell killing by immune cells [41,42]. Details about the model variables as well as the baseline and initial values of the model parameters are given in Tables B and C in S1 Text. Results Model validation and determination of model parameters values The values of the model parameters that could not be obtained from previous studies (Table C in S1 Text) were determined by fitting the model to tumor growth data from two published studies [17,35]. These studies included a control group that did not receive any treatment (control) and a group that received intratumoral injection of conjugated-cytokines as a drug. For the control groups, the variables related to the injected conjugated-cytokines become zero so that the pro-inflammatory cytokines in the tissue are produced only by the immune cells (Eq (30) in S1 Text). We did not consider any other variation of model parameters between the control and injected cytokines group. All tumor growth curves were fitted simultaneously to optimize the global fit. An optimization algorithm in MATLAB (The Mathworks, Inc., Natick, MA, United States) using the COMSOL with MATLAB interface was employed for the fitting. More information about the optimization and the optimized parameters can be found in the S1 Text. As shown in Fig 2 the model can reproduce tumor growth data with a good accuracy (R2 ~ 1). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Experimental data (circles) of tumor growth and model predictions (solid line) for control tumors (blue) and tumors treated with intratumoral injection of conjugated-cytokines (red) by Momin et al. [17] and Agarwal et al. [35]. https://doi.org/10.1371/journal.pcbi.1011740.g002 Due to the complexity of the model that includes various interactions and mechanisms, the behavior of the model variables is not intuitive. Thus, we generated plots to further investigate the changes in the model variables that led to the reduction of the tumor growth after the injection of therapy. Model predictions for the spatial distribution of cytokines are presented in Fig 3, whereas predictions for IFP, antigen concentration, CD8+ T cells and NK cells are presented in Fig 4 for both studies. Day 0 corresponds to the time of the intratumoral injection of the conjugated-cytokines. The concentration of the total cytokines decreased after the injection as expected. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Results for the total amount of cytokines and the bound conjugated-cytokines for various time points for each study. The plots represent the distribution in the radial direction. The value 0 in the x axis corresponds to the tumor center. As we move along the x axis, we move away from the tumor center towards the host tissue. Plots include both the tumor region and host tissue that surrounds the tumor. The vertical dashed lines show the tumor boundary at the given time points. https://doi.org/10.1371/journal.pcbi.1011740.g003 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Results at various time points for each optimization case. The plots represent the distribution in the radial direction. The value 0 in the x axis corresponds to the tumor center. As we move along the x axis, we move away from the tumor center towards the host tissue. Plots include both the tumor region and host tissue that surrounds the tumor. The vertical dashed lines represent the tumor boundary. https://doi.org/10.1371/journal.pcbi.1011740.g004 The IFP was elevated within the tumor, reaching the levels of microvascular fluid pressure at the tumor center and droped to normal values at the tumor margin (Fig 4, control). This spatial distribution of IFP created a fluid flux at the tumor margin towards the host tissue, resulting in increased concentration of antigen, effector CD8+ T cells and NK cells at the interface of the tumor with the host tissue compared to the tumor interior (control group). Intratumoral injection of cytokines can reduce the IFP levels, which is more evident in the case of Momin et al.[17] where the efficacy of the treatment is more pronounced and induced considerably higher amounts of innate and adaptive immune cells compared to the respective control cases. In the treatment case, the spatial distribution of immune cells changed compared to the control and most immune cells can be found at the center of the tumor where the concentration of cytokines and antigens is the highest. Dependence of treatment efficacy on conjugated-cytokines properties Subsequently, we aimed to investigate how changing the properties of the conjugated-cytokines can affect the efficacy of treatment. Specifically, we varied the size and binding affinity of the drug and the model predictions are presented in Fig 5 for varying the conjugated-cytokines radius, rs, from 1 to 8 nm and when the binding rate constant, kon, is increased/decreased by an order of magnitude. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. The impact of various model components to tumor growth by varying a single parameter. Figure presents the tumor growth through time and the number of innate cells that induce cytolysis (NK cells), antigen presenting cells and effector CD8+ T cells when varying: the injected conjugate radius, the conjugate binding rate constant, the vascular density inside the tumor region, the vessel wall pore radius inside tumor, and the hydraulic conductivity inside the tumor region. The baseline values of the parameters for these simulations are: rs = 3.85[nm], kon = 100 [m3/mol/s], Sv = 50[1/cm], r0 = 100 [nm], kh = 4.13e-8 [cm2/mmHg/s]. https://doi.org/10.1371/journal.pcbi.1011740.g005 Changes in both the size of conjugated-cytokines from 1 to 8 nm in radius and the binding rate constant from 2 to 200 m3/mol/s altered the tumor growth rate and induced significant changes in the number of immune cells. Cytokine conjugates of small size were cleared fast from the tumor owing to increased diffusion within the tumor and intravasation into blood vessels and thus, cannot induce a significant anti-tumor immune response. Increasing the size of the drug to 4–8 nm in radius dramatically reduced tumor volume and even eliminated tumor. Increases in binding rate constant hindered the clearance of the cytokines and thus, improved anti-tumor immune responses, by increasing the number of intratumoral CD8+ T cells soon after intratumoral administration of cytokines. Role of the tumor microenvironment in treatment efficacy Next, we set out to study how varying the physical and physiological properties of the TME can improve the efficacy of injected conjugated-cytokines. Specifically, we varied the vascular density and tumor vessel wall permeability (i.e., the size of the pores in the tumor vessel walls) as well as the hydraulic conductivity of the tumor. The tumor functional vascular density was varied from 50 to 100 cm-1 [43], the radius of the pores of the tumor vessel walls from 20 nm to 120 nm [44,45], and the tumor hydraulic conductivity from 5x10-9 to 5x10-5 cm2/mmHg-s [45]. As shown in Fig 5, a 50% increase in the functional vascular density and thus, tumor perfusion, was sufficient to potentiate anti-tumor immunity. In the model and in agreement with the literature, increase in perfusion increased the number of immune cells in the tumor at early times after cytokines injection (Fig 5), which led to complete tumor elimination. Subsequently the immune cells left the tumor and their numbers go down to zero. Elimination of tumor is also predicted when the hydraulic conductivity of the tumor was increased. The increase in the tumor hydraulic conductivity increased the interstitial velocity and thus, allowed for better penetration of the conjugated cytokines in all regions of the tumor. This resulted in a robust anti-tumor immune response and a dramatic reduction in tumor volume. Finally, the vessel wall pore size determined the transport of the conjugates across the tumor vessel wall. Tumors hinder the transport of nano-sized drugs across the tumor vessels [27]. Model predictions agree with previous findings in that tumors with more permeable vessels allowed the transvascular transport of nano-sized therapeutics and in our case allowed the clearance of the conjugated cytokines, which reduced treatment efficacy (Fig 5). Interestingly, the model predicted that even though the tumor responded to therapy at early times after cytokines administration and thus, the tumor volume decreased, at longer times the tumor regrew, which implies the need for repeated intratumoral administration of cytokines. Interestingly, vascular normalization strategies aim to reduce vessel permeability to large molecule/nanoparticles, whereas stroma normalization strategies improve tumor hydraulic conductivity, in both cases improving perfusion [46]. To further investigate the effect of the properties of the TME and the injected conjugated-cytokines, we varied two parameters simultaneously to generate tumor volume diagrams as shown in Fig 6. From these diagrams, firstly, we conclude that increasing the tumor hydraulic conductivity enhanced the efficacy of conjugated cytokines even of small size and low binding affinity (Fig 6A and 6B). Furthermore, increasing the size of the drug and thus, decreasing both the diffusion of the conjugated-cytokines within the tumor tissue and their extravasation into the blood vessels results in reduced tumor volumes for various values of the hydraulic conductivity. Interestingly, increasing the drug size for a tumor with low hydraulic conductivity can induce a similar effect with a smaller drug in a tumor environment with high hydraulic conductivity (Fig 6B). Additionally, reduced tumor volumes can be achieved for lower binding capabilities of the conjugated-cytokines by decreasing the vessel wall pores. Also, increasing the binding rate constant to more than 50 m3/mol/s can reduce tumor volume independent of the vessel wall pore size (Fig 6C). By also increasing the drug size we can achieve improved therapeutic efficacy independently from the vessel wall pore size as well (Fig 6D). Finally, increasing vascular density, while also increasing either the binding affinity or the size of the conjugated cytokines can enhance the efficacy of the treatment (Fig 6E and 6F). From all the analysis, can be inferred that conjugated-cytokines larger than 5 nm in radius with binding rate constant above 50 m3/mol/s can induce better therapeutic outcomes. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. Diagrams of the efficacy of conjugated-cytokines injection as a function of tumor physiological properties and conjugate radius and binding affinity. Each point in the diagrams represents the tumor volume of a different simulation. The tumor volume is measured either at the end of the simulations (day 10) or at the time point where at least one of the simulations reached complete cure (i.e., tumor volume becomes zero). For each simulation only the parameters shown in the two axes were varied. (a) The hydraulic conductivity in the tumor region was varied relative to the binding of the injected conjugate (day 7.5) and (b) the conjugate radius (day 5.2). (c) The tumor vessel wall pore radius was varied relative to the binding of the injected conjugate (day 10) and (d) the conjugate radius (day 6.0). (e) The tumor vascular density was varied relative to the binding of the injected conjugate (day 3.2) and (f) the conjugate radius (day 2.9). https://doi.org/10.1371/journal.pcbi.1011740.g006 Model validation and determination of model parameters values The values of the model parameters that could not be obtained from previous studies (Table C in S1 Text) were determined by fitting the model to tumor growth data from two published studies [17,35]. These studies included a control group that did not receive any treatment (control) and a group that received intratumoral injection of conjugated-cytokines as a drug. For the control groups, the variables related to the injected conjugated-cytokines become zero so that the pro-inflammatory cytokines in the tissue are produced only by the immune cells (Eq (30) in S1 Text). We did not consider any other variation of model parameters between the control and injected cytokines group. All tumor growth curves were fitted simultaneously to optimize the global fit. An optimization algorithm in MATLAB (The Mathworks, Inc., Natick, MA, United States) using the COMSOL with MATLAB interface was employed for the fitting. More information about the optimization and the optimized parameters can be found in the S1 Text. As shown in Fig 2 the model can reproduce tumor growth data with a good accuracy (R2 ~ 1). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Experimental data (circles) of tumor growth and model predictions (solid line) for control tumors (blue) and tumors treated with intratumoral injection of conjugated-cytokines (red) by Momin et al. [17] and Agarwal et al. [35]. https://doi.org/10.1371/journal.pcbi.1011740.g002 Due to the complexity of the model that includes various interactions and mechanisms, the behavior of the model variables is not intuitive. Thus, we generated plots to further investigate the changes in the model variables that led to the reduction of the tumor growth after the injection of therapy. Model predictions for the spatial distribution of cytokines are presented in Fig 3, whereas predictions for IFP, antigen concentration, CD8+ T cells and NK cells are presented in Fig 4 for both studies. Day 0 corresponds to the time of the intratumoral injection of the conjugated-cytokines. The concentration of the total cytokines decreased after the injection as expected. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Results for the total amount of cytokines and the bound conjugated-cytokines for various time points for each study. The plots represent the distribution in the radial direction. The value 0 in the x axis corresponds to the tumor center. As we move along the x axis, we move away from the tumor center towards the host tissue. Plots include both the tumor region and host tissue that surrounds the tumor. The vertical dashed lines show the tumor boundary at the given time points. https://doi.org/10.1371/journal.pcbi.1011740.g003 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Results at various time points for each optimization case. The plots represent the distribution in the radial direction. The value 0 in the x axis corresponds to the tumor center. As we move along the x axis, we move away from the tumor center towards the host tissue. Plots include both the tumor region and host tissue that surrounds the tumor. The vertical dashed lines represent the tumor boundary. https://doi.org/10.1371/journal.pcbi.1011740.g004 The IFP was elevated within the tumor, reaching the levels of microvascular fluid pressure at the tumor center and droped to normal values at the tumor margin (Fig 4, control). This spatial distribution of IFP created a fluid flux at the tumor margin towards the host tissue, resulting in increased concentration of antigen, effector CD8+ T cells and NK cells at the interface of the tumor with the host tissue compared to the tumor interior (control group). Intratumoral injection of cytokines can reduce the IFP levels, which is more evident in the case of Momin et al.[17] where the efficacy of the treatment is more pronounced and induced considerably higher amounts of innate and adaptive immune cells compared to the respective control cases. In the treatment case, the spatial distribution of immune cells changed compared to the control and most immune cells can be found at the center of the tumor where the concentration of cytokines and antigens is the highest. Dependence of treatment efficacy on conjugated-cytokines properties Subsequently, we aimed to investigate how changing the properties of the conjugated-cytokines can affect the efficacy of treatment. Specifically, we varied the size and binding affinity of the drug and the model predictions are presented in Fig 5 for varying the conjugated-cytokines radius, rs, from 1 to 8 nm and when the binding rate constant, kon, is increased/decreased by an order of magnitude. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. The impact of various model components to tumor growth by varying a single parameter. Figure presents the tumor growth through time and the number of innate cells that induce cytolysis (NK cells), antigen presenting cells and effector CD8+ T cells when varying: the injected conjugate radius, the conjugate binding rate constant, the vascular density inside the tumor region, the vessel wall pore radius inside tumor, and the hydraulic conductivity inside the tumor region. The baseline values of the parameters for these simulations are: rs = 3.85[nm], kon = 100 [m3/mol/s], Sv = 50[1/cm], r0 = 100 [nm], kh = 4.13e-8 [cm2/mmHg/s]. https://doi.org/10.1371/journal.pcbi.1011740.g005 Changes in both the size of conjugated-cytokines from 1 to 8 nm in radius and the binding rate constant from 2 to 200 m3/mol/s altered the tumor growth rate and induced significant changes in the number of immune cells. Cytokine conjugates of small size were cleared fast from the tumor owing to increased diffusion within the tumor and intravasation into blood vessels and thus, cannot induce a significant anti-tumor immune response. Increasing the size of the drug to 4–8 nm in radius dramatically reduced tumor volume and even eliminated tumor. Increases in binding rate constant hindered the clearance of the cytokines and thus, improved anti-tumor immune responses, by increasing the number of intratumoral CD8+ T cells soon after intratumoral administration of cytokines. Role of the tumor microenvironment in treatment efficacy Next, we set out to study how varying the physical and physiological properties of the TME can improve the efficacy of injected conjugated-cytokines. Specifically, we varied the vascular density and tumor vessel wall permeability (i.e., the size of the pores in the tumor vessel walls) as well as the hydraulic conductivity of the tumor. The tumor functional vascular density was varied from 50 to 100 cm-1 [43], the radius of the pores of the tumor vessel walls from 20 nm to 120 nm [44,45], and the tumor hydraulic conductivity from 5x10-9 to 5x10-5 cm2/mmHg-s [45]. As shown in Fig 5, a 50% increase in the functional vascular density and thus, tumor perfusion, was sufficient to potentiate anti-tumor immunity. In the model and in agreement with the literature, increase in perfusion increased the number of immune cells in the tumor at early times after cytokines injection (Fig 5), which led to complete tumor elimination. Subsequently the immune cells left the tumor and their numbers go down to zero. Elimination of tumor is also predicted when the hydraulic conductivity of the tumor was increased. The increase in the tumor hydraulic conductivity increased the interstitial velocity and thus, allowed for better penetration of the conjugated cytokines in all regions of the tumor. This resulted in a robust anti-tumor immune response and a dramatic reduction in tumor volume. Finally, the vessel wall pore size determined the transport of the conjugates across the tumor vessel wall. Tumors hinder the transport of nano-sized drugs across the tumor vessels [27]. Model predictions agree with previous findings in that tumors with more permeable vessels allowed the transvascular transport of nano-sized therapeutics and in our case allowed the clearance of the conjugated cytokines, which reduced treatment efficacy (Fig 5). Interestingly, the model predicted that even though the tumor responded to therapy at early times after cytokines administration and thus, the tumor volume decreased, at longer times the tumor regrew, which implies the need for repeated intratumoral administration of cytokines. Interestingly, vascular normalization strategies aim to reduce vessel permeability to large molecule/nanoparticles, whereas stroma normalization strategies improve tumor hydraulic conductivity, in both cases improving perfusion [46]. To further investigate the effect of the properties of the TME and the injected conjugated-cytokines, we varied two parameters simultaneously to generate tumor volume diagrams as shown in Fig 6. From these diagrams, firstly, we conclude that increasing the tumor hydraulic conductivity enhanced the efficacy of conjugated cytokines even of small size and low binding affinity (Fig 6A and 6B). Furthermore, increasing the size of the drug and thus, decreasing both the diffusion of the conjugated-cytokines within the tumor tissue and their extravasation into the blood vessels results in reduced tumor volumes for various values of the hydraulic conductivity. Interestingly, increasing the drug size for a tumor with low hydraulic conductivity can induce a similar effect with a smaller drug in a tumor environment with high hydraulic conductivity (Fig 6B). Additionally, reduced tumor volumes can be achieved for lower binding capabilities of the conjugated-cytokines by decreasing the vessel wall pores. Also, increasing the binding rate constant to more than 50 m3/mol/s can reduce tumor volume independent of the vessel wall pore size (Fig 6C). By also increasing the drug size we can achieve improved therapeutic efficacy independently from the vessel wall pore size as well (Fig 6D). Finally, increasing vascular density, while also increasing either the binding affinity or the size of the conjugated cytokines can enhance the efficacy of the treatment (Fig 6E and 6F). From all the analysis, can be inferred that conjugated-cytokines larger than 5 nm in radius with binding rate constant above 50 m3/mol/s can induce better therapeutic outcomes. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. Diagrams of the efficacy of conjugated-cytokines injection as a function of tumor physiological properties and conjugate radius and binding affinity. Each point in the diagrams represents the tumor volume of a different simulation. The tumor volume is measured either at the end of the simulations (day 10) or at the time point where at least one of the simulations reached complete cure (i.e., tumor volume becomes zero). For each simulation only the parameters shown in the two axes were varied. (a) The hydraulic conductivity in the tumor region was varied relative to the binding of the injected conjugate (day 7.5) and (b) the conjugate radius (day 5.2). (c) The tumor vessel wall pore radius was varied relative to the binding of the injected conjugate (day 10) and (d) the conjugate radius (day 6.0). (e) The tumor vascular density was varied relative to the binding of the injected conjugate (day 3.2) and (f) the conjugate radius (day 2.9). https://doi.org/10.1371/journal.pcbi.1011740.g006 Discussion Our model simulations support the hypothesis that intratumoral injection of tethered cytokines is a promising strategy to control tumor growth. Previous mathematical models showed that by increasing molecular size and/or matrix-targeting affinity of the injected cytokines improves therapeutic efficacy [18]. Our study agrees with these findings, predicting that by increasing the molecular size, the effective diffusion of the injected conjugated-cytokines decreases and thus, they remain within the tumor at higher concentrations. Also, the exposure within the tumor region increases when increasing the binding affinity and thus, making it more difficult for the cytokines to escape from the tumor. Therefore, both molecular weight and binding will lower the effective diffusion rate of the injected drug and only convection can distribute the drug uniformly from the injection site to throughout the tumor. Additionally, our study extends the modeling framework by adding spatiotemporal variations in model parameters, pathophysiological properties of the TME, IFP gradients, convection-diffusion within the tumor and host tissue and across the vessel walls, and cancer-immune cells interactions. Our results suggest that these additional considerations shed further light on the outcome of the treatment. For example, incorporation of the immune system revealed that the injected conjugated-cytokines boost the activation of the adaptive immune cells and also support innate immune cells to further activate the adaptive immune system. Our results also highlight the fact that normalizing pathophysiological features of the TME can improve therapeutic effects. Abnormal blood vessels is a hallmark of solid tumors [47]. Blood vessel abnormalities include hyperpermeability of the tumor vessel wall, as a result of increased levels of proangiogenic factors released under tumor hypoxic conditions, and/or vessel compression/collapse due to the accumulation of mechanical forces in the tumor [23,48]. In both cases, tumor vessel perfusion is reduced. Tumor hydraulic conductivity is often low in fibrotic, desmoplastic tumors, such as triple-negative breast cancer, pancreatic cancer and sarcomas. The excessive collagen matrix and hyaluronan in these tumors hinder the transport of fluid within the tumor interstitial space and thus, decrease the hydraulic conductivity. Stroma normalization strategies aim to target these components of the extracellular matrix either directly or by reprogramming cancer-associated fibroblasts. Therefore, stroma normalization can decompress vessels, improving functional vascular density and increasing the hydraulic conductivity of the tumor [23,48]. Increase in the hydraulic conductivity also enhances convective transport and makes the distribution of the conjugated-cytokines in the tumor more uniform. Our model simulations show that modulation of the TME to reduce vascular permeability, improves perfusion and increases hydraulic conductivity. These strategies should be considered to improve therapeutic outcomes of intratumorally injected cytokines. The strategy to normalize the TME should be tailored to its specific pathophysiological characteristics: abundant hyperpermeable vessels or abundant extracellular matrix or both. Our model simulations also agree with published data, highlighting that the conjugate size and binding capability have a large impact on the outcome of therapy. This is promising because by designing the optimal conjugate, the treatment could be improved. Furthermore, combination with a TME-normalizing strategy would further add to the efficacy of the treatment. Although the model predicted reduced tumor growth due to the administration of conjugated-cytokines, at longer times the tumor recovered. Repeating intratumoral administration might further maintain therapeutic effects and increase efficacy. However, multiple injections might increase systemic accumulation of the conjugated-cytokines, leading to toxic effects [49]. Modulation of the TME and designing conjugated-cytokines with increased molecular size and/or matrix-targeting affinity reduces toxic accumulation and might increase the number of the permiting injections without causing toxic effects. In general, there might be a minimum time of exposure of a certain concentration of the conjugated-cytokines inside the tumor, for the therapy to be effective. This threshold could be akin to the Allee effect [50–52], where below a certain exposure time of this minimum concentration, the treatment is not effective enough to trigger a sufficient immune response to combat the cancer cells. There might be also a minimum exposure time of a certain concentration of the cytokines in the blood that causes toxicity. Thus, when considering intratumoral injection of conjugated-cytokines this level should not be exceeded. Both these thresholds may vary from patient to patient, which makes the development of a personalized adaptive therapy framework that includes the monitoring of the individual’s tumor and immune response a promising approach to optimize therapeutic effects. Our model also has some limitations as we made several assumptions to keep the model simple. The tumor was assumed to grow as a sphere, which is not usually the case. In addition, the model did not account for the drug-conjugate surface charge and configuration, which along with the conjugate size, can affect its transport properties [48,53,54]. Furthermore, the vessel wall pore radius was assumed uniform, while there must be a distribution. Transport properties, such as the interstitial diffusivity of the conjugates, depend not only on their size but also on the density (i.e., porosity) of the tumor interstitial space that varies among tumor types [36]. In this study we did not consider changes in the diffusion coefficient of the conjugates due to variations among tumor types. We also assumed very few intratumor immune cells and none of them activated at the beginning of the simulation. This may not be the case for many tumors. Also, our model did not account for the fact that immune cells can secrete immunosuppressive cytokines. Furthermore, our model does not explicitly incorporate the draining lymph node and effector T cell priming or the cancer cells leaving the tumor via the blood vessels and peri-tumoral lymphatics. In principle, we can relax these assumptions by incorporating additional parameters into our model. However, this is likely to change the results only quantitatively, whereas the conclusions reached in this study related to the parameters that affect the efficacy of intratumoral injection of conjugated-cytokines would remain unchanged. Supporting information S1 Text. Detailed description of the equations that form the mathematical model. It contains the following Figures and Tables. Fig A Computational domain with axial symmetry. The domain includes the tumor region and the host tissue. The needle reaches the center of a spherical tumor. Fig B Computational domain with spherical symmetry. The domain includes the tumor region and the host tissue. The tumor grows as a sphere and deforms the host tissue. Table A: Mathematical model characteristics compared to other models. Table B: Table of model variables. Table C: Table of model parameters. https://doi.org/10.1371/journal.pcbi.1011740.s001 (PDF) Acknowledgments We thank Dr. James W. Baish and Dr. Lance L. Munn for their insightful comments and helpful suggestions on the manuscript.

Natesan, Gunalan;Hamilton, Timothy;Deeds, Eric J.;Shah, Pavak K.

doi: 10.1371/journal.pcbi.1011733pmid: 38113280

Introduction The differentiation of cell types in the developing embryo depends on both cell autonomous processes and signaling from neighbors, diffusible cues, and mechanical forces. In metazoa, lineal history plays an important role in patterning many of these factors and thus in establishing the basic animal body plan [1]. The study of cell lineages in eutelic organisms, which possess a fixed number of somatic cells and thus exhibit stereotypical cell lineages, has been a powerful driving force in our understanding of fundamental developmental and biological processes [2,3]. Cell lineages in these organisms represent valuable scientific resources, providing a spatial and temporal index of the animal onto which multimodal measurements can be aligned [4]. Aligning measurements such as gene expression [5,6], chromatin accessibility [7], cell size and shape [8], and the effects of genetic perturbations [9–12] onto the Caenorhabditis elegans lineage has contributed to an increasingly holistic view of development. Advances in light microscopy and computer vision have dramatically expanded the reach of these approaches, and datasets are now available containing measurements aligned to thousands of embryonic cell lineages [10,13]. The scale of these data poses interesting challenges for data exploration and analysis. Cell lineages map intuitively to mathematical graphs, and the alignment of cell divisions along body axes in C. elegans allows for unambiguous names to be assigned to each cell produced from a division [3]. This allows C. elegans cell lineages, and those of any other eutelic species, to be considered as ordered binary trees, a type of graph that allows straightforward one-to-one alignments to be made between any pair of lineages within or between individual embryos. This property dramatically simplifies the application of metrics computed on lineage trees, since a single unique value can be calculated for each comparison. While the inference of lineage relationships is common practice in the study of evolutionary relationships [14,15], the distinct problem in comparing phenotypic measurements aligned to lineages has been less extensively explored as relevant studies of developmental timing use summary statistics based on linear regressions [11,12,16,17]. Comparisons of the topology of cell lineages has been previously performed using the Robinson-Foulds distance and triplet distance, which each rely on the generation and comparison of sub-trees, accumulating a count of shared sub-trees between lineages normalized against the total number of possible sub-trees to arrive at a metric [18]. In this work, we developed several metrics that operate on the topology of lineages and applied them to the analysis of the C. elegans embryonic cell lineage. The first of these is the tree edit distance [19], which has previously been applied to the comparison of neuron morphology [20] and RNA secondary structure [21]. The tree edit distance allows one to quantify the topological changes in lineage between different embryos, different sublineages, or different experimental conditions. For ordered binary trees, the tree edit distance can be computed more efficiently and is more directly interpretable than previously used sub-tree-based metrics [19]. The second metric that we developed is the “branch distance,” which measures the similarity between lineages based on quantitative measurements of properties of either cells or branches within the lineage. In this case, our particular focus was on the timing of cell division events in the lineage. Both metrics represent intuitive notions of distance, which greatly aids their use in downstream analyses such as unsupervised clustering and hypothesis testing. We benchmarked these metrics using a published database of wild type and RNAi-perturbed C. elegans embryonic cell lineages [10]. We first demonstrate the statistical effects of using cell birth/division timing as a measure of developmental time, motivating the use of cell cycle duration [11,12]. Our analysis also describes previously uncharacterized heterogeneity in wild type lineages and in the phenotypic consequences of RNAi variability on developmental timing. Others have previously demonstrated that sibling asynchrony in division timing reflects the signaling history of that lineage [11]. We apply the branch distance to measure lineage-wide patterns in cell cycle timing and show that Notch signaling is responsible for producing a striking pattern of similarity among the anterior cell lineages of the embryo. Finally, we apply this approach to a systematic analysis of RNAi perturbations that result in cell fate transformations where we find that, while developmental timing appears to be highly sensitive to genetic perturbation, RNAi against genes in a subset of important developmental regulators generate transformations that preserve lineage-specific developmental clocks. Results Defining metrics on spaces of cell lineages Multicellular organisms develop from a single cell through a sequence of divisions. The stereotypical nature of C. elegans development makes it possible to uniquely identify every cell in its somatic lineage based on the orientation of the division of its predecessor relative to the embryo’s body axes [3]. This feature of C. elegans development has been a major advantage of its use as a model system, enabling systematic and quantitative studies of developmental processes. Here, we take advantage of the structured nature of the C. elegans cell lineage to represent it digitally as an ordered binary tree, such that nodes and edges represent cells and division events respectively. The cells in the embryo and the corresponding nodes in the tree can be labeled using the convention based on the orientation of cell divisions along body axes [3] and can be associated with quantitative measurements on a cell-by-cell basis. This natural representation of the lineage as a binary tree suggests several straightforward metrics for comparing lineages to quantify how, say, a gene knockout impacts development (Fig 1A). The first is the tree edit distance, which is derived from the graph edit distance in graph theory and is based on counting the minimum number of operations (such as adding or removing a node or edge) that is needed to convert one tree into another. Since C. elegans development is stereotypical, there is a natural alignment between any two trees based on the naming convention described above [16]. This makes computing the tree edit distance very straightforward, essentially reducing the calculation to determining the number of nodes that are different between the two trees (Fig 1B), a notion similar to other measures, such as the Robinson Foulds metric (S1 Appendix). This metric captures how perturbations like gene knockouts influence the topology of the lineage. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Defining Distance Metrics on Lineage Based Tree Structures. (A) Cell lineages can be expressed as binary trees, with parent, sibling, and child cells relationship reflected in node topology. C. elegans has a naming convention that allows for direct comparisons between cells in distinct lineages. Here we show schematics of two lineages with different topologies, corresponding to embryos “X” and “Y.” The canonical names of the cells are shown either next to or at the end of the corresponding edges. Lineage tracing data also provides information about how long each cell persists between when it is “born” through division and when it divides itself. The numerical values next to each edge indicate these cell cycle times in this schematic. (B) The tree edit distance describes the topological differences between trees by counting the number of additive/subtractive operations required to transform one tree into another. In the case of C. elegans lineage trees, this corresponds to the size of the symmetric difference between the set of nodes present in one embryo vs. another. (C) The intersection branch distance is the Euclidean or L2 norm between measurements associated with shared nodes or edges of trees, disregarding topological differences between trees by only considering nodes/edges present in both trees. (D) The union branch distance is the Euclidean or L2 norm between values on the union set of nodes or edges between trees. Nodes or edges that are absent from one tree in any comparison are given a 0 value. https://doi.org/10.1371/journal.pcbi.1011733.g001 The second is a class of metrics that can be computed on any measurement associated with individual cells within the tree, such as gene expression or time. Again, here we use the stereotyped nature of C. elegans development to directly align any two lineage trees on a cell-by-cell basis, meaning that a single and unique distance can be calculated for any comparison between lineages. Given this alignment, any numerical property of cells during development can be unambiguously converted into a vector representation (Fig 1C). This allows us to use any metric on such vectors to compare the trees. The most straightforward metric that might be used is simply the Euclidean distance (i.e., the L2 norm). Here we focus on the application of the L2 norm to comparisons of cellular division timing between embryos, which has been shown to vary under genetic perturbation [16], and is a measurement produced by every method for lineage tracing by cell tracking. We call this metric the “branch distance” since these division timings represent the length of the branches in the tree (Fig 1C). Of course, a gene knockout or mutation might impact both the topology of the tree and the timing of cell divisions. To account for this, we need to have a way for dealing with cases where a cell/node exists in one tree, but not in the other (Fig 1C). Here, we define two different types of branch distance to account for this problem. The first is the intersection branch distance, where the vector of division timings is constructed only on the basis of cells that are shared between two lineages (i.e., they are in the intersection of the list of cells in the two) (Fig 1D). The second is the union branch distance, where we simply set the cell cycle timing of any cell that is missing from any embryo to 0 and calculate the distance in the normal way (Fig 1D). The union branch distance thus captures differences in both timing and topology, as topological differences are reflected in the absence of cells and thus cycle times. Unpaired cell cycle times are squared and added to the branch distance, as the existence of a pair to that cell would only add the square of the difference of those values. This is functionally equivalent to imputing a zero for missing nodes. As described below, these two metrics capture different aspects of variation between trees. The branch distance reveals unexpected batch effects in WT embryos The most straightforward method for cell lineage tracing is via direct observation and cell tracking. Even in the absence of visible reporters, this approach inherently generates both spatial (3D cell positions and cell trajectories) and temporal (the timing of cell divisions) measurements of the embryo. The distribution of individual cell cycle times within lineages have been deeply explored at a single cell resolution [11,12,22], making hierarchically structured lineage data an attractive target for analysis via our branch metrics. During lineage tracing experiments, every cell in developing embryos is tracked, and each cell division can be mapped to a particular time t, with t = 0 corresponding to, say, the first division of the zygote. There are thus two ways of thinking about the “branch length” value for each cell in the tree (Fig 1C). In one scenario, we could label each cell with its “birth time,” which is just the time t at which the cell was generated through a division event. Another alternative is to consider the “cycle time” for that cell, which is just the length of time between when the cell is born through a division event until it divides itself. Prior work has claimed that developmental timing in C. elegans is highly coordinated, drawn primarily on comparisons of cell birth times [16,17]. In particular, Bao et al. showed that birth times for cognate cells from different embryos are highly correlated, with R2 values that range between 0.995 and 0.997 (Fig 2A) [16]. While comparing birth times between embryos seems natural, there is a potential issue with that approach. In particular, the birth time of a given cell is the sum of the cell cycle times of all the previous division events (Fig 1C). Since there is some randomness in these cycle times, we can think of those times as random variables, noting that summing over random variables always reduces variation [23]. In other words, the “birth time” is essentially equivalent to averaging the previous cycle times, and averaging generally suppresses variation (i.e., the standard error of the mean is generally less than the standard deviation). In addition, because the birth times are a sum of previous cycle times, birth times for cells born later in development will always be larger than birth times for cells born earlier. Both effects can spuriously increase the correlation in cell birth times between embryos. To demonstrate this, we completely randomized the cell cycle times in the embryo, intentionally destroying any correlation in the length of cell cycles for the same cells across each randomized embryo (see Methods). After randomizing these cycle times, we found birth time correlations with R2 values between 0.65 and 0.85 (Fig 2B), despite a complete absence of correlation in the individual cell cycle times (Fig 2C). Thus, while the cycle times are still highly correlated between WT embryos (R2 between 0.97 and 0.99, Fig 2D), the correlation is less than we observe with birth times. Since using the cycle time avoids spurious correlations and reveals more variation and structure in the data (Fig 2D), we focused on using the cycle time to calculate branch distances in this work. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Summation of Cycle Times into Birth Times Suppresses Variation. (A) A comparison between the birth time of each cell (calculated as the sum of cell cycle times of each cell’s ancestors) in two wild type C. elegans embryos. (B) Comparison between birth times calculated from two randomly shuffled wild type embryos, where each cell is assigned another random cell’s birth time from within the lineage of the same embryo. Note that a significant correlation in birth times exists even in this shuffled data. (C) Comparison between shuffled cell cycle times rather than birth times. In this case, there is no correlation, as would be expected. (D) Comparison between the cell cycle times of each cell in two wild type embryos. Note that the same two embryos were used for all comparisons in panels A-D. https://doi.org/10.1371/journal.pcbi.1011733.g002 We used the cycle times to calculate the branch distance between each pair of 30 WT embryos with cycle times taken from lineage tracing data from Du et al. [10] We then hierarchically clustered the embryos based on these distances. Surprisingly, clustering on the branch distance revealed two distinct previously unknown populations of WT embryos in the published dataset, representing differences in cell cycle timing between these two groups (Fig 3A). If we calculate the R2 value between each pair of embryos, the two different clusters vanish (Fig 3B). To understand how these embryos could be highly correlated and still cluster into two groups based on the branch distance, we considered not just the correlation of the birth time relationship, but also its slope m (see Methods). Intuitively, this slope quantifies the systematic variation in developmental timing between two embryos and can be thought of as the slope of the best-fit line to the data in Fig 2D. When this slope “m = 1”, it means that on average, each cell has similar cycle times between the two embryos; if the slope “m < 1”, that means that on average, cells in the embryo on the x-axis have cycle times that are systematically longer than cells in the embryo on the y-axis (Fig 3C). We can interpret changes in this slope as representing changes in the relative “global clock” that times cell divisions in the two embryos. We calculated this slope for each pair of WT embryos in the data (Fig 3C). Note that here the embryo on the x-axis of the heat map is also used for the x-axis of the slope calculation. In this case, we used Principal Component Analysis (PCA) rather than linear regression to estimate the slope, since in any given comparison between embryos the choice of dependent vs. independent variable would be arbitrary (see Methods). These slope calculations, along with the high correlations in Fig 3B, indicate that the primary difference between these two groups of embryos is indeed the global rate of development. In particular, the larger group of “Cluster 1” embryos develop systematically slower than the “Cluster 2” embryos. This effect is unlikely to be a result of temperature differences as some embryos in Cluster 1 were imaged at the same time as some embryos that were found to be a part of Cluster 2. Some other epigenetic factor, such as a maternal effect, may have been responsible for this difference between the two populations [24]. This analysis exemplifies how the branch distance can reveal systematic differences in the data, such as this batch effect, that a focus on correlations alone cannot identify (Fig 3). As such, the Branch Distance provides a new graph-based metric that can identify differences that regression analysis alone cannot. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. The Branch Distance Reveals previously undetected Batch Effects in WT Embryo Cell Cycle Timing. (A) Heatmap showing the union branch distance calculated between each pair of wild type embryos in the dataset. The ordering of embryos was sorted based on their assignment to two clusters computed using hierarchical clustering. (B) Heatmap showing the R2 in cell cycle times between all pairs of WT embryos, sorted as in (A). (C) The slope calculated between cell cycle times between all pairs of WT embryos, sorted as in (A). https://doi.org/10.1371/journal.pcbi.1011733.g003 The branch distance reveals heterogeneity between RNAi replicates We then computed the tree edit and branch distances between the 1352 embryos treated with RNAi against 204 genes described by Du et al. [10] We hierarchically clustered these embryos based on the union branch distance into 4 major groups (Fig 4A), where the number of partitions was decided by analysis of the union branch distance dendrogram (S1 Fig). Of these, 2 clusters shown in the upper right corner and lower left corner of Fig 4A likely represent many outliers, as these embryos are approximately as different from one another as they are from the other 2 groups. Even among the remaining 2 clusters, we observe a significant degree of heterogeneity (S2 Fig). This heterogeneity exists not just between embryos treated with RNAi against different gene targets, but also between embryos treated with RNAi against the same gene (Fig 4B). The examples in Fig 4B highlight just two patterns that we observed. In the case of embryos treated with RNAi against suf-1, three pairs of embryos exhibit distinct levels of divergence from wild type lineage topologies (as indicated by the tree edit distance, Fig 4Bi) as well as from wild type patterns of cell cycle timing (as indicated by the branch distance, Fig 4Bii). RNAi against skr-2, on the other hand, induces minor defects in lineage topology (Fig 4Biii) but a broad spectrum of defects in the distribution of cell cycle times (Fig 4Biv). Surprisingly, this variability isn’t a simple manifestation of variable phenotypic severity, as these embryos often differ from one another as much as they differ from wild type, as embryos with the same gene knocked down are not necessarily in the same cluster (S1, S2, and S3 Datasets and S2 Fig). Prior work has demonstrated that some mutant phenotypes manifest variable penetrance due to underlying variation in endogenous gene expression [25], which may contribute to the variability we observe in combination with embryo-to-embryo variation in RNAi penetrance. We note that the wide degree of heterogeneity shown illustrates that the notion of “RNAi penetrance” as a continuous variation in the efficacy of gene knock-down does not translate to a similar interpretation of phenotypic severity, as downstream effects can be complex and heterogeneous. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. The union branch distance reveals Heterogeneity in RNAi Cell Cycle Timing Coordination. (A) Heatmap showing the union branch distance between all 30 WT embryos and 1322 RNAi Embryos in the dataset. Embryos were hierarchically clustered and sorted into 4 clusters shown along the axes of the heatmap, with WT embryos visible as a “white block” in cluster 3. (B) i. Distribution of the tree edit distance between 6 SUF-1 embryos and 30 WT embryos. ii. Distribution of the intersection branch distance between 6 SUF-1 embryos and 30 WT embryos. iii. Distribution of the tree edit distance between 10 SKR-2 embryos and 30 WT embryos. iv. Distribution of the intersection branch distance between 10 SKR-2 embryos and 30 WT embryos. (C) Comparison between the tree edit distance and intersection branch distance for each of the 1322 RNAi embryos relative to a single WT reference embryo. https://doi.org/10.1371/journal.pcbi.1011733.g004 Our findings in Fig 4A and 4B show that perturbations through RNAi can impact both lineage topology and the timing of cell cycle events. To separate these effects, we chose a single representative WT embryo from Cluster 1 (Fig 3A) and used this embryo to calculate both the tree edit distance between each RNAi embryo and WT and the intersection branch distance between each RNAi embryo and WT. We chose to focus here on the intersection branch distance because it focuses on just the duration of the cell cycle events among cells that are present in both the WT and RNAi-treated embryos; the union branch distance reflects both changes in timing and topology (Fig 1). In Fig 4C, we plot the tree edit distance to a WT reference embryo for each RNAi-treated embryo on the x-axis, and the intersection branch distance to WT on the y-axis. It is immediately clear that there is a bimodal distribution of tree edit distances, with a smaller subset of RNAi embryos having WT-like lineage topologies (with tree edit distances near 0) and most RNAi perturbations having a large impact on the structure of the lineage. Interestingly, we see that there is a general lack of correlation between tree edit distance and intersection branch distance, indicating that some RNAi perturbations have a large impact on topology, but the duration of the cell cycle is similar to WT amongst lineages with preserved topologies, while other perturbations leave the topology of the lineage almost intact but have a relatively large impact on cycle duration (Fig 4C). We then examined whether RNAi against genes with related functions generated similar phenotypes based on our graph metrics. We grouped RNAi embryos together based on their functions as annotated by Du et al. [10] and observed a weak correlation between tree edit distance and intersection branch distance relative to WT (S3 Fig) although for most groups of genes, intra-class variability is greater than inter-class. We then asked whether any genes or functional classifications might be enriched in each cluster, finding that transcription factors, kinase/phosphatase, signaling and polarity related genes are overrepresented in the cluster nearest to the WT, while DNA replication/repair and mitochondria/stress genes are overrepresented in a cluster of highly heterogeneous embryos that are also the most distant from the WT embryos (S1 Fig). This is consistent with the findings of S3 Fig, which illustrates that kinase/phosphatase, signaling and polarity related genes are topologically and temporally closest to the WT, while DNA replication/repair and mitochondria/stress genes are furthest. While we cannot rule out the possibility that a portion of the observed heterogeneity among RNAi treated embryos may be a consequence of environmental or epigenetic factors as we found among WT embryos in Fig 3, the differences between WT clusters are dramatically smaller than the typical differences between RNAi treated embryos and at least a subset of treatments appear to cluster together reliably. Application of the branch distance to sublineages in WT and RNAi embryos In all the work above, we applied our metrics to the cell lineage of entire embryos. While informative, this approach ignores the fact that certain developmental processes are specific to certain sublineages and might be lost in a global analysis. For instance, previous work on developmental timing in C. elegans focused on cell-by-cell comparisons and found that while cell birth times were globally well correlated [16], the specific ordering of cell divisions within the AB lineage was variable [26]. Given these observations, we wondered whether any structure existed in the distribution of cell cycle durations within the sublineages of each individual embryo. We computed the intersection branch distance, which does not reflect differences in lineage topology, between each pair of canonical founder lineages in the early C. elegans embryo (Fig 5A). Cells in the C. elegans embryo are named based on their lineage history. A few founding cells in the early embryo possess unique names, but all cells derived from these are incrementally named according to the body axis it was born along. Cells with names containing the same number of characters following the unique name of the originating cell in the early embryo are thus born in the same generation of cell divisions and cells whose name only differs in the last character are siblings. The distribution of the intersection branch distance between each of the major differentiated lineages of the embryo show consistent patterns across the wild type samples that match the intuitive prediction that the posterior mesodermal and endodermal lineages derived from P1 are quite different from each other and from the AB lineage. We also found the same patterns reflected in the union branch distance which is sensitive to differences in topology and statistically significant differences (S4 Fig) between every pair of lineages derived from AB except for ABplp and ABprp. ABa and ABp derived lineages show distinct patterns of similarity, with the two lineages rooted at ABpl and ABpr being closer to one another than ABal and ABar, reflected in the left/right symmetric pattern of similarity in the lineages rooted at ABpxx and the lack of any such symmetry in ABaxx lineages. What is the origin of these patterns in cell cycle timing among the AB-derived sub-lineages? Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Branch Distance reveals structure in the AB lineage. A) Heatmap showing the intersection branch distance between every pair of sublineages in every pair of 21 wild type embryos. B) Illustration of the first two Notch signaling events in the early AB lineage. C) Heatmap showing a zoomed in view of the intersection branch distance between the 21 wild type embryos for each pair of AB-derived sublineages. Colormap is scaled from 0 to the max intersection branch distances between same-generation AB sublineages. D) Heatmap showing a zoomed in view of the intersection branch distance between AB-derived sublineages of 6 embryos treated with RNAi against glp-1 for each pair of AB-derived sublineages. Colormap is scaled from 0 to the max intersection branch distances between same-generation AB sublineages. E) Distributions of intersection branch distances between subsets of AB-derived sublineages in WT embryos and embryos treated with RNAi against glp-1. P-values calculated using 106 iterations of a permutation test. https://doi.org/10.1371/journal.pcbi.1011733.g005 Notch signaling modulates cell cycle duration in a lineage-specific manner Since Notch is responsible for breaking fate symmetry between the ABa and ABp lineages in the 4-cell embryo [27,28] and the pattern of branch distance distributions between the wild type AB lineages align with known Notch signaling events in the early embryo [29] (Fig 5B and 5C), we were interested in whether this pattern might be generated by Notch signaling. The role of Notch in the patterning of C. elegans developmental timing has been studied in respect to asynchrony in the timing of divisions of sister cells [11]. The patterns we observe in the WT lineage based on the branch distance suggest that this may be a broader phenomenon affecting timing patterns throughout entire lineages. In glp-1 RNAi embryos, the structure visible in the intersection branch distance between AB sublineages in wild embryos is clearly lost (Fig 5D). At the AB4 stage, the two left/right symmetric lineages produced by ABp, which received a Notch signal from P2, are closer to one another by the branch distance than the two lineages produced by ABa are to each other (Fig 5E). Lineages derived from cells that independently receive Notch induction (ABalp and ABara by MS) also have a smaller branch distance between each other than between their direct siblings (Fig 5E). Embryos treated with RNAi against glp-1 lose these differences and glp-1 RNAi produces a nearly uniform pattern of branch distances among the AB lineages. Both ABp derived lineages in the wild type and all AB derived lineages in glp-1 RNAi embryos exhibit a distinct pattern where left/right homologous lineages are closer to each other based on the branch distance than sibling lineages (Fig 5E). Perhaps Wnt signaling, which has been shown to incrementally accumulate in the posterior child of each cell division [30], continues to act to break fate symmetries between sibling cells in glp-1 depleted embryos. Does the decreased intersection branch distance between Notch-stimulated sublineages represent a consistent effect on the duration of the cell cycle or increased variability between cognate cells across sub-lineages? To answer this, we compared the overall clock speeds of the AB8 lineages, indicating that Notch affected lineages were faster compared to unstimulated lineages (S5 Fig) suggesting a consistent impact on cell cycle durations within sub-lineages, even when the stimulated sublineages are derived from separate founder cells that independently receive Notch stimulation. Preservation of cell cycle timing structure through lineage transformations A key process in the early embryo is the differentiation of cell lineages needed for the formation of different organs and body parts. The large-scale RNAi screen performed by Du et al. systematically explored these phenomena using genetic markers of tissue fate [10]. They characterized diverse genes whose depletion results in homeotic transformations, where some cell lineages adopt the pattern of tissue fates normally produced by another lineage, and genes whose loss results in patterns of tissue fates not normally seen in any wild type lineage. Most of the lineages in the wild type embryo have both qualitatively and, as we showed above in Fig 5, quantitatively distinct patterns of cell cycle times. We wondered whether patterns in cell cycle timing are a product of the same differentiation processes that define the tissue fate of these lineages. In other words, in an embryo where one lineage adopts the fate of another, does the pattern of cell cycle lengths in the transformed lineage change to match the pattern of the newly acquired fate? This question builds on a prior study which found that differences in cell cycle time between pairs of sibling cells indicated perturbations to cell fate in specific lineages [11]. We wondered whether lineage-wide transformations in fate might be detectable as a shift in cell cycle timing to match the pattern normally expressed by the acquired fate using the union branch distance. We designed a heuristic based on the branch distance to search for cases where this is true. For each homeotically transformed lineage identified by Du et al. [10] we refer to the transformed lineage as the origin and the acquired lineage fate as the destination. To account for natural variation in each of the wild type lineages, we first define a diameter D equal to the maximum pairwise branch distance between wild type examples of the destination fate (Fig 6A). We then assigned a transformation score to each origin lineage based on how many of the wild type destination lineages lie within D D minutes of the origin sub-lineage in any particular RNAi embryo (Fig 6B) and normalizing by dividing by the number of WT embryos (n = 21) used in the analysis. While most lineage transformations do not adopt the pattern of cell cycle times normally expressed by the acquired fate, we identified 95 cases where they do. Interestingly, cases where the transformed lineage falls within the neighborhood of all 21 wild type examples of the destination fate are more common than cases where the transformed lineage is proximal but not fully overlapping the neighborhood of the destination fate. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. Evidence for cell fate control over cell cycle timing. A) Illustration of the transformation heuristic. For each WT destination lineage (black dots) a diameter D is calculated as the maximum intragroup intersection branch distance. The transformation efficiency is then defined as the fraction of WT destination lineages that fall within diameter D of each RNAi origin lineage (colored squares). In some cases, the transformation efficiency is 0 but the RNAi lineage has WT origin neighbors (green square) suggesting that the RNAi perturbed lineage maintained its original fate in terms of cell cycle timing. In other cases, this value is 0 and the RNAi origin lineage has no WT neighbors (orange square) suggesting that the RNAi perturbed lineage has both lost its original fate and failed to acquire the pattern of cell cycle timing of the destination lineage. In a minority of cases the RNAi origin lineage is within D of 1 or more WT destination sublineages and a transformation efficiency is reported (magenta square). B) Histogram of the number of WT destination neighbors that homeotically transformed RNAi lineages have, using the heuristic defined in A. C) Histogram of the number of new WT neighbors that perturbed RNAi lineages have. D) Heatmaps representing the transformation heuristic in A. for homeotically transformed lineages with at least 1 WT destination neighbor. The genes that induce these transformations and functions are listed alongside the corresponding heatmap of transformation. https://doi.org/10.1371/journal.pcbi.1011733.g006 One advantage of using cell cycle timing and the branch distance as a phenotypic marker of lineage identity is that it can be assessed even in the absence of visible markers of cell fates, so we generalized our approach to measure the frequency of transformations between all possible pairs of lineages (Fig 6C). In this case, we find the nearest neighbor among possible destination fates for each origin lineage and count the number of wild type examples of the destination that fall within D minutes of the origin lineage in the RNAi treated embryo. For genes with or without homeotic transformations identified on the basis of marker gene expression by Du et al. [10], the majority of origin lineages fall outside the range of variation of all wild type destination lineages, suggesting that the patterns of cell cycle times in the wild type lineage are very sensitive to genetic perturbation. In both figures, the two most populated bins are the bin with 0 WT neighbors followed by the bin with 21 WT neighbors (Fig 6B and 6C) noting that RNAi lineages aggregate more closely region around WT neighborhoods than regions further away (S6 Fig). Still, we identified 12 genes for which RNAi generates homeotic transformations based on both marker expression by Du et al. [10] and our approach of using the branch distance. These genes belong to a small set of key pathways that have well-known roles in specifying cell fate in the early embryo including Notch, Wnt, PAR polarity genes, and the maternally derived transcription factors pie-1 and skn-1 (Fig 6D). These pathways operate to break symmetry in the early embryo. This set is likely an underestimate since we have not accounted for two common types of perturbations to cell cycle timing and lineage structure: changes to the “global clock” (since the branch distance is not scale invariant), and the partial transformation of lineages since we examined only lineages rooted at the major founder cells in the early embryo. The nature of these transformations is consistent with prior work analyzing the change of cell cycle timing in the daughters of specific Notch receptors [11]. Specifically, each Notch induced homeotic transformation produces downstream timing perturbations that shift the entire transformed lineage to match the acquired fate. Interestingly, examining each RNAi lineage on an embryo-by-embryo basis reveals striking diversity in penetrance and phenotypic consistency in homeotic transformations detected by Du et al. [10] (Fig 6D) and previously uninvestigated lineages (S4 Dataset), which list WT neighbors for RNAi lineages without annotated marker-based transformations. Several genes, such as wwp-1, pop-1, and skn-1, have sublineages that are within 1 diameter of their original and acquired fates, suggesting a degree of mixture in the neighborhoods of the third-generation descendants of the AB lineage. Furthermore, the variance in the number of embryos transformed and relative strength of each transformation suggests that RNAi penetrance and phenotypic severity are separable phenomena. In our case, we use penetrance to refer to the number of RNAi lineages that are transformed to the neighborhood of at least 1 WT reference lineage, while severity correlates with our measure of the transformation efficiency for each transformed lineage, a reflection of how close each transformed lineage is to the set of all WT reference lineages. For example, Notch pathway components apx-1, glp-1, and lag-1 all have transformations from ABp to ABa, where the transformed lineage lies within 1 diameter of a similar number of wild type examples. The more consistent presence of ABp to ABa transformations in apx-1 might imply more complete penetrance of apx-1 RNAi than that of glp-1, and lag-1, or that the knockdown of apx-1 more consistently induces temporal transformations, suggesting a more central role for apx-1 in fate specification for first-generation AB lineages. In contrast, for the E3 ligase wwp-1, the penetrance of the ABa to ABp transformation is both fairly high and occurs with a higher degree of transformation (as indicated by the fraction of wild type ABp sublineages within 1 diameter of the transformed lineage), or the transformation from MS to E in pop-1 RNAi which is only observed in 1 embryo but is a perfect match for the neighborhood of all wild type E lineages, suggesting an extremely precise transformation of identity. Generalizing the branch distance to unlabeled binary trees reveals robustness in WT lineages The principal novelty of our work, the branch distance, operates on labeled binary trees which allows one to trivially determine whether a node in one exists in another tree, and what the corresponding weights are. Node labels allow for easy comparison of corresponding nodes between trees, as without it, a space of alignments must be considered in order to calculate a distance. In the case of the generalized tree edit distance, the minimum distance of all possible alignments is considered [19]. Computing the branch distance between all possible alignments of two binary trees, selecting the smallest distance computed yields a simple generalization of the branch metric that preserves its desirable properties. The value of a generalized branch distance would be its applicability to non-invariant lineages, yet few large scale datasets exist for species beyond C. elegans that include lineage topologies and phenotypic measurements for a large number of individuals. Thus, we benchmark this generalized metric using the C. elegans AB lineage. We applied the generalized branch distance between WT C. elegans AB8 lineages, disregarding cell identity labels while iterating through all possible alignments and reporting the minimum computed branch distance. In comparing the pattern of distances produced by computing branch distances based on cell identity alignments (Fig 7A) to the pattern produced by computing generalized branch distances (Fig 7B), we find a high degree of similarity, demonstrating that the generalized branch distance is able to capture the same structure we found by computing the branch distance based on alignments that follow the C. elegans lineage. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. The Generalized Branch Distance recapitulates structure in lineage coordination. (A) Heatmap showing the intersection branch distance calculated between pairs of wild type AB8 lineages (B) Heatmap showing the generalized branch distance calculated between pairs of wild type AB8 lineages (C) Heatmap showing the percentage of cells in the lineage of one embryo that map to cells in a different position in the lineage of another embryo under the alignment produced by the generalized branch distance in (B). https://doi.org/10.1371/journal.pcbi.1011733.g007 If the minimum distance between all possible alignments is similar to the lineage-based alignment, we then asked whether the minimized alignment is itself the lineage-based alignment. To do this, we counted the percentage of cells in the embryo which have different matches between the alignment that produces the minimum branch distance, and the lineage-based alignment (Fig 7C). For pairwise alignments between lineages that have short branch distances between them, we find that the minimized alignment matches the lineage alignment very closely. Defining metrics on spaces of cell lineages Multicellular organisms develop from a single cell through a sequence of divisions. The stereotypical nature of C. elegans development makes it possible to uniquely identify every cell in its somatic lineage based on the orientation of the division of its predecessor relative to the embryo’s body axes [3]. This feature of C. elegans development has been a major advantage of its use as a model system, enabling systematic and quantitative studies of developmental processes. Here, we take advantage of the structured nature of the C. elegans cell lineage to represent it digitally as an ordered binary tree, such that nodes and edges represent cells and division events respectively. The cells in the embryo and the corresponding nodes in the tree can be labeled using the convention based on the orientation of cell divisions along body axes [3] and can be associated with quantitative measurements on a cell-by-cell basis. This natural representation of the lineage as a binary tree suggests several straightforward metrics for comparing lineages to quantify how, say, a gene knockout impacts development (Fig 1A). The first is the tree edit distance, which is derived from the graph edit distance in graph theory and is based on counting the minimum number of operations (such as adding or removing a node or edge) that is needed to convert one tree into another. Since C. elegans development is stereotypical, there is a natural alignment between any two trees based on the naming convention described above [16]. This makes computing the tree edit distance very straightforward, essentially reducing the calculation to determining the number of nodes that are different between the two trees (Fig 1B), a notion similar to other measures, such as the Robinson Foulds metric (S1 Appendix). This metric captures how perturbations like gene knockouts influence the topology of the lineage. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Defining Distance Metrics on Lineage Based Tree Structures. (A) Cell lineages can be expressed as binary trees, with parent, sibling, and child cells relationship reflected in node topology. C. elegans has a naming convention that allows for direct comparisons between cells in distinct lineages. Here we show schematics of two lineages with different topologies, corresponding to embryos “X” and “Y.” The canonical names of the cells are shown either next to or at the end of the corresponding edges. Lineage tracing data also provides information about how long each cell persists between when it is “born” through division and when it divides itself. The numerical values next to each edge indicate these cell cycle times in this schematic. (B) The tree edit distance describes the topological differences between trees by counting the number of additive/subtractive operations required to transform one tree into another. In the case of C. elegans lineage trees, this corresponds to the size of the symmetric difference between the set of nodes present in one embryo vs. another. (C) The intersection branch distance is the Euclidean or L2 norm between measurements associated with shared nodes or edges of trees, disregarding topological differences between trees by only considering nodes/edges present in both trees. (D) The union branch distance is the Euclidean or L2 norm between values on the union set of nodes or edges between trees. Nodes or edges that are absent from one tree in any comparison are given a 0 value. https://doi.org/10.1371/journal.pcbi.1011733.g001 The second is a class of metrics that can be computed on any measurement associated with individual cells within the tree, such as gene expression or time. Again, here we use the stereotyped nature of C. elegans development to directly align any two lineage trees on a cell-by-cell basis, meaning that a single and unique distance can be calculated for any comparison between lineages. Given this alignment, any numerical property of cells during development can be unambiguously converted into a vector representation (Fig 1C). This allows us to use any metric on such vectors to compare the trees. The most straightforward metric that might be used is simply the Euclidean distance (i.e., the L2 norm). Here we focus on the application of the L2 norm to comparisons of cellular division timing between embryos, which has been shown to vary under genetic perturbation [16], and is a measurement produced by every method for lineage tracing by cell tracking. We call this metric the “branch distance” since these division timings represent the length of the branches in the tree (Fig 1C). Of course, a gene knockout or mutation might impact both the topology of the tree and the timing of cell divisions. To account for this, we need to have a way for dealing with cases where a cell/node exists in one tree, but not in the other (Fig 1C). Here, we define two different types of branch distance to account for this problem. The first is the intersection branch distance, where the vector of division timings is constructed only on the basis of cells that are shared between two lineages (i.e., they are in the intersection of the list of cells in the two) (Fig 1D). The second is the union branch distance, where we simply set the cell cycle timing of any cell that is missing from any embryo to 0 and calculate the distance in the normal way (Fig 1D). The union branch distance thus captures differences in both timing and topology, as topological differences are reflected in the absence of cells and thus cycle times. Unpaired cell cycle times are squared and added to the branch distance, as the existence of a pair to that cell would only add the square of the difference of those values. This is functionally equivalent to imputing a zero for missing nodes. As described below, these two metrics capture different aspects of variation between trees. The branch distance reveals unexpected batch effects in WT embryos The most straightforward method for cell lineage tracing is via direct observation and cell tracking. Even in the absence of visible reporters, this approach inherently generates both spatial (3D cell positions and cell trajectories) and temporal (the timing of cell divisions) measurements of the embryo. The distribution of individual cell cycle times within lineages have been deeply explored at a single cell resolution [11,12,22], making hierarchically structured lineage data an attractive target for analysis via our branch metrics. During lineage tracing experiments, every cell in developing embryos is tracked, and each cell division can be mapped to a particular time t, with t = 0 corresponding to, say, the first division of the zygote. There are thus two ways of thinking about the “branch length” value for each cell in the tree (Fig 1C). In one scenario, we could label each cell with its “birth time,” which is just the time t at which the cell was generated through a division event. Another alternative is to consider the “cycle time” for that cell, which is just the length of time between when the cell is born through a division event until it divides itself. Prior work has claimed that developmental timing in C. elegans is highly coordinated, drawn primarily on comparisons of cell birth times [16,17]. In particular, Bao et al. showed that birth times for cognate cells from different embryos are highly correlated, with R2 values that range between 0.995 and 0.997 (Fig 2A) [16]. While comparing birth times between embryos seems natural, there is a potential issue with that approach. In particular, the birth time of a given cell is the sum of the cell cycle times of all the previous division events (Fig 1C). Since there is some randomness in these cycle times, we can think of those times as random variables, noting that summing over random variables always reduces variation [23]. In other words, the “birth time” is essentially equivalent to averaging the previous cycle times, and averaging generally suppresses variation (i.e., the standard error of the mean is generally less than the standard deviation). In addition, because the birth times are a sum of previous cycle times, birth times for cells born later in development will always be larger than birth times for cells born earlier. Both effects can spuriously increase the correlation in cell birth times between embryos. To demonstrate this, we completely randomized the cell cycle times in the embryo, intentionally destroying any correlation in the length of cell cycles for the same cells across each randomized embryo (see Methods). After randomizing these cycle times, we found birth time correlations with R2 values between 0.65 and 0.85 (Fig 2B), despite a complete absence of correlation in the individual cell cycle times (Fig 2C). Thus, while the cycle times are still highly correlated between WT embryos (R2 between 0.97 and 0.99, Fig 2D), the correlation is less than we observe with birth times. Since using the cycle time avoids spurious correlations and reveals more variation and structure in the data (Fig 2D), we focused on using the cycle time to calculate branch distances in this work. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Summation of Cycle Times into Birth Times Suppresses Variation. (A) A comparison between the birth time of each cell (calculated as the sum of cell cycle times of each cell’s ancestors) in two wild type C. elegans embryos. (B) Comparison between birth times calculated from two randomly shuffled wild type embryos, where each cell is assigned another random cell’s birth time from within the lineage of the same embryo. Note that a significant correlation in birth times exists even in this shuffled data. (C) Comparison between shuffled cell cycle times rather than birth times. In this case, there is no correlation, as would be expected. (D) Comparison between the cell cycle times of each cell in two wild type embryos. Note that the same two embryos were used for all comparisons in panels A-D. https://doi.org/10.1371/journal.pcbi.1011733.g002 We used the cycle times to calculate the branch distance between each pair of 30 WT embryos with cycle times taken from lineage tracing data from Du et al. [10] We then hierarchically clustered the embryos based on these distances. Surprisingly, clustering on the branch distance revealed two distinct previously unknown populations of WT embryos in the published dataset, representing differences in cell cycle timing between these two groups (Fig 3A). If we calculate the R2 value between each pair of embryos, the two different clusters vanish (Fig 3B). To understand how these embryos could be highly correlated and still cluster into two groups based on the branch distance, we considered not just the correlation of the birth time relationship, but also its slope m (see Methods). Intuitively, this slope quantifies the systematic variation in developmental timing between two embryos and can be thought of as the slope of the best-fit line to the data in Fig 2D. When this slope “m = 1”, it means that on average, each cell has similar cycle times between the two embryos; if the slope “m < 1”, that means that on average, cells in the embryo on the x-axis have cycle times that are systematically longer than cells in the embryo on the y-axis (Fig 3C). We can interpret changes in this slope as representing changes in the relative “global clock” that times cell divisions in the two embryos. We calculated this slope for each pair of WT embryos in the data (Fig 3C). Note that here the embryo on the x-axis of the heat map is also used for the x-axis of the slope calculation. In this case, we used Principal Component Analysis (PCA) rather than linear regression to estimate the slope, since in any given comparison between embryos the choice of dependent vs. independent variable would be arbitrary (see Methods). These slope calculations, along with the high correlations in Fig 3B, indicate that the primary difference between these two groups of embryos is indeed the global rate of development. In particular, the larger group of “Cluster 1” embryos develop systematically slower than the “Cluster 2” embryos. This effect is unlikely to be a result of temperature differences as some embryos in Cluster 1 were imaged at the same time as some embryos that were found to be a part of Cluster 2. Some other epigenetic factor, such as a maternal effect, may have been responsible for this difference between the two populations [24]. This analysis exemplifies how the branch distance can reveal systematic differences in the data, such as this batch effect, that a focus on correlations alone cannot identify (Fig 3). As such, the Branch Distance provides a new graph-based metric that can identify differences that regression analysis alone cannot. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. The Branch Distance Reveals previously undetected Batch Effects in WT Embryo Cell Cycle Timing. (A) Heatmap showing the union branch distance calculated between each pair of wild type embryos in the dataset. The ordering of embryos was sorted based on their assignment to two clusters computed using hierarchical clustering. (B) Heatmap showing the R2 in cell cycle times between all pairs of WT embryos, sorted as in (A). (C) The slope calculated between cell cycle times between all pairs of WT embryos, sorted as in (A). https://doi.org/10.1371/journal.pcbi.1011733.g003 The branch distance reveals heterogeneity between RNAi replicates We then computed the tree edit and branch distances between the 1352 embryos treated with RNAi against 204 genes described by Du et al. [10] We hierarchically clustered these embryos based on the union branch distance into 4 major groups (Fig 4A), where the number of partitions was decided by analysis of the union branch distance dendrogram (S1 Fig). Of these, 2 clusters shown in the upper right corner and lower left corner of Fig 4A likely represent many outliers, as these embryos are approximately as different from one another as they are from the other 2 groups. Even among the remaining 2 clusters, we observe a significant degree of heterogeneity (S2 Fig). This heterogeneity exists not just between embryos treated with RNAi against different gene targets, but also between embryos treated with RNAi against the same gene (Fig 4B). The examples in Fig 4B highlight just two patterns that we observed. In the case of embryos treated with RNAi against suf-1, three pairs of embryos exhibit distinct levels of divergence from wild type lineage topologies (as indicated by the tree edit distance, Fig 4Bi) as well as from wild type patterns of cell cycle timing (as indicated by the branch distance, Fig 4Bii). RNAi against skr-2, on the other hand, induces minor defects in lineage topology (Fig 4Biii) but a broad spectrum of defects in the distribution of cell cycle times (Fig 4Biv). Surprisingly, this variability isn’t a simple manifestation of variable phenotypic severity, as these embryos often differ from one another as much as they differ from wild type, as embryos with the same gene knocked down are not necessarily in the same cluster (S1, S2, and S3 Datasets and S2 Fig). Prior work has demonstrated that some mutant phenotypes manifest variable penetrance due to underlying variation in endogenous gene expression [25], which may contribute to the variability we observe in combination with embryo-to-embryo variation in RNAi penetrance. We note that the wide degree of heterogeneity shown illustrates that the notion of “RNAi penetrance” as a continuous variation in the efficacy of gene knock-down does not translate to a similar interpretation of phenotypic severity, as downstream effects can be complex and heterogeneous. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. The union branch distance reveals Heterogeneity in RNAi Cell Cycle Timing Coordination. (A) Heatmap showing the union branch distance between all 30 WT embryos and 1322 RNAi Embryos in the dataset. Embryos were hierarchically clustered and sorted into 4 clusters shown along the axes of the heatmap, with WT embryos visible as a “white block” in cluster 3. (B) i. Distribution of the tree edit distance between 6 SUF-1 embryos and 30 WT embryos. ii. Distribution of the intersection branch distance between 6 SUF-1 embryos and 30 WT embryos. iii. Distribution of the tree edit distance between 10 SKR-2 embryos and 30 WT embryos. iv. Distribution of the intersection branch distance between 10 SKR-2 embryos and 30 WT embryos. (C) Comparison between the tree edit distance and intersection branch distance for each of the 1322 RNAi embryos relative to a single WT reference embryo. https://doi.org/10.1371/journal.pcbi.1011733.g004 Our findings in Fig 4A and 4B show that perturbations through RNAi can impact both lineage topology and the timing of cell cycle events. To separate these effects, we chose a single representative WT embryo from Cluster 1 (Fig 3A) and used this embryo to calculate both the tree edit distance between each RNAi embryo and WT and the intersection branch distance between each RNAi embryo and WT. We chose to focus here on the intersection branch distance because it focuses on just the duration of the cell cycle events among cells that are present in both the WT and RNAi-treated embryos; the union branch distance reflects both changes in timing and topology (Fig 1). In Fig 4C, we plot the tree edit distance to a WT reference embryo for each RNAi-treated embryo on the x-axis, and the intersection branch distance to WT on the y-axis. It is immediately clear that there is a bimodal distribution of tree edit distances, with a smaller subset of RNAi embryos having WT-like lineage topologies (with tree edit distances near 0) and most RNAi perturbations having a large impact on the structure of the lineage. Interestingly, we see that there is a general lack of correlation between tree edit distance and intersection branch distance, indicating that some RNAi perturbations have a large impact on topology, but the duration of the cell cycle is similar to WT amongst lineages with preserved topologies, while other perturbations leave the topology of the lineage almost intact but have a relatively large impact on cycle duration (Fig 4C). We then examined whether RNAi against genes with related functions generated similar phenotypes based on our graph metrics. We grouped RNAi embryos together based on their functions as annotated by Du et al. [10] and observed a weak correlation between tree edit distance and intersection branch distance relative to WT (S3 Fig) although for most groups of genes, intra-class variability is greater than inter-class. We then asked whether any genes or functional classifications might be enriched in each cluster, finding that transcription factors, kinase/phosphatase, signaling and polarity related genes are overrepresented in the cluster nearest to the WT, while DNA replication/repair and mitochondria/stress genes are overrepresented in a cluster of highly heterogeneous embryos that are also the most distant from the WT embryos (S1 Fig). This is consistent with the findings of S3 Fig, which illustrates that kinase/phosphatase, signaling and polarity related genes are topologically and temporally closest to the WT, while DNA replication/repair and mitochondria/stress genes are furthest. While we cannot rule out the possibility that a portion of the observed heterogeneity among RNAi treated embryos may be a consequence of environmental or epigenetic factors as we found among WT embryos in Fig 3, the differences between WT clusters are dramatically smaller than the typical differences between RNAi treated embryos and at least a subset of treatments appear to cluster together reliably. Application of the branch distance to sublineages in WT and RNAi embryos In all the work above, we applied our metrics to the cell lineage of entire embryos. While informative, this approach ignores the fact that certain developmental processes are specific to certain sublineages and might be lost in a global analysis. For instance, previous work on developmental timing in C. elegans focused on cell-by-cell comparisons and found that while cell birth times were globally well correlated [16], the specific ordering of cell divisions within the AB lineage was variable [26]. Given these observations, we wondered whether any structure existed in the distribution of cell cycle durations within the sublineages of each individual embryo. We computed the intersection branch distance, which does not reflect differences in lineage topology, between each pair of canonical founder lineages in the early C. elegans embryo (Fig 5A). Cells in the C. elegans embryo are named based on their lineage history. A few founding cells in the early embryo possess unique names, but all cells derived from these are incrementally named according to the body axis it was born along. Cells with names containing the same number of characters following the unique name of the originating cell in the early embryo are thus born in the same generation of cell divisions and cells whose name only differs in the last character are siblings. The distribution of the intersection branch distance between each of the major differentiated lineages of the embryo show consistent patterns across the wild type samples that match the intuitive prediction that the posterior mesodermal and endodermal lineages derived from P1 are quite different from each other and from the AB lineage. We also found the same patterns reflected in the union branch distance which is sensitive to differences in topology and statistically significant differences (S4 Fig) between every pair of lineages derived from AB except for ABplp and ABprp. ABa and ABp derived lineages show distinct patterns of similarity, with the two lineages rooted at ABpl and ABpr being closer to one another than ABal and ABar, reflected in the left/right symmetric pattern of similarity in the lineages rooted at ABpxx and the lack of any such symmetry in ABaxx lineages. What is the origin of these patterns in cell cycle timing among the AB-derived sub-lineages? Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Branch Distance reveals structure in the AB lineage. A) Heatmap showing the intersection branch distance between every pair of sublineages in every pair of 21 wild type embryos. B) Illustration of the first two Notch signaling events in the early AB lineage. C) Heatmap showing a zoomed in view of the intersection branch distance between the 21 wild type embryos for each pair of AB-derived sublineages. Colormap is scaled from 0 to the max intersection branch distances between same-generation AB sublineages. D) Heatmap showing a zoomed in view of the intersection branch distance between AB-derived sublineages of 6 embryos treated with RNAi against glp-1 for each pair of AB-derived sublineages. Colormap is scaled from 0 to the max intersection branch distances between same-generation AB sublineages. E) Distributions of intersection branch distances between subsets of AB-derived sublineages in WT embryos and embryos treated with RNAi against glp-1. P-values calculated using 106 iterations of a permutation test. https://doi.org/10.1371/journal.pcbi.1011733.g005 Notch signaling modulates cell cycle duration in a lineage-specific manner Since Notch is responsible for breaking fate symmetry between the ABa and ABp lineages in the 4-cell embryo [27,28] and the pattern of branch distance distributions between the wild type AB lineages align with known Notch signaling events in the early embryo [29] (Fig 5B and 5C), we were interested in whether this pattern might be generated by Notch signaling. The role of Notch in the patterning of C. elegans developmental timing has been studied in respect to asynchrony in the timing of divisions of sister cells [11]. The patterns we observe in the WT lineage based on the branch distance suggest that this may be a broader phenomenon affecting timing patterns throughout entire lineages. In glp-1 RNAi embryos, the structure visible in the intersection branch distance between AB sublineages in wild embryos is clearly lost (Fig 5D). At the AB4 stage, the two left/right symmetric lineages produced by ABp, which received a Notch signal from P2, are closer to one another by the branch distance than the two lineages produced by ABa are to each other (Fig 5E). Lineages derived from cells that independently receive Notch induction (ABalp and ABara by MS) also have a smaller branch distance between each other than between their direct siblings (Fig 5E). Embryos treated with RNAi against glp-1 lose these differences and glp-1 RNAi produces a nearly uniform pattern of branch distances among the AB lineages. Both ABp derived lineages in the wild type and all AB derived lineages in glp-1 RNAi embryos exhibit a distinct pattern where left/right homologous lineages are closer to each other based on the branch distance than sibling lineages (Fig 5E). Perhaps Wnt signaling, which has been shown to incrementally accumulate in the posterior child of each cell division [30], continues to act to break fate symmetries between sibling cells in glp-1 depleted embryos. Does the decreased intersection branch distance between Notch-stimulated sublineages represent a consistent effect on the duration of the cell cycle or increased variability between cognate cells across sub-lineages? To answer this, we compared the overall clock speeds of the AB8 lineages, indicating that Notch affected lineages were faster compared to unstimulated lineages (S5 Fig) suggesting a consistent impact on cell cycle durations within sub-lineages, even when the stimulated sublineages are derived from separate founder cells that independently receive Notch stimulation. Preservation of cell cycle timing structure through lineage transformations A key process in the early embryo is the differentiation of cell lineages needed for the formation of different organs and body parts. The large-scale RNAi screen performed by Du et al. systematically explored these phenomena using genetic markers of tissue fate [10]. They characterized diverse genes whose depletion results in homeotic transformations, where some cell lineages adopt the pattern of tissue fates normally produced by another lineage, and genes whose loss results in patterns of tissue fates not normally seen in any wild type lineage. Most of the lineages in the wild type embryo have both qualitatively and, as we showed above in Fig 5, quantitatively distinct patterns of cell cycle times. We wondered whether patterns in cell cycle timing are a product of the same differentiation processes that define the tissue fate of these lineages. In other words, in an embryo where one lineage adopts the fate of another, does the pattern of cell cycle lengths in the transformed lineage change to match the pattern of the newly acquired fate? This question builds on a prior study which found that differences in cell cycle time between pairs of sibling cells indicated perturbations to cell fate in specific lineages [11]. We wondered whether lineage-wide transformations in fate might be detectable as a shift in cell cycle timing to match the pattern normally expressed by the acquired fate using the union branch distance. We designed a heuristic based on the branch distance to search for cases where this is true. For each homeotically transformed lineage identified by Du et al. [10] we refer to the transformed lineage as the origin and the acquired lineage fate as the destination. To account for natural variation in each of the wild type lineages, we first define a diameter D equal to the maximum pairwise branch distance between wild type examples of the destination fate (Fig 6A). We then assigned a transformation score to each origin lineage based on how many of the wild type destination lineages lie within D D minutes of the origin sub-lineage in any particular RNAi embryo (Fig 6B) and normalizing by dividing by the number of WT embryos (n = 21) used in the analysis. While most lineage transformations do not adopt the pattern of cell cycle times normally expressed by the acquired fate, we identified 95 cases where they do. Interestingly, cases where the transformed lineage falls within the neighborhood of all 21 wild type examples of the destination fate are more common than cases where the transformed lineage is proximal but not fully overlapping the neighborhood of the destination fate. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. Evidence for cell fate control over cell cycle timing. A) Illustration of the transformation heuristic. For each WT destination lineage (black dots) a diameter D is calculated as the maximum intragroup intersection branch distance. The transformation efficiency is then defined as the fraction of WT destination lineages that fall within diameter D of each RNAi origin lineage (colored squares). In some cases, the transformation efficiency is 0 but the RNAi lineage has WT origin neighbors (green square) suggesting that the RNAi perturbed lineage maintained its original fate in terms of cell cycle timing. In other cases, this value is 0 and the RNAi origin lineage has no WT neighbors (orange square) suggesting that the RNAi perturbed lineage has both lost its original fate and failed to acquire the pattern of cell cycle timing of the destination lineage. In a minority of cases the RNAi origin lineage is within D of 1 or more WT destination sublineages and a transformation efficiency is reported (magenta square). B) Histogram of the number of WT destination neighbors that homeotically transformed RNAi lineages have, using the heuristic defined in A. C) Histogram of the number of new WT neighbors that perturbed RNAi lineages have. D) Heatmaps representing the transformation heuristic in A. for homeotically transformed lineages with at least 1 WT destination neighbor. The genes that induce these transformations and functions are listed alongside the corresponding heatmap of transformation. https://doi.org/10.1371/journal.pcbi.1011733.g006 One advantage of using cell cycle timing and the branch distance as a phenotypic marker of lineage identity is that it can be assessed even in the absence of visible markers of cell fates, so we generalized our approach to measure the frequency of transformations between all possible pairs of lineages (Fig 6C). In this case, we find the nearest neighbor among possible destination fates for each origin lineage and count the number of wild type examples of the destination that fall within D minutes of the origin lineage in the RNAi treated embryo. For genes with or without homeotic transformations identified on the basis of marker gene expression by Du et al. [10], the majority of origin lineages fall outside the range of variation of all wild type destination lineages, suggesting that the patterns of cell cycle times in the wild type lineage are very sensitive to genetic perturbation. In both figures, the two most populated bins are the bin with 0 WT neighbors followed by the bin with 21 WT neighbors (Fig 6B and 6C) noting that RNAi lineages aggregate more closely region around WT neighborhoods than regions further away (S6 Fig). Still, we identified 12 genes for which RNAi generates homeotic transformations based on both marker expression by Du et al. [10] and our approach of using the branch distance. These genes belong to a small set of key pathways that have well-known roles in specifying cell fate in the early embryo including Notch, Wnt, PAR polarity genes, and the maternally derived transcription factors pie-1 and skn-1 (Fig 6D). These pathways operate to break symmetry in the early embryo. This set is likely an underestimate since we have not accounted for two common types of perturbations to cell cycle timing and lineage structure: changes to the “global clock” (since the branch distance is not scale invariant), and the partial transformation of lineages since we examined only lineages rooted at the major founder cells in the early embryo. The nature of these transformations is consistent with prior work analyzing the change of cell cycle timing in the daughters of specific Notch receptors [11]. Specifically, each Notch induced homeotic transformation produces downstream timing perturbations that shift the entire transformed lineage to match the acquired fate. Interestingly, examining each RNAi lineage on an embryo-by-embryo basis reveals striking diversity in penetrance and phenotypic consistency in homeotic transformations detected by Du et al. [10] (Fig 6D) and previously uninvestigated lineages (S4 Dataset), which list WT neighbors for RNAi lineages without annotated marker-based transformations. Several genes, such as wwp-1, pop-1, and skn-1, have sublineages that are within 1 diameter of their original and acquired fates, suggesting a degree of mixture in the neighborhoods of the third-generation descendants of the AB lineage. Furthermore, the variance in the number of embryos transformed and relative strength of each transformation suggests that RNAi penetrance and phenotypic severity are separable phenomena. In our case, we use penetrance to refer to the number of RNAi lineages that are transformed to the neighborhood of at least 1 WT reference lineage, while severity correlates with our measure of the transformation efficiency for each transformed lineage, a reflection of how close each transformed lineage is to the set of all WT reference lineages. For example, Notch pathway components apx-1, glp-1, and lag-1 all have transformations from ABp to ABa, where the transformed lineage lies within 1 diameter of a similar number of wild type examples. The more consistent presence of ABp to ABa transformations in apx-1 might imply more complete penetrance of apx-1 RNAi than that of glp-1, and lag-1, or that the knockdown of apx-1 more consistently induces temporal transformations, suggesting a more central role for apx-1 in fate specification for first-generation AB lineages. In contrast, for the E3 ligase wwp-1, the penetrance of the ABa to ABp transformation is both fairly high and occurs with a higher degree of transformation (as indicated by the fraction of wild type ABp sublineages within 1 diameter of the transformed lineage), or the transformation from MS to E in pop-1 RNAi which is only observed in 1 embryo but is a perfect match for the neighborhood of all wild type E lineages, suggesting an extremely precise transformation of identity. Generalizing the branch distance to unlabeled binary trees reveals robustness in WT lineages The principal novelty of our work, the branch distance, operates on labeled binary trees which allows one to trivially determine whether a node in one exists in another tree, and what the corresponding weights are. Node labels allow for easy comparison of corresponding nodes between trees, as without it, a space of alignments must be considered in order to calculate a distance. In the case of the generalized tree edit distance, the minimum distance of all possible alignments is considered [19]. Computing the branch distance between all possible alignments of two binary trees, selecting the smallest distance computed yields a simple generalization of the branch metric that preserves its desirable properties. The value of a generalized branch distance would be its applicability to non-invariant lineages, yet few large scale datasets exist for species beyond C. elegans that include lineage topologies and phenotypic measurements for a large number of individuals. Thus, we benchmark this generalized metric using the C. elegans AB lineage. We applied the generalized branch distance between WT C. elegans AB8 lineages, disregarding cell identity labels while iterating through all possible alignments and reporting the minimum computed branch distance. In comparing the pattern of distances produced by computing branch distances based on cell identity alignments (Fig 7A) to the pattern produced by computing generalized branch distances (Fig 7B), we find a high degree of similarity, demonstrating that the generalized branch distance is able to capture the same structure we found by computing the branch distance based on alignments that follow the C. elegans lineage. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. The Generalized Branch Distance recapitulates structure in lineage coordination. (A) Heatmap showing the intersection branch distance calculated between pairs of wild type AB8 lineages (B) Heatmap showing the generalized branch distance calculated between pairs of wild type AB8 lineages (C) Heatmap showing the percentage of cells in the lineage of one embryo that map to cells in a different position in the lineage of another embryo under the alignment produced by the generalized branch distance in (B). https://doi.org/10.1371/journal.pcbi.1011733.g007 If the minimum distance between all possible alignments is similar to the lineage-based alignment, we then asked whether the minimized alignment is itself the lineage-based alignment. To do this, we counted the percentage of cells in the embryo which have different matches between the alignment that produces the minimum branch distance, and the lineage-based alignment (Fig 7C). For pairwise alignments between lineages that have short branch distances between them, we find that the minimized alignment matches the lineage alignment very closely. Methods Wild type and RNAi treated embryonic lineage data was retrieved as text files with each row corresponding to an individual cell in the lineage tree. Lineage relationships were reconstructed from the cell names, which are structured according to a common convention where a unique root cell ID indicates the identity of the founder cell of the lineage and each subsequent pair of cells is named according to the body axis along which its division was polarized. Cell cycle duration was extracted based on the number of columns associated with each cell from the data provided by Du et al. [10] used to report tissue specific transgenic reporter signal intensity. Each column represents the intensity measure made for each timepoint of imaging where the corresponding cell existed, thus we calculate the duration of each cell’s cell cycle as 1.25 minutes per time point, based on the imaging frequency reported. A total of 30 wild type and 1322 RNAi treated embryos were retrieved and time-resolved lineage trees were generated from these raw data. While all wild type embryos covered a uniform set of cells, RNAi treated embryos were only partially curated by Du et al. [10] to validate reporter expression. In order to address these discrepancies, we truncated each of these lineage trees based on the time cutoffs provided alongside the raw data and wild type embryos were pruned similarly for distance calculations. Each WT lineage is truncated based on the annotations provided by Du et al. [10] with RNAi lineages following gene and lineage specific cutoff instructions. The implementation of our data import and pre-processing is available alongside a complete codebase implementing our distance metrics and analysis routines at (https://github.com/shahlab-ucla/graph_distances)) Graph based distance metrics applied to C. elegans lineages In order to store the lineage data for each embryo as a binary tree, we take advantage of the naming convention for each cell using a standard hash table (or “dictionary” in python) data structure. A cell would be stored in the dictionary with name/reference X (i.e. the cell’s name) and an element value representing its cell cycle time. The children of X would have the name/reference of X followed by suffix ‘a’, ‘l’, or ‘d’, representing anterior, left, and dorsal orientations of division respectively. This cell would have a corresponding sibling name/reference of X with a suffix ‘p’, ‘r’, or ‘v’ representing a posterior, right, or ventral division relative to its sibling. For instance, a cell in the data set might have the name “ABal.” This represents a cell descended from the AB cell in the embryo, where it is the left daughter of the anterior cell of the first AB division. With all cells in the embryo following this convention in the dictionary, any cell and all of its ancestors can be referenced by looking at the cell name and truncating its suffix one letter at a time. The tree edit distance is a metric defined by counting “the minimum number of node deletions, insertions, and replacements that are necessary to transform one tree into another” as a measure of topological distance between trees (Fig 1A). This can be applied to the dictionaries that we use as proxies for graph structures. If a tree has a specific node not in another embryo, then a corresponding node must be inserted into the lacking embryo as a descendant of the appropriate shared node to produce topologically identical trees. Thus, a single operation has taken place to transform the structure of one tree into another. This approach can be generalized to describe any tree-based topological differences, as discussed in S1 Appendix. Using the dictionary format allows us to take advantage of the naming convention. Any cell that is added contains information pertaining to the connection to its parent nodes, allowing for trivial checks of hierarchical and topological relationships. Indeed, this can be expressed further by noting that addition/subtraction operations of nodes can be represented by the absence/presence of nodes in one embryo that is not in the other. Extending this concept allows us to calculate the number of transformation operations as the number of nodes that are in one, but not both, dictionary sets. This means that tree edit distance between two dictionaries with nodes under the naming convention of Sulston et al [3]. is defined as the magnitude of the intersection set subtracted from the magnitude of the union set of the dictionaries (in other words, it is size of the symmetric difference between the sets of nodes). In terms of the python implementation, this is calculated as the length of the XOR set of cells between the two embryos. It is utilized in Fig 4B (Looking at SUF-1 and SKR-2 RNAi embryos tree edit distance to WT stereotype) and in Fig 4C (Tree edit distance from each RNA embryo to WT stereotype Plotted on x axis). In order to compare the trees in terms of the division timings, we introduce the concept of the branch distance. We define the branch distance as the Euclidean or L2 norm of a vectorized representation of each lineage under comparison. To generate the vector, cells within each lineage are aligned first on the basis of depth from the root cell of each lineage and then on the lineage name derived from division orientation. In other words, we determine the components of each vector in such a way that division times for one cell are always being compared to division times for that same cell in a different embryo. When we calculate the L2 norm, the difference between the values ascribed to a cell in one embryo and the corresponding cell in another embryo is taken and squared. Summing up these values and then taking the square root allows for an extension of the Euclidean norm to these weighted graphs. To compensate for alternate topologies, we computed one of two variants of the branch distance. The Intersection branch distance only computes the distance on the intersection set of cells contained in both lineages (Fig 1B), treating values that are not shared as absent from the comparison. It is used in Fig 3A to look at distances between all 30 WT embryos, and hierarchically cluster them into 2 groups, with the larger group of 21 embryos representing the Wild Type in all future calculations (unless otherwise noted). It is also used in Fig 4B (Looking at SUF-1 and SKR-2 RNAi embryos intersection branch distance to 21 WT embryos), Fig 4C (Average branch distance from each RNAi embryos to 21 WT embryos Plotted on x axis), and Fig 7A (computing intersection branch distance between AB8 sublineages). Meanwhile, the Union branch distance treats any missing cells as having a cell cycle duration of 0 (Fig 1C). Thus, the Union branch distance compensates for differences in topology by directly adding the squares of values of cells without counterparts to the distance value, increasing it depending on topological variance and the value of the missing node. It is used to calculate the distance matrix between all WT and RNAi embryos (Fig 4A). In comparing any two trees with any of these metrics, we note that the metric should work on subtrees or trees with different root nodes. This necessitates a change to the naming convention in cases where we compared different sublineages. This is done by finding the root node of both subtrees and assigning them an arbitrary letter. In cases where descendants of a root node are to be compared but have different orientations of division, we treat ‘a’, ‘l’, and ‘d’ suffix letters as equivalent as well as ‘p’, ‘r’, and ‘v’. For example, the values of subtree [‘A’, ‘Aa’, ‘Ap’] and the values of subtree [‘B’,’Bl’,’Br’], if roots were normalized, would both have the naming convention [‘Q’,’Qa’,’Qp’]. Utilizing this convention allows us to apply the metrics described above to compute the distances between distinct sublineages. In Fig 5, this is used to compute the intersection branch distance between different sublineages in 21 selected embryos WT and all 7 glp-1 knocked down embryos. In Fig 6, this is used to compute union branch distances between sublineages of 21 WT embryos and sublineages of RNAi embryos. Our initial implementation of the branch distance assumes a unique alignment between the trees under comparison, as is enabled by the invariant lineage of C. elegans. In order to generalize this metric to cases where the optimal alignment is unknown, we adopt the approach utilized by the generalized graph edit distance [19]. Simply put, we compute the branch distance for all possible tree alignments and select the minimum computed value as the generalized distance. Note that there are several key constraints on this alignment. For one, we require that nodes only be aligned to nodes of the same depth in the other tree; a leaf node of one tree can’t be aligned with the root node of the other tree. Secondly, if a node in one tree is aligned to a node in another, the child nodes must be aligned with one another as well. In other words, if we align node X from tree T1 to node Y in tree T2, then the child nodes of X can only be aligned to the child nodes of Y. This ensures that the alignments respect the topological structure of the tree. Proof of the generalizability of the branch distance is included in S1 Appendix. We thus apply the generalized branch distance to exhaustively search through all alignments of the WT AB8 in Fig 7B. Calculating correlation of timing events between embryos In previous work, some authors have used an alternative to the cell cycle time for comparing the timing of division events between embryos. Specifically, Bao et al. [16] compared embryos using the “cell birth time,” defined as the time from the fertilization of the embryo to the birth time of a cell. It can be calculated as the sum of the cell cycle times of the ancestors of a cell. Previous authors have found extremely high correlations between different embryos using this birth time definition. Since a cell’s birth time is the sum of all previous division timings, comparing embryos using this parameter could suppress variation and introduce spurious correlations between embryos. A sum of random variables will often show less variation than the underlying variables themselves–this is the reason the “standard error of the mean” is generally less than the underlying standard deviation in the population. To test this, we shuffled all of the division times in the embryos in question. Specifically, we randomly assigned each cell in an embryo to the cell timing parameter of a different cell in the same embryo, effectively removing any correlation between the division timing of cells in any embryo while still preserving the underlying distribution of cell cycle times that can be produced. A simple method of comparing the differences in cell timing events (Fig 2) is by plotting the times for each cell of one embryo against the times for each corresponding cell of another embryo. We then calculated the linear correlation coefficient between the cell cycle times between the cells of two embryos (Figs 2D and 3B) as well as the correlation coefficient between shuffled cell cycle times (Fig 2C). Shuffled birth times are computed by calculating the sum of the Shuffled cycle times of all ancestors of a particular cell and were also compared using the correlation coefficient (Fig 2B). Our analysis clearly demonstrates a significant correlation in cell birth times even in the shuffled data. As such, our subsequent analyses focused on comparing embryos using the cell cycle times. Computing the time between WT embryos Our analysis in Figs 2 and 3A suggested that there are two distinct groups of WT embryos in the Du et al. data [10]. While the correlation between cell cycle times is lower than cell birth times (Fig 2), we nonetheless saw fairly high correlations between embryos of the two groups, despite their distinct branch distances (Fig 3A and 3B). We thus hypothesized that the difference between the two groups was due to a uniform rescaling of time–in other words, all of the division events in one group of embryos were likely slower than the events in the other group of embryos by a constant factor. The plots in Fig 2 suggest a straightforward way to quantify this difference in timing: the slope of the timings in one embryo vs. another. If this slope is less than 1, this suggests that the embryo whose times are plotted on the x-axis develops slower than the one plotted on the y-axis; if the slope is greater than 1, that suggests the reverse. A natural way to estimate this slope would be to simply perform a linear regression between the two data sets. Doing so, however, involves selecting one set of timings as the “independent variable.” Since both sets of timings in any comparison is subject to random variation, however, we chose a slightly different approach to calculating the slope. To do this, we employed simple Principal Component Analysis (PCA) on each pair of embryos. The eigenvector corresponding to the largest eigenvalue corresponds to a line that best fits the principal axis of variation in the data. In all the embryo comparisons, this axis of variation corresponds naturally to the line that compares the cell cycle times between the two embryos (e.g., Fig 2D). We thus performed PCA on each pair of embryos with cell cycle times plotted against one another as in Fig 2D. The slope of this best fit line was then calculated by comparing the resulting principal eigenvector to the standard basis (i.e., calculating the “rise over run” for the eigenvector in the plane of Fig 2D). This method is used in Fig 3C to find the cell cycle scaling by comparing the cell cycle times of all 30 WT embryos against each other and partitioning the embryos into the clusters indicated in Fig 3A. These findings confirmed our hypothesis, indicating that the “group 1” embryos develop about 20% slower than the group 2 embryos. Clustering wild type and mutant embryos We generated a distance matrix consisting of all pairwise union branch distances between WT and mutant embryos. We then performed single linkage hierarchical clustering on this distance matrix to generate a dendrogram between the embryos Since the number of clusters must be selected before the clustering is performed in hierarchical clustering, we analyzed the dendrogram to find a point with a large distance between generations [see S1 Fig for further details]. In the case of Fig 3, this approach partitioned the WT embryos into two groups. In the case of Fig 4, this approach resulted in 4 distinct clusters. Note that the distance matrices in Fig 5 were not clustered in order to show the pattern of variation between sublineages. Nonparametric permutation significance testing for distributions of distances We found that the intersection branch distances between certain sublineages of WT embryos were generally smaller than the intersection branch distances between other sublineages. This difference seemed to be related to Notch signaling events during development (Fig 5A, 5B, and 5C). We used a simple permutation test to evaluate the statistical significance of this observation. In this test, we had two sets of distances: for instance, we compared the distances between ABal and ABar to the distances between ABpl and ABpr. This data corresponded to an observed difference between the means of the distribution. We then pooled the datasets together and generated randomized datasets of the same sizes by sampling without replacement. We calculated the difference of means between these randomized datasets. The p-values reported in Fig 5 represent the number of random cases where the absolute value of the difference in the means in these randomized datasets was greater than or equal to the observed difference. Detecting fate transformations for mutant embryos Homeotic transformations that were identified by Du et al. [10] indicate cases where a sublineage of a RNAi embryo combinations of marker genes present more consistent with a different sublineage of the WT embryo. This transformation indicates that the RNAi treatment has transformed the cell-fate pattern from that typical of one sublineage (the “origin”) into that of a different sublineage (the “destination”). A list of the transformations that take place is available at digital-development.org/download.html under the name “Excel spreadsheet of all homeotic transformation phenotypes”. In our work, we considered whether these cell fate transformations also had an impact on cell cycle timing. To do this, we developed an approach to see whether one sublineage in an RNAi embryo was “close” to a different lineage in the WT embryos. Our approach represents the WT sublineage populations as point clouds in high dimensional space. Each such point cloud has associated with it the maximum distance between WT embryos in the population; we call this maximum distance the “diameter” of that sublineage (Fig 6A). In this case, we chose the larger group of 21 embryos with similar developmental timings that naturally cluster together in Fig 3A to avoid higher diameters that might arise from systematic differences in experimental conditions. For any given RNAi embryo, we then compute the distance between each of its sublineages and each sublineage from each of the 21 WT embryos. Note that in this case we use the intersection branch distance. If the distance between an RNAi sublineage and a WT sublineage from any embryo is less than the diameter of the WT sublineage, we say that the RNAi sublineage is in the neighborhood of the WT sublineage from that embryo. If the RNAi sublineage is in the neighborhood of that same lineage in the WT embryos, then we say that the sublineage is “unperturbed.” In other words, the origin and destination lineage for that sublineage is the same (in the sense of the neighborhood described above). If the RNAi lineage is in the neighborhood of a different lineage in the WT embryo, we say that that sublineage has been transformed (i.e., the origin and destination are different). If an RNAi sublineage is not in the neighborhood of any WT sublineage, then we say the sublineage is “perturbed” (Fig 6A). The degree of transformation/perturbation is quantified based on the number of origin/destination WT sublineages that are neighbors of each RNAi sublineage (Fig 6D). Using the transformation framework outlined in Fig 6A, we then used a bootstrapping method to determine the significance of the distributions of Fig 6B and 6C. Specifically, we tried to determine whether lineages with randomly chosen cell cycle times lengths contain a number of WT neighbors comparable to homeotically transformed and other RNAi lineages. In other words, we iteratively created “Null embryos” with all cell cycle times randomly assigned to a different cell (Fig 2) and computed the number of WT neighbors to the null embryos to find the degree to which completely randomized embryo times correspond with the Wild Type Embryo. In every homeotically transformed lineage, the length of every cell’s cycle is shuffled among all cells of the same name across all embryos treated with RNAi against genes that produce homeotic transformations. For each of these shuffled lineages, the number of WT lineage neighbors (out of 21) is counted (S6B Fig). Along with homeotic transformations, we also shuffled all other RNAi lineages, where the length of every cell’s cycle is shuffled among all cells of the same name across all embryos treated with RNAi, and counted the WT neighbors for each lineage (S6F Fig). We then repeated these shuffling 10000 times and counted the number of lineages with 21 WT neighbors (S6C and S6G Fig) along with the number of lineages with at least 1 WT neighbor (S6D and S6H Fig). Graph based distance metrics applied to C. elegans lineages In order to store the lineage data for each embryo as a binary tree, we take advantage of the naming convention for each cell using a standard hash table (or “dictionary” in python) data structure. A cell would be stored in the dictionary with name/reference X (i.e. the cell’s name) and an element value representing its cell cycle time. The children of X would have the name/reference of X followed by suffix ‘a’, ‘l’, or ‘d’, representing anterior, left, and dorsal orientations of division respectively. This cell would have a corresponding sibling name/reference of X with a suffix ‘p’, ‘r’, or ‘v’ representing a posterior, right, or ventral division relative to its sibling. For instance, a cell in the data set might have the name “ABal.” This represents a cell descended from the AB cell in the embryo, where it is the left daughter of the anterior cell of the first AB division. With all cells in the embryo following this convention in the dictionary, any cell and all of its ancestors can be referenced by looking at the cell name and truncating its suffix one letter at a time. The tree edit distance is a metric defined by counting “the minimum number of node deletions, insertions, and replacements that are necessary to transform one tree into another” as a measure of topological distance between trees (Fig 1A). This can be applied to the dictionaries that we use as proxies for graph structures. If a tree has a specific node not in another embryo, then a corresponding node must be inserted into the lacking embryo as a descendant of the appropriate shared node to produce topologically identical trees. Thus, a single operation has taken place to transform the structure of one tree into another. This approach can be generalized to describe any tree-based topological differences, as discussed in S1 Appendix. Using the dictionary format allows us to take advantage of the naming convention. Any cell that is added contains information pertaining to the connection to its parent nodes, allowing for trivial checks of hierarchical and topological relationships. Indeed, this can be expressed further by noting that addition/subtraction operations of nodes can be represented by the absence/presence of nodes in one embryo that is not in the other. Extending this concept allows us to calculate the number of transformation operations as the number of nodes that are in one, but not both, dictionary sets. This means that tree edit distance between two dictionaries with nodes under the naming convention of Sulston et al [3]. is defined as the magnitude of the intersection set subtracted from the magnitude of the union set of the dictionaries (in other words, it is size of the symmetric difference between the sets of nodes). In terms of the python implementation, this is calculated as the length of the XOR set of cells between the two embryos. It is utilized in Fig 4B (Looking at SUF-1 and SKR-2 RNAi embryos tree edit distance to WT stereotype) and in Fig 4C (Tree edit distance from each RNA embryo to WT stereotype Plotted on x axis). In order to compare the trees in terms of the division timings, we introduce the concept of the branch distance. We define the branch distance as the Euclidean or L2 norm of a vectorized representation of each lineage under comparison. To generate the vector, cells within each lineage are aligned first on the basis of depth from the root cell of each lineage and then on the lineage name derived from division orientation. In other words, we determine the components of each vector in such a way that division times for one cell are always being compared to division times for that same cell in a different embryo. When we calculate the L2 norm, the difference between the values ascribed to a cell in one embryo and the corresponding cell in another embryo is taken and squared. Summing up these values and then taking the square root allows for an extension of the Euclidean norm to these weighted graphs. To compensate for alternate topologies, we computed one of two variants of the branch distance. The Intersection branch distance only computes the distance on the intersection set of cells contained in both lineages (Fig 1B), treating values that are not shared as absent from the comparison. It is used in Fig 3A to look at distances between all 30 WT embryos, and hierarchically cluster them into 2 groups, with the larger group of 21 embryos representing the Wild Type in all future calculations (unless otherwise noted). It is also used in Fig 4B (Looking at SUF-1 and SKR-2 RNAi embryos intersection branch distance to 21 WT embryos), Fig 4C (Average branch distance from each RNAi embryos to 21 WT embryos Plotted on x axis), and Fig 7A (computing intersection branch distance between AB8 sublineages). Meanwhile, the Union branch distance treats any missing cells as having a cell cycle duration of 0 (Fig 1C). Thus, the Union branch distance compensates for differences in topology by directly adding the squares of values of cells without counterparts to the distance value, increasing it depending on topological variance and the value of the missing node. It is used to calculate the distance matrix between all WT and RNAi embryos (Fig 4A). In comparing any two trees with any of these metrics, we note that the metric should work on subtrees or trees with different root nodes. This necessitates a change to the naming convention in cases where we compared different sublineages. This is done by finding the root node of both subtrees and assigning them an arbitrary letter. In cases where descendants of a root node are to be compared but have different orientations of division, we treat ‘a’, ‘l’, and ‘d’ suffix letters as equivalent as well as ‘p’, ‘r’, and ‘v’. For example, the values of subtree [‘A’, ‘Aa’, ‘Ap’] and the values of subtree [‘B’,’Bl’,’Br’], if roots were normalized, would both have the naming convention [‘Q’,’Qa’,’Qp’]. Utilizing this convention allows us to apply the metrics described above to compute the distances between distinct sublineages. In Fig 5, this is used to compute the intersection branch distance between different sublineages in 21 selected embryos WT and all 7 glp-1 knocked down embryos. In Fig 6, this is used to compute union branch distances between sublineages of 21 WT embryos and sublineages of RNAi embryos. Our initial implementation of the branch distance assumes a unique alignment between the trees under comparison, as is enabled by the invariant lineage of C. elegans. In order to generalize this metric to cases where the optimal alignment is unknown, we adopt the approach utilized by the generalized graph edit distance [19]. Simply put, we compute the branch distance for all possible tree alignments and select the minimum computed value as the generalized distance. Note that there are several key constraints on this alignment. For one, we require that nodes only be aligned to nodes of the same depth in the other tree; a leaf node of one tree can’t be aligned with the root node of the other tree. Secondly, if a node in one tree is aligned to a node in another, the child nodes must be aligned with one another as well. In other words, if we align node X from tree T1 to node Y in tree T2, then the child nodes of X can only be aligned to the child nodes of Y. This ensures that the alignments respect the topological structure of the tree. Proof of the generalizability of the branch distance is included in S1 Appendix. We thus apply the generalized branch distance to exhaustively search through all alignments of the WT AB8 in Fig 7B. Calculating correlation of timing events between embryos In previous work, some authors have used an alternative to the cell cycle time for comparing the timing of division events between embryos. Specifically, Bao et al. [16] compared embryos using the “cell birth time,” defined as the time from the fertilization of the embryo to the birth time of a cell. It can be calculated as the sum of the cell cycle times of the ancestors of a cell. Previous authors have found extremely high correlations between different embryos using this birth time definition. Since a cell’s birth time is the sum of all previous division timings, comparing embryos using this parameter could suppress variation and introduce spurious correlations between embryos. A sum of random variables will often show less variation than the underlying variables themselves–this is the reason the “standard error of the mean” is generally less than the underlying standard deviation in the population. To test this, we shuffled all of the division times in the embryos in question. Specifically, we randomly assigned each cell in an embryo to the cell timing parameter of a different cell in the same embryo, effectively removing any correlation between the division timing of cells in any embryo while still preserving the underlying distribution of cell cycle times that can be produced. A simple method of comparing the differences in cell timing events (Fig 2) is by plotting the times for each cell of one embryo against the times for each corresponding cell of another embryo. We then calculated the linear correlation coefficient between the cell cycle times between the cells of two embryos (Figs 2D and 3B) as well as the correlation coefficient between shuffled cell cycle times (Fig 2C). Shuffled birth times are computed by calculating the sum of the Shuffled cycle times of all ancestors of a particular cell and were also compared using the correlation coefficient (Fig 2B). Our analysis clearly demonstrates a significant correlation in cell birth times even in the shuffled data. As such, our subsequent analyses focused on comparing embryos using the cell cycle times. Computing the time between WT embryos Our analysis in Figs 2 and 3A suggested that there are two distinct groups of WT embryos in the Du et al. data [10]. While the correlation between cell cycle times is lower than cell birth times (Fig 2), we nonetheless saw fairly high correlations between embryos of the two groups, despite their distinct branch distances (Fig 3A and 3B). We thus hypothesized that the difference between the two groups was due to a uniform rescaling of time–in other words, all of the division events in one group of embryos were likely slower than the events in the other group of embryos by a constant factor. The plots in Fig 2 suggest a straightforward way to quantify this difference in timing: the slope of the timings in one embryo vs. another. If this slope is less than 1, this suggests that the embryo whose times are plotted on the x-axis develops slower than the one plotted on the y-axis; if the slope is greater than 1, that suggests the reverse. A natural way to estimate this slope would be to simply perform a linear regression between the two data sets. Doing so, however, involves selecting one set of timings as the “independent variable.” Since both sets of timings in any comparison is subject to random variation, however, we chose a slightly different approach to calculating the slope. To do this, we employed simple Principal Component Analysis (PCA) on each pair of embryos. The eigenvector corresponding to the largest eigenvalue corresponds to a line that best fits the principal axis of variation in the data. In all the embryo comparisons, this axis of variation corresponds naturally to the line that compares the cell cycle times between the two embryos (e.g., Fig 2D). We thus performed PCA on each pair of embryos with cell cycle times plotted against one another as in Fig 2D. The slope of this best fit line was then calculated by comparing the resulting principal eigenvector to the standard basis (i.e., calculating the “rise over run” for the eigenvector in the plane of Fig 2D). This method is used in Fig 3C to find the cell cycle scaling by comparing the cell cycle times of all 30 WT embryos against each other and partitioning the embryos into the clusters indicated in Fig 3A. These findings confirmed our hypothesis, indicating that the “group 1” embryos develop about 20% slower than the group 2 embryos. Clustering wild type and mutant embryos We generated a distance matrix consisting of all pairwise union branch distances between WT and mutant embryos. We then performed single linkage hierarchical clustering on this distance matrix to generate a dendrogram between the embryos Since the number of clusters must be selected before the clustering is performed in hierarchical clustering, we analyzed the dendrogram to find a point with a large distance between generations [see S1 Fig for further details]. In the case of Fig 3, this approach partitioned the WT embryos into two groups. In the case of Fig 4, this approach resulted in 4 distinct clusters. Note that the distance matrices in Fig 5 were not clustered in order to show the pattern of variation between sublineages. Nonparametric permutation significance testing for distributions of distances We found that the intersection branch distances between certain sublineages of WT embryos were generally smaller than the intersection branch distances between other sublineages. This difference seemed to be related to Notch signaling events during development (Fig 5A, 5B, and 5C). We used a simple permutation test to evaluate the statistical significance of this observation. In this test, we had two sets of distances: for instance, we compared the distances between ABal and ABar to the distances between ABpl and ABpr. This data corresponded to an observed difference between the means of the distribution. We then pooled the datasets together and generated randomized datasets of the same sizes by sampling without replacement. We calculated the difference of means between these randomized datasets. The p-values reported in Fig 5 represent the number of random cases where the absolute value of the difference in the means in these randomized datasets was greater than or equal to the observed difference. Detecting fate transformations for mutant embryos Homeotic transformations that were identified by Du et al. [10] indicate cases where a sublineage of a RNAi embryo combinations of marker genes present more consistent with a different sublineage of the WT embryo. This transformation indicates that the RNAi treatment has transformed the cell-fate pattern from that typical of one sublineage (the “origin”) into that of a different sublineage (the “destination”). A list of the transformations that take place is available at digital-development.org/download.html under the name “Excel spreadsheet of all homeotic transformation phenotypes”. In our work, we considered whether these cell fate transformations also had an impact on cell cycle timing. To do this, we developed an approach to see whether one sublineage in an RNAi embryo was “close” to a different lineage in the WT embryos. Our approach represents the WT sublineage populations as point clouds in high dimensional space. Each such point cloud has associated with it the maximum distance between WT embryos in the population; we call this maximum distance the “diameter” of that sublineage (Fig 6A). In this case, we chose the larger group of 21 embryos with similar developmental timings that naturally cluster together in Fig 3A to avoid higher diameters that might arise from systematic differences in experimental conditions. For any given RNAi embryo, we then compute the distance between each of its sublineages and each sublineage from each of the 21 WT embryos. Note that in this case we use the intersection branch distance. If the distance between an RNAi sublineage and a WT sublineage from any embryo is less than the diameter of the WT sublineage, we say that the RNAi sublineage is in the neighborhood of the WT sublineage from that embryo. If the RNAi sublineage is in the neighborhood of that same lineage in the WT embryos, then we say that the sublineage is “unperturbed.” In other words, the origin and destination lineage for that sublineage is the same (in the sense of the neighborhood described above). If the RNAi lineage is in the neighborhood of a different lineage in the WT embryo, we say that that sublineage has been transformed (i.e., the origin and destination are different). If an RNAi sublineage is not in the neighborhood of any WT sublineage, then we say the sublineage is “perturbed” (Fig 6A). The degree of transformation/perturbation is quantified based on the number of origin/destination WT sublineages that are neighbors of each RNAi sublineage (Fig 6D). Using the transformation framework outlined in Fig 6A, we then used a bootstrapping method to determine the significance of the distributions of Fig 6B and 6C. Specifically, we tried to determine whether lineages with randomly chosen cell cycle times lengths contain a number of WT neighbors comparable to homeotically transformed and other RNAi lineages. In other words, we iteratively created “Null embryos” with all cell cycle times randomly assigned to a different cell (Fig 2) and computed the number of WT neighbors to the null embryos to find the degree to which completely randomized embryo times correspond with the Wild Type Embryo. In every homeotically transformed lineage, the length of every cell’s cycle is shuffled among all cells of the same name across all embryos treated with RNAi against genes that produce homeotic transformations. For each of these shuffled lineages, the number of WT lineage neighbors (out of 21) is counted (S6B Fig). Along with homeotic transformations, we also shuffled all other RNAi lineages, where the length of every cell’s cycle is shuffled among all cells of the same name across all embryos treated with RNAi, and counted the WT neighbors for each lineage (S6F Fig). We then repeated these shuffling 10000 times and counted the number of lineages with 21 WT neighbors (S6C and S6G Fig) along with the number of lineages with at least 1 WT neighbor (S6D and S6H Fig). Discussion Analyses of the structure of lineages in biology have focused principally on the construction of phylogenies [14,15] and measurements of inter-node distances within individual trees [31]. This approach has been shaped by the requirements of taxonomic work, where no ground truth topology exists, and multiple measures of distance might be employed. Cell lineages, on the other hand, have clearly defined structure, and recent work has explored strategies for measuring differences in tree topology [18]. Recently, techniques from spectral analysis were applied to phenotypic measures aligned to cell lineages, including in C. elegans, but with an emphasis on characterizing these phenotypes in the context of lineages with variable structure [32]. Automated cell lineage tracing is an increasingly mature technology, having been applied to C. elegans [12,32,33] Drosophila [34], zebrafish [35], and mouse [36] development as well as to the study of lineage relationships in stem cell [37] and immune cell [38] culture systems. A limited set of metrics have been applied to the comparison of cell lineages [18,32], in part driven by the unordered nature of cell lineages reconstructed in most organisms. In the case of C. elegans lineages, for which there are now public repositories containing measured lineages from thousands of wild type and perturbed embryos [10,13,39], the ordered and stereotypical nature of its somatic lineage removes the need to align the lineages, thus dramatically simplifying the application of graph-based approaches to the problem of quantitatively comparing lineage trees. We thus applied the intuitive graph-theoretic notion of the tree edit distance [19], and its extension in the branch distance, to dissect the structure of C. elegans embryonic lineage. We note that the tree edit distance, when computed on labeled binary trees, is operationally identical to the Robinson-Foulds metric, as each of these quantifies the magnitude of the exclusive disjunction between elements in two trees [40]. These metrics allowed us to uncover previously unknown heterogeneity between populations of wild type embryos, to quantify the variability of RNAi-induced lineage phenotypes, and to shed light on key mechanisms of patterning in early embryogenesis, and expand on analyses of RNAi induced phenotypic changes [11,12]. Most prior analyses of developmental timing in the C. elegans embryo used the time of a cell’s birth relative to an early reference point (ex. the first division of the zygote) and correlation between the birth time of the same cell across embryos as a measure of developmental similarity. We showed here that these two choices mask heterogeneity present in previously published records of wild type development. We wondered then whether these effects, combined with 1-to-1 comparisons of timing between the same cell across multiple embryos may have obscured patterns in developmental timing across lineages within the embryo. Using the tremendous volume of existing lineage data available from the work of Du et al [10], we sought to benchmark our metrics on wild type and RNAi-perturbed embryos and explore whether a lineage-centric view of developmental timing may reveal previously unappreciated patterns. It is well known that RNAi, especially by feeding in C. elegans, induces phenotypes with variable penetrance [41]. It has been shown that variable penetrance in mutants may occur due to underlying heterogeneity in gene expression [25]. Our graph metrics show that phenotypic variability under RNAi can exhibit a wide range of patterns of severity. This includes patterns that correspond to linear gradients of severity, multimodal distributions of phenotypes, and apparently random variation between individual embryos (Fig 4B). These measurements show that, for many genes, RNAi induces variability between individual embryos that is often on par with the phenotypic distance between wild type and individual RNAi embryos. Our approach expands on prior work which tracks changes through phenotypes in individual cells [11,12] noting that our approach allows for multimodal measurements, allows for rigorous distance-based bioinformatics analyses, and consolidates lineage descent data. Taking advantage of the ordered nature of cell divisions in the C. elegans embryo to align arbitrary pairs lineages within the embryo, we sought to characterize the structure of cell cycle duration in the wild type lineage. We were especially surprised to find reproducible patterns within the lineages derived from AB, the larger of two cells born from the first asymmetric division of the zygote. In C. elegans, the major patterns of cell fate that are established by intercellular Notch signaling are well known, and the pattern of branch distances between the AB-derived lineages we observed aligns perfectly to the first two Notch signaling events in the early embryo. RNAi against Notch/glp-1 abolishes this structure, demonstrating that this pattern of cell cycle timing in the AB lineage is a product of Notch signaling. Lineages that receive Notch signals also exhibit on average shorter cell cycle lengths than lineages that do not (S5 Fig). Biophysical parameters such as cell volume affect cell cycle duration [42–46], but genetic regulation of subtle differences in cell cycle timing may occur via many potential mechanisms. Du et al. [9,10] demonstrated using transcriptional reporters of tissue fate that the loss of any one of many genes essential for development can induce homeotic transformations between the major founder lineages in the early embryo [9,10]. We set out to determine, using the branch distance, whether developmental timing in transformed lineages is independent of lineage fate, is transformed along with fate, or is lost upon fate perturbation. We devised a simple heuristic to assess the proximity of RNAi-treated origin lineages to the wild type destination lineage that expresses the closest pattern of cell fates as defined by Du et al. [10] by counting the number of wild type examples of the destination lineage that are less than the maximum inter-wild type branch distance away from the RNAi origin lineage. Using this conservative approach, which would fail to detect transformations in cases where the global embryo clock is altered or where subsets of individual lineages are transformed, we find that only a handful of genes (12 genes out of 204 characterized) induce homeotic fate transformations where developmental timing in the transformed lineages also transforms to match that of the newly acquired fate. This set is composed of genes in the Notch and Wnt pathways, two PAR polarity genes, and the maternally derived transcription factors skn-1 and pie-1. The fact that most perturbations produce homeotic transformations generate patterns of cell cycle duration that match neither that of the original wild type lineage or the newly acquired fate suggests that lineage-specific developmental timing is likely quite sensitive to genetic perturbation. When homeotic transformations in cell cycle timing do occur, a perfect match to wild type lineages outnumbers incomplete matches suggesting that, despite its sensitivity to perturbation, the wild type patterns of cell cycle timing may represent stable states. This tool could be extended to any quantitative cell specific measurement in any tracked lineage, extending the definition and modalities used to measure cell fate. Our generalized formulation of the branch distance expands this capability by allowing for its use in cases where no intuitive alignment exists between pairs of cell lineages, for example in the development of non-eutelic animals. Our analysis demonstrates that the genetic identity of cell lineages can reproducibly and finely tune the distribution of cell cycle duration within cell lineages. It is interesting that the pathways that preserve lineage-specific developmental timing across homeotic transformations are known to play a critical role in cell fate specifications upstream of most tissue-specific transcriptional programs. It is thus likely that either a specific subset of factors downstream of fate regulators or finely tuned expression levels of tissue-specific genes are required for the proper patterning of cell cycle duration within lineages. Whether this tuning is itself a functional element of the developmental program remains unclear. Perturbations to key cell cycle regulators generate dramatic changes in cell cycle duration as well as homeotic fate transformations in C. elegans [47–50], and changes in the duration of the cell cycle of stem cells in other systems are correlated with specific cell fates such as in the generation of bipolar cells during retinal development [51]. Our results demonstrate a precise relationship between cell fate and developmental timing that motivates revisiting gaps in our understanding of links between cell cycle regulation and cell fate control. More broadly, our findings highlight the ways in which quantitative analysis of phenotypic similarity can reveal unexpected structure in animal development. In particular, the use of pairwise distance metrics applied to lineage-resolve metrics allows for an intuitive extension of notions of cell state and identity. Reducing these multidimensional data types using such intuitive measures of distance simplifies the application of common data exploration and visualization strategies. Supporting information S1 Fig. Structure in the distance between all WT and RNAi embryos. Shown is a dendrogram constructed using the union branch distance measured between all WT and RNAi treated embryos along with the 4 classes we partition the dataset into. The WT embryos and WT-like RNAi-treated embryos are highlighted in cluster 3. Significantly Overrepresented Genes and Functional Classifications in each cluster are listed here. P values were calculated with Boschloo’s test and significance was determined based on a Bonferroni corrected threshold of 1.5 * 10(-4). Raw data are available in supplemental datasets S1 and S3. https://doi.org/10.1371/journal.pcbi.1011733.s001 (TIF) S2 Fig. RNAi embryos are far more dispersed than WT embryos. (A) The distribution of union branch distances between wild type embryos (red) and between embryos treated with RNAi against the same gene (blue). Densities were generated using a kernel density estimator. (B) Strip plot of all RNAi embryos showing the branch distances between embryos treated with RNAi against the same gene. Embryos treated with RNAi against genes with shared functions are grouped together. The median branch distance within each set of embryos is plotted as a large circle. https://doi.org/10.1371/journal.pcbi.1011733.s002 (TIF) S3 Fig. Correlation between the tree edit distance and branch distance between RNAi treated embryos and a single WT reference embryo. The mean (circle) and minimum/maximum values (blue lines) of the distance between embryos treated with RNAi against genes with common function and a single WT reference embryo are shown. https://doi.org/10.1371/journal.pcbi.1011733.s003 (TIF) S4 Fig. All but one WT sublineages possess unique patterns of cell cycle timing. (A) A heatmap showing the union branch distance between each pair of sub-lineages across 21 WT reference embryos. (B) Heatmap showing the p-value calculated using 441 permutation tests (see methods) against the null hypothesis that sublineages share a common distribution of intersection branch distances. All but a single comparison (ABprp vs ABpla) are significant based on a Bonferroni corrected threshold of 2 * 10−5. https://doi.org/10.1371/journal.pcbi.1011733.s004 (TIF) S5 Fig. Notch induction systematically shortens cell cycles in affected sublineages. (A) A heatmap showing the fraction of cells in the lineage listed along the Y axis that have a shorter cell cycle than the corresponding cell in the lineage along the X axis. Each sublineage contains 32 cells so a lineage in which >16 cells possess a shorter cell cycle is considered to be “faster” (B) The distribution of comparisons from the heatmap in panel A grouped based on lineages that do not have a history of Notch activation (green, ABala and ABarp), lineages derived from ABa in which Notch is activated (red, ABalp and ABara), and all lineages derived from ABp. https://doi.org/10.1371/journal.pcbi.1011733.s005 (TIF) S6 Fig. Significantly more transformed lineages match the cell cycle timing of their ectopic fate than random. (A) The number of WT destination lineages that fall within the transformed neighborhood of lineage annotated as homeotically transformed by Du et al. [10]. See Main Fig 6B. (B) The number of WT destination lineages that fall within the transformed neighborhood of homeotically transformed lineages where the length of every cell’s cycle is shuffled among all cells of the same name across all embryos treated with RNAi against genes that produce homeotic transformations. Note there are no WT neighbors to any of these shuffled lineages. (C) The number of cases where lineages from RNAi treated embryos that are shuffled as in (B) fall within the neighborhood of all 21 WT samples of the destination lineage. The red line shows the corresponding value for the unshuffled data. (D) The number of cases where lineages from RNAi treated embryos that are shuffled as in (B) fall within the neighborhood of 1 or more WT samples of the destination lineage. The red line shows the corresponding value for the unshuffled data. (E) The number of WT destination lineages that fall within the transformed neighborhood of all sublineages from all RNAi treated embryos. See Main Fig 6C. (F) The number of WT destination lineages that fall within the transformed neighborhood of all lineages from RNAi treated embryos where the length of every cell’s cycle is shuffled among all cells of the same name. (G) The number of cases where lineages from RNAi treated embryos that are shuffled as in (F) fall within the neighborhood of all 21 WT samples of any lineage. The red line shows the corresponding value for the unshuffled data. (H) The number of cases where lineages from RNAi treated embryos that are shuffled as in (F) fall within the neighborhood of 1 or more sample of any lineage. The red line shows the corresponding value for the unshuffled data. https://doi.org/10.1371/journal.pcbi.1011733.s006 (TIF) S1 Dataset. Number of RNAi embryos in each Cluster. Text file listing the number, type, and p-value analyzing whether RNAi embryos with specific genes knocked down were significantly overrepresented in each cluster, from the hierarchically clustered union distance matrix (Fig 4A) between all embryos and its corresponding dendrogram (S1 Fig). P-values were calculated with Boschloo’s Test. https://doi.org/10.1371/journal.pcbi.1011733.s007 (TXT) S2 Dataset. List of Genes and the corresponding clusters. Text file listing each gene and the Clusters its corresponding RNAi embryo belongs to, from the hierarchically clustered union distance matrix (Fig 4A) between all embryos and its corresponding dendrogram (S1 Fig). https://doi.org/10.1371/journal.pcbi.1011733.s008 (TXT) S3 Dataset. List of clusters and significant functional classifications. Text file listing each cluster, the functional classifications of all genes in the cluster, and an enrichment analysis analyzing whether each functional classification is significantly overrepresented in that cluster. P-values were calculated with Boschloo’s Test (S1 Fig.) https://doi.org/10.1371/journal.pcbi.1011733.s009 (TXT) S4 Dataset. List of All Found Transformations. Text file listing each Gene, its corresponding RNAi embryos and sublineages, along with listing each RNAi sublineages WT neighbor names and frequencies (Fig 6C). https://doi.org/10.1371/journal.pcbi.1011733.s010 (TXT) S1 Appendix. Proof that the generalized branch distance is a metric and comparisons between the tree edit distance and the Robinson-Foulds distance. https://doi.org/10.1371/journal.pcbi.1011733.s011 (PDF) Acknowledgments The authors would like to thank Dr. Zhou Du and Dr. Anthony Santella for guidance in parsing their lineage data, and Dr. Roy Wollman and Dr. Alex Hoffman for their feedback and advice, along with present and former members of the Deeds and Shah Labs.