PLoS Computational Biology

Cascante-Vega, Jaime;Galanti, Marta;Schley, Katharina;Pei, Sen;Shaman, Jeffrey

doi: 10.1371/journal.pcbi.1011564pmid: 37889910

Introduction Invasive meningococcal disease (IMD) is caused by the bacterium Neisseria meningitidis (N. meningitidis). IMD has a rapid progression and can cause pneumonia, meningitis, and bloodstream infection. The case-fatality rate of IMD is estimated between 10% and 15%, and 20% of individuals who survive infection have lifelong disabilities, including vision and hearing loss, neurological deficits, and limb loss [1,2]. While meningococcal disease affects all age groups, infections are reported predominantly in infants, young adults, and adults over 85 years old [3], and, in temperate regions, cases are predominant in winter and spring months [4]. Nasopharyngeal colonization with N. meningitidis in healthy individuals is relatively common: reports show 5% to 35% of the population are carriers [5,6]. The frequency of carriage depends on age and peaks in young adults [3]. In the vast majority of cases, carriage is harmless to the host, but in some instances, shortly after colonization, the pathogen enters the bloodstream and causes invasive disease [5]. Transmission of N. meningitidis to a susceptible individual happens through contact with the respiratory droplets or saliva of a colonized or infected host. Due to the sensitivity of the bacteria to atmospheric conditions, transmission requires close contact [7]. There are 12 identified serogroups of N. meningitidis, but only 5 of them—A, B, C, W, Y—are responsible for almost all cases of invasive disease. Two types of vaccines are currently available in the United States and recommended by the US Advisory Committee on Immunization Practices (ACIP) of the Centers for Disease Control and Prevention (CDC): Meningococcal conjugate (MenACWY) vaccine, routinely recommended for primary immunization in 11–12 year olds since 2005 [8] followed by a booster at 16 years of age, which was introduced in 2011 [9]; and vaccines that protect against serogroup B meningococcal (MenB) bacteria and are recommended as a shared-clinical decision making recommendation in 2015 [10]. Rates of meningococcal disease have declined in the US during the last two decades and have remained low in recent years (0.11 cases per 100,000 population in 2019) [11]. Given the severity of IMD, it is extremely important to monitor its epidemic trends and to identify changes in carriage prevalence and vaccination rates that might lead to rapid disease resurgence. Here we develop a suite of mechanistic and statistical models to simulate the transmission of N. meningitidis and forecast IMD incidence. We evaluate the retrospective accuracy of forecasting IMD at the country scale using case reports from 2006 to 2020 for the US. Specifically, we show that a model-inference system based on a combination of mechanistic models and Bayesian inference methods successfully captures IMD dynamics during the last 14 years in the US and is able to forecast future disease outcomes. Similar model-inference systems have been used for parameter estimation, evaluation of counterfactual interventions, and forecast for a variety of diseases, including influenza, SARS-CoV-2, West Nile Virus [12], malaria, dengue [13], and methicillin-resistant Staphylococcus aureus [14–20]. The optimized models developed here can be used to forecast IMD cases at the country scale and to estimate the effects of control measures, human behavior, or pathogen biology, such as drops in vaccine uptake, changes in mixing patterns across a population, or the emergence of a more virulent meningococcal strain. The present analysis is intended as validation and assessment of the N. meningitidis model performance, which will be leveraged in future work to study the impact of the SARS-COV-2 pandemic and vaccinations on IMD incidence. Materials and methods Data description Weekly IMD incidence data at the state and national level in the US were compiled from the CDC Wonder dataset [21]. We used data from 2006 to 2022, as only yearly cumulative counts were reported before 2006. Classification of cases by serogroup (ACWY and B) only began in 2020 and was not used in this study; rather, incidence includes all serogroups. Spectral analysis We used wavelet time series analysis to capture the temporal properties of IMD in the US [22]. The objective of this analysis is to represent the IMD time series in both the time and frequency domains and reveal shifts in seasonality or other periodicities. We used the Morlet wavelet function as the basis. Similar analyses have been previously used to explore the seasonality of measles and influenza [23]. We investigate the fit of the inverse wavelet transform (IWT), as well as study the periodicity of the system averaging the local wavelet power spectrum (LWPS) across the study period. Model description We developed three mathematical mechanistic models and one purely statistical model–the autoregressive integrated moving average model (ARIMA). The three mechanistic models aim to represent the underlying transmission process of the disease, have the same core model structure shown in Fig 1, and are described in detail in S1 Text. Briefly, for the mechanistic models, we compartmentalize the population (N) into 3 groups: susceptible/non-carrier individuals (S), carriers of the bacteria (C) that are colonized with N. meningitidis but not infected with IMD, and carriers who have become infected with IMD (I) (Table 1). Fig 1 shows the transition between the compartments: susceptible individuals (S) become carriers with a force of infection λ (S→C), a fraction θ of carriers become infected with IMD at the rate α2 (C→I) and the proportion of C that don’t develop infection become susceptible again with the decolonization rate α1 (C→S). We model the force of infection λ using the law of mass action with contact rate β, and assume both carriers (C) and infected (I) contribute to transmission: λ = β(C+I)/N. Description of the parameters, units and value used in presented in Table 2. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Model diagram. Compartmental model of Neisseria Meningitidis transmission. S is the susceptible/non-carrier population, C is the carrier population, and I is the infected (IMD) population. γ is the rate of recovery after infection, θ the likelihood of infection given carriage, α1 is the decolonization rate and α2 is the infection rate. Further information on parameter ranges and units can be found in Table 2; variables are consigned in Table 1. https://doi.org/10.1371/journal.pcbi.1011564.g001 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Variables of the transmission dynamical model, description and its units. https://doi.org/10.1371/journal.pcbi.1011564.t001 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Parameters of the transmission dynamical model, descriptions, range or value used and its units. https://doi.org/10.1371/journal.pcbi.1011564.t002 We segregated this core form into three separate, mechanistic models representing alternate transmission dynamics: i) a constant contact rate, β = β0, ii) a seasonally varying contact rate, β = β(t), and iii) a seasonally varying likelihood of infection θ = θ(t). Models ii and iii assume different mechanisms behind the observed seasonality of IMD. Model form ii assumes that the mechanism driving seasonality is related to the transmission rate or the pattern of contact between non-carriers and carriers. Model form iii assumes seasonality in the likelihood of transitioning from carriage to infection, i.e. the probability θ that a given carrier becomes infected with IMD. This assumption is supported by evidence that unlike disease prevalence, carriage prevalence does not show a seasonal pattern [3,24]. We computed the average power across the study period from the wavelet analysis and used the frequency with the maximum power to impose seasonality in models ii and iii (see Fig B in S1 Text lower subplot). Model-inference framework The three mechanistic models (i-iii) were coupled with Bayesian inference methods that assimilate IMD surveillance data into the model simulations. For each model, an ensemble of simulations was first initialized with parameters and state variables values drawn from ranges consistent with observations and estimates reported in previous studies. Specifically, for the initialization step, we derived a prior range of parameters and variables by imposing at equilibrium that the basic reproductive number R0~1 and the carriage prevalence is around 20%, we study the sensitivity of the restriction to different decolonization periods (Fig A in S1 Text). As the ensembles were integrated through time, a statistical filter was used to iteratively assimilate monthly observations and adjust the prior, model-simulated distribution of variables and parameters into posterior distributions that better represent the observed dynamics. We used two alternative data assimilation (Bayesian inference) algorithms: 1) a single run of the Ensemble Adjustment Kalman Filter (EAKF) and 2) an iterated Filtering framework (IF-EAKF) [4,5], which gradually adjusts the parameters through multiple iterations of the EAKF and provide point estimates of the parameters (see S1 Text section The ensemble adjustment Kalman filter for details). We tested each model-inference combination by 1) evaluating the posterior fit of the observable variable and 2) comparing free simulations of IMD incidence, run with the posterior parameters estimated with either EAKF or IF-EAKF, to the observed trajectory of cases (Figs C-E in S1 Text). The free simulations with the IF-EAKF point estimated parameters matched the IMD trajectory better than those generated with the EAKF alone, except for model 2 (See Figs C2-C3 and D2-D3 and E2-E3 in S1 Text for the 3 models, respectively). ARIMA We optimized an autoregressive integrated moving average (ARIMA) model by computing both autocorrelation and partial autocorrelation with a maximum lag of 40 months. The model and model-fit are described in section ‘The ARIMA model’ in the S1 Text. We optimized the number of lag observations in the model, i.e., lag order p, and the size of the moving average window, i.e. order of moving average q, using the significant (p<0.05) lags from the autocorrelation and partial autocorrelation. We fixed the number of times that the time series is differenced, d, as 1 (See SI section The ARIMA for details in the implementation). SARIMA We optimized a seasonal autoregressive integrated moving average (SARIMA) model by computing both autocorrelation and partial autocorrelation with a maximum lag of 20 months. The model and model-fit are described in section ‘The SARIMA model’ in the S1 Text. We optimized the number of lag observations in the model, i.e., lag order p, and the size of the moving average window, i.e. order of moving average q, using the significant (p<0.05) lags from the autocorrelation and partial autocorrelation. We fixed the number of times that the time series is differenced, d, as 1. We used the same order of seasonal moving average Q and seasonal lag P, as obtained from the autocorrelation and partial autocorrelation (See S1 Text section The ARIMA for details in the implementation). Mechanistic retrospective forecasting framework We evaluated the forecasting skill of the different models by generating retrospective forecasts of monthly IMD cases between 2006 and 2020. Specifically, we sequentially assimilated IMD incidence data within the EAKF framework to generate posterior fits up to each (monthly) forecast initiation date and then integrated the model into the future to generate probabilistic forecasts without further training. We also used the IF-EAKF framework to estimate parameters before initiating the forecasts and found that the EAKF alone performs better. We evaluate the forecast by plotting a subset in Fig F in S1 Text Evaluation of retrospective forecasting We used one evaluation metric for point predictions—mean absolute error (MAE) and one proper scoring rule to evaluate probabilistic predictions—the Weighted Interval Score (WIS) [25]. We examined the forecast accuracy of predictions for monthly IMD cases 1 to 6 months in the future. MAE is calculated as the absolute value of the difference between the mean prediction of the probabilistic forecast and reported IMD incidence. The WIS accounts for the probabilistic distributions of predicted values specified by 20 quantile intervals [25] as described in the S1 Text (See S1 Text on the WIS computation) Multi-model ensemble of forecasting models and evaluation Aggregating probabilistic forecasts generated by different model systems in a multi-model ensemble (MME) often produces more accurate ‘multi-model’ predictions than the individual component model systems [26,27]. We used two methods to aggregate the forecasts from different models into MME predictions. i) We equally weighted each model (a simple average for each quantile). ii) We used an expectation maximization (EM) algorithm based on a probabilistic marginal distribution to draw from the model space [26]. Here we considered two approaches for computing the marginal distribution of each model forecast for the EM algorithm: a) the WIS of each model for all prior (historical) predictions and b) the WIS of each model for a fixed window (K months) of past predictive performance. See S1 Text for further information on the implementation of the MME methods. We examined the performance of the MME predictions using the WIS, as described for the component models. Transmission dynamics during the study period were possibly impacted by changes in vaccination policy, specifically, the introduction of a booster shot for 16-year-olds after 2011. To account for this exogenous factor, we investigated the performance of the models during three different study periods: the entire study period, before 2011, and after 2011. Data description Weekly IMD incidence data at the state and national level in the US were compiled from the CDC Wonder dataset [21]. We used data from 2006 to 2022, as only yearly cumulative counts were reported before 2006. Classification of cases by serogroup (ACWY and B) only began in 2020 and was not used in this study; rather, incidence includes all serogroups. Spectral analysis We used wavelet time series analysis to capture the temporal properties of IMD in the US [22]. The objective of this analysis is to represent the IMD time series in both the time and frequency domains and reveal shifts in seasonality or other periodicities. We used the Morlet wavelet function as the basis. Similar analyses have been previously used to explore the seasonality of measles and influenza [23]. We investigate the fit of the inverse wavelet transform (IWT), as well as study the periodicity of the system averaging the local wavelet power spectrum (LWPS) across the study period. Model description We developed three mathematical mechanistic models and one purely statistical model–the autoregressive integrated moving average model (ARIMA). The three mechanistic models aim to represent the underlying transmission process of the disease, have the same core model structure shown in Fig 1, and are described in detail in S1 Text. Briefly, for the mechanistic models, we compartmentalize the population (N) into 3 groups: susceptible/non-carrier individuals (S), carriers of the bacteria (C) that are colonized with N. meningitidis but not infected with IMD, and carriers who have become infected with IMD (I) (Table 1). Fig 1 shows the transition between the compartments: susceptible individuals (S) become carriers with a force of infection λ (S→C), a fraction θ of carriers become infected with IMD at the rate α2 (C→I) and the proportion of C that don’t develop infection become susceptible again with the decolonization rate α1 (C→S). We model the force of infection λ using the law of mass action with contact rate β, and assume both carriers (C) and infected (I) contribute to transmission: λ = β(C+I)/N. Description of the parameters, units and value used in presented in Table 2. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Model diagram. Compartmental model of Neisseria Meningitidis transmission. S is the susceptible/non-carrier population, C is the carrier population, and I is the infected (IMD) population. γ is the rate of recovery after infection, θ the likelihood of infection given carriage, α1 is the decolonization rate and α2 is the infection rate. Further information on parameter ranges and units can be found in Table 2; variables are consigned in Table 1. https://doi.org/10.1371/journal.pcbi.1011564.g001 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Variables of the transmission dynamical model, description and its units. https://doi.org/10.1371/journal.pcbi.1011564.t001 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Parameters of the transmission dynamical model, descriptions, range or value used and its units. https://doi.org/10.1371/journal.pcbi.1011564.t002 We segregated this core form into three separate, mechanistic models representing alternate transmission dynamics: i) a constant contact rate, β = β0, ii) a seasonally varying contact rate, β = β(t), and iii) a seasonally varying likelihood of infection θ = θ(t). Models ii and iii assume different mechanisms behind the observed seasonality of IMD. Model form ii assumes that the mechanism driving seasonality is related to the transmission rate or the pattern of contact between non-carriers and carriers. Model form iii assumes seasonality in the likelihood of transitioning from carriage to infection, i.e. the probability θ that a given carrier becomes infected with IMD. This assumption is supported by evidence that unlike disease prevalence, carriage prevalence does not show a seasonal pattern [3,24]. We computed the average power across the study period from the wavelet analysis and used the frequency with the maximum power to impose seasonality in models ii and iii (see Fig B in S1 Text lower subplot). Model-inference framework The three mechanistic models (i-iii) were coupled with Bayesian inference methods that assimilate IMD surveillance data into the model simulations. For each model, an ensemble of simulations was first initialized with parameters and state variables values drawn from ranges consistent with observations and estimates reported in previous studies. Specifically, for the initialization step, we derived a prior range of parameters and variables by imposing at equilibrium that the basic reproductive number R0~1 and the carriage prevalence is around 20%, we study the sensitivity of the restriction to different decolonization periods (Fig A in S1 Text). As the ensembles were integrated through time, a statistical filter was used to iteratively assimilate monthly observations and adjust the prior, model-simulated distribution of variables and parameters into posterior distributions that better represent the observed dynamics. We used two alternative data assimilation (Bayesian inference) algorithms: 1) a single run of the Ensemble Adjustment Kalman Filter (EAKF) and 2) an iterated Filtering framework (IF-EAKF) [4,5], which gradually adjusts the parameters through multiple iterations of the EAKF and provide point estimates of the parameters (see S1 Text section The ensemble adjustment Kalman filter for details). We tested each model-inference combination by 1) evaluating the posterior fit of the observable variable and 2) comparing free simulations of IMD incidence, run with the posterior parameters estimated with either EAKF or IF-EAKF, to the observed trajectory of cases (Figs C-E in S1 Text). The free simulations with the IF-EAKF point estimated parameters matched the IMD trajectory better than those generated with the EAKF alone, except for model 2 (See Figs C2-C3 and D2-D3 and E2-E3 in S1 Text for the 3 models, respectively). ARIMA We optimized an autoregressive integrated moving average (ARIMA) model by computing both autocorrelation and partial autocorrelation with a maximum lag of 40 months. The model and model-fit are described in section ‘The ARIMA model’ in the S1 Text. We optimized the number of lag observations in the model, i.e., lag order p, and the size of the moving average window, i.e. order of moving average q, using the significant (p<0.05) lags from the autocorrelation and partial autocorrelation. We fixed the number of times that the time series is differenced, d, as 1 (See SI section The ARIMA for details in the implementation). SARIMA We optimized a seasonal autoregressive integrated moving average (SARIMA) model by computing both autocorrelation and partial autocorrelation with a maximum lag of 20 months. The model and model-fit are described in section ‘The SARIMA model’ in the S1 Text. We optimized the number of lag observations in the model, i.e., lag order p, and the size of the moving average window, i.e. order of moving average q, using the significant (p<0.05) lags from the autocorrelation and partial autocorrelation. We fixed the number of times that the time series is differenced, d, as 1. We used the same order of seasonal moving average Q and seasonal lag P, as obtained from the autocorrelation and partial autocorrelation (See S1 Text section The ARIMA for details in the implementation). Mechanistic retrospective forecasting framework We evaluated the forecasting skill of the different models by generating retrospective forecasts of monthly IMD cases between 2006 and 2020. Specifically, we sequentially assimilated IMD incidence data within the EAKF framework to generate posterior fits up to each (monthly) forecast initiation date and then integrated the model into the future to generate probabilistic forecasts without further training. We also used the IF-EAKF framework to estimate parameters before initiating the forecasts and found that the EAKF alone performs better. We evaluate the forecast by plotting a subset in Fig F in S1 Text Evaluation of retrospective forecasting We used one evaluation metric for point predictions—mean absolute error (MAE) and one proper scoring rule to evaluate probabilistic predictions—the Weighted Interval Score (WIS) [25]. We examined the forecast accuracy of predictions for monthly IMD cases 1 to 6 months in the future. MAE is calculated as the absolute value of the difference between the mean prediction of the probabilistic forecast and reported IMD incidence. The WIS accounts for the probabilistic distributions of predicted values specified by 20 quantile intervals [25] as described in the S1 Text (See S1 Text on the WIS computation) Multi-model ensemble of forecasting models and evaluation Aggregating probabilistic forecasts generated by different model systems in a multi-model ensemble (MME) often produces more accurate ‘multi-model’ predictions than the individual component model systems [26,27]. We used two methods to aggregate the forecasts from different models into MME predictions. i) We equally weighted each model (a simple average for each quantile). ii) We used an expectation maximization (EM) algorithm based on a probabilistic marginal distribution to draw from the model space [26]. Here we considered two approaches for computing the marginal distribution of each model forecast for the EM algorithm: a) the WIS of each model for all prior (historical) predictions and b) the WIS of each model for a fixed window (K months) of past predictive performance. See S1 Text for further information on the implementation of the MME methods. We examined the performance of the MME predictions using the WIS, as described for the component models. Transmission dynamics during the study period were possibly impacted by changes in vaccination policy, specifically, the introduction of a booster shot for 16-year-olds after 2011. To account for this exogenous factor, we investigated the performance of the models during three different study periods: the entire study period, before 2011, and after 2011. Results Wavelet time-series analysis A local wavelet power spectrum (LWPS) of the weekly national IMD incidence time series for the US is shown in Fig 2. The inverse transform is presented in Fig A in S1 Text. The start of the complete vaccination regimen in adolescents (primary dose + booster) is indicated by the vertical line corresponding to year 2011 [9]. Prior to that, the LWPS shows consistently high power at 1-year and 0.3-year (4-month) periodicities. After introduction of the complete adolescent vaccination program, the period with maximum power decreases to around 0.9 years and the magnitude of the maximum power decreases. After 2011, the power decreased consistently until the end of the study period (see Fig 2C at a period equal to y = 1.1 years). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Wavelet timeseries analysis. A) Time series of Invasive Meningococcal disease (IMD) monthly incidence for the US, 2006–2022 B) Normalized detrended IMD incidence (black line) and inverse wavelet transform (salmon dashed line) C) Local wavelet power spectrum; power is color coded with lower magnitudes shown in darker red and higher magnitudes in lighter yellow. https://doi.org/10.1371/journal.pcbi.1011564.g002 Fig B in S1 Text shows the average power across the study period, which maximizes at 0.98 years. This periodicity was used to modulate the contact rate and the likelihood of infection in mechanistic model structures ii and iii, respectively (see Methods section). Posterior fit and free simulation with MLE We evaluated the posterior fit of the model-inference framework for the three mechanistic model structures. Model iii (seasonality in the likelihood of infection given carriage) consistently performed better both for its EAKF posterior incidence fit and in free simulations run with estimated parameters (Fig 3C and Figs C-E in S1 Text.). The posterior incidence of model iii) simulates IMD data well across the entire study period except for the spike with exceptionally high reported levels of IMD during February 2008. The EAKF posterior estimates for all 3 models are shown in Figs E and H and K in S1 Text. For all models, the posterior susceptibility profile increases as a function of time, but the posterior estimates of model iii capture observed seasonal patterns. In particular, free simulation using the posterior estimates of the IF-EAKF spans observed IMD incidence (Figs C-E in S1 Text), indicating that the model can reproduce transmission dynamics. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Posterior fit. Model iii) (seasonality in the likelihood of infection). A) Simulated susceptibility; the solid line shows the median and the darker and lighter ribbons show 50% and 95% CI. B) Carriage prevalence; the solid line shows the median and the darker and lighter ribbons show 50% and 95% CI. C) Invasive Meningococcal Disease (IMD) incidence. The teal line, darker and lighter ribbons show the simulated median and 50% and 95% CI, respectively. Red dots are observations. https://doi.org/10.1371/journal.pcbi.1011564.g003 We estimated that before the start of the complete vaccination regimen in 2011 carriage prevalence was 6.00% (95% CI: 3.20–10.29%); after 2011, this estimate dropped to 1.64% (95% CI: 1.18–2.18) (Fig 3, right plot). The prevalence of the infected population was 0.019 (95% CI: 0.014–0.025) per 100,000 population before 2011 and dropped to 0.0072 (95% CI: 0.0049–0.0098) per 100,000 population after 2011 for the US, following the same decreasing trend. Retrospective forecasting using individual models We generated retrospective forecasts using the four individual models in order to quantify the performance of each model using out-of-sample predictions (Fig F in S1 Text). In general, model iii possesses a narrower prediction interval than the purely statistical autoregressive ARIMA. Relative performance across models remained similar as the forecast horizon increased from 1 to 6 months. Point predictions across the study period indicate that the mechanistic models consistently outperformed the ARIMA and that differences among the three mechanistic models were negligible, consistent across forecast horizons (Figs G-H in S1 Text). The mean error for each forecast date showed substantial underprediction of the unusual peak during February 2008. The overbroad probabilistic forecasts of the ARIMA and SARIMA were penalized by the WIS score, resulting in higher WIS scores than the mechanistic models (i.e., a worse performance). Finally, we compared the mean performance of the models across the entire study period and before and after the beginning of the complete adolescent vaccine regimen in 2011. Overall, the mechanistic models outperformed the ARIMA and SARIMA considering the average performance during the study period (Fig I1 in S1 Text); we used the Wilcoxon signed-rank test to assess statistical of the distribution of WIS between the ARIMA and each mechanistic model and conclude that mechanistic models performed better, and this finding was consistent across forecast horizons (see Tables A-B in S1 Text for the p-values and Fig I2 in S1 Text for the WIS distribution). Among the three mechanistic models, model iii outperformed the others in the average performance during the study period; however, model i produced better forecasts for the period prior to 2011 (Fig I in S1 Text). Forecasting with a multi-model ensemble (MME) We found that MME constructed using all past predictions performed better across all periods and horizons than MME constructed using only more recent predictions. We also found that forecast performance worsened (WIS increases) as the size of the training window increased (Fig J in S1 Text). Finally, we compared the best individual model iii, the trained MME using performance from the preceding 2 months, the MME using all past predictions, and the equally weighted MME (see Fig 4). We found that the MME constructed with all past predictions outperformed the other MME approaches for the entire study period; however, for the sub-periods before and after the start of the complete vaccine regimen, mean WIS was lower for both model iii and the MME trained with 2 months of prior performance. The untrained MME had the worst performance across all data splits (Fig 4). To further examine statistically significant differences, we plotted the distribution of WIS and used the Wilcoxon signed-rank test to assess differences, we only found statistical significance differences between the equally weighted ensemble and the rest of the models, model 3, MME with 2 months and all past performance (See Table C and Fig K in S1 Text). We also compared the performance of the MME after 2011 (onset of the complete vaccine regime) and found that the MME trained with performance during the prior 2 months was the best across all past performances. Finally, to understand the importance assigned to each model we present the weights from the MME (Fig M in S1 Text in general, the MME principally weighted the mechanistic models with the greatest weight alternating among model 1 and model 3 for some study periods. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Performance of model and ensemble forecasts. Each bar shows the mean performance for the indicated split of the data; forecast horizon is color-coded and indicated in the legend. A) The entire study period B) Pre 2011. C) Post 2011. https://doi.org/10.1371/journal.pcbi.1011564.g004 Wavelet time-series analysis A local wavelet power spectrum (LWPS) of the weekly national IMD incidence time series for the US is shown in Fig 2. The inverse transform is presented in Fig A in S1 Text. The start of the complete vaccination regimen in adolescents (primary dose + booster) is indicated by the vertical line corresponding to year 2011 [9]. Prior to that, the LWPS shows consistently high power at 1-year and 0.3-year (4-month) periodicities. After introduction of the complete adolescent vaccination program, the period with maximum power decreases to around 0.9 years and the magnitude of the maximum power decreases. After 2011, the power decreased consistently until the end of the study period (see Fig 2C at a period equal to y = 1.1 years). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Wavelet timeseries analysis. A) Time series of Invasive Meningococcal disease (IMD) monthly incidence for the US, 2006–2022 B) Normalized detrended IMD incidence (black line) and inverse wavelet transform (salmon dashed line) C) Local wavelet power spectrum; power is color coded with lower magnitudes shown in darker red and higher magnitudes in lighter yellow. https://doi.org/10.1371/journal.pcbi.1011564.g002 Fig B in S1 Text shows the average power across the study period, which maximizes at 0.98 years. This periodicity was used to modulate the contact rate and the likelihood of infection in mechanistic model structures ii and iii, respectively (see Methods section). Posterior fit and free simulation with MLE We evaluated the posterior fit of the model-inference framework for the three mechanistic model structures. Model iii (seasonality in the likelihood of infection given carriage) consistently performed better both for its EAKF posterior incidence fit and in free simulations run with estimated parameters (Fig 3C and Figs C-E in S1 Text.). The posterior incidence of model iii) simulates IMD data well across the entire study period except for the spike with exceptionally high reported levels of IMD during February 2008. The EAKF posterior estimates for all 3 models are shown in Figs E and H and K in S1 Text. For all models, the posterior susceptibility profile increases as a function of time, but the posterior estimates of model iii capture observed seasonal patterns. In particular, free simulation using the posterior estimates of the IF-EAKF spans observed IMD incidence (Figs C-E in S1 Text), indicating that the model can reproduce transmission dynamics. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Posterior fit. Model iii) (seasonality in the likelihood of infection). A) Simulated susceptibility; the solid line shows the median and the darker and lighter ribbons show 50% and 95% CI. B) Carriage prevalence; the solid line shows the median and the darker and lighter ribbons show 50% and 95% CI. C) Invasive Meningococcal Disease (IMD) incidence. The teal line, darker and lighter ribbons show the simulated median and 50% and 95% CI, respectively. Red dots are observations. https://doi.org/10.1371/journal.pcbi.1011564.g003 We estimated that before the start of the complete vaccination regimen in 2011 carriage prevalence was 6.00% (95% CI: 3.20–10.29%); after 2011, this estimate dropped to 1.64% (95% CI: 1.18–2.18) (Fig 3, right plot). The prevalence of the infected population was 0.019 (95% CI: 0.014–0.025) per 100,000 population before 2011 and dropped to 0.0072 (95% CI: 0.0049–0.0098) per 100,000 population after 2011 for the US, following the same decreasing trend. Retrospective forecasting using individual models We generated retrospective forecasts using the four individual models in order to quantify the performance of each model using out-of-sample predictions (Fig F in S1 Text). In general, model iii possesses a narrower prediction interval than the purely statistical autoregressive ARIMA. Relative performance across models remained similar as the forecast horizon increased from 1 to 6 months. Point predictions across the study period indicate that the mechanistic models consistently outperformed the ARIMA and that differences among the three mechanistic models were negligible, consistent across forecast horizons (Figs G-H in S1 Text). The mean error for each forecast date showed substantial underprediction of the unusual peak during February 2008. The overbroad probabilistic forecasts of the ARIMA and SARIMA were penalized by the WIS score, resulting in higher WIS scores than the mechanistic models (i.e., a worse performance). Finally, we compared the mean performance of the models across the entire study period and before and after the beginning of the complete adolescent vaccine regimen in 2011. Overall, the mechanistic models outperformed the ARIMA and SARIMA considering the average performance during the study period (Fig I1 in S1 Text); we used the Wilcoxon signed-rank test to assess statistical of the distribution of WIS between the ARIMA and each mechanistic model and conclude that mechanistic models performed better, and this finding was consistent across forecast horizons (see Tables A-B in S1 Text for the p-values and Fig I2 in S1 Text for the WIS distribution). Among the three mechanistic models, model iii outperformed the others in the average performance during the study period; however, model i produced better forecasts for the period prior to 2011 (Fig I in S1 Text). Forecasting with a multi-model ensemble (MME) We found that MME constructed using all past predictions performed better across all periods and horizons than MME constructed using only more recent predictions. We also found that forecast performance worsened (WIS increases) as the size of the training window increased (Fig J in S1 Text). Finally, we compared the best individual model iii, the trained MME using performance from the preceding 2 months, the MME using all past predictions, and the equally weighted MME (see Fig 4). We found that the MME constructed with all past predictions outperformed the other MME approaches for the entire study period; however, for the sub-periods before and after the start of the complete vaccine regimen, mean WIS was lower for both model iii and the MME trained with 2 months of prior performance. The untrained MME had the worst performance across all data splits (Fig 4). To further examine statistically significant differences, we plotted the distribution of WIS and used the Wilcoxon signed-rank test to assess differences, we only found statistical significance differences between the equally weighted ensemble and the rest of the models, model 3, MME with 2 months and all past performance (See Table C and Fig K in S1 Text). We also compared the performance of the MME after 2011 (onset of the complete vaccine regime) and found that the MME trained with performance during the prior 2 months was the best across all past performances. Finally, to understand the importance assigned to each model we present the weights from the MME (Fig M in S1 Text in general, the MME principally weighted the mechanistic models with the greatest weight alternating among model 1 and model 3 for some study periods. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Performance of model and ensemble forecasts. Each bar shows the mean performance for the indicated split of the data; forecast horizon is color-coded and indicated in the legend. A) The entire study period B) Pre 2011. C) Post 2011. https://doi.org/10.1371/journal.pcbi.1011564.g004 Discussion In this work we develop and test different model structures for reproducing and forecasting invasive meningococcal disease dynamics, resulting from infection by the bacterium Neisseria meningitidis. We used IMD incidence in US from 2006 to 2022 to calibrate and test the forecasting skill of 5 different models and their ensemble combinations. The findings provide a foundation for conducting future analyses to investigate the impact of vaccination and changes in mixing patterns resulting from the impact of the SARS-COV-2 pandemic. Here, we found a marked seasonality of IMD before 2011 with a 1-year period followed by a decreasing seasonal signal after 2011. This qualitative change coincided with changes in vaccine policy and uptake: teenager vaccination for N meningitidis was introduced in 2005 and was extended with an additional booster in 2011. It is possible that vaccination contributed to this shift in seasonality and to the decrease in overall IMD incidence observed in the last 2 decades (11). Evidence suggests that the ACWY conjugate vaccine reduces carriage [28], impacting transmission dynamics by reducing the force of infection. However, it is also possible that the observed change in seasonality is explained by changes in the age distribution of infection [29]. We explored different mechanisms through which seasonality could affect transmission by testing different transmission model forms. Different climatic, socio-demographical, and behavioural factors may affect transmission in different ways and drive seasonal patterns. Influenza transmission is modulated by absolute humidity [14,30,31,31]; malaria transmission is modulated by rain, temperature, and humidity [32]; dengue, and other arboviruses outbreaks are modulated by the synergistic effects of temperature and population density [33,34]; human mobility shapes SARS-CoV2 transmission [35]; and rainfall modulates cholera dynamics [36–38]. Researchers can represent these drivers by designing mathematical models in which the contact or transmission rate is a function of these relevant factors. Here we show that forcing the IMD transmission model via a seasonal likelihood of infection given carriage better explains observed incidence and had the best performance in retrospective forecast (Figs K-M in S1 Text). This finding is potentially supported by carriage surveys studies in the meningitis belt of Sub-Saharan Africa, that have reported a large increase of the disease to carrier ratio during the dry season [39]. Despite the different climate conditions in the US, evidence from the meningitis Belt supports a potential effect of climate on disease mechanisms. This result suggests important future areas for research to improve understanding of the mechanisms behind this forcing (e.g. climatic, contact, phenotypical, etc) [4,24]. That is, while IMD rates display seasonal trends, peaking in winter months, carriage prevalence does not show seasonality [4]. There are several mechanisms that might increase the likelihood of infection with IMD in winter months: 1) cold/dry air can damage the nasopharyngeal mucosa of the host facilitating bacterial invasion, as proposed by studies set in the meningitis belt of Sub-Saharan Africa, where dry Harmattan winds are believed to be responsible for increased disease-to-carriage ratios [40]; 2) previous infections with some seasonal viruses (i.e. influenza) can predispose the host to IMD infection; and 3) seasonal factors may affect the host immune system, making some more prone to disease [4]. However, current IMD data availability limits the possibility of investigating these mechanisms. Stratification of the IMD incidence dataset by age, serogroup and vaccination status could support testing these hypotheses through more detailed IMD modeling. Lastly, future research, perhaps leveraging a more complete IMD dataset resolved at finer spatial scales, could shed some light on the link between influenza and IMD. If climate is shown to be a driver of the seasonality observed [41,42], the effect of a changing climate would need to be incorporated into IMD models. We also reviewed the literature for other possible determinants of transmission and found there have been multiple outbreaks of IMD among men who have sex with men (MSM) in the last 20 years. Transmission of IMD in the MSM community requires further study. In our mechanistic models, we assume homogeneous mixing of the population, so specific mixing patterns among subpopulations are not represented; however, the model could be elaborated in the future to represent subpopulations. Additionally, we do not explicitly model the effect of vaccination and its possible impact on the carriage acquisition or the likelihood of infection. It also has been shown that prior influenza infection is a risk for IMD [24], which could also make model iii the best just by seasonally adjusting the likelihood of infection θ. The posterior estimates of susceptibility and carriage for the 3 models (Fig 3C and Figs C-E in S1 Text) show that susceptibility increased during the study period, and in consequence, carriage decreased. This result could be due to the combined effects of immunity acquired via infection and vaccination. Additionally, the posterior estimates for all models show a substantial drop in carriage during 2011 that we think is a consequence of the introduction of a booster in 2011 combined with immunity acquired via natural infection. We also investigated the effect of system initial conditions, which assumed prevalence between 5–30% (see S1 Text Equilibrium section and Fig A in S1 Text), on posterior estimates. We found that the posterior estimates remained unchanged, suggesting the system is correctly identifying susceptibility and prevalence. Our models also estimated a low fraction of the population as susceptible by the end of the data record, suggesting that rebounds caused by possible increases in susceptibility during 2020 due to non-pharmaceutical intervention to control the spread of SARS-CoV2 were not of substantial magnitude. However, our modelling approach does not account for spatial heterogeneity within the US. As a consequence, we cannot describe any geographical spots that might have a substantial pocket of susceptibles and therefore be where IMD rebound might be probable. A modeling study from the UK, accounting for the effects of decreased vaccination during the pandemic, suggests a long term effect of NPIs on carriage prevalence [43]. However, further modeling, validated with recent, local data are needed to better assess the effects of the COVID-19 pandemic on IMD. The model-inference structures developed here can retrospectively predict the transmission of Neisseria meningitidis in the continental US (See Fig 4, and Figs G-I in S1 Text) We showed that in all periods the purely statistical models, ARIMA and SARIMA, performed the worst, whereas mechanistic models 1 and 3 were best across study periods (Fig I in S1 Text). In addition to individual models, we evaluated an MME forecasting system comprised of four models—one statistical and three mechanistic. For the MME forecasting system based on the past performance of the individual models, the form using all past predictions for establishing component model weights outperformed forms using only recent predictions (Figs J-K in S1 Text). This trained MME also outperforms all the individual models and the equally weighted MME (Fig 4). This result is consistent with research on endemic diseases [26]. We also showed that all individual models outperformed the equally weighted MME model (Fig 4 and Fig K in S1 Text). This finding contradicts previous research showing that equally weighted ensembles usually outperform individual models for an endemic respiratory disease [26]. Limitations in this study arise principally from the geographical resolution considered and the assumption of complete mixing across the US. Additionally, data on carriage are poor, so inference was only informed by incidence of IMD. We also didn’t find any available data on vaccination with the exception of data from NIS teen surveys (Fig O in S1 Text); however, these survey data are not representative of vaccine hesitancy for the US. Limitations affecting the forecasting system include that real-time predictions are compromised by delays in reporting IMD. The implications of the change of seasonality after 2011 might be confounded by vaccination patterns or outbreaks in certain subpopulations, such as MSM [44]. The fact that influenza infection is a risk for IMD (causing a possible increase of θ) is not modeled explicitly and a model representing both infections with influenza and IMD might better explain transmission dynamics [24]. Supporting information S1 Text. Supplementary information with sections as follows: Description of the time-series analysis using wavelets, description of the process-based models, calculation of the disease-free equilibrium (DFE), non-DFE and basic reproductive number, description of the Bayesian inference method the Ensemble Adjustment Kalman Filter (EAKF) and a description of the retrospective forecasting and the algorithm to produce the Multi-model Ensemble (MME). We included 2 last sections with the Supplementary Tables and Figures, a description of these is listed below. Table A. Tables with Wilxonxon signed rank significant statistical tests for the ARIMA vs each individual process-based model. Table B. Tables with Wilxonxon signed rank significant statistical tests for the SARIMA vs each individual process-based model. Table C. Tables with Wilxonxon signed rank significant statistical tests for the MME vs process-based model 3. Fig A. Heatmap of R0 and carriage prevalence for varying values of the contact rate and likelihood of infection upon carriage. In Figs A1-A3 in s1 Text, we varied the recovery rate from 3 to 60 days, as indicated in the legend. Fig B. Inverse Wavelet Transform and detrended IMD incident cases. Fig C. C1. Posterior estimate of model 1 state variables from an EAKF. C2. Simulation of model 1 with time-varying parameters estimates from an EAKF. C3. Simulation of model 1 with point parameters estimates from an IF-EAKF. Fig D. D1. Posterior estimate of model 2 state variables from an EAKF. D2. Simulation of model 2 with time-varying parameters estimates from an EAKF. D3. Simulation of model 2 with point parameters estimates from an IF-EAKF. Fig E. E1. Posterior estimate of model 3 state variables from an EAKF. E2. Simulation of model 3 with time-varying parameters estimates from an EAKF. E3. Simulation of model 3 with point parameters estimates from an IF-EAKF. Fig F. Visualization of the forecast at a 6-month forecast horizon for the individual models. Fig G. G1. Time series of the WIS at 1-month forecast horizon for the 5 individual models. G2. Time series of the WIS at 3-month forecast horizon for the 5 individual models. G3. Time series of the WIS at 6-month forecast horizon for the 5 individual models. Fig H. Time series of the WIS at 6-month forecast horizon for the dynamical mechanistic models. Fig I. I1. Mean weighted interval score for the 5 individual models. I2. Boxplot of the WIS for the 5 individual models. Fig J. Mean WIS for the ensembles trained with different information of previous models’ performance. Fig K. Boxplot of the WIS for the ensembles trained will all past performance with the previous 2 months and naively and best individual model. Fig L. MME weights assigned to each model trained with the performance of the previous K months. L1. K: 3 months, L2. K: 6 months, L3. K: all past performance. Fig M. Coverage estimates of Meningococcal vaccination for teenagers. https://doi.org/10.1371/journal.pcbi.1011564.s001 (PDF)

Albantakis, Larissa;Barbosa, Leonardo;Findlay, Graham;Grasso, Matteo;Haun, Andrew M.;Marshall, William;Mayner, William G. P.;Zaeemzadeh, Alireza;Boly, Melanie;Juel, Bjørn E.;Sasai, Shuntaro;Fujii, Keiko;David, Isaac;Hendren, Jeremiah;Lang, Jonathan P.;Tononi, Giulio

doi:

Posani, Lorenzo;Rizzato, Francesca;Monasson, Rémi;Cocco, Simona

doi: 10.1371/journal.pcbi.1011521pmid: 37883593

Introduction Predictability of evolution of organisms in fitness landscape has been a driving concept in the development of evolutionary biology since the origins of the field [1–5]. In particular, our capability to predict the effects of detrimental mutations has enormous practical impact on the diagnosis of genetic variances causing diseases [6–10]. This issue can now be quantitatively investigated, thanks to high-throughput sequencing and mutagenesis experiments, which allow for in-vivo and in-vitro measurements of the effects of many mutants [1, 5, 11–14, 14–25]. However, despite the impressive progress of these large-scale techniques, the number of possible mutations, growing exponentially with the protein length, is so huge that measuring the fitness landscape in its entirety is out of reach, with the exception of short protein regions [1]. Computational approaches, in particular machine-learning-based models exploiting the large corpus of available sequence data [26, 27] are needed for the full reconstruction and prediction of fitness landscapes. Briefly speaking, these methods are based on the assumption that statistically rare mutations (in homologous sequence data) are likely to be deleterious [6, 28]. Such conservation-based methods can be combined with structural [7, 29], physico-chemical [8], as well as phylogenetic [30, 31] information. Graphical Potts models, also called direct coupling analysis (DCA) [32–34], have pushed further the approaches based on sequence conservation by including statistical couplings capturing pairwise amino-acid covariation. These couplings allow DCA to account for background effects on the mutations depending on the wild-type (wt) sequence under consideration. DCA is thought to approximate the fitness landscape reflecting the structural and functional properties common to homologous proteins. As sketched in Fig 1, natural sequences are assumed to lie at, or close to the different peaks of the fitness landscape explored during evolution. The scores of sequences around the wt protein provide predictions for the effects of single or multiple mutations on fitness, in good agreement with mutational effects measured through mutagenesis experiments [34–38]. Other approaches for fitness prediction exploit deep learning (DL) architectures, at the origin of recent progress in image or natural language processing, as well as in protein folding [9, 10, 39–41]. DL models have much higher expressive power than pairwise graphical models, but demand massive sequence data to be trained. Recent applications of DL to protein fitness modelling combine unsupervised learning of hundreds of millions of sequences with supervised learning of mutagenesis experimental data [38, 42, 43]. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Schematic visualization of the fitness landscape over the sequence space (green curve). Two models (red and blue curves) are inferred to assign high fitness values to sequences found in the Multi-Sequence Alignment (MSA) of a protein family. A complex model (red curve) can be a better predictor of the landscape globally while scoring poorly in predicting single-point mutations around a specific wild-type sequence, see local fitness landscape in the zoomed area. Conversely, a simple model (blue line) fitted on a local subset of sequences can give a better local approximation of the landscape, but will likely fail in distant regions of the fitness landscape. https://doi.org/10.1371/journal.pcbi.1011521.g001 Depending on the protein family under consideration, multi-sequence alignments (MSA) show huge variations in sizes, with tens to hundred of thousands sequences, and in homology, ranging from ∼30%, for alignments of orthologous sequences [27, 37, 44], to 90%, for HIV sequences of the same clade [35, 45]. The quantity and diversity of the data, as well as the models considered are empirically known to strongly impact the performances for fitness prediction. As pointed out in [46], classical methods based on homology detection, such as SIFT [6], PolyPhen-2 [7], Align-GVGD [8], rely on different empirical procedures in selecting the alignments, and are not always optimal. Remarkably, single mutations effects are predicted with comparable accuracy by graphical models inferred from a small number of highly similar sequences of the HIV envelope protein [35] and from a much larger number of diverse sequences of Betalactamases, while the two proteins have comparable lengths [37]. Gemme, a recently introduced algorithm based only on conservation and phylogenetic tracing of mutations [31] was shown to outperform deep neural networks models [39] in predicting the effect of mutations in viral sequences, all characterized by a large degree of similarity. Furthermore, the performance of models trained from Uniprot sequences with high pairwise alignment score to a fixed wt sequence considerably vary with the threshold used for alignment [37, 46]. These examples suggest the existence of a compromise between taking into account many sequence data to get statistics and removing far away sequences, whose relation to fitness may be very different from wt due to complex epistatic effects. This compromise, in turn, depends on the expression power of the model considered, which can be tuned at will, and on the complexity of the fitness landscape, which is generally unknown. As sketched in Fig 1, on the one hand, predicting mutations around the wt requires local reconstruction of the landscape only, a task within reach of simple models with few defining parameters. These models are however unreliable for sequences far away from the wt sequence; hence, only few data points, concentrated around the latter can be actually used for training. On the other hand, powerful models able to capture the complex features, such as high-order epistasis that characterize the global fitness landscape on large scales can, in principle, exploit at best sequence data. However, even if the available data are sufficient to infer their huge number of parameters with enough accuracy, it is unclear whether the global description they offer allows for an accurate local reconstruction of the fitness landscape around the wt protein. The scope of the present work is to provide theoretical foundation to address this question. Careful analysis of the different contributions to the prediction error allows us to quantitatively understand how fitness prediction performance depend on both model complexity and on the sequence data, and to estimate the amount of ‘complexity’ in the fitness landscape that is not captured by the model. Our theory is in full agreement with the analysis of sequence data and mutagenesis experiments for 7 protein families we have studied. We also validate our approach in silico on Lattice-Protein models [47–50], for which the ground truth for the fitness is mathematically well defined. Last of all, we demonstrate how our framework allows us, in practice, to optimally tune sequence alignments and models to maximize the performance in fitness prediction. Results Quantity-relevance trade-off in MSA sequence selection We consider a reference sequence, hereafter referred to as wt. We denote by the variation of fitness resulting from the mutation wti → a on the ith site of wt. This quantity can be estimated experimentally, either in vivo (relative enrichment of organisms with mutated gene compared to wt), or in vitro (measurement of appropriate biochemical property). A computational model provides a predictor, , for the difference of fitness between the mutant and the wt. The overall quality of the predictor will be assessed through the Spearman coefficient ρ between the mutation effects computed with the model and with the experimental data . Using Spearman correlations allows one to capture monotonous relations, irrespective of non-linearities. The computational model is generally trained from homologous sequences to wt, i.e. belonging to the same protein family. The similarities between the wt and these sequences, sampled from evolutionary diverse organisms, can vary significantly. As an illustration, we consider the RNA binding domain of the nuclear poly(A)-binding protein (PABPN1), involved in the synthesis of the mRNA poly(A) tails in eukaryotes [14]. Any two sequences in the corresponding MSA (as used in [37]) generally have few amino acids in common (mean Hamming distance -normalized by sequence length- between pairs of sequences in the MSA = 0.75). As a result, a specific sequence, such as the wt of Saccharomyces cerevisiae, is generally surrounded by a small number of similar sequences and is far away from most of the MSA (RNA-bind protein: mean normalized Hamming distance between wt and MSA sequences = 0.73, Fig 2A; see S1 Fig for similar results on other families). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Behaviour of model predictive performance with different selections of training data. A. Distribution of Hamming distances to the wt sequence (RNA-binding domain of Pab1-Yeast) in the MSA of [37]. Note the log scale on the y axis. The three colored lines correspond to three possible sequence selections performed by excluding sequences farther than a certain threshold dcut from wt. A smaller dcut corresponds to fewer sequences with a lower mean Hamming distance to the wt, denoted as D. B Comparison between predicted and experimental fitness mutational effects for an independent-site model trained on the three sub-MSAs corresponding to, respectively, dcut = 32 (orange), 43 (purple), and 82 (green). The Spearman correlation coefficient ρ between predicted and experimental values defines the predictive performance of the model. C Same analysis as panel B repeated for all possible cutoffs between dcut = 32 and dcut = 82 (the sequence length). The non monotonous behavior of the predictive performance indicates that a trade-off between number of sequences (denoted as B) and proximity to wt is controlling the predictive performance of the inferred model. D. Systematic analysis of the predictive power ρ as a function of the mean Hamming distance D of sub-alignments with fixed size B (top), and of the sub-alignment size B at fixed Hamming distance D (bottom). Each individual point shows the average over n = 5 sub-samples obtained at the corresponding values of D and B (see Methods). The dashed curves and error bars are computed by binned average and standard deviation over the displayed individual points. All significance levels refer to Spearman rank correlation of the individual points. *** P < 0.001. https://doi.org/10.1371/journal.pcbi.1011521.g002 Hereafter, we show that sequences far away from wt are not relevant for fitness prediction. To do so we train independent-site Potts models (Methods) on shorter MSAs obtained by discarding sequences further than a certain cut-off distance dcut from wt. As dcut becomes smaller, fewer sequences with higher proximity are selected (Fig 2A). We see that the performance consistently increases when decreasing the cut-off distance, up to a peak ρ = 0.56 at dcut = 43, a 33% increase with respect to the full MSA (ρ(dcut = 82) = 0.42), see Fig 2B and 2C). After peaking, the performance starts decreasing again due to the increasingly-lower number of sequences in the MSA, see Fig 2C. The non-monotonous behavior of the predictive performance indicates that a trade-off between the number of sequences and their proximity to wt is controlling the predictive performance of the inferred model. To investigate the respective effects of these two quantities, we create sub-alignments of the original MSA with controlled sizes B (effective number of sequence taking into account sequence redundancy, see [51] and Methods) and average Hamming distances to wt, which we denote as D. We then test how the performance of the independent-site Potts model trained on these sub-alignments relates to these two quantities. This analysis showed that the predictive performance strongly depends on the mean Hamming distance D and on the number B of sequences (P < 0.001 for all Spearman correlations between ρ and B or D, Fig 2D). The performance significantly decreases with D at fixed B, i.e., when the relevance of the data deteriorates and their quantity is kept fixed, and increases with B at fixed D, i.e., when the quantity of data increases at fixed similarity with wt. Similar results are found for the six other protein families under study (see S1 Fig). Theoretical investigation of the quantity-relevance trade-off To study the trade-off between relevance and quantity we draw our inspiration from the bias-variance framework developed in statistics [52, 53]. Let us consider the error between the statistical predictor and the experimental fitness . This error can be decomposed into the sum of two contributions: (1) a systematic bias in the prediction, due to the inability of the model to capture the exact relation between sequence mutation and fitness, (2) a statistical error coming from the fact that the predictive model has been trained on a particular data set; the value of this contribution fluctuates when the data set changes, and is expected to be smaller and smaller for larger and larger data sets. Consequently, the mean squared error on the single-point mutation wti → a can be written as the sum of a squared bias and a variance contributions, (1) where averages [] are taken on repetitions of the prediction process in fixed conditions (relevance and quantity of data). Notably, these two quantities are hard to minimize together. For instance, powerful models with many parameters will accurately fit the data and thus achieve small squared biases , but will result in large variances due to the statistical errors on the many parameters to be inferred. As we will see below, we can directly relate our descriptors of relevance (D) and quantity (B) of the sequence data to, respectively, the squared bias and the variance as defined in (1). Furthermore, we will introduce a class of increasingly powerful Potts models to investigate the effect of model complexity on these two quantities. In addition to its theoretical appeal and close connection with the bias-variance decomposition, considering the mean–squared error is ultimately justified by the empirically observed monotonic relation with the predictive performance as measured through the Spearman coefficient ρ. K–link Potts model. We consider hereafter the class of sparse Potts models, which include K pairwise couplings between the protein sites, Jij(a, b), whose values depend on the amino acids they carry and a field (position weight matrix) hi(a) on each site; These parameters are learned from the MSA (Methods). The choice of the K pairs of sites carrying couplings is decided based on heuristics, which aim at capturing interrelations between the residues (Methods). By tuning the value of K, we can interpolate between the independent-site model (K = 0, i.e. no coupling) and the full Potts model ( couplings, where N is the protein length). Imposing small values of K is a way to regularize the inferred network of interactions. Notice that the number of parameters to be inferred, Npar = NQ + KQ2, where Q = 20 is the number of amino acids, grows quickly with K since Q2 = 400. For the K-link Potts model the predictor of the fitness difference resulting from the mutation wti → a reads (2) where the sums runs on the sites j in the neighborhood of site i, i.e. coupled to i (Methods). This neighborhood is empty for the independent-site model. Estimation of variance. For the Potts model, expressions for the uncertainties on the inferred fields hi(a) and couplings Jij(a, b) can be formally derived from sampling errors due to the finite size of the data set. The resulting variance of the predictor for a specific K-link model can then be estimated from (2) [38, 54], see Appendix A in S1 Text. Averaging over the sites i and mutations a, we obtain a single global variance, (3) where Q = 20 is the number of amino-acid types, and ki is the cardinality of , i.e. the number of sites interacting with i in the model. The global variance depends on the statistics of the data through the probabilities pi(a) of finding amino acid a on the i-th site and pij(a, b) of finding simultaneously a on site i and b on site j computed on the sub-alignment. Thus, σ2 increases with residue conservation, due to the contributions of amino acids that are rarely observed on some sites in the sub-alignment and have low pi(a), and with the number K of coupling parameters in the model. We also see that σ2 is inversely proportional to the number of sequences, B. The variance therefore decreases with the quantity of data. Estimation of squared bias. Computing the squared bias μ2 in (1) is generally hard, not to say impossible, as it requires detailed knowledge of the fitness landscape. We rely below on simplifying assumptions to gather insights on the value and meaning of the bias. Assume first that we use the independent-site model for fitness prediction. If the ‘true’ fitness landscape shows no epistasis, this model is exact (up to statistical fluctuations due to the finite amount of training data, taken care of by σ2), and the bias vanishes. Therefore, a non zero bias would signal the presence of epistatic interactions between residues not captured by the simple model used for predictions. We stress that this statement is true in an idealized setting, in which the only source of bias is the mismatch between the model power and the ground-truth fitness landscape. In reality, biases can have multiple origins, including non-uniform sampling of sequence data (resulting from preferential choices of organisms or from evolutionary correlations), discrepancies between in vivo fitness reflected by sequence data and in vitro biochemical measurements, etc. Let us now turn to more complex landscapes and models. We assume that the fitness landscape is characterized by pairwise epistasis only, i.e. the fitness differences are exactly described by a full Potts model with Kmax interactions through an equation analogous to (2). The K–link Potts model used for fitness prediction will not be powerful enough to account for the complexity of this landscape and of the sequence data if K < Kmax. As a result a non-zero squared bias will appear, whose expression is derived in Appendix B in S1 Text, and reads (4) where D is the mean Hamming distance of the sub-alignment sequences to wt, and the bias factor J0 is the product of a multiplicative factor depending on the background distribution of amino acids in the MSA and of the variance of the epistatic couplings JFnot included in the prediction model. J0 is thus a decreasing function of K. This expression of μ2 confirms that the Hamming distance D is related to the notion of relevance (similarity to the wt) of the sequence data, as varying D affects the systematic error (bias) of the predictive model. Validation of the theory on Lattice Proteins To validate the key role of the squared bias and of the variance in explaining performance, as well as their approximate expressions above and the interpretation of the bias factor J0 as reflecting un-modeled epistasis, we resort to an in silico model for proteins folding on a 27-site cubic lattice [47, 49, 50, 55, 56], see Fig 3A. In the model, the fitness represents the propensity of a protein sequence to fold into one specific conformation, called native, out of the ≃ 105 folds on the cube [49]. Following [50], the native fold and wildtype sequence were chosen such that the fitness of the wildtype was high enough to be stable but low enough to allow for positive mutations (Pnat ≃ 0.995, see Methods). As we can precisely compute the exact value of the fitness, the ground-truth values of the squared bias and of the variance defined in (1) can be computed with great accuracy (see Methods); we hereafter denote these ground truth values by and . Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Quantity-relevance trade-off for lattice proteins. A: Cubic fold that defines the protein family in the lattice model. Amino acids on sites that are in proximity to each other interact and define the energy of the protein (Methods). B: Predictive performance ρ for single mutations of 5 Sparse Potts models with different degrees of sparsity (defined by K, the number of pairwise links included in the energy function; K = 0 is the independent model) vs. . The collapse of the results is in agreement with Eq (5). C: Squared bias vs. mean Hamming distance in the sequence data, see Eq (4), for the same sparse Potts models as in panel B. Line plots and error bars show mean and standard deviation at a given D and different Bs. D: Variance σ2 vs. estimated variance σ2 in Eq (3) for the same Sparse Potts models as in panel B. E: Bias factor J0(K) (divided by J0(0)) obtained by fitting the squared bias as a linear function of the mean Hamming distance for the various K-link models in panel C. F: Visualization of pairwise couplings inferred by a fully-connected Potts model, highlighting the larger variance of couplings associated to structural contacts (in orange) compared to non-structural ones (in blue)—note the log scale on the y-axis. G: Normalized value of J0(K) (divided by J0(0)) obtained with an effective theory using the variance of couplings associated to modeled and un-modeled structural contacts, see Appendix A in S1 Text. H: scaling for predictive performance ρ of our statistical models for single point mutations as a function of the sum of the estimated squared bias J0D and of the variance σ2 in Eq (3). J0(K) (denoted as in the plot axis label) is fitted to for each value of K by maximizing the scaling correlation as explained in the main text. I: Bias factor J0(K) (normalized by J0(0)) inferred from maximizing the scaling correlation as in panel H. https://doi.org/10.1371/journal.pcbi.1011521.g003 Bias and variance are sufficient to explain model performance. Eq (1) stipulates that the mean squared error over fitness prediction depends on the sum of squared bias and variance of the fitness predictors. If the performance ρ is, in turn, controlled by this mean squared error, we expect a relation such as (5) where F is a decreasing function of its argument. To test the validity of (5), we compare the values of ρ obtained with the independent-site Potts models (K = 0) and different K-link Potts models (K = 4, 8, 16, 24) trained from various sub-alignments with different B, D to the sums of the squared bias and variance, see Fig 3B. We obtain an excellent anti-correlation between ρ and across a large range of values of B and D, in full agreement with (5) (R ∼ 1 for every K–link model). The sum of squared bias and variance is by far the biggest factor in determining the predicting performance of the models. Bias and variance are related to the relevance and the quantity of data as predicted by theory. We then test the relation between the squared bias and the Hamming distance in (4), by generating MSAs at a given D and numerically computing for several K-link Potts model of increasing complexity. As shown in Fig 3C, the linear relation between the true squared bias and D is confirmed for every value of K (R ≃ 1 for every tested K-link model). Similarly, we find a good agreement between the numerical variance and our theoretical estimate in (3), see Fig 3D (R ≃1 for every K-link Potts model). J0 reflects the un-modeled epistasis. The slope of the numerical bias μ2 with D (Fig 3B) gives access to an estimate for J0. We plot in Fig 3F the corresponding J0 as a function of the number K of links in the Potts model, from K = 0 (independent model) to K = 40. We find that J0(K) decreases almost linearly with K before reaching a saturation point around K = 20. This decrease is in accordance with the notion of J0 as reflecting the un-modeled epistasis. In the context of Lattice Proteins, this saturation behavior is expected to reflect the presence of two distinct classes of un-modeled epistatic couplings. Strong pairwise interactions correspond to the Nc = 28 contacts on the 3D fold (Fig 3A). These “structural” couplings are expected to be largely responsible for the magnitude of epistatic effects in the fitness function, therefore contributing the most to the value of J0. The remaining Kmax − Nc are weaker, and may be due to the need to avoid other folds (negative design) or to higher-order interactions [50]. To verify this hypothesis, we retrieve a pairwise approximation of the real fitness function by inferring a fully-connected Potts model from a very large alignment (B ∼ 106 sequences). We then separate the inferred Potts couplings into structural and non-structural and compute their variance as a proxy for their expected contribution to the value of J0 (see Appendix A in S1 Text). As shown in Fig 3F, structural couplings have a much larger variance than the other ones. We can devise an effective theoretical approximation of the behavior for J0(K) by assuming that all structural and non-structural couplings are uniformly drawn from two distributions with the two variances above, and that the sparse model progressively includes structural couplings in its energy function up to K = Nc. The expected behavior of J0(K) under this effective model, shown in Fig 3G, agrees with Fig 3E, and saturates to its lowest value around K = 28, which corresponds to the total number of structural couplings. J0 can be inferred from mutational scan data. Last, we propose an alternative approach to estimate the bias factor J0, which is applicable to real protein data, where the sequence-to-fitness mapping is unknown but mutational scans are available. For fixed model complexity (value of K), we subsample the MSA, infer the corresponding K-link Potts models, and estimate the predictive performances ρ. The procedure is repeated by varying the quantity (B) and relevance (D) of the sub-MSAs. We then consider J0 as a free parameter and infer its value by maximizing the Spearman correlation between the two sides of (5), where σ2 is estimated from Eq (3) and μ2 = J0D. We call this approach the “best scaling fit”. We apply this procedure to the same lattice protein data shown above. Results for the performance ρ vs. J0D + σ2 are shown in Fig 3H for all K-link Potts models (R ≃ 1 for every tested K-link model), in excellent agreement with the ground truth results of Fig 3B. The fitted values of J0(K) are reported in Fig 3I, in excellent agreement with Fig 3E and 3G. Performance vs. quantity and relevance of sequence data for real proteins Trade-off explains the predictive performance in mutagenesis experiments. The relation in (1), which we verified on in-silico proteins, postulates that the performance ρ of the predictive model is controlled by the sum of the squared bias J0D, as an inverse proxy for the relevance of the sequence data, and of the variance σ2, which inversely depends on the quantity of data. To test our theory on real data, we consider 7 different mutagenesis experiments on 7 proteins. For each protein, we sub-sample the corresponding MSA as done in Fig 2, to obtain sub-MSAs with a large range of values of D and B, from which we can compute the estimated variance σ2. We then compute the two descriptors D and σ2 from each sub-MSA, and compare them with the predictive performance inferred from the data. As reported in Fig 4A and 4D, is a fairly good predictor for the performance of an independent-site Potts model (RNA-binding domain—absolute value of Spearman correlation coefficient rS between D and ρ = 0.70), while the variance alone correlates more weakly with the predictive performance (RNA-binding domain—absolute value of Spearman correlation coefficient rS between σ2 and ρ = 0.25). However, when the performance is compared to the sum of the squared bias and the variance, J0D + σ2, the correlation can be made much higher through fitting of J0 (RNA-binding domain—absolute value of Spearman correlation coefficient rS between J0D + σ2 and ρ = 0.95, Fig 4B). This strong correlation is confirmed for the 7 protein families (rS > 0.9 for all 7 families, Fig 4C and S2 Fig), providing a strong verification of the theoretical and numerical framework developed above. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Relevance-quantity trade-off explains the predictive performance of statistical modelling. A predictive performance of single-point mutations using the Independent-site on the RNA-bind protein, shown as a function of the mean Hamming distance of the MSA (top) and variance estimated from the alignments (bottom). B predictive performance of single-point mutations as a function of the linear sum of squared bias and variance. The scaling correlation rS is computed as the absolute value of the Spearman correlation coefficient of J0D + σ2 vs. ρ. The bias factor J0 is inferred by maximizing rS, as done in Fig 2E. C scaling correlation rS for the seven protein families, compared to chance levels. The chance distribution is built by destroying the relationship between the performance ρ and the two descriptors by random order shuffling, then repeating the J0 inference procedure to account for the scaling optimization during its estimation. Error bars show standard deviations over n = 100 repetitions of the random shuffling. D top: RNA-bind family, predictive performance ρ as a function of the cutoff distance dcut, showing the existence of an optimal cutoff dopt (black dashed line). Bottom: individual contributions of squared bias (J0 D, purple line), variance (σ2, green line) and their sum (blue line). The red dashed line indicates the minimum of J0D + σ2, which corresponds to the predicted maximum performance cutoff dbv. E Values of predictive performance ρ at the optimal cutoffs compared to the full alignments for the 7 protein families. F ratio between performance increase at cutoffs of interest and at the optimal cutoff for the 7 protein families. https://doi.org/10.1371/journal.pcbi.1011521.g004 Optimization of performance through a focusing procedure. We may now exploit our understanding of how performance depends on the number B and on the mean Hamming distance D of the sequences in the MSA to find the optimal sub-alignments maximizing ρ. As we see in Fig 2, we can start from the full MSA and progressively focus around wt by excluding all sequences of “low relevance”, i.e., at Hamming distances higher than a given cutoff dcut. As we lower dcut from its maximal value (N, number of sites) down to 0, this focusing procedure increases the variance while decreasing the bias, as we select fewer sequences with higher homology to wt. As already seen in Fig 2C, the predictive performance ρ has a maximum at a certain optimal cutoff dopt (Fig 4D (top panel)), highlighting the trade-off between bias and variance in controlling the performance. In Fig 4E, we report the performance of the independent-site model at the optimal cutoff dopt. We find notable improvements in the predictive performance for 6 out of 7 protein families with respect to the full MSA (mean improvement Δρ(dopt) = 0.081). Importantly, for 3 families out of 7 (DNA-bind, RL401, WW), the value of ρ at the optimal cutoff exceeded the best performance reported in [37] and obtained with PLM-DCA, a standard approach to learn the Potts model parameters [57]. This result is striking, as both the number of parameters and the number of training sequences involved in the inference at dopt are greatly reduced compared to fully-connected Potts models on large MSAs. The most outstanding illustration is the DNA-bind family, where top performance (Δρ = 0.26) is found for dopt = 29, corresponding to only B = 37 effective sequences in the MSA (see S3 Fig). Cutoff for optimal focusing can be reliably predicted from heuristics. According to (5) the best performance is reached for the alignment that minimizes the sum J0D + σ2. We call this optimal predicted cutoff dbv, as for bias-variance, (6) As reported in Fig 4D(top) (red line) and Fig 4F, this procedure allows us to predict the optimal cutoff with good precision (mean relative error = 0.08). Importantly, the performance increase at the predicted cutoff dbv captures most of the total possible improvement (mean guessed relative increase for the 7 families Δρ(dopt)/Δρ(dopt) = 0.86 ± 0.08, see Fig 4F). Globally, the performance at the predicted cutoff dopt is systematically higher than the performance with the full MSA (mean Δρ(dopt) = 0.073, paired Wilcoxon test over the n = 7 families: P = 0.018). However, knowledge of the bias factor J0 entering Eq (6) is not always available, as it requires a systematic analysis of predictive performance relying on the outcome of mutagenesis experiments as a reference. We propose below a simple heuristics for predicting the optimal cutoff, requiring no experimental input and based on a signal-to-noise ratio (SNR) comparing the spread of inferred fitness values across sites and mutations and the statistical variance σ2, Fig 4D(bottom): (7) Setting for instance the cutoff dsnr corresponding to a threshold of SNR = 3, we again find systematic improvements in the predictive performance (mean guessed relative increase for the 7 families Δρ(dsnr)/Δρ(dopt) = 0.71 ± 0.10, see Fig 4F, S3 and S4(b) Figs), providing an unsupervised, parameter-free criterion to select the optimal MSA for the predictive analysis. Notice that the choice of the value SNR = 3 above is arbitrary; A consistent improvement of performance can be found for SNR in the range ∼ 2 to ∼ 4, see S4(a) Fig. The bias factor J0 depends on the model expressivity. We repeat in Fig 5A the approach of Fig 4B, using the K–link Potts model rather than the independent-site model for fitness predictions. The number of couplings, K, is chosen to be a fraction of N, and is much smaller than Kmax, implying that the Potts model is very sparse. For each sub-alignment of the RNA-binding domain data we determine the best scaling fit bias J0(K). We observe very high correlations between ρ and J0(K)D + σ2. We also observe that top performances are found for a non-zero value of K, e.g. K = 0.1N in Fig 5A. The optimal value of K generally varies from family to family, as reported below. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. The bias factor J0 depends on the model expressivity. A Scaling correlation between predictive performance ρ and J0D + σ2 for the RNA-bind protein, modeled with the Sparse Potts Model with different numbers K of couplings. N is the length of the protein (82 sites). B: values of the bias factor J0 as a function of the number of modelled couplings in the Sparse Potts Model for the RNA-bind protein. C: same as B for the seven protein families combined; the black line and the blue area represent the mean and the standard deviation over the seven protein families. D Relation between bias factor J0(K) and improvement at best cutoff Δρ(dopt) for the RNA-bind protein. E same of D for the seven families combined. Values of K range from K = 0 to K = N. Each color corresponds to a different protein family as reported in the legend. https://doi.org/10.1371/journal.pcbi.1011521.g005 The value of the bias factor J0(K) is shown as a function of the number of links per site in Fig 5B for the RNA-binding domain and for all 7 protein families in Fig 5C. The general behaviour is similar to the one observed for lattice proteins (Fig 3), and shows that J0(K) decreases with K until saturation is reached. As the expressive power of the predictive model increases, the squared bias decreases and is less affected by the relevance of the sequence data. The saturation indicates that, above some critical K, adding more pairwise couplings does not help to reduce the bias. A possible explanation for this residual bias is the presence of higher-order epistasis, e.g. 3-site couplings between residues, which cannot be accounted for by the K–link Potts model. Empirically, we expect that the focusing procedure should provide substantial improvement if the bias strongly decreases with D, that is, if the bias factor J0 is large, e.g. in the case of the independent-site model. The intuition is that, when the bias quickly decrease with the relevance of the data, there is a margin for improvement of performances by removing some low-relevance data, while not increasing too much the statistical variance of the inferred model parameters. We report in Fig 5D the gain in performance ρ (compared to the independent Potts model, with K = 0) for the RNA-binding domain as a function of the bias factor J0 when K is varied. Results show a strong positive correlation between the two quantities. The same correlation is found across all 7 protein families, see Fig 5E and S5 Fig. Quantity-relevance trade-off in MSA sequence selection We consider a reference sequence, hereafter referred to as wt. We denote by the variation of fitness resulting from the mutation wti → a on the ith site of wt. This quantity can be estimated experimentally, either in vivo (relative enrichment of organisms with mutated gene compared to wt), or in vitro (measurement of appropriate biochemical property). A computational model provides a predictor, , for the difference of fitness between the mutant and the wt. The overall quality of the predictor will be assessed through the Spearman coefficient ρ between the mutation effects computed with the model and with the experimental data . Using Spearman correlations allows one to capture monotonous relations, irrespective of non-linearities. The computational model is generally trained from homologous sequences to wt, i.e. belonging to the same protein family. The similarities between the wt and these sequences, sampled from evolutionary diverse organisms, can vary significantly. As an illustration, we consider the RNA binding domain of the nuclear poly(A)-binding protein (PABPN1), involved in the synthesis of the mRNA poly(A) tails in eukaryotes [14]. Any two sequences in the corresponding MSA (as used in [37]) generally have few amino acids in common (mean Hamming distance -normalized by sequence length- between pairs of sequences in the MSA = 0.75). As a result, a specific sequence, such as the wt of Saccharomyces cerevisiae, is generally surrounded by a small number of similar sequences and is far away from most of the MSA (RNA-bind protein: mean normalized Hamming distance between wt and MSA sequences = 0.73, Fig 2A; see S1 Fig for similar results on other families). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Behaviour of model predictive performance with different selections of training data. A. Distribution of Hamming distances to the wt sequence (RNA-binding domain of Pab1-Yeast) in the MSA of [37]. Note the log scale on the y axis. The three colored lines correspond to three possible sequence selections performed by excluding sequences farther than a certain threshold dcut from wt. A smaller dcut corresponds to fewer sequences with a lower mean Hamming distance to the wt, denoted as D. B Comparison between predicted and experimental fitness mutational effects for an independent-site model trained on the three sub-MSAs corresponding to, respectively, dcut = 32 (orange), 43 (purple), and 82 (green). The Spearman correlation coefficient ρ between predicted and experimental values defines the predictive performance of the model. C Same analysis as panel B repeated for all possible cutoffs between dcut = 32 and dcut = 82 (the sequence length). The non monotonous behavior of the predictive performance indicates that a trade-off between number of sequences (denoted as B) and proximity to wt is controlling the predictive performance of the inferred model. D. Systematic analysis of the predictive power ρ as a function of the mean Hamming distance D of sub-alignments with fixed size B (top), and of the sub-alignment size B at fixed Hamming distance D (bottom). Each individual point shows the average over n = 5 sub-samples obtained at the corresponding values of D and B (see Methods). The dashed curves and error bars are computed by binned average and standard deviation over the displayed individual points. All significance levels refer to Spearman rank correlation of the individual points. *** P < 0.001. https://doi.org/10.1371/journal.pcbi.1011521.g002 Hereafter, we show that sequences far away from wt are not relevant for fitness prediction. To do so we train independent-site Potts models (Methods) on shorter MSAs obtained by discarding sequences further than a certain cut-off distance dcut from wt. As dcut becomes smaller, fewer sequences with higher proximity are selected (Fig 2A). We see that the performance consistently increases when decreasing the cut-off distance, up to a peak ρ = 0.56 at dcut = 43, a 33% increase with respect to the full MSA (ρ(dcut = 82) = 0.42), see Fig 2B and 2C). After peaking, the performance starts decreasing again due to the increasingly-lower number of sequences in the MSA, see Fig 2C. The non-monotonous behavior of the predictive performance indicates that a trade-off between the number of sequences and their proximity to wt is controlling the predictive performance of the inferred model. To investigate the respective effects of these two quantities, we create sub-alignments of the original MSA with controlled sizes B (effective number of sequence taking into account sequence redundancy, see [51] and Methods) and average Hamming distances to wt, which we denote as D. We then test how the performance of the independent-site Potts model trained on these sub-alignments relates to these two quantities. This analysis showed that the predictive performance strongly depends on the mean Hamming distance D and on the number B of sequences (P < 0.001 for all Spearman correlations between ρ and B or D, Fig 2D). The performance significantly decreases with D at fixed B, i.e., when the relevance of the data deteriorates and their quantity is kept fixed, and increases with B at fixed D, i.e., when the quantity of data increases at fixed similarity with wt. Similar results are found for the six other protein families under study (see S1 Fig). Theoretical investigation of the quantity-relevance trade-off To study the trade-off between relevance and quantity we draw our inspiration from the bias-variance framework developed in statistics [52, 53]. Let us consider the error between the statistical predictor and the experimental fitness . This error can be decomposed into the sum of two contributions: (1) a systematic bias in the prediction, due to the inability of the model to capture the exact relation between sequence mutation and fitness, (2) a statistical error coming from the fact that the predictive model has been trained on a particular data set; the value of this contribution fluctuates when the data set changes, and is expected to be smaller and smaller for larger and larger data sets. Consequently, the mean squared error on the single-point mutation wti → a can be written as the sum of a squared bias and a variance contributions, (1) where averages [] are taken on repetitions of the prediction process in fixed conditions (relevance and quantity of data). Notably, these two quantities are hard to minimize together. For instance, powerful models with many parameters will accurately fit the data and thus achieve small squared biases , but will result in large variances due to the statistical errors on the many parameters to be inferred. As we will see below, we can directly relate our descriptors of relevance (D) and quantity (B) of the sequence data to, respectively, the squared bias and the variance as defined in (1). Furthermore, we will introduce a class of increasingly powerful Potts models to investigate the effect of model complexity on these two quantities. In addition to its theoretical appeal and close connection with the bias-variance decomposition, considering the mean–squared error is ultimately justified by the empirically observed monotonic relation with the predictive performance as measured through the Spearman coefficient ρ. K–link Potts model. We consider hereafter the class of sparse Potts models, which include K pairwise couplings between the protein sites, Jij(a, b), whose values depend on the amino acids they carry and a field (position weight matrix) hi(a) on each site; These parameters are learned from the MSA (Methods). The choice of the K pairs of sites carrying couplings is decided based on heuristics, which aim at capturing interrelations between the residues (Methods). By tuning the value of K, we can interpolate between the independent-site model (K = 0, i.e. no coupling) and the full Potts model ( couplings, where N is the protein length). Imposing small values of K is a way to regularize the inferred network of interactions. Notice that the number of parameters to be inferred, Npar = NQ + KQ2, where Q = 20 is the number of amino acids, grows quickly with K since Q2 = 400. For the K-link Potts model the predictor of the fitness difference resulting from the mutation wti → a reads (2) where the sums runs on the sites j in the neighborhood of site i, i.e. coupled to i (Methods). This neighborhood is empty for the independent-site model. Estimation of variance. For the Potts model, expressions for the uncertainties on the inferred fields hi(a) and couplings Jij(a, b) can be formally derived from sampling errors due to the finite size of the data set. The resulting variance of the predictor for a specific K-link model can then be estimated from (2) [38, 54], see Appendix A in S1 Text. Averaging over the sites i and mutations a, we obtain a single global variance, (3) where Q = 20 is the number of amino-acid types, and ki is the cardinality of , i.e. the number of sites interacting with i in the model. The global variance depends on the statistics of the data through the probabilities pi(a) of finding amino acid a on the i-th site and pij(a, b) of finding simultaneously a on site i and b on site j computed on the sub-alignment. Thus, σ2 increases with residue conservation, due to the contributions of amino acids that are rarely observed on some sites in the sub-alignment and have low pi(a), and with the number K of coupling parameters in the model. We also see that σ2 is inversely proportional to the number of sequences, B. The variance therefore decreases with the quantity of data. Estimation of squared bias. Computing the squared bias μ2 in (1) is generally hard, not to say impossible, as it requires detailed knowledge of the fitness landscape. We rely below on simplifying assumptions to gather insights on the value and meaning of the bias. Assume first that we use the independent-site model for fitness prediction. If the ‘true’ fitness landscape shows no epistasis, this model is exact (up to statistical fluctuations due to the finite amount of training data, taken care of by σ2), and the bias vanishes. Therefore, a non zero bias would signal the presence of epistatic interactions between residues not captured by the simple model used for predictions. We stress that this statement is true in an idealized setting, in which the only source of bias is the mismatch between the model power and the ground-truth fitness landscape. In reality, biases can have multiple origins, including non-uniform sampling of sequence data (resulting from preferential choices of organisms or from evolutionary correlations), discrepancies between in vivo fitness reflected by sequence data and in vitro biochemical measurements, etc. Let us now turn to more complex landscapes and models. We assume that the fitness landscape is characterized by pairwise epistasis only, i.e. the fitness differences are exactly described by a full Potts model with Kmax interactions through an equation analogous to (2). The K–link Potts model used for fitness prediction will not be powerful enough to account for the complexity of this landscape and of the sequence data if K < Kmax. As a result a non-zero squared bias will appear, whose expression is derived in Appendix B in S1 Text, and reads (4) where D is the mean Hamming distance of the sub-alignment sequences to wt, and the bias factor J0 is the product of a multiplicative factor depending on the background distribution of amino acids in the MSA and of the variance of the epistatic couplings JFnot included in the prediction model. J0 is thus a decreasing function of K. This expression of μ2 confirms that the Hamming distance D is related to the notion of relevance (similarity to the wt) of the sequence data, as varying D affects the systematic error (bias) of the predictive model. K–link Potts model. We consider hereafter the class of sparse Potts models, which include K pairwise couplings between the protein sites, Jij(a, b), whose values depend on the amino acids they carry and a field (position weight matrix) hi(a) on each site; These parameters are learned from the MSA (Methods). The choice of the K pairs of sites carrying couplings is decided based on heuristics, which aim at capturing interrelations between the residues (Methods). By tuning the value of K, we can interpolate between the independent-site model (K = 0, i.e. no coupling) and the full Potts model ( couplings, where N is the protein length). Imposing small values of K is a way to regularize the inferred network of interactions. Notice that the number of parameters to be inferred, Npar = NQ + KQ2, where Q = 20 is the number of amino acids, grows quickly with K since Q2 = 400. For the K-link Potts model the predictor of the fitness difference resulting from the mutation wti → a reads (2) where the sums runs on the sites j in the neighborhood of site i, i.e. coupled to i (Methods). This neighborhood is empty for the independent-site model. Estimation of variance. For the Potts model, expressions for the uncertainties on the inferred fields hi(a) and couplings Jij(a, b) can be formally derived from sampling errors due to the finite size of the data set. The resulting variance of the predictor for a specific K-link model can then be estimated from (2) [38, 54], see Appendix A in S1 Text. Averaging over the sites i and mutations a, we obtain a single global variance, (3) where Q = 20 is the number of amino-acid types, and ki is the cardinality of , i.e. the number of sites interacting with i in the model. The global variance depends on the statistics of the data through the probabilities pi(a) of finding amino acid a on the i-th site and pij(a, b) of finding simultaneously a on site i and b on site j computed on the sub-alignment. Thus, σ2 increases with residue conservation, due to the contributions of amino acids that are rarely observed on some sites in the sub-alignment and have low pi(a), and with the number K of coupling parameters in the model. We also see that σ2 is inversely proportional to the number of sequences, B. The variance therefore decreases with the quantity of data. Estimation of squared bias. Computing the squared bias μ2 in (1) is generally hard, not to say impossible, as it requires detailed knowledge of the fitness landscape. We rely below on simplifying assumptions to gather insights on the value and meaning of the bias. Assume first that we use the independent-site model for fitness prediction. If the ‘true’ fitness landscape shows no epistasis, this model is exact (up to statistical fluctuations due to the finite amount of training data, taken care of by σ2), and the bias vanishes. Therefore, a non zero bias would signal the presence of epistatic interactions between residues not captured by the simple model used for predictions. We stress that this statement is true in an idealized setting, in which the only source of bias is the mismatch between the model power and the ground-truth fitness landscape. In reality, biases can have multiple origins, including non-uniform sampling of sequence data (resulting from preferential choices of organisms or from evolutionary correlations), discrepancies between in vivo fitness reflected by sequence data and in vitro biochemical measurements, etc. Let us now turn to more complex landscapes and models. We assume that the fitness landscape is characterized by pairwise epistasis only, i.e. the fitness differences are exactly described by a full Potts model with Kmax interactions through an equation analogous to (2). The K–link Potts model used for fitness prediction will not be powerful enough to account for the complexity of this landscape and of the sequence data if K < Kmax. As a result a non-zero squared bias will appear, whose expression is derived in Appendix B in S1 Text, and reads (4) where D is the mean Hamming distance of the sub-alignment sequences to wt, and the bias factor J0 is the product of a multiplicative factor depending on the background distribution of amino acids in the MSA and of the variance of the epistatic couplings JFnot included in the prediction model. J0 is thus a decreasing function of K. This expression of μ2 confirms that the Hamming distance D is related to the notion of relevance (similarity to the wt) of the sequence data, as varying D affects the systematic error (bias) of the predictive model. Validation of the theory on Lattice Proteins To validate the key role of the squared bias and of the variance in explaining performance, as well as their approximate expressions above and the interpretation of the bias factor J0 as reflecting un-modeled epistasis, we resort to an in silico model for proteins folding on a 27-site cubic lattice [47, 49, 50, 55, 56], see Fig 3A. In the model, the fitness represents the propensity of a protein sequence to fold into one specific conformation, called native, out of the ≃ 105 folds on the cube [49]. Following [50], the native fold and wildtype sequence were chosen such that the fitness of the wildtype was high enough to be stable but low enough to allow for positive mutations (Pnat ≃ 0.995, see Methods). As we can precisely compute the exact value of the fitness, the ground-truth values of the squared bias and of the variance defined in (1) can be computed with great accuracy (see Methods); we hereafter denote these ground truth values by and . Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Quantity-relevance trade-off for lattice proteins. A: Cubic fold that defines the protein family in the lattice model. Amino acids on sites that are in proximity to each other interact and define the energy of the protein (Methods). B: Predictive performance ρ for single mutations of 5 Sparse Potts models with different degrees of sparsity (defined by K, the number of pairwise links included in the energy function; K = 0 is the independent model) vs. . The collapse of the results is in agreement with Eq (5). C: Squared bias vs. mean Hamming distance in the sequence data, see Eq (4), for the same sparse Potts models as in panel B. Line plots and error bars show mean and standard deviation at a given D and different Bs. D: Variance σ2 vs. estimated variance σ2 in Eq (3) for the same Sparse Potts models as in panel B. E: Bias factor J0(K) (divided by J0(0)) obtained by fitting the squared bias as a linear function of the mean Hamming distance for the various K-link models in panel C. F: Visualization of pairwise couplings inferred by a fully-connected Potts model, highlighting the larger variance of couplings associated to structural contacts (in orange) compared to non-structural ones (in blue)—note the log scale on the y-axis. G: Normalized value of J0(K) (divided by J0(0)) obtained with an effective theory using the variance of couplings associated to modeled and un-modeled structural contacts, see Appendix A in S1 Text. H: scaling for predictive performance ρ of our statistical models for single point mutations as a function of the sum of the estimated squared bias J0D and of the variance σ2 in Eq (3). J0(K) (denoted as in the plot axis label) is fitted to for each value of K by maximizing the scaling correlation as explained in the main text. I: Bias factor J0(K) (normalized by J0(0)) inferred from maximizing the scaling correlation as in panel H. https://doi.org/10.1371/journal.pcbi.1011521.g003 Bias and variance are sufficient to explain model performance. Eq (1) stipulates that the mean squared error over fitness prediction depends on the sum of squared bias and variance of the fitness predictors. If the performance ρ is, in turn, controlled by this mean squared error, we expect a relation such as (5) where F is a decreasing function of its argument. To test the validity of (5), we compare the values of ρ obtained with the independent-site Potts models (K = 0) and different K-link Potts models (K = 4, 8, 16, 24) trained from various sub-alignments with different B, D to the sums of the squared bias and variance, see Fig 3B. We obtain an excellent anti-correlation between ρ and across a large range of values of B and D, in full agreement with (5) (R ∼ 1 for every K–link model). The sum of squared bias and variance is by far the biggest factor in determining the predicting performance of the models. Bias and variance are related to the relevance and the quantity of data as predicted by theory. We then test the relation between the squared bias and the Hamming distance in (4), by generating MSAs at a given D and numerically computing for several K-link Potts model of increasing complexity. As shown in Fig 3C, the linear relation between the true squared bias and D is confirmed for every value of K (R ≃ 1 for every tested K-link model). Similarly, we find a good agreement between the numerical variance and our theoretical estimate in (3), see Fig 3D (R ≃1 for every K-link Potts model). J0 reflects the un-modeled epistasis. The slope of the numerical bias μ2 with D (Fig 3B) gives access to an estimate for J0. We plot in Fig 3F the corresponding J0 as a function of the number K of links in the Potts model, from K = 0 (independent model) to K = 40. We find that J0(K) decreases almost linearly with K before reaching a saturation point around K = 20. This decrease is in accordance with the notion of J0 as reflecting the un-modeled epistasis. In the context of Lattice Proteins, this saturation behavior is expected to reflect the presence of two distinct classes of un-modeled epistatic couplings. Strong pairwise interactions correspond to the Nc = 28 contacts on the 3D fold (Fig 3A). These “structural” couplings are expected to be largely responsible for the magnitude of epistatic effects in the fitness function, therefore contributing the most to the value of J0. The remaining Kmax − Nc are weaker, and may be due to the need to avoid other folds (negative design) or to higher-order interactions [50]. To verify this hypothesis, we retrieve a pairwise approximation of the real fitness function by inferring a fully-connected Potts model from a very large alignment (B ∼ 106 sequences). We then separate the inferred Potts couplings into structural and non-structural and compute their variance as a proxy for their expected contribution to the value of J0 (see Appendix A in S1 Text). As shown in Fig 3F, structural couplings have a much larger variance than the other ones. We can devise an effective theoretical approximation of the behavior for J0(K) by assuming that all structural and non-structural couplings are uniformly drawn from two distributions with the two variances above, and that the sparse model progressively includes structural couplings in its energy function up to K = Nc. The expected behavior of J0(K) under this effective model, shown in Fig 3G, agrees with Fig 3E, and saturates to its lowest value around K = 28, which corresponds to the total number of structural couplings. J0 can be inferred from mutational scan data. Last, we propose an alternative approach to estimate the bias factor J0, which is applicable to real protein data, where the sequence-to-fitness mapping is unknown but mutational scans are available. For fixed model complexity (value of K), we subsample the MSA, infer the corresponding K-link Potts models, and estimate the predictive performances ρ. The procedure is repeated by varying the quantity (B) and relevance (D) of the sub-MSAs. We then consider J0 as a free parameter and infer its value by maximizing the Spearman correlation between the two sides of (5), where σ2 is estimated from Eq (3) and μ2 = J0D. We call this approach the “best scaling fit”. We apply this procedure to the same lattice protein data shown above. Results for the performance ρ vs. J0D + σ2 are shown in Fig 3H for all K-link Potts models (R ≃ 1 for every tested K-link model), in excellent agreement with the ground truth results of Fig 3B. The fitted values of J0(K) are reported in Fig 3I, in excellent agreement with Fig 3E and 3G. Bias and variance are sufficient to explain model performance. Eq (1) stipulates that the mean squared error over fitness prediction depends on the sum of squared bias and variance of the fitness predictors. If the performance ρ is, in turn, controlled by this mean squared error, we expect a relation such as (5) where F is a decreasing function of its argument. To test the validity of (5), we compare the values of ρ obtained with the independent-site Potts models (K = 0) and different K-link Potts models (K = 4, 8, 16, 24) trained from various sub-alignments with different B, D to the sums of the squared bias and variance, see Fig 3B. We obtain an excellent anti-correlation between ρ and across a large range of values of B and D, in full agreement with (5) (R ∼ 1 for every K–link model). The sum of squared bias and variance is by far the biggest factor in determining the predicting performance of the models. Bias and variance are related to the relevance and the quantity of data as predicted by theory. We then test the relation between the squared bias and the Hamming distance in (4), by generating MSAs at a given D and numerically computing for several K-link Potts model of increasing complexity. As shown in Fig 3C, the linear relation between the true squared bias and D is confirmed for every value of K (R ≃ 1 for every tested K-link model). Similarly, we find a good agreement between the numerical variance and our theoretical estimate in (3), see Fig 3D (R ≃1 for every K-link Potts model). J0 reflects the un-modeled epistasis. The slope of the numerical bias μ2 with D (Fig 3B) gives access to an estimate for J0. We plot in Fig 3F the corresponding J0 as a function of the number K of links in the Potts model, from K = 0 (independent model) to K = 40. We find that J0(K) decreases almost linearly with K before reaching a saturation point around K = 20. This decrease is in accordance with the notion of J0 as reflecting the un-modeled epistasis. In the context of Lattice Proteins, this saturation behavior is expected to reflect the presence of two distinct classes of un-modeled epistatic couplings. Strong pairwise interactions correspond to the Nc = 28 contacts on the 3D fold (Fig 3A). These “structural” couplings are expected to be largely responsible for the magnitude of epistatic effects in the fitness function, therefore contributing the most to the value of J0. The remaining Kmax − Nc are weaker, and may be due to the need to avoid other folds (negative design) or to higher-order interactions [50]. To verify this hypothesis, we retrieve a pairwise approximation of the real fitness function by inferring a fully-connected Potts model from a very large alignment (B ∼ 106 sequences). We then separate the inferred Potts couplings into structural and non-structural and compute their variance as a proxy for their expected contribution to the value of J0 (see Appendix A in S1 Text). As shown in Fig 3F, structural couplings have a much larger variance than the other ones. We can devise an effective theoretical approximation of the behavior for J0(K) by assuming that all structural and non-structural couplings are uniformly drawn from two distributions with the two variances above, and that the sparse model progressively includes structural couplings in its energy function up to K = Nc. The expected behavior of J0(K) under this effective model, shown in Fig 3G, agrees with Fig 3E, and saturates to its lowest value around K = 28, which corresponds to the total number of structural couplings. J0 can be inferred from mutational scan data. Last, we propose an alternative approach to estimate the bias factor J0, which is applicable to real protein data, where the sequence-to-fitness mapping is unknown but mutational scans are available. For fixed model complexity (value of K), we subsample the MSA, infer the corresponding K-link Potts models, and estimate the predictive performances ρ. The procedure is repeated by varying the quantity (B) and relevance (D) of the sub-MSAs. We then consider J0 as a free parameter and infer its value by maximizing the Spearman correlation between the two sides of (5), where σ2 is estimated from Eq (3) and μ2 = J0D. We call this approach the “best scaling fit”. We apply this procedure to the same lattice protein data shown above. Results for the performance ρ vs. J0D + σ2 are shown in Fig 3H for all K-link Potts models (R ≃ 1 for every tested K-link model), in excellent agreement with the ground truth results of Fig 3B. The fitted values of J0(K) are reported in Fig 3I, in excellent agreement with Fig 3E and 3G. Performance vs. quantity and relevance of sequence data for real proteins Trade-off explains the predictive performance in mutagenesis experiments. The relation in (1), which we verified on in-silico proteins, postulates that the performance ρ of the predictive model is controlled by the sum of the squared bias J0D, as an inverse proxy for the relevance of the sequence data, and of the variance σ2, which inversely depends on the quantity of data. To test our theory on real data, we consider 7 different mutagenesis experiments on 7 proteins. For each protein, we sub-sample the corresponding MSA as done in Fig 2, to obtain sub-MSAs with a large range of values of D and B, from which we can compute the estimated variance σ2. We then compute the two descriptors D and σ2 from each sub-MSA, and compare them with the predictive performance inferred from the data. As reported in Fig 4A and 4D, is a fairly good predictor for the performance of an independent-site Potts model (RNA-binding domain—absolute value of Spearman correlation coefficient rS between D and ρ = 0.70), while the variance alone correlates more weakly with the predictive performance (RNA-binding domain—absolute value of Spearman correlation coefficient rS between σ2 and ρ = 0.25). However, when the performance is compared to the sum of the squared bias and the variance, J0D + σ2, the correlation can be made much higher through fitting of J0 (RNA-binding domain—absolute value of Spearman correlation coefficient rS between J0D + σ2 and ρ = 0.95, Fig 4B). This strong correlation is confirmed for the 7 protein families (rS > 0.9 for all 7 families, Fig 4C and S2 Fig), providing a strong verification of the theoretical and numerical framework developed above. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Relevance-quantity trade-off explains the predictive performance of statistical modelling. A predictive performance of single-point mutations using the Independent-site on the RNA-bind protein, shown as a function of the mean Hamming distance of the MSA (top) and variance estimated from the alignments (bottom). B predictive performance of single-point mutations as a function of the linear sum of squared bias and variance. The scaling correlation rS is computed as the absolute value of the Spearman correlation coefficient of J0D + σ2 vs. ρ. The bias factor J0 is inferred by maximizing rS, as done in Fig 2E. C scaling correlation rS for the seven protein families, compared to chance levels. The chance distribution is built by destroying the relationship between the performance ρ and the two descriptors by random order shuffling, then repeating the J0 inference procedure to account for the scaling optimization during its estimation. Error bars show standard deviations over n = 100 repetitions of the random shuffling. D top: RNA-bind family, predictive performance ρ as a function of the cutoff distance dcut, showing the existence of an optimal cutoff dopt (black dashed line). Bottom: individual contributions of squared bias (J0 D, purple line), variance (σ2, green line) and their sum (blue line). The red dashed line indicates the minimum of J0D + σ2, which corresponds to the predicted maximum performance cutoff dbv. E Values of predictive performance ρ at the optimal cutoffs compared to the full alignments for the 7 protein families. F ratio between performance increase at cutoffs of interest and at the optimal cutoff for the 7 protein families. https://doi.org/10.1371/journal.pcbi.1011521.g004 Optimization of performance through a focusing procedure. We may now exploit our understanding of how performance depends on the number B and on the mean Hamming distance D of the sequences in the MSA to find the optimal sub-alignments maximizing ρ. As we see in Fig 2, we can start from the full MSA and progressively focus around wt by excluding all sequences of “low relevance”, i.e., at Hamming distances higher than a given cutoff dcut. As we lower dcut from its maximal value (N, number of sites) down to 0, this focusing procedure increases the variance while decreasing the bias, as we select fewer sequences with higher homology to wt. As already seen in Fig 2C, the predictive performance ρ has a maximum at a certain optimal cutoff dopt (Fig 4D (top panel)), highlighting the trade-off between bias and variance in controlling the performance. In Fig 4E, we report the performance of the independent-site model at the optimal cutoff dopt. We find notable improvements in the predictive performance for 6 out of 7 protein families with respect to the full MSA (mean improvement Δρ(dopt) = 0.081). Importantly, for 3 families out of 7 (DNA-bind, RL401, WW), the value of ρ at the optimal cutoff exceeded the best performance reported in [37] and obtained with PLM-DCA, a standard approach to learn the Potts model parameters [57]. This result is striking, as both the number of parameters and the number of training sequences involved in the inference at dopt are greatly reduced compared to fully-connected Potts models on large MSAs. The most outstanding illustration is the DNA-bind family, where top performance (Δρ = 0.26) is found for dopt = 29, corresponding to only B = 37 effective sequences in the MSA (see S3 Fig). Cutoff for optimal focusing can be reliably predicted from heuristics. According to (5) the best performance is reached for the alignment that minimizes the sum J0D + σ2. We call this optimal predicted cutoff dbv, as for bias-variance, (6) As reported in Fig 4D(top) (red line) and Fig 4F, this procedure allows us to predict the optimal cutoff with good precision (mean relative error = 0.08). Importantly, the performance increase at the predicted cutoff dbv captures most of the total possible improvement (mean guessed relative increase for the 7 families Δρ(dopt)/Δρ(dopt) = 0.86 ± 0.08, see Fig 4F). Globally, the performance at the predicted cutoff dopt is systematically higher than the performance with the full MSA (mean Δρ(dopt) = 0.073, paired Wilcoxon test over the n = 7 families: P = 0.018). However, knowledge of the bias factor J0 entering Eq (6) is not always available, as it requires a systematic analysis of predictive performance relying on the outcome of mutagenesis experiments as a reference. We propose below a simple heuristics for predicting the optimal cutoff, requiring no experimental input and based on a signal-to-noise ratio (SNR) comparing the spread of inferred fitness values across sites and mutations and the statistical variance σ2, Fig 4D(bottom): (7) Setting for instance the cutoff dsnr corresponding to a threshold of SNR = 3, we again find systematic improvements in the predictive performance (mean guessed relative increase for the 7 families Δρ(dsnr)/Δρ(dopt) = 0.71 ± 0.10, see Fig 4F, S3 and S4(b) Figs), providing an unsupervised, parameter-free criterion to select the optimal MSA for the predictive analysis. Notice that the choice of the value SNR = 3 above is arbitrary; A consistent improvement of performance can be found for SNR in the range ∼ 2 to ∼ 4, see S4(a) Fig. The bias factor J0 depends on the model expressivity. We repeat in Fig 5A the approach of Fig 4B, using the K–link Potts model rather than the independent-site model for fitness predictions. The number of couplings, K, is chosen to be a fraction of N, and is much smaller than Kmax, implying that the Potts model is very sparse. For each sub-alignment of the RNA-binding domain data we determine the best scaling fit bias J0(K). We observe very high correlations between ρ and J0(K)D + σ2. We also observe that top performances are found for a non-zero value of K, e.g. K = 0.1N in Fig 5A. The optimal value of K generally varies from family to family, as reported below. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. The bias factor J0 depends on the model expressivity. A Scaling correlation between predictive performance ρ and J0D + σ2 for the RNA-bind protein, modeled with the Sparse Potts Model with different numbers K of couplings. N is the length of the protein (82 sites). B: values of the bias factor J0 as a function of the number of modelled couplings in the Sparse Potts Model for the RNA-bind protein. C: same as B for the seven protein families combined; the black line and the blue area represent the mean and the standard deviation over the seven protein families. D Relation between bias factor J0(K) and improvement at best cutoff Δρ(dopt) for the RNA-bind protein. E same of D for the seven families combined. Values of K range from K = 0 to K = N. Each color corresponds to a different protein family as reported in the legend. https://doi.org/10.1371/journal.pcbi.1011521.g005 The value of the bias factor J0(K) is shown as a function of the number of links per site in Fig 5B for the RNA-binding domain and for all 7 protein families in Fig 5C. The general behaviour is similar to the one observed for lattice proteins (Fig 3), and shows that J0(K) decreases with K until saturation is reached. As the expressive power of the predictive model increases, the squared bias decreases and is less affected by the relevance of the sequence data. The saturation indicates that, above some critical K, adding more pairwise couplings does not help to reduce the bias. A possible explanation for this residual bias is the presence of higher-order epistasis, e.g. 3-site couplings between residues, which cannot be accounted for by the K–link Potts model. Empirically, we expect that the focusing procedure should provide substantial improvement if the bias strongly decreases with D, that is, if the bias factor J0 is large, e.g. in the case of the independent-site model. The intuition is that, when the bias quickly decrease with the relevance of the data, there is a margin for improvement of performances by removing some low-relevance data, while not increasing too much the statistical variance of the inferred model parameters. We report in Fig 5D the gain in performance ρ (compared to the independent Potts model, with K = 0) for the RNA-binding domain as a function of the bias factor J0 when K is varied. Results show a strong positive correlation between the two quantities. The same correlation is found across all 7 protein families, see Fig 5E and S5 Fig. Trade-off explains the predictive performance in mutagenesis experiments. The relation in (1), which we verified on in-silico proteins, postulates that the performance ρ of the predictive model is controlled by the sum of the squared bias J0D, as an inverse proxy for the relevance of the sequence data, and of the variance σ2, which inversely depends on the quantity of data. To test our theory on real data, we consider 7 different mutagenesis experiments on 7 proteins. For each protein, we sub-sample the corresponding MSA as done in Fig 2, to obtain sub-MSAs with a large range of values of D and B, from which we can compute the estimated variance σ2. We then compute the two descriptors D and σ2 from each sub-MSA, and compare them with the predictive performance inferred from the data. As reported in Fig 4A and 4D, is a fairly good predictor for the performance of an independent-site Potts model (RNA-binding domain—absolute value of Spearman correlation coefficient rS between D and ρ = 0.70), while the variance alone correlates more weakly with the predictive performance (RNA-binding domain—absolute value of Spearman correlation coefficient rS between σ2 and ρ = 0.25). However, when the performance is compared to the sum of the squared bias and the variance, J0D + σ2, the correlation can be made much higher through fitting of J0 (RNA-binding domain—absolute value of Spearman correlation coefficient rS between J0D + σ2 and ρ = 0.95, Fig 4B). This strong correlation is confirmed for the 7 protein families (rS > 0.9 for all 7 families, Fig 4C and S2 Fig), providing a strong verification of the theoretical and numerical framework developed above. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Relevance-quantity trade-off explains the predictive performance of statistical modelling. A predictive performance of single-point mutations using the Independent-site on the RNA-bind protein, shown as a function of the mean Hamming distance of the MSA (top) and variance estimated from the alignments (bottom). B predictive performance of single-point mutations as a function of the linear sum of squared bias and variance. The scaling correlation rS is computed as the absolute value of the Spearman correlation coefficient of J0D + σ2 vs. ρ. The bias factor J0 is inferred by maximizing rS, as done in Fig 2E. C scaling correlation rS for the seven protein families, compared to chance levels. The chance distribution is built by destroying the relationship between the performance ρ and the two descriptors by random order shuffling, then repeating the J0 inference procedure to account for the scaling optimization during its estimation. Error bars show standard deviations over n = 100 repetitions of the random shuffling. D top: RNA-bind family, predictive performance ρ as a function of the cutoff distance dcut, showing the existence of an optimal cutoff dopt (black dashed line). Bottom: individual contributions of squared bias (J0 D, purple line), variance (σ2, green line) and their sum (blue line). The red dashed line indicates the minimum of J0D + σ2, which corresponds to the predicted maximum performance cutoff dbv. E Values of predictive performance ρ at the optimal cutoffs compared to the full alignments for the 7 protein families. F ratio between performance increase at cutoffs of interest and at the optimal cutoff for the 7 protein families. https://doi.org/10.1371/journal.pcbi.1011521.g004 Optimization of performance through a focusing procedure. We may now exploit our understanding of how performance depends on the number B and on the mean Hamming distance D of the sequences in the MSA to find the optimal sub-alignments maximizing ρ. As we see in Fig 2, we can start from the full MSA and progressively focus around wt by excluding all sequences of “low relevance”, i.e., at Hamming distances higher than a given cutoff dcut. As we lower dcut from its maximal value (N, number of sites) down to 0, this focusing procedure increases the variance while decreasing the bias, as we select fewer sequences with higher homology to wt. As already seen in Fig 2C, the predictive performance ρ has a maximum at a certain optimal cutoff dopt (Fig 4D (top panel)), highlighting the trade-off between bias and variance in controlling the performance. In Fig 4E, we report the performance of the independent-site model at the optimal cutoff dopt. We find notable improvements in the predictive performance for 6 out of 7 protein families with respect to the full MSA (mean improvement Δρ(dopt) = 0.081). Importantly, for 3 families out of 7 (DNA-bind, RL401, WW), the value of ρ at the optimal cutoff exceeded the best performance reported in [37] and obtained with PLM-DCA, a standard approach to learn the Potts model parameters [57]. This result is striking, as both the number of parameters and the number of training sequences involved in the inference at dopt are greatly reduced compared to fully-connected Potts models on large MSAs. The most outstanding illustration is the DNA-bind family, where top performance (Δρ = 0.26) is found for dopt = 29, corresponding to only B = 37 effective sequences in the MSA (see S3 Fig). Cutoff for optimal focusing can be reliably predicted from heuristics. According to (5) the best performance is reached for the alignment that minimizes the sum J0D + σ2. We call this optimal predicted cutoff dbv, as for bias-variance, (6) As reported in Fig 4D(top) (red line) and Fig 4F, this procedure allows us to predict the optimal cutoff with good precision (mean relative error = 0.08). Importantly, the performance increase at the predicted cutoff dbv captures most of the total possible improvement (mean guessed relative increase for the 7 families Δρ(dopt)/Δρ(dopt) = 0.86 ± 0.08, see Fig 4F). Globally, the performance at the predicted cutoff dopt is systematically higher than the performance with the full MSA (mean Δρ(dopt) = 0.073, paired Wilcoxon test over the n = 7 families: P = 0.018). However, knowledge of the bias factor J0 entering Eq (6) is not always available, as it requires a systematic analysis of predictive performance relying on the outcome of mutagenesis experiments as a reference. We propose below a simple heuristics for predicting the optimal cutoff, requiring no experimental input and based on a signal-to-noise ratio (SNR) comparing the spread of inferred fitness values across sites and mutations and the statistical variance σ2, Fig 4D(bottom): (7) Setting for instance the cutoff dsnr corresponding to a threshold of SNR = 3, we again find systematic improvements in the predictive performance (mean guessed relative increase for the 7 families Δρ(dsnr)/Δρ(dopt) = 0.71 ± 0.10, see Fig 4F, S3 and S4(b) Figs), providing an unsupervised, parameter-free criterion to select the optimal MSA for the predictive analysis. Notice that the choice of the value SNR = 3 above is arbitrary; A consistent improvement of performance can be found for SNR in the range ∼ 2 to ∼ 4, see S4(a) Fig. The bias factor J0 depends on the model expressivity. We repeat in Fig 5A the approach of Fig 4B, using the K–link Potts model rather than the independent-site model for fitness predictions. The number of couplings, K, is chosen to be a fraction of N, and is much smaller than Kmax, implying that the Potts model is very sparse. For each sub-alignment of the RNA-binding domain data we determine the best scaling fit bias J0(K). We observe very high correlations between ρ and J0(K)D + σ2. We also observe that top performances are found for a non-zero value of K, e.g. K = 0.1N in Fig 5A. The optimal value of K generally varies from family to family, as reported below. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. The bias factor J0 depends on the model expressivity. A Scaling correlation between predictive performance ρ and J0D + σ2 for the RNA-bind protein, modeled with the Sparse Potts Model with different numbers K of couplings. N is the length of the protein (82 sites). B: values of the bias factor J0 as a function of the number of modelled couplings in the Sparse Potts Model for the RNA-bind protein. C: same as B for the seven protein families combined; the black line and the blue area represent the mean and the standard deviation over the seven protein families. D Relation between bias factor J0(K) and improvement at best cutoff Δρ(dopt) for the RNA-bind protein. E same of D for the seven families combined. Values of K range from K = 0 to K = N. Each color corresponds to a different protein family as reported in the legend. https://doi.org/10.1371/journal.pcbi.1011521.g005 The value of the bias factor J0(K) is shown as a function of the number of links per site in Fig 5B for the RNA-binding domain and for all 7 protein families in Fig 5C. The general behaviour is similar to the one observed for lattice proteins (Fig 3), and shows that J0(K) decreases with K until saturation is reached. As the expressive power of the predictive model increases, the squared bias decreases and is less affected by the relevance of the sequence data. The saturation indicates that, above some critical K, adding more pairwise couplings does not help to reduce the bias. A possible explanation for this residual bias is the presence of higher-order epistasis, e.g. 3-site couplings between residues, which cannot be accounted for by the K–link Potts model. Empirically, we expect that the focusing procedure should provide substantial improvement if the bias strongly decreases with D, that is, if the bias factor J0 is large, e.g. in the case of the independent-site model. The intuition is that, when the bias quickly decrease with the relevance of the data, there is a margin for improvement of performances by removing some low-relevance data, while not increasing too much the statistical variance of the inferred model parameters. We report in Fig 5D the gain in performance ρ (compared to the independent Potts model, with K = 0) for the RNA-binding domain as a function of the bias factor J0 when K is varied. Results show a strong positive correlation between the two quantities. The same correlation is found across all 7 protein families, see Fig 5E and S5 Fig. Materials and methods Multiple sequence alignments Proteins families and the corresponding alignments were taken from [37]. The alignment procedure of EVmutation (https://github.com/debbiemarkslab/EVmutation) is based on a query against the UniRef100 database of nonredundant protein sequences (release 11/2015) [58] from the wild-type sequence, using the profile HMM homology search tool jackhmmer [59] and choosing a default score in the alignment depth of about 0.5 bits/residue; the threshold was adjusted if the alignment had not enough coverage or number of sequences [37]. Redundant sequences were removed from the alignments, as well as sites with more than 50% of gaps in the alignment. The list of families and wild type, the sequence length, the number of sequences, and the reference to the mutational scans are given in Table 1. Alignments are made available in Supplementary Information. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. From left to right: Numbers N of sites, B of sequences (after removal of redundant sequences from the alignment), of tested single mutations, M1 of possible single mutations, and corresponding references. https://doi.org/10.1371/journal.pcbi.1011521.t001 Sequence re-weighting and MSA descriptors We partially corrected for sampling biases by using a re-weighting procedure with 80% homology threshold in all statistical estimates on sequence data [32]. We therefore define the weight of a sequence s to be (8) where δ(X) = 1 if condition X is satisfied, and 0 otherwise,and d(s, s′) is the normalized Hamming distance between sequences s and s′. The MSA descriptors were then computed as weighted averages: (9) Fitness predictions and comparison with experiments With the K-sparse Potts model the probability of the sequence s reads (10) up to a normalization constant. denotes the set of sites connected to site i on the interaction graph. The predicted fitness difference is defined as the difference in the log probabilities of the wild-type sequence (wt) and of the one where the amino acid at site i in is substituted with the amino acid a as wti→a: (11) giving Eq (2). The predictive performance ρ of the model is then computed as the Spearman rank correlation between experimental measures of delta-fitness for single-point mutations and the corresponding predictions. Inference of sparse Potts models Following a number of recent works [34, 36–38, 61], we predicted the effects of single point mutations by inferring a Potts model from sequence data in the alignment. Here, we employed a K-link Potts model introduced in [38], where we constrain the model to have non-zero couplings only on K statistically-relevant links (i, j) (K = 0 being the independent model, K = N(N − 1)/2 the fully connected Potts model). We chose the K links by scoring each link (i, j) as done for contact prediction in DCA analysis: we inferred a fully-connected Potts model with parameters optimized to perform contact prediction by pseudo-likelihood maximization [57]; from the resulting couplings JPLM we defined a score for each link (i, j) based on the Frobenius norm of the two-sites coupling matrix . Finally, we selected those K pairs (i, j) that showed the highest Frobenius score. We then used a two-site approximation to re-infer the value of the Jij matrix for each of these K pairs given the sparsity constraint [38]. Sub-sampling the MSA allows for varying data relevance and quantity To create new MSAs of different degrees of relevance and quantity for real protein families, we sub-sampled the corresponding MSA using the following procedure. We first chose a target Hamming distance D and a number of sequences B0 (before re-weighting). We then randomly sampled B0 sequences s from the full MSA (without repetition) with a probability decreasing with the Hamming distances d(s, wt) between the sequences s and the wildtype: (12) where Z = ∑s′∈MSAe−αd(s′,wt). The parameter α was optimized to reach the defined D within a given precision (here set to 0.01). From each sampled sub-MSA, we then computed the effective number of sequences B as described above, as well as the variance σ2 for each sparse Potts model with K links through (3). We repeated this procedure for all combinations of 16 values of D ∈ [0.4N, 0.8N], where N is the protein length, and 10 values of B0 in a range that depended on the protein family and its initial MSA size. Doing so, we obtained a population of 160 sub-MSAs with as many corresponding values of D and B. Lattice proteins Lattice proteins are in silico proteins of fixed sequence length (N = 27) folding on the sites of a 3 × 3 × 3 cube [47, 49, 55]. The protein family attached to a specific fold F is defined as the set of sequences s with low (favorable) folding energies ϵ(F, s) in F and unfavorable folding energies ϵ(F′, s) for all other possible folding structures F′ (little competition) [56]; ϵ(F, s) is defined as the sum of Miyazawa-Jernigan interactions [62, 63] between residues si, sj in contact on structure F. The fitness of a protein s (with respect to the native fold F) is defined as (13) An MSA for the family F can then be obtained by sampling from the effective Hamiltonian . To control for the mean Hamming distance from a given wildtype sequence wt of the sampled MSA, we follow the procedure of [50] and sample from a biased Hamiltonian (14) As in [50], Monte Carlo sampling is performed with the Metropolis rule at effective temperature β = 1000 and with T = 1000 thermalization steps between each sampled sequence. Precise values of D were obtained by sub-sampling and mixing four large alignments obtained with γ = 0, 0.025, 0.050, and 0.075. From each MSA, the computation of descriptors σ2 and D as well as training and performance assessment of Potts models, were performed as explained below for real proteins, with the difference that no re-weighting procedure was applied to lattice proteins data. Numerical estimation of bias and variance in Lattice Proteins In the case of Lattice Proteins, we numerically computed the real fitness difference caused by single-point mutations as . For a given inferred Potts model, we then computed the bias and variance of its delta-fitness predictors as (15) (16) where averages are computed over n = 10 inferences performed on as many sampled alignments with fixed number B of sequences and mean Hamming distance D to wt. To relate these quantities to the single predictive performance value ρ of the inferred model, we defined two global measures that account for all single-point mutations (i, a): (17) (18) where [⋅]i,a denotes the averages over sites and mutations, and the global shift in the bias is removed, as the Spearman rank correlation ρ is invariant under the addition of a constant to . In these numerical settings, some mutations are so deleterious that will never be observed in the data, and their effect is systematically estimated by regularization only. To avoid that these outliers dominate the averages above, we restricted our analysis to those mutations that satisfy an “observability” criterion of . Unless specified differently, we use θ = 5.0 throughout all Lattice Protein results. Multiple sequence alignments Proteins families and the corresponding alignments were taken from [37]. The alignment procedure of EVmutation (https://github.com/debbiemarkslab/EVmutation) is based on a query against the UniRef100 database of nonredundant protein sequences (release 11/2015) [58] from the wild-type sequence, using the profile HMM homology search tool jackhmmer [59] and choosing a default score in the alignment depth of about 0.5 bits/residue; the threshold was adjusted if the alignment had not enough coverage or number of sequences [37]. Redundant sequences were removed from the alignments, as well as sites with more than 50% of gaps in the alignment. The list of families and wild type, the sequence length, the number of sequences, and the reference to the mutational scans are given in Table 1. Alignments are made available in Supplementary Information. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. From left to right: Numbers N of sites, B of sequences (after removal of redundant sequences from the alignment), of tested single mutations, M1 of possible single mutations, and corresponding references. https://doi.org/10.1371/journal.pcbi.1011521.t001 Sequence re-weighting and MSA descriptors We partially corrected for sampling biases by using a re-weighting procedure with 80% homology threshold in all statistical estimates on sequence data [32]. We therefore define the weight of a sequence s to be (8) where δ(X) = 1 if condition X is satisfied, and 0 otherwise,and d(s, s′) is the normalized Hamming distance between sequences s and s′. The MSA descriptors were then computed as weighted averages: (9) Fitness predictions and comparison with experiments With the K-sparse Potts model the probability of the sequence s reads (10) up to a normalization constant. denotes the set of sites connected to site i on the interaction graph. The predicted fitness difference is defined as the difference in the log probabilities of the wild-type sequence (wt) and of the one where the amino acid at site i in is substituted with the amino acid a as wti→a: (11) giving Eq (2). The predictive performance ρ of the model is then computed as the Spearman rank correlation between experimental measures of delta-fitness for single-point mutations and the corresponding predictions. Inference of sparse Potts models Following a number of recent works [34, 36–38, 61], we predicted the effects of single point mutations by inferring a Potts model from sequence data in the alignment. Here, we employed a K-link Potts model introduced in [38], where we constrain the model to have non-zero couplings only on K statistically-relevant links (i, j) (K = 0 being the independent model, K = N(N − 1)/2 the fully connected Potts model). We chose the K links by scoring each link (i, j) as done for contact prediction in DCA analysis: we inferred a fully-connected Potts model with parameters optimized to perform contact prediction by pseudo-likelihood maximization [57]; from the resulting couplings JPLM we defined a score for each link (i, j) based on the Frobenius norm of the two-sites coupling matrix . Finally, we selected those K pairs (i, j) that showed the highest Frobenius score. We then used a two-site approximation to re-infer the value of the Jij matrix for each of these K pairs given the sparsity constraint [38]. Sub-sampling the MSA allows for varying data relevance and quantity To create new MSAs of different degrees of relevance and quantity for real protein families, we sub-sampled the corresponding MSA using the following procedure. We first chose a target Hamming distance D and a number of sequences B0 (before re-weighting). We then randomly sampled B0 sequences s from the full MSA (without repetition) with a probability decreasing with the Hamming distances d(s, wt) between the sequences s and the wildtype: (12) where Z = ∑s′∈MSAe−αd(s′,wt). The parameter α was optimized to reach the defined D within a given precision (here set to 0.01). From each sampled sub-MSA, we then computed the effective number of sequences B as described above, as well as the variance σ2 for each sparse Potts model with K links through (3). We repeated this procedure for all combinations of 16 values of D ∈ [0.4N, 0.8N], where N is the protein length, and 10 values of B0 in a range that depended on the protein family and its initial MSA size. Doing so, we obtained a population of 160 sub-MSAs with as many corresponding values of D and B. Lattice proteins Lattice proteins are in silico proteins of fixed sequence length (N = 27) folding on the sites of a 3 × 3 × 3 cube [47, 49, 55]. The protein family attached to a specific fold F is defined as the set of sequences s with low (favorable) folding energies ϵ(F, s) in F and unfavorable folding energies ϵ(F′, s) for all other possible folding structures F′ (little competition) [56]; ϵ(F, s) is defined as the sum of Miyazawa-Jernigan interactions [62, 63] between residues si, sj in contact on structure F. The fitness of a protein s (with respect to the native fold F) is defined as (13) An MSA for the family F can then be obtained by sampling from the effective Hamiltonian . To control for the mean Hamming distance from a given wildtype sequence wt of the sampled MSA, we follow the procedure of [50] and sample from a biased Hamiltonian (14) As in [50], Monte Carlo sampling is performed with the Metropolis rule at effective temperature β = 1000 and with T = 1000 thermalization steps between each sampled sequence. Precise values of D were obtained by sub-sampling and mixing four large alignments obtained with γ = 0, 0.025, 0.050, and 0.075. From each MSA, the computation of descriptors σ2 and D as well as training and performance assessment of Potts models, were performed as explained below for real proteins, with the difference that no re-weighting procedure was applied to lattice proteins data. Numerical estimation of bias and variance in Lattice Proteins In the case of Lattice Proteins, we numerically computed the real fitness difference caused by single-point mutations as . For a given inferred Potts model, we then computed the bias and variance of its delta-fitness predictors as (15) (16) where averages are computed over n = 10 inferences performed on as many sampled alignments with fixed number B of sequences and mean Hamming distance D to wt. To relate these quantities to the single predictive performance value ρ of the inferred model, we defined two global measures that account for all single-point mutations (i, a): (17) (18) where [⋅]i,a denotes the averages over sites and mutations, and the global shift in the bias is removed, as the Spearman rank correlation ρ is invariant under the addition of a constant to . In these numerical settings, some mutations are so deleterious that will never be observed in the data, and their effect is systematically estimated by regularization only. To avoid that these outliers dominate the averages above, we restricted our analysis to those mutations that satisfy an “observability” criterion of . Unless specified differently, we use θ = 5.0 throughout all Lattice Protein results. Discussion In this work, we have investigated, through a combination of analytical and numerical approaches, how the nature (quantity, similarity to wt) of sequence data determine the capability of statistical models, with variable expressive power, to predict the fitness effects of single-point mutations. As expected from the bias-variance trade-off of statistics, simple models require few data to be inferred, but result in systematic prediction errors (bias μ). Conversely, powerful models are in principle capable of expressing complex sequence-to-fitness relationships but their many defining parameters are subject to more statistical errors due to the limited amount of available data (variance σ2). We have shown that a good predictor of performances was given by the sum μ2 + σ2, and have analytically related the variance to the number of sequences B in the alignment and the squared bias μ2 to the evolutionary depth, estimated through the mean Hamming distance D to the mutated wild type sequence. Our theory was quantitatively confirmed by extensive tests on in silico lattice proteins for which the ground-truth fitness is known, and on mutagenesis datasets of 7 proteins families we have analyzed. Based on the results above, we then proposed a “focusing” procedure to optimally select the best subset of sequences from a multi-sequence alignment, and tested it on the 7 mutagenesis experiments. With this procedure, the least powerful, independent sites model, showed performances higher than fully connected graphical models trained on the same data for 4 out of 7 studied protein families, and comparable performances for the remaining ones. An important finding of the present work is the so-called bias factor J0, which relates the squared bias μ2 to the mean Hamming distance D of the sequence data to the wild-type sequence: μ2 ≃ J0 D. In our idealized theoretical framework, confirmed by simulations on in silico proteins, J0 accounts for un-modelled epistasis, i.e., for the statistical properties of the fitness landscape that cannot be reproduced by the class of models considered. Though other sources of biases can be present in real data or in the inferred models due to regularization, and contribute to J0, our result has two consequences, both conceptual and practical. First, it explicitly demonstrates that key information about the unmodeled features of fitness landscapes are, in principle, accessible even with models with limited complexity, constrained by data availability. From a practical point of view J0 can be estimated through a regression of the performance ρ vs. a linear combination of D –chosen at will through subsampling of the multi-sequence alignment– and σ2 –given by Eq (3)–, see Fig 5A; this procedure can therefore be applied to any protein family, for which sequence and mutagenesis data are available. Second, the meaning of J0 emphasizes the role of the expressive power of the model in the relative importance of the bias and variance terms, and to what extent each one of these factors affect performances. The value of J0 is a good predictor of how much can be gained in performance by pruning the sequence data and focusing around the wt sequence (Fig 5E). This result entails that simpler models have higher potential for improvement in fitting a local neighborhood through focusing, and can overcome complex models when training data is appropriately selected. Determining the optimal cutoff distance for focusing can theoretically be done following the quantity-relevance trade-off analysis presented above, e.g. using some already available mutagenesis experiment. We proposed an empirical rule that did not require any mutational information and was based on a signal-to-noise criterion. This empirical cutoff led to systematic improvement of performance for all tested families (S3 Fig). Our focusing and modeling procedures could be further improved along several directions. First, in the the K-links Potts model considered here we have selected relevant links according to the Frobenius norms of the couplings of the inferred Potts model (equivalently, in the Direct Coupling Analysis, DCA). The rationale for this criterion is that the coupling norm is a good proxy for coevolution and contact between residues. Sparsity of the interaction graph can be enforced, within DCA, through L1 regularization over the couplings [54, 61, 64]. However, in a related work [38], we have shown that DCA-based ranking is not an optimal predictor of relevance of couplings for protein function. Couplings can be better selected using a semi-supervised procedure, which exploits a subset of mutational data. Such optimally selected K-Links Potts models achieve a clear increase of the performances in predicting the effect of mutations. Second, we have estimated, so far, the closeness of an alignment to the wild-type sequence through the average Hamming distance D. This choice is justified both by its simplicity, and the deep relation between D and the (squared) bias. However it would be worth considering more refined estimates for the distances, taking into account the phylogeny of the sequence data. Residue conservation can be assessed according to mutational history [30, 31], or to their relevance in the functionality under consideration. In addition, our focusing procedure could make use of alignment methods based on local homology, recently used to discover specific functionality proper to some protein subfamilies [65]. Our theoretical study could help improve models and alignment processing for predicting the effects of missense mutations and their impact in genetic diseases [6–10]. Natural alignments of sampled missense mutations are limited in depth and naturally focused around the human genome, making independent-site models (or K-link Potts models with small K values) especially adequate. It would be very interesting to apply our focusing approach to understand how to best select sequence data in this context. The capability of deriving optimal independent-site models, whose parameters are tuned according to the region in the sequence space under focus, could be also be important for phylogeny studies. Inferring phylogenetic relations between a set of sequences requires the capability to compute transition probabilities under a mutation-selection process. Independent-site models are particularly attractive in this regard as they lead to mathematically tractable expressions for the transitions [66], but cannot describe complex sequence-to-fitness mappings. An alternative would be to use multiple focused independent-site models to compute transitions, adapted to the multiple portions of the sequence space explored by the phylogenetic tree. Last of all, we stress that the question of how to select the best subset of data ensuring optimal performance given a statistical model is of interest in the field of proteins beyond fitness predictions, and, more generally, in machine learning. In the context of structural predictions, it is known that AlphaFold performances are sensitive to the input multi-sequence alignment; in CASP15 some methods found improved predictions by changing the way sequence data were generated [67]. From a general machine-learning point of view, the present work bears some similarity with classical issues in statistics, in particular, the dependence of performance on the quantity of training data. In theoretical consistency frameworks, data are assumed to be generated independently at random from a fixed model distribution (sometimes referred to as the teacher), and then used to train another model with the same architecture (the student), see for instance [68–70] for applications to graphical model reconstruction. However, in our case, the teacher (fitness landscape) is of high and unknown complexity, while the student is much simpler (independent-site or sparse Potts models). Our goal was to provide theoretical support for a pruning strategy, in which data likely to be poorly modeled by the student are explicitly filtered out in the training phase. Our focusing procedure is, in this sense, conceptually related to local regression methods, such as moving least squares approaches, which aim at locally fitting a function from data. It is therefore expected that it will find applications beyond the prediction of fitness considered here. For instance, focused independent-site models could be useful in the context of gene expression, where microarray data generally suffer from missing values, impeding the use of many multivariate statistical methods [71]. Supporting information S1 Text. Supplementary information. Contains Appendices A and B. https://doi.org/10.1371/journal.pcbi.1011521.s001 (PDF) S1 Fig. Supplementary figure 1. Same as Main text Fig 2D for all protein families except RNA-Bind (shown in Main text Fig 2D): systematic analysis of the predictive power ρ as a function of the mean Hamming distance D of sub-alignments with fixed size B (left panels), and of the sub-alignment size B at fixed Hamming distance D (right panels). Each point represents the binned average and standard deviation of several sub-samples obtained at the corresponding values of D and B (see Methods). All significance levels refer to Spearman rank correlation. * P < 0.05; ** P < 0.01; *** P < 0.001. https://doi.org/10.1371/journal.pcbi.1011521.s002 (PDF) S2 Fig. Supplementary figure 2. Same as Main text Fig 4A&4B for all protein families except RNA-Bind (shown in Main text Fig 4A&4B): predictive performance of single-point mutations using the independent-site models, as a function of the squared bias and variance estimated from the alignments, separately (left and right panels) and combined (central panel). https://doi.org/10.1371/journal.pcbi.1011521.s003 (PDF) S3 Fig. Supplementary figure 3. Top: single mutation prediction performance of the independent Potts model (K = 0) along the focusing axis (as a function of the cutoff distance D0) for the 7 studied protein families. Black dashed lines indicate the optimal cutoffs dopt; blue lines indicate the predicted cutoffs dbv by minimizing the linear sum of bias and variance; the light blue lines indicate the predicted cutoff from the signal-to-noise heuristic dsnr. Green areas highlight the performance increase from the full alignment (dc = N) to the predicted cutoff dbv. Yellow areas indicate the remaining performance increase to the optimal cutoff dopt. Horizontal dashed grey lines indicate the performance reported in [37] with a fully connected Potts model inferred by pseudo-likelihood. Bottom: distribution of the hamming distance to the wildtype D of sequences in the MSA. Black dashed lines indicate the optimal cutoff at which the best performance is reached. is the effective number of sequences remaining in the MSA at the optimal cutoff. Refer to Table 1 in Methods for the original number of sequences in the MSA. https://doi.org/10.1371/journal.pcbi.1011521.s004 (PDF) S4 Fig. Supplementary figure 4. a: Single mutation prediction performance of the independent model at the predicted optimal cutoff using the SNR method, as a function of the SNR threshold, for the 7 protein families. b comparison between performance without any cutoff (MSA full) and performance at the cutoff predicted by using the rule of thumb SNR = 3. https://doi.org/10.1371/journal.pcbi.1011521.s005 (PDF) S5 Fig. Supplementary figure 5. Relation between the bias factor J0(K) and improvement Δρ(dopt) for the optimal focusing cutoff for the 7 studied protein families. For each family, K is varied between 0 and N (number of sites in the alignment). https://doi.org/10.1371/journal.pcbi.1011521.s006 (PDF) Acknowledgments The authors are grateful to J. Tubiana and M. Molari for insightful discussions, and to J. Fernandez de Cossio Diaz and E. Mauri for a careful reading of the manuscript.

Ojala, Fanni;Sater, Mohamad R. Abdul;Miller, Loren G.;McKinnell, James A.;Hayden, Mary K.;Huang, Susan S.;Grad, Yonatan H.;Marttinen, Pekka

doi: 10.1371/journal.pcbi.1010898pmid: 37883601

Introduction Staphylococcus aureus colonizes approximately 30% of the population [1] and is a leading cause of healthcare and community associated infections [2]. Healthcare-associated infections with MRSA are associated with higher mortality rates as well as increased cost and hospitalization duration compared to infection with methicillin-susceptible S. aureus. MRSA carriers have a higher predisposition for infection with a 35% risk of MRSA infection within one year following colonization [3–6]. The anterior nares are the main reservoir of S. aureus, and the skin, particularly the axilla and groin, and pharynx are also often colonized. The risk of infection is correlated with the extent of colonization, as determined by the number of body sites found to be colonized [7]. MRSA infections are most often caused by the colonizing strain [8]. Infection prevention and control strategies include reducing spread by preventing colonization of new individuals as well as decolonization of MRSA carriers. Decolonization reduces carriage rates and subsequent infection by 30% [9–12]. However, the effectiveness of decolonization protocols varies. The extent to which this variation is due to the protocol, the features of the MRSA strains colonizing the study subjects, characteristics of the colonized individuals, and the interaction among these factors has been unclear. Moreover, most studies have lacked appropriate controls and/or have had limited sample sizes [13]; additionally, most studies of decolonization protocols have had limited if any analysis of the colonizing MRSA strains [14]. The CLEAR (Changing Lives by Eradicating Antibiotic Resistance) Trial is a randomized controlled clinical trial of MRSA carriers comparing hygiene education to education plus decolonization after hospital discharge. The intervention arm underwent repeated decolonization involving a 5-day decolonization regimen applied every other week for six months. The decolonization regimen involved chlorhexidine antiseptic for daily showering and twice-daily mouthwash plus twice-daily mupirocin (a topical antibiotic) treatment of bilateral nares. The education (control) arm received hygiene education alone. Body site samples were collected five times: at enrollment, at one, three- and six-months post-enrollment during the intervention phase, and at nine months, which was three months after the end of intervention. Swabs were obtained from multiple body sites including nares, throat, skin (axilla and groin), as well as accessible wounds, if present. While the trial demonstrated the benefit of the decolonization protocol with chlorhexidine and mupirocin, persistent colonization was noted in a subset of both trial arms [15]. The factors that contributed to persistent colonization were not addressed in the primary manuscript. The goal for the current investigation was to model the association between antibiotic resistance genes and the persistence of MRSA strains during a decolonization protocol. To study this, we used whole genome sequencing of 3901 isolates from 880 study subjects from CLEAR who had completed all follow-up visits. We used a Bayesian statistical framework (BaeMBac software) to define persistent strains [16] and formulated a Bayesian mixed effects survival model [17], where the survival outcome was the clearance of MRSA during a given study interval, and the lack of clearance represented persistent colonization. Resistance to different antibiotics as predicted by Mykrobe predictor [18] were the covariates in the fixed effects, and the study subject- and strain-specific random effects were included to quantify the impact of other subject and strain related factors on clearance. Our approach was fully Bayesian, which allowed characterization of uncertainty of all quantities of interest and incorporation of prior knowledge [19]. Materials and methods Data See [9] for the details of the study protocol. In brief, subjects were selected for the study based on an MRSA positive culture within the hospitalization prior to enrollment (0-month). Isolates were collected from the study subjects at discharge, 1-month, 3-month, 6-month and 9-month visits from the start of the study. Per protocol, the decolonization regimen was stopped six months after discharge, and therefore data from the 9-month interval were excluded from the subsequent analysis. Whole genome sequencing was performed as described in [16], briefly Paired end DNA libraries were constructed via Illumina Nextera according to the manufacturer guidelines. Libraries were sequenced on the Illumina Hiseq platform. Sequencing reads were assembled de novo using SPAdes-3.10.1. in silico MLST typing was performed using PubMLST (https://pubmlst.org/saureus/), grouping samples by sequence type (ST). Single nucleotide polymorphism (SNP) analysis was performed using a mapping based approach relative to a reference genome of matching ST for each group. Pairwise SNP distance was inferred from SNP-alignments using custom scripts. Genotype-based resistance prediction was performed using presence or absence of resistance markers associated with high-level resistance to mupirocin and quaternary ammonium compounds. Mykrobe predictor [18] was used to predict mupirocin resistance based on the presence of mupA and mupB genes associated with high-level resistance. Chlorhexidine resistance prediction based on the presence of qac genes (qacA, qacC was performed with BLAST [20]. In addition, mupirocin and chlorhexidine resistance phenotyping was performed on the first and last collected samples per patient. CHG susceptibility testing was performed using broth microdilution and a complete inhibition endpoint. Starting with a 20% (wt/vol) CHG solution (Sigma-Aldrich, St. Louis, MO), a 2-fold dilution series (from 32 to 0.0625 μg/ml) was prepared daily. An isolate was classified as nonsusceptible to CHG if the MIC was >4 μg/ml, which is outside the wild-type distribution of CHG MICs for S. aureus. Susceptibility to mupirocin was determined by the Etest method (bioMérieux, Durham, NC) according to the manufacturer’s instructions. LLMR was defined as a MIC of 8 to 256 μg/ml and HLMR as a MIC of ≥512 μg/ml. [21], [22], [23]. The MRSA isolates from a single study subject were divided into strains using the software BaeMBac [16], where a ‘strain’ is defined as a population of genetically closely related isolates. The software uses a Bayesian model based on the single nucleotide polymorphism (SNP) distance and time between consecutive visits to estimate the probability that a pair of isolates collected from a study subject represent the same strain. The SNP distance of 45, estimated by BaeMBac using 10 percent of the education arm data, was used as a threshold (see S1 Fig) to divide the isolates from each subject into strains. The MRSA isolates were primarily from ST5 (N = 1337) and ST8 (N = 1968) [16]; isolates from the remaining STs (N = 533) were excluded because of the small number of samples. Most subjects were colonized with only one strain over the course of the study, but some were colonized with multiple strains (Table 1). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Distribution of the number of strains colonizing a study subject. https://doi.org/10.1371/journal.pcbi.1010898.t001 Preprocessing Our goal was to study the clearance of MRSA using survival models. For this purpose, we defined observations (yi, xi, ti) in our survival data as follows. One observation i corresponded to one study interval (between consecutive follow-up visits) from one subject such that the subject was colonized by MRSA in the beginning of the interval. If the study subject cleared MRSA carriage during a given interval (e.g., from 1-month visit to 3-month visit), then yi = 1, otherwise yi = 0. The vector xi specified the characteristics of the strain in the beginning of the interval, and it included the vector of indicators for resistance to different antibiotics. The time ti was simply the length of the interval in months. We assessed clearance at 1-month, 3-month and 6-month visits. We denoted the starting visit by v0 of the interval of interest (for example the 1-month visit) and v1 the end visit (for example the 3-month visit). The covariates xi included the presence of genetic markers for resistance of the colonizing strain at v0 to the following antibiotics: ciprofloxacin, clindamycin, erythromycin, gentamicin, mupirocin, rifampicin, tetracycline, trimethoprim and chlorhexidine. Penicillin and methicillin resistance were excluded, as all isolates were expected to be resistant. Vancomycin and fusidic acid were also excluded, because there was no resistance to these antibiotics. Statistics of the survival data are given in Tables 2 and 3. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Summary of the survival data. https://doi.org/10.1371/journal.pcbi.1010898.t002 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Summary of the resistance profiles in the survival data. https://doi.org/10.1371/journal.pcbi.1010898.t003 The data were formulated for survival analysis in two ways, as illustrated in Fig 1. First, in a strain-specific analysis, clearance was defined such that at v1 the subject was not colonized by the strain at any site (see Fig 1B for illustration). Furthermore, in the strain-specific analysis there may have been multiple colonizing isolates at v0 belonging to the same strain, and the covariate corresponding to a certain resistance was defined as present if at least one of these isolates was resistant; in practice, the isolates of the same strain were so closely related that their predicted full resistance profiles were identical in 86% of cases. Second, in a site-specific analysis, clearance of a strain was defined as the absence of the strain at v1 on a body site of interest (either no strain was observed at v1 or a strain different from the one at v0 was observed). If no swab was collected on v1, the observation was excluded from the survival analysis, except if the MRSA-positive v0 swab was taken from wound, when it was considered cleared by v1 (wound healed), see Fig 1C. The preprocessing pipeline is summarized in S1 Appendix. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Strains in a multiply-colonized study subject. a) An example of a study subject colonized by four separate strains, A, B, C, and D, over the study period. b) Observations in the strain-specific survival data formulation for the subject. The subject contributed three intervals to the survival data, since the 9-month visit was excluded. Strains B and C were cleared immediately after acquisition, whereas strain A was persistent throughout the study. c) Observations in the site-specific survival data formulation (see text for details). https://doi.org/10.1371/journal.pcbi.1010898.g001 Bayesian survival model In survival analysis, the goal is to characterize time-to-event data in terms of the hazard of event or survival time until an event, affected by some covariates of interest. In our study, the fixed covariates were the presence or absence of resistance to each antibiotic, and the model included also subject- and strain-specific random effects. The parameters of interest were the fixed effect coefficients β, which denoted the magnitude of increase or decrease in risk or survival time for the covariates. Hence, formally, we were modeling the ‘hazard’ of clearance of an MRSA strain in a study interval from v0 to v1 when the strain was known to be resistant to some antibiotics at v0. Consequently, an estimated hazard ratio exp(β) of 1.5, for example, indicated that a unit increase in the corresponding covariate resulted in a 1.5-fold risk of the clearance. Our observations were either interval- or right censored: observations corresponding to study intervals where the clearance occurred (y = 1) between v0 and v1 were modeled as interval censored, as the exact event time of the clearance within the interval was not known. Right censoring was used for observations corresponding to study intervals in which the clearance event did not occur (y = 0) by the end of the interval v1. The data consisted of (1) where i is an index for a visit interval such that i = 1, …, N comprised all visit intervals from all participants in the data set where the subject was colonized at the beginning of the interval, according to either the strain- or site-specific formulation, as described under the section Preprocessing. The response variable yi indicated whether the clearance happened within the interval, ti was the length of the interval, and vector xi held the resistance profile at the beginning of the interval. The data in the decolonization and education arms were analysed separately. We used an exponential survival model with the proportional hazards parameterization. We assumed that the clearance rate was constant during a given interval. In addition, conditionally on the fixed covariates, the subject, the strain and the fact that the study subject was colonized in the beginning of the interval, the clearance probability in the interval was independent of clearances on other intervals (this assumption follows from the ‘memorylessness’ property of the exponential distribution). Hence, the hazard was given by (2) where ηi was the linear predictor, defined as (3) In Eq 3, γs, s = 1, …, S, and ρh, h = 1, …, H, were the the strain- and subject-specific random effects and S and H were the numbers of strains and study subjects, respectively. Functions h(i) and s(i) specified the subject (i.e. ‘host’) and the strain corresponding to interval i, respectively. The priors for the random effects were defined as (4) The survival function for interval i was thus (5) and it represented the probability that the study subject was still colonized by the same strain (i.e., “survived from clearance”) at the end of the interval. By letting θ denote jointly all model parameters, the log-likelihood function was defined as (6) The priors for the coefficient β for the fixed effects and the intercept term β0 were defined to be relatively non-informative, i.e., to have a variance that exceeded the range of effects expected in the data, as follows: (7) where D was the dimension of the fixed effects and I the identity matrix. The priors on the hyperparameters for the random effects (σγ and σρ) were determined from the decomposition of the covariance matrix of the random effects into a correlation matrix Ω, a simplex π, and a scale parameter τ. Details of this decomposition can be found in rstanarm documentation and the Stan user guide [24, 25]. We set the hyperparameters as follows: (8) We estimated the posterior distributions for the parameters of interest by drawing samples from the posterior with an MCMC sampler, implemented using rstanarm’s function stan_surv. The R package rstanarm is an extension of the Stan programming language developed specifically as a platform for statistical analysis and Bayesian inference [24]. We ran the No-U-Turn sampler (a variant of Hamiltonian Monte Carlo) [26] for 7500 iterations for the strain-specific models and for 10000 iterations for the site-specific models over four chains. We increased the target average acceptance probability in the presence of divergent transitions as suggested by rstanarm documentation [27]. We assessed the convergence with values and by visual inspection of traceplots. The full model included all antibiotics and both strain and study subject random effects. We compared the full model with different random effect configurations using the 10-fold cross-validation (CV). The coefficients β were estimated separately for the decolonization and education arms. The coefficient β represents the logarithm of the hazard ratio, which here means that the ‘instantaneous rate’ of clearance happening for a resistant strain is exp(β) times the rate for non-resistant strains. Data See [9] for the details of the study protocol. In brief, subjects were selected for the study based on an MRSA positive culture within the hospitalization prior to enrollment (0-month). Isolates were collected from the study subjects at discharge, 1-month, 3-month, 6-month and 9-month visits from the start of the study. Per protocol, the decolonization regimen was stopped six months after discharge, and therefore data from the 9-month interval were excluded from the subsequent analysis. Whole genome sequencing was performed as described in [16], briefly Paired end DNA libraries were constructed via Illumina Nextera according to the manufacturer guidelines. Libraries were sequenced on the Illumina Hiseq platform. Sequencing reads were assembled de novo using SPAdes-3.10.1. in silico MLST typing was performed using PubMLST (https://pubmlst.org/saureus/), grouping samples by sequence type (ST). Single nucleotide polymorphism (SNP) analysis was performed using a mapping based approach relative to a reference genome of matching ST for each group. Pairwise SNP distance was inferred from SNP-alignments using custom scripts. Genotype-based resistance prediction was performed using presence or absence of resistance markers associated with high-level resistance to mupirocin and quaternary ammonium compounds. Mykrobe predictor [18] was used to predict mupirocin resistance based on the presence of mupA and mupB genes associated with high-level resistance. Chlorhexidine resistance prediction based on the presence of qac genes (qacA, qacC was performed with BLAST [20]. In addition, mupirocin and chlorhexidine resistance phenotyping was performed on the first and last collected samples per patient. CHG susceptibility testing was performed using broth microdilution and a complete inhibition endpoint. Starting with a 20% (wt/vol) CHG solution (Sigma-Aldrich, St. Louis, MO), a 2-fold dilution series (from 32 to 0.0625 μg/ml) was prepared daily. An isolate was classified as nonsusceptible to CHG if the MIC was >4 μg/ml, which is outside the wild-type distribution of CHG MICs for S. aureus. Susceptibility to mupirocin was determined by the Etest method (bioMérieux, Durham, NC) according to the manufacturer’s instructions. LLMR was defined as a MIC of 8 to 256 μg/ml and HLMR as a MIC of ≥512 μg/ml. [21], [22], [23]. The MRSA isolates from a single study subject were divided into strains using the software BaeMBac [16], where a ‘strain’ is defined as a population of genetically closely related isolates. The software uses a Bayesian model based on the single nucleotide polymorphism (SNP) distance and time between consecutive visits to estimate the probability that a pair of isolates collected from a study subject represent the same strain. The SNP distance of 45, estimated by BaeMBac using 10 percent of the education arm data, was used as a threshold (see S1 Fig) to divide the isolates from each subject into strains. The MRSA isolates were primarily from ST5 (N = 1337) and ST8 (N = 1968) [16]; isolates from the remaining STs (N = 533) were excluded because of the small number of samples. Most subjects were colonized with only one strain over the course of the study, but some were colonized with multiple strains (Table 1). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Distribution of the number of strains colonizing a study subject. https://doi.org/10.1371/journal.pcbi.1010898.t001 Preprocessing Our goal was to study the clearance of MRSA using survival models. For this purpose, we defined observations (yi, xi, ti) in our survival data as follows. One observation i corresponded to one study interval (between consecutive follow-up visits) from one subject such that the subject was colonized by MRSA in the beginning of the interval. If the study subject cleared MRSA carriage during a given interval (e.g., from 1-month visit to 3-month visit), then yi = 1, otherwise yi = 0. The vector xi specified the characteristics of the strain in the beginning of the interval, and it included the vector of indicators for resistance to different antibiotics. The time ti was simply the length of the interval in months. We assessed clearance at 1-month, 3-month and 6-month visits. We denoted the starting visit by v0 of the interval of interest (for example the 1-month visit) and v1 the end visit (for example the 3-month visit). The covariates xi included the presence of genetic markers for resistance of the colonizing strain at v0 to the following antibiotics: ciprofloxacin, clindamycin, erythromycin, gentamicin, mupirocin, rifampicin, tetracycline, trimethoprim and chlorhexidine. Penicillin and methicillin resistance were excluded, as all isolates were expected to be resistant. Vancomycin and fusidic acid were also excluded, because there was no resistance to these antibiotics. Statistics of the survival data are given in Tables 2 and 3. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Summary of the survival data. https://doi.org/10.1371/journal.pcbi.1010898.t002 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Summary of the resistance profiles in the survival data. https://doi.org/10.1371/journal.pcbi.1010898.t003 The data were formulated for survival analysis in two ways, as illustrated in Fig 1. First, in a strain-specific analysis, clearance was defined such that at v1 the subject was not colonized by the strain at any site (see Fig 1B for illustration). Furthermore, in the strain-specific analysis there may have been multiple colonizing isolates at v0 belonging to the same strain, and the covariate corresponding to a certain resistance was defined as present if at least one of these isolates was resistant; in practice, the isolates of the same strain were so closely related that their predicted full resistance profiles were identical in 86% of cases. Second, in a site-specific analysis, clearance of a strain was defined as the absence of the strain at v1 on a body site of interest (either no strain was observed at v1 or a strain different from the one at v0 was observed). If no swab was collected on v1, the observation was excluded from the survival analysis, except if the MRSA-positive v0 swab was taken from wound, when it was considered cleared by v1 (wound healed), see Fig 1C. The preprocessing pipeline is summarized in S1 Appendix. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Strains in a multiply-colonized study subject. a) An example of a study subject colonized by four separate strains, A, B, C, and D, over the study period. b) Observations in the strain-specific survival data formulation for the subject. The subject contributed three intervals to the survival data, since the 9-month visit was excluded. Strains B and C were cleared immediately after acquisition, whereas strain A was persistent throughout the study. c) Observations in the site-specific survival data formulation (see text for details). https://doi.org/10.1371/journal.pcbi.1010898.g001 Bayesian survival model In survival analysis, the goal is to characterize time-to-event data in terms of the hazard of event or survival time until an event, affected by some covariates of interest. In our study, the fixed covariates were the presence or absence of resistance to each antibiotic, and the model included also subject- and strain-specific random effects. The parameters of interest were the fixed effect coefficients β, which denoted the magnitude of increase or decrease in risk or survival time for the covariates. Hence, formally, we were modeling the ‘hazard’ of clearance of an MRSA strain in a study interval from v0 to v1 when the strain was known to be resistant to some antibiotics at v0. Consequently, an estimated hazard ratio exp(β) of 1.5, for example, indicated that a unit increase in the corresponding covariate resulted in a 1.5-fold risk of the clearance. Our observations were either interval- or right censored: observations corresponding to study intervals where the clearance occurred (y = 1) between v0 and v1 were modeled as interval censored, as the exact event time of the clearance within the interval was not known. Right censoring was used for observations corresponding to study intervals in which the clearance event did not occur (y = 0) by the end of the interval v1. The data consisted of (1) where i is an index for a visit interval such that i = 1, …, N comprised all visit intervals from all participants in the data set where the subject was colonized at the beginning of the interval, according to either the strain- or site-specific formulation, as described under the section Preprocessing. The response variable yi indicated whether the clearance happened within the interval, ti was the length of the interval, and vector xi held the resistance profile at the beginning of the interval. The data in the decolonization and education arms were analysed separately. We used an exponential survival model with the proportional hazards parameterization. We assumed that the clearance rate was constant during a given interval. In addition, conditionally on the fixed covariates, the subject, the strain and the fact that the study subject was colonized in the beginning of the interval, the clearance probability in the interval was independent of clearances on other intervals (this assumption follows from the ‘memorylessness’ property of the exponential distribution). Hence, the hazard was given by (2) where ηi was the linear predictor, defined as (3) In Eq 3, γs, s = 1, …, S, and ρh, h = 1, …, H, were the the strain- and subject-specific random effects and S and H were the numbers of strains and study subjects, respectively. Functions h(i) and s(i) specified the subject (i.e. ‘host’) and the strain corresponding to interval i, respectively. The priors for the random effects were defined as (4) The survival function for interval i was thus (5) and it represented the probability that the study subject was still colonized by the same strain (i.e., “survived from clearance”) at the end of the interval. By letting θ denote jointly all model parameters, the log-likelihood function was defined as (6) The priors for the coefficient β for the fixed effects and the intercept term β0 were defined to be relatively non-informative, i.e., to have a variance that exceeded the range of effects expected in the data, as follows: (7) where D was the dimension of the fixed effects and I the identity matrix. The priors on the hyperparameters for the random effects (σγ and σρ) were determined from the decomposition of the covariance matrix of the random effects into a correlation matrix Ω, a simplex π, and a scale parameter τ. Details of this decomposition can be found in rstanarm documentation and the Stan user guide [24, 25]. We set the hyperparameters as follows: (8) We estimated the posterior distributions for the parameters of interest by drawing samples from the posterior with an MCMC sampler, implemented using rstanarm’s function stan_surv. The R package rstanarm is an extension of the Stan programming language developed specifically as a platform for statistical analysis and Bayesian inference [24]. We ran the No-U-Turn sampler (a variant of Hamiltonian Monte Carlo) [26] for 7500 iterations for the strain-specific models and for 10000 iterations for the site-specific models over four chains. We increased the target average acceptance probability in the presence of divergent transitions as suggested by rstanarm documentation [27]. We assessed the convergence with values and by visual inspection of traceplots. The full model included all antibiotics and both strain and study subject random effects. We compared the full model with different random effect configurations using the 10-fold cross-validation (CV). The coefficients β were estimated separately for the decolonization and education arms. The coefficient β represents the logarithm of the hazard ratio, which here means that the ‘instantaneous rate’ of clearance happening for a resistant strain is exp(β) times the rate for non-resistant strains. Results Exploratory data analysis Before estimating the hazard ratios using the Bayesian approach, we conducted an exploratory analysis by calculating the clearance probability at v1 given resistance at v0 directly from the observation counts (intervals in the survival data) for resistant and non-resistant strains. This approach only considered one type of resistance at a time and neglected the different interval lengths. We saw that only 32% of the mupirocin-resistant observations were cleared at v1 in the decolonization arm while 67% of the non-resistant cases were cleared (Fig 2). The difference was significant with non-overlapping 95% confidence intervals. In the education arm, there was no difference in the clearance probability between resistant and non-resistant strains. Further, the clearance probability of the non-resistant strains is considerably larger in the decolonization arm than in the education arm, reflecting the overall efficiency of the protocol [9]. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Clearance probabilities calculated from the counts of observations. Clearance probabilities given mupirocin resistance, computed directly from the counts of intervals colonized with resistant or non-resistant observations. On the y-axis, we have the clearance probability at the end of an interval, i.e., at v1, and the x-axis shows the resistance status at v0. The probability of clearance was calculated by dividing the numbers of persistent and cleared cases with the numbers of resistant or non-resistant observations in the data. The probability of clearance was lower for mupirocin-resistant strains than for non-resistant strains in the decolonization arm (D; blue). In the education arm (E; lavender), the probability of clearance (i.e., spontaneous loss of carriage) was the same regardless of the resistance status. https://doi.org/10.1371/journal.pcbi.1010898.g002 Model comparison We used the 10-fold cross-validation to compare the prediction accuracy of the different random effect combinations (no random effects, study subject, strain, study subject and strain). We quantified the results using the expected log-predictive density (elpd) [28], which is a metric for prediction accuracy. In both education and decolonization arms, including strain random effects improved the model considerably (Table 4). In contrast, including the subject-specific random effects did not improve the model, but instead slightly decreased the elpd value in the education arm. Because this decrease was minor and not significant, we decided to use the complete model to characterize both the strain and study subject random effects in the following section. The estimates for the fixed effects representing the impact of antibiotic resistance types on clearance were approximately the same regardless of whether the study subject random effects were included in the model (compare Fig 3 and S2 Fig). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Credible intervals for the effects of antibiotic resistance types in the decolonization and education arms. 95% posterior credible intervals for the β parameters, representing the impact of each antibiotic resistance type on clearance. The model has study subject and strain random effects included and resistance types as fixed effects. A lower coefficient indicates a decreased rate (hazard) of clearance. https://doi.org/10.1371/journal.pcbi.1010898.g003 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 4. Model comparison. https://doi.org/10.1371/journal.pcbi.1010898.t004 Impact of antibiotic resistance on persistence In the decolonization arm, the mupirocin resistance coefficient was -2.6 (95% CI is -4.0 to -1.3), indicating that the clearance rate of resistant strains was approximately 0.07 times the clearance rate of the non-resistant strains (Fig 3). Mupirocin resistance thus was correlated with greater MRSA persistence in the decolonization arm. However, this effect was not observed in the education arm (i.e., spontaneous loss was similar regardless of resistance). In contrast, chlorhexidine-resistant strains were not more persistent in the decolonization arm than the non-resistant strains, despite the use of chlorhexidine mouth and body-wash as part of the decolonization protocol. Ciprofloxacin (–0.71, 95% CI: [-1.31, -0.12]) and erythromycin (–0.97, 95% CI: [-1.67, -0.29]) resistances were weakly associated with increased persistence in the education arm. Resistance to other antibiotics was not significantly associated with clearance, but the number of samples corresponding to some resistance types was limited (see Table 3), leading to wide credible intervals. In addition to using the genetic resistance determinants, we analysed phenotypic resistance based on MIC thresholds. We include the comparison of genotypic and phenotypic resistance as a supplementary table (S1 Table). The model is otherwise the same except that a distinction is made between high-level (HLR) and low-level (LLR) phenotypic mupirocin resistance. The results (S3 Fig) show that increased persistence is associated with high-level mupirocin resistance, but not with low-level resistance in the decolonization arm. Study subject and strain random effects There was more variation in the strain random effects than in the study subject random effects in both the decolonization and education arms (Fig 4), which means that antibiotic resistance alone does not fully explain the variability in persistence. Furthermore, the variation in the strain random effects was larger in the education arm than in the decolonization arm. The study subject random effects were small in both arms. However, we note that many subjects were colonized by one strain only (see Table 1) and for those cases the effects of the strain and subject are statistically indistinguishable. Sequence type did not correlate with strain random effects (S4 Fig). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Estimated study subject and strain random effects. The figure shows histograms of the estimated strain and study subject-specific random effects. In both the decolonization (D) and education (E) arms, there was more variability in the strain random effects than in the study subject random effects. https://doi.org/10.1371/journal.pcbi.1010898.g004 Site-specific analysis In the decolonization arm, mupirocin resistance was again strongly associated with a reduced rate of clearance (i.e., increased persistence) in the nares (-2.26, 95% CI: [-3.8, -0.87]) but did not significantly correlate with clearance at other body sites (Fig 5, S5 Fig). Ciprofloxacin resistance and gentamicin resistance were weakly associated with increased persistence (–0.99, 95% CI: [-1.67, -0.32] and -1.12, [-2.06, -0.21]) in the nares in the education arm. In addition, we saw possible weak associations between chlorhexidine resistance and decreased persistence in the throat in the decolonization arm (2.07, 95% CI: [0.13, 4.00]), and between tetracycline resistance and increased persistence in the wound in the education arm (-1.65, 95% CI: [-3.32, -0.11]) (S5 Fig). The variation in the strain random effects was again greater than in the subject random effects (S6 Fig). Furthermore, this effect was clearest in the nares, from which most of the samples were obtained. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Results of the site-specific analysis. The figure shows 95% credible intervals for the effect of each antibiotic resistance type on clearance in both study arms. The results for the nares are shown here, as it had the largest effect, and for the other sites in S5 Fig. https://doi.org/10.1371/journal.pcbi.1010898.g005 Sensitivity analysis As a sensitivity analysis, we conducted Bayesian logistic regression and Cox porportional hazards (PH) analysis to compare wiht our Bayesian survival analysis results. Here, we used a model with no random effects, for strain-specific survival data. The main observation of mupirocin resistance leading to a lower probability of clearance is visible with both methods (S7 Fig), indicating robustness of our approach. We observed some correlation between mupirocin and gentamicin resistance (S8 Fig). To account for this correlation, we ran the survival model separately for each antimicrobial, observing that the effect of gentamicin resistance shifted towards zero (S9 Fig). Exploratory data analysis Before estimating the hazard ratios using the Bayesian approach, we conducted an exploratory analysis by calculating the clearance probability at v1 given resistance at v0 directly from the observation counts (intervals in the survival data) for resistant and non-resistant strains. This approach only considered one type of resistance at a time and neglected the different interval lengths. We saw that only 32% of the mupirocin-resistant observations were cleared at v1 in the decolonization arm while 67% of the non-resistant cases were cleared (Fig 2). The difference was significant with non-overlapping 95% confidence intervals. In the education arm, there was no difference in the clearance probability between resistant and non-resistant strains. Further, the clearance probability of the non-resistant strains is considerably larger in the decolonization arm than in the education arm, reflecting the overall efficiency of the protocol [9]. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Clearance probabilities calculated from the counts of observations. Clearance probabilities given mupirocin resistance, computed directly from the counts of intervals colonized with resistant or non-resistant observations. On the y-axis, we have the clearance probability at the end of an interval, i.e., at v1, and the x-axis shows the resistance status at v0. The probability of clearance was calculated by dividing the numbers of persistent and cleared cases with the numbers of resistant or non-resistant observations in the data. The probability of clearance was lower for mupirocin-resistant strains than for non-resistant strains in the decolonization arm (D; blue). In the education arm (E; lavender), the probability of clearance (i.e., spontaneous loss of carriage) was the same regardless of the resistance status. https://doi.org/10.1371/journal.pcbi.1010898.g002 Model comparison We used the 10-fold cross-validation to compare the prediction accuracy of the different random effect combinations (no random effects, study subject, strain, study subject and strain). We quantified the results using the expected log-predictive density (elpd) [28], which is a metric for prediction accuracy. In both education and decolonization arms, including strain random effects improved the model considerably (Table 4). In contrast, including the subject-specific random effects did not improve the model, but instead slightly decreased the elpd value in the education arm. Because this decrease was minor and not significant, we decided to use the complete model to characterize both the strain and study subject random effects in the following section. The estimates for the fixed effects representing the impact of antibiotic resistance types on clearance were approximately the same regardless of whether the study subject random effects were included in the model (compare Fig 3 and S2 Fig). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Credible intervals for the effects of antibiotic resistance types in the decolonization and education arms. 95% posterior credible intervals for the β parameters, representing the impact of each antibiotic resistance type on clearance. The model has study subject and strain random effects included and resistance types as fixed effects. A lower coefficient indicates a decreased rate (hazard) of clearance. https://doi.org/10.1371/journal.pcbi.1010898.g003 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 4. Model comparison. https://doi.org/10.1371/journal.pcbi.1010898.t004 Impact of antibiotic resistance on persistence In the decolonization arm, the mupirocin resistance coefficient was -2.6 (95% CI is -4.0 to -1.3), indicating that the clearance rate of resistant strains was approximately 0.07 times the clearance rate of the non-resistant strains (Fig 3). Mupirocin resistance thus was correlated with greater MRSA persistence in the decolonization arm. However, this effect was not observed in the education arm (i.e., spontaneous loss was similar regardless of resistance). In contrast, chlorhexidine-resistant strains were not more persistent in the decolonization arm than the non-resistant strains, despite the use of chlorhexidine mouth and body-wash as part of the decolonization protocol. Ciprofloxacin (–0.71, 95% CI: [-1.31, -0.12]) and erythromycin (–0.97, 95% CI: [-1.67, -0.29]) resistances were weakly associated with increased persistence in the education arm. Resistance to other antibiotics was not significantly associated with clearance, but the number of samples corresponding to some resistance types was limited (see Table 3), leading to wide credible intervals. In addition to using the genetic resistance determinants, we analysed phenotypic resistance based on MIC thresholds. We include the comparison of genotypic and phenotypic resistance as a supplementary table (S1 Table). The model is otherwise the same except that a distinction is made between high-level (HLR) and low-level (LLR) phenotypic mupirocin resistance. The results (S3 Fig) show that increased persistence is associated with high-level mupirocin resistance, but not with low-level resistance in the decolonization arm. Study subject and strain random effects There was more variation in the strain random effects than in the study subject random effects in both the decolonization and education arms (Fig 4), which means that antibiotic resistance alone does not fully explain the variability in persistence. Furthermore, the variation in the strain random effects was larger in the education arm than in the decolonization arm. The study subject random effects were small in both arms. However, we note that many subjects were colonized by one strain only (see Table 1) and for those cases the effects of the strain and subject are statistically indistinguishable. Sequence type did not correlate with strain random effects (S4 Fig). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Estimated study subject and strain random effects. The figure shows histograms of the estimated strain and study subject-specific random effects. In both the decolonization (D) and education (E) arms, there was more variability in the strain random effects than in the study subject random effects. https://doi.org/10.1371/journal.pcbi.1010898.g004 Site-specific analysis In the decolonization arm, mupirocin resistance was again strongly associated with a reduced rate of clearance (i.e., increased persistence) in the nares (-2.26, 95% CI: [-3.8, -0.87]) but did not significantly correlate with clearance at other body sites (Fig 5, S5 Fig). Ciprofloxacin resistance and gentamicin resistance were weakly associated with increased persistence (–0.99, 95% CI: [-1.67, -0.32] and -1.12, [-2.06, -0.21]) in the nares in the education arm. In addition, we saw possible weak associations between chlorhexidine resistance and decreased persistence in the throat in the decolonization arm (2.07, 95% CI: [0.13, 4.00]), and between tetracycline resistance and increased persistence in the wound in the education arm (-1.65, 95% CI: [-3.32, -0.11]) (S5 Fig). The variation in the strain random effects was again greater than in the subject random effects (S6 Fig). Furthermore, this effect was clearest in the nares, from which most of the samples were obtained. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Results of the site-specific analysis. The figure shows 95% credible intervals for the effect of each antibiotic resistance type on clearance in both study arms. The results for the nares are shown here, as it had the largest effect, and for the other sites in S5 Fig. https://doi.org/10.1371/journal.pcbi.1010898.g005 Sensitivity analysis As a sensitivity analysis, we conducted Bayesian logistic regression and Cox porportional hazards (PH) analysis to compare wiht our Bayesian survival analysis results. Here, we used a model with no random effects, for strain-specific survival data. The main observation of mupirocin resistance leading to a lower probability of clearance is visible with both methods (S7 Fig), indicating robustness of our approach. We observed some correlation between mupirocin and gentamicin resistance (S8 Fig). To account for this correlation, we ran the survival model separately for each antimicrobial, observing that the effect of gentamicin resistance shifted towards zero (S9 Fig). Discussion We applied Bayesian survival analysis on a dataset of sequenced MRSA samples collected from colonized patients after hospital discharge at given intervals during a follow-up period. Our results showed that mupirocin-resistant MRSA strains were more persistent than non-resistant strains in the decolonization arm, but not in the education only arm. We additionally analyzed the data using high- and low-level phenotypic mupirocin resistance based on MIC values. The results on high-level mupirocin resistance confirmed the findings based on genetic resistance determinants. When we looked at each body site separately, the effect of mupirocin was detected only in the nares, and not in the skin, throat, or wound. Since mupirocin is administered intranasally as part of the decolonization protocol and nares is the most prominent site of MRSA colonization [29], this result seems expected. However, despite chlorhexidine also being part of the decolonization protocol, chlorhexidine resistance did not seem to be associated with decolonization failure. This could be because chlorhexidine is applied to the throat (mouth wash), and skin and wound (baths), but not intranasally, but is most likely because genetic correlates of chlorhexidine resistance are quite poor. In the site-specific analysis, we saw a possible association between chlorhexidine and decreased persistence in the throat; however, the credible interval only slightly excluded zero, and hence this could be due to noise or otherwise reflect an unknown interaction between resistance types and persistence. Our study also confirmed the previously reported overall increased probability of clearance in the decolonization arm [9]. In the education arm, resistance to ciprofloxacin and erythromycin were found associated with increased persistence. A plausible biological mechanism is that some antibiotics had been used by the study subjects, giving resistant strains an advantage over non-resistant strains. A similar explanation could apply to the findings from the site-specific analysis, where we observed an increase in persistence in the education arm for gentamicin and ciprofloxacin-resistant strains in the nares and for tetracycline-resistant strains in wounds. Another possible explanation is that the order of causation is reversed: a strain that is persistent might be more likely to become antibiotic-resistant than a non-persistent strain, perhaps due to increased antibiotic exposure or shared biological mechanism. The same mechanism might contribute to the association between persistence and resistance for mupirocin and/or chlorhexidine. Further studies are needed to distinguish among these hypotheses. Strain-specific random effects improved the ability of the model to predict persistence, which suggests that another genomic factor beyond the resistance determinants may affect persistence. The sequence type (ST5 vs ST8) of the MRSA strain did not seem to be associated with persistence S4 Fig. Including the study subject-specific random effects in addition to the strain-specific effects did not improve the model, in contrast with reports that study subject related factors affect MRSA colonization (for example the nasal microbiota [30]). However, we note that some strain and subject-specific effects were overlapping (when there was only one strain from one subject), and consequently some subject effects might be explained as part of the strain effects. In future studies, a larger dataset could confirm associations between resistance and persistence that did not reach statistical significance in this study. This could also help to identify in more detail the genetic determinants of persistence that were represented here by the strain-specific random effects. While the random effects for study subjects were not significant in our study, including explicit characteristics of the subjects might add power to find some features that affect persistence. In addition, the use of antibiotics other than those that were part of the protocol during the study period was not considered in this study, but including them as covariates might reveal further insights about the relationship between resistance and persistence. Conclusion We showed that genetic variants for mupirocin-resistance in MRSA were associated with a large drop in the efficiency of a decolonization protocol that includes intranasal mupirocin. Therefore, alternative decolonization protocols for patients with mupirocin-resistant MRSA colonization should be considered, such as nasal iodophor in place of mupirocin, although mupirocin is a superior treatment of the two in general [31]. However, we did not see a similar effect for chlorhexidine body and mouth wash, another part of the decolonization protocol, nor for any other antibiotic, which supports chlorhexidine as a reliable component of a skin decontamination protocol even when genetic correlates of chlorhexidine resistance are identified. In general, these findings point to the potential utility of improving the efficiency of decolonization protocols by characterizing an individual’s colonizing strain to determine its resistance profile. Supporting information S1 Appendix. Preprocessing pipeline. Preprocessing steps to format the data for the survival analysis. ng represents the number of genomes (isolates), ni the number of intervals and ns the number of study subjects with sequencing data for the colonizing strains available. The number of observations is smaller after “Restructure data”, because the survival data are considered by interval: recruitment to 1-month, 1-month to 3-month and 3-month to 6-month. For example, in the original data one body site could have an isolate at each of the visits, contributing in total five observations in the original data. However, we only have three intervals to consider in the case of survival data, because we are interested in the clearance status of an MRSA strain between or at the end interval of two consecutive isolates (excluding the 6-month to 9-month interval). https://doi.org/10.1371/journal.pcbi.1010898.s001 (PDF) S1 Table. Phenotypic and genetic resistance. Discrepancy between phenotypic and genetic resistance profiles in mupirocin and chlorhexidine. https://doi.org/10.1371/journal.pcbi.1010898.s002 (PDF) S1 Fig. BaeMBac output for same strain probability. Contour plot detailing the same strain probability with SNP distance d* on the x-axis and the time between consecutive visits (in generations) on the y-axis. The probability of 0.5 was used to decide the threshold distance of 45 SNPs that was used to classify a pair of MRSA isolates observed in consecutive visits as the same or different strain. The BaeMBac software was run using 10 percent of randomly selected isolates from the education arm. The threshold was not sensitive to the amount of data, and the decolonization arm was not used as the BaeMBac software assumes a model of neutral evolution when calculating the same strain probability. https://doi.org/10.1371/journal.pcbi.1010898.s003 (PDF) S2 Fig. Estimated fixed effects with only strain random effects in the model. The 95% credible intervals for parameters β for all antibiotics with strain random effects included in the model, but excluding the subject-specific random effects. https://doi.org/10.1371/journal.pcbi.1010898.s004 (PDF) S3 Fig. Phenotypic low- and high-level mupirocin resistance. We included additional analysis of the phenotypic mupirocin resistance to complement our results on genetic resistance, and to separately consider low-level mupirocin resistance in our results. An MIC threshold of 512 ug/mL was used to distinguish between low-level and high-level resistance—a MIC value below 8 ug/mL indicated no resistance. Genetic resistance was used for other antimicrobials. The median and 50/95% credible intervals for each covariate are shown. In addition, a table comparing the resistance profiles of phenotypic and genotypic resistance for mupirocin and chlorhexidine is included. We can see that the results support the main analysis, with high-level mupirocin resistance contributing to persistence in the decolonization arm. We also note that low-level resistance is not significantly associated with persistence. Further studies could include characterization of low-level and high-level mupirocin resistance on a genetic level, for example by including a marker for the IleS gene. https://doi.org/10.1371/journal.pcbi.1010898.s005 (PDF) S4 Fig. Estimated strain random effects for sequence types ST5 and ST8. The strain-specific survival model was used to estimate the strain random effects. Histograms show the distributions of the strain random effect posterior means in the decolonization and education arms. https://doi.org/10.1371/journal.pcbi.1010898.s006 (PDF) S5 Fig. Throat, skin and wound posterior intervals 95% posterior intervals. https://doi.org/10.1371/journal.pcbi.1010898.s007 (PDF) S6 Fig. Estimated study subject and strain random effects by site. Strain random effects have more variation in the posterior means than study subject random effects, most notably in the nares. https://doi.org/10.1371/journal.pcbi.1010898.s008 (PDF) S7 Fig. Cox proportional hazards and Bayesian logistic regression models. In addition to the Bayesian survival model, we considered the standard Cox proportional hazards (PH) model with the antimicrobials as covariates. As another comparison, we included Bayesian logistic regression, conducted using the rstanarm package. Logistic regression does not consider the time difference between consecutive observations or censoring. Further, each of these is applied without the random effects. The figure shows 95% CIs for the coefficients of each antimicrobial. We see that qualitatively the results are similar to our model that includes the random effects, which highlights the robustness of our results. However, our model allows estimation of the host and strain specific contributions using the random effects, and leads to larger effect estimates for resistance types, which could be caused by the fact that effects are easier to estimate when additional noise due to random effects is first explained away. https://doi.org/10.1371/journal.pcbi.1010898.s009 (PDF) S8 Fig. Collinearity between mupirocin and gentamicin and correlation matrix. Collinearity represents the correlation between the effect sizes in the posterior samples for the given pair of antimicrobials, and it was visualized using the bayesplot [32] package. Furthermore, we included a heatmap of the Spearman correlation between each pair of antimicrobials. We see a small positive correlation between gentamicin and mupirocin resistance. Consequently, their effects are weakly negatively correlated in the posterior distribution. https://doi.org/10.1371/journal.pcbi.1010898.s010 (PDF) S9 Fig. Separate models for each antimicrobial. We repeated the Bayesian survival analysis for each antimicrobial in a separate model to account for the weak correlation between gentamicin and mupirocin (see S8 Fig). The figure shows the median and 50/95% CIs of the coefficient of each model. https://doi.org/10.1371/journal.pcbi.1010898.s011 (PDF)

Loman, Torkel E.;Ma, Yingbo;Ilin, Vasily;Gowda, Shashi;Korsbo, Niklas;Yewale, Nikhil;Rackauckas, Chris;Isaacson, Samuel A.

doi: 10.1371/journal.pcbi.1011530pmid: 37851697

1 Introduction Chemical reaction network (CRN) models are used across a variety of fields, including the biological sciences, epidemiology, physical chemistry, combustion modeling, and pharmacology [1–7]. At their core, they combine a set of species (defining a system’s state) with a set of reaction events (rates for reactions occurring combined with rules for altering the system’s state when a reaction occurs). One advantage of formulating a biological model as a CRN is that these can be simulated according to several well-defined mathematical representations, representing different physical scales at which reaction processes can be studied. For example, the reaction rate equation (RRE) is a macroscopic system of ordinary differential equations (ODEs), providing a deterministic model of chemical reaction processes. Similarly, the chemical Langevin equation (CLE) is a system of stochastic differential equations (SDEs), providing a more microscopic model that can capture certain types of fluctuations in reaction processes [8]. Finally, stochastic chemical kinetics, typically simulated with the Gillespie algorithm (as well as modifications to, and improvements of, it), provides an even more microscopic model, that captures both stochasticity and discreteness of populations in chemical reaction processes [9, 10]. That a CRN can be unambiguously represented using these models forms the basis of several CRN modeling tools [11–26]. Here we present a new modeling tool for CRNs, Catalyst.jl, which we believe offers a unique set of advantages for both inexperienced and experienced modelers. Catalyst’s defining trait, which sets it apart from other popular CRN modeling packages, is that it represents models in an entirely symbolic manner, accessible via standard Julia language programs. This permits algebraic manipulation and simplification of the models, either by the user, or by other tools. Once a CRN has been defined, it is stored in a symbolic intermediate representation (IR). This IR is the target of methods that provide functionality to Catalyst, including numerical solvers for both continuous ODEs and SDEs, as well as discrete Gillespie-style stochastic simulation algorithms (SSAs). As Catalyst’s symbolic representations can be converted to compiled Julia functions, it can be easily used with a variety of Julia libraries. These include packages for parameter fitting, sensitivity analysis, steady state finding, and bifurcation analysis. To simplify model implementation, Catalyst provides a domain-specific language (DSL) that allows users to declare CRN models using classic chemical reaction notation. Finally, Catalyst also provides a comprehensive API to enable programmatic manipulation and combination of models, combined with functionality for analyzing and simplifying CRNs (such as detection of conservation laws and elimination of conserved species). Catalyst is implemented in Julia, a relatively recent (version 1.0 released in August 2018) open-source programming language for scientific computing. Its combination of high performance and user-friendliness makes it highly promising [27, 28]. Julia has grown quickly, with a highly developed ecosystem of packages for scientific simulation. This includes the many packages provide by the Scientific Machine Learning (SciML) organization, of which Catalyst is a part. SciML, through its ModelingToolkit.jl package, provides the IR on which Catalyst is based [29]. This IR is used across the organization’s projects, providing a target structure both for model-generation tools (such as Catalyst), and tools that provide additional functionality. ModelingToolkit symbolic models leverage the Symbolics.jl [30] computer algebraic system (CAS), enabling them to be represented in a symbolic manner. Simulations of ModelingToolkit-based models are typically carried out using DifferentialEquations.jl, perhaps the largest software package of state-of-the-art, high-performance numerical solvers for ODEs, SDEs, and jump processes [31]. Several existing modeling packages provide overlapping functionality with Catalyst. COPASI is a well known and popular software that enables both deterministic and stochastic CRN modeling, as well as many auxiliary features (such as parameter fitting and sensitivity analysis) [12]. BioNetGen is another such software suite, currently available as a Visual Studio Code extension, that is built around the popular BioNetGen language for easily specifying complex reaction network models [21]. It provides options for model creation, network simulation, and network free-modeling. Another popular tool, VCell, provides extensive functionality, via an intuitive graphical interface [11]. Finally, Tellurium ties together a range of tools to be used in a Python environment, allowing CRN models to be created using the Antimony DSL, and simulated using the libRoadRunner numeric solver suite [15, 23, 24, 32]. Other modeling softwares include GINsim, CellNOpt, GillesPy2, and Matlab’s SimBiology toolbox [13, 16, 33]. Several of these packages are primarily designed around a GUI-based workflow (BioNetGen, COPASI, and VCell). In constrast, Catalyst is DSL and API-based, with simulation and analysis of models carried out via Julia scripts. A typical Catalyst workflow therefore requires users to write Julia language scripts instead of using a GUI-based interface, but also enables users to easily integrate Catalyst models with a large variety of other Julia libraries. Catalyst also has immediate access to a more extensive set of numerical solvers for ODEs, SDEs, and SSAs. In this paper, we demonstrate that using these solvers, Catalyst’s simulation speed often outperforms the other tools by more than one order of magnitude. Catalyst has the ability to include Julia-native functions within rate laws and stoichiometric expressions, and to include coupled ODE or algebraic constraint equations for reaction rate equation models (potentially resulting in differential-algebraic equation (DAE) models). For example, to encode bursty reactions stoichiometric coefficients can be defined using standard Julia functions that sample from a random variable distribution. Similarly, rate-laws can include data-driven modeling terms (e.g. neural networks) constructed via Julia libraries such as Surrogates.jl, SciMLSensitivity.jl, and DiffEqFlux.jl. Moreover, Catalyst generates differentiable models, which can be easily incorporated into higher-level Julia codes that require automatic differentiation [34] and composed with other Julia libraries. One current limitation of Catalyst is that in contrast to BioNetGen, COPASI, and GillesPy2, Catalyst can not generate inputs for hybrid and τ-leaping solvers, though adding support for these features is planned. In the next sections we overview a basic workflow for using Catalyst to define and simulate CRNs; overview how Catalyst performs relative to several popular CRN modeling packages for solving ODEs and simulating stochastic chemical kinetics models; discuss Catalyst’s symbolic representation of CRNs, Catalyst’s network analysis functionality, and how it can compose with other Julia packages; and introduce some of the higher-level applications in which Catalyst models can be easily embedded. 2 Results 2.1 The Catalyst DSL enables models to be created using chemical reaction notation Catalyst offers several ways to define a CRN model, with the most effortless option being the @reaction_network DSL. This feature extends the natural Julia syntax via a macro, allowing users to declare CRN models using classic chemical reaction notation (as opposed to declaring models using equations, or by declaring reactions implicitly or through functions). This alternative notation makes scripts more human-legible, and greatly reduces code length (simplifying both script writing and debugging). Using the DSL, the CRN’s chemical reactions are listed, each preceded by its reaction rate (Fig 1). From this, the system’s species and parameters are automatically extracted and a ReactionSystem IR structure is created (which can be used as input to e.g. numerical simulators). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Catalyst connects an intuitive domain-specific language with a well-supported intermediate representation. The extracellular signal-regulated kinase (ERK) network is important to the regulation of many cellular functions, and its disruption has been implicated in cancer [35]. (a) A CRN representation of the ERK network. (b) A model of the ERK CRN can be implemented in Julia through the Catalyst DSL, using code very similar to the actual CRN representation. (c) From this code, the DSL generates a ReactionSystem intermediate representation (IR) that is the target structure for a range of supported simulation and analysis methods. https://doi.org/10.1371/journal.pcbi.1011530.g001 To facilitate a more concise notation, similar reactions (e.g. several degradation events) can be bundled together. Each reaction rate can either be a constant, a parameter, or a function. Predefined Michaelis–Menten and Hill functions are provided by Catalyst, but any user-defined Julia function can be used to define a rate. Both parametric and non-integer stoichiometric coefficients are possible. There are also several non-DSL methods for model creation. They include loading networks from files via SBMLToolkit.jl [36] (for SBML files) and ReactionNetworkImporters.jl [37] (for BioNetGen generated .net files). CRNs can also be created via defining symbolic variables via the combined Catalyst/ModelingToolkit API, and directly building ReactionSystems from collections of Reaction structures. This enables programmatic definition of CRNs, making it possible to create large models by iterating through a relatively small number of rules within standard Julia scripts. 2.2 Catalyst models can be simulated using a wide range of high-performance methods Numerical simulations of Catalyst models are generally carried out using the DifferentialEquations.jl package [31]. It contains a large number of numerical solvers and a wide range of additional features (such as event handling, support for GPUs and threading, flexibility in choice of linear solvers for stiff integrators, and more). The package is highly competitive, often outperforming packages written in C and Fortran [31]. Simulation syntax is straightforward, and output solutions can be plotted using the Plots.jl package [38] via a recipe that allows users to easily select the species and times to display. CRNs can be translated and simulated using the ODE-based RREs, the SDE-based CLE, and through discrete SSAs (Fig 2). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Catalyst models can be simulated using both deterministic and stochastic interpretations. (a) The Brusselator network contains two species (X and Y) and two parameters (A and B, in practical implementation these are species present in excess, but they can in practice be considered parameters) [39, 40]. Here, we show the four reactions of the Brusselator CRN, and its implementation using the Catalyst DSL. (b-d) Simulations of models for the Brusselator at the three physical scales supported by Catalyst (RRE, CLE, SSA). Post-processing has been carried out on the plots to improve their visualization in this article’s format. (b) While B > 1 + A2, the deterministic model exhibits a limit cycle. This is confirmed using ODE RRE simulations. (c) The model can also be simulated using the stochastic CLE interpretation. (d) Finally, the discrete, stochastic, jump process interpretation is simulated via Gillespie’s direct method. The system displays a limit cycle even though B < 1 + A2, confirming the well known phenomenon of noise induced oscillations [41]. https://doi.org/10.1371/journal.pcbi.1011530.g002 To demonstrate the performance of these solvers, we benchmarked simulations of CRN models using a range of CRN modeling tools (BioNetGen, Catalyst, COPASI, GillesPy2, and Matlab’s SimBiology toolbox). These tools were selected as they are popular and highly cited, well documented, scriptable for running benchmark studies, and actively maintained. The Matlab SimBiology toolbox was selected due to the enduring popularity of the Matlab language. Overall, they provide a representative sample of the broader chemical reaction network modeling software ecosystem. We used both ODE simulations and discrete SSAs. Fewer packages permit SDE simulations, hence such simulations were not benchmarked. We note, however, that DifferentialEquations’ SDE solvers are highly performative [42]. When comparing a range of models, from small to large, we see that Catalyst typically outperforms the other packages, often by at least an order of magnitude (Fig 3). For the ODE benchmarks, to try to provide as fair a comparison as possible, identical absolute and relative tolerances were used for all simulations. Furthermore, in Fig C in S1 Text we demonstrate the relation between simulation time and actual error across the Julia solvers, lsoda, and CVODE (with the native Julia solvers typically having smaller errors as compared to lsoda and CVODE for any given tolerance). All SSA methods tested generate exact realizations, in the sense that they should each give statistics consistent with the underlying Chemical Master Equation of the model [43], and their simulation times are hence directly comparable. Here, the wide range of methods provided by the JumpProcesses.jl package [44], a component of DifferentialEquations, enables Catalyst to outperform the other packages (most of which only provide Gillespie’s direct method or its sorting direct variant [45]). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Simulations of Catalyst models outperform those of other modeling packages. Benchmarks of simulation runtimes for Catalyst and four other modeling packages (BioNetGen, COPASI, GillesPy2, and Matlab SimBiology). The benchmarks were run on the multi-state (Multistate, 9 species and 18 reactions [47]), multi-site (Multisite 2, 66 species and 288 reactions [48]), epidermal growth factor receptor signalling (Egfr_net, 356 species and 3749 reactions [49]), B-cell receptor (1122 species and 24388 reactions [50]), and high-affinity human IgE receptor signalling (Fceri_gamma2, 3744 species and 58276 reactions [51]) models. (a-e) Benchmarks of deterministic RRE ODE simulations of the five models. Each bar shows, for a given method, the runtime to simulate the model (to steady-state for those that approach a steady-state). For Catalyst, we show the three best-performing native Julia methods, as well as the performance of lsoda and CVODE. For each of the other tools, we show its best-performing method. Identical values for absolute and relative tolerance are used across all packages and methods. For each benchmark, the method options used can be found in Section 4.1, the exact benchmark times in Table A in S1 Text, and further details on the solver options for each tool in Section B in S1 Text. While this figure only contains the most performant methods, a full list of methods investigated can be found in Section B in S1 Text, with their results described in Figs A and B in S1 Text. (f-j) Benchmarks of stochastic chemical kinetics SSA simulations of the five models. Via JumpProcesses.jl, Catalyst can use several different algorithms (e.g. Direct, Sorting Direct, RSSA, and RSSACR above) for exact Gillespie simulations. Here, the simulation runtime is plotted against the (physical) final time of the simulation. Due to their long runtimes, some tools were not benchmarked for the largest models. We note that, in [52], it was remarked that BioNetGen (dashed orange lines) use a pseudo-random number generator in SSAs that, while fast, is of lower quality than many (slower) modern generators such as Mersenne Twister. For full details on benchmarks, see Section 4.1. https://doi.org/10.1371/journal.pcbi.1011530.g003 In contrast to the exact SSA methods, timestep-based ODE integrators typically provide a variety of numerical parameters, such as error tolerances and configuration options for implicit solvers (i.e. how to calculate Jacobians, how to solve linear and nonlinear systems, etc). ODE simulation performance then depends on which combinations of options are used with a given solver. Here, we limit ourselves to trying combinations of numeric solvers (Julia-native solvers for comparing performance of Catalyst-generated models, and lsoda and/or CVODE for comparisons between tools), methods for Jacobian computation and representation (automatic differentiation, finite differences, or symbolic computation, and dense vs. sparse representations), linear solvers (LU, GMRES, or KLU), and whether to use a preconditioner or not when using GMRES. The non-Catalyst simulators generally provide limited ability to change these options, in which case only the default was used in benchmarking. In contrast, the DifferentialEquations.jl solvers that Catalyst utilise, while they do not require the user to set these options, do give them full control to do so. Full documentation is available at [46]. The details of the most performant options we used for each tool and model are provided in Section 4.1. A list of all benchmarks we carried out (for various combinations of tool, method, and options) is provided in Section B in S1 Text, with their results described in Figs A and B in S1 Text. Finally, the benchmarking process is described in more detail in Section 4.1. The observed performance results for Catalyst-generated models arise from a variety of factors. For example, Catalyst inlines all mass action reaction terms for ODE models within a single generated function that evaluates the ODE derivative. This provides opportunities for the compiler to optimise expression evaluation, and avoids the overhead of repeatedly calling non-inlined functions to evaluate such terms. For the largest ODE models, Catalyst and ModelingToolkit’s support for generating explicit sparse Jacobians led to significant performance improvements when using the CVODE solver, see Section 4.1 and S1 Text. For jump process SSA simulations, Catalyst uses a sparse reaction specification that automatically analyses each reaction, and then classifies the reaction into the most performant but physically valid representation supported by JumpProcesses.jl (corresponding to jumps with general time-varying intensities, jumps with general rate expressions but for which the intensity is constant between the occurrence of two jumps, and jumps for which the intensity is a mass action type rate law). This enables JumpProcesses.jl to avoid the overhead of calling a large collection of user-provided functions via pointers, by using a single pre-defined and inlined function to evaluate individual mass action reaction intensities, while still supporting calling general user rate functions via pointers (for non-mass action rate laws). These Catalyst-specific features, when coupled to the large variety of solvers in DifferentialEquations.jl and broad flexibility in tuning solver components (i.e. different Jacobian and jump representations, flexibility in choice of linear solvers, etc.), help enable Catalyst’s observed performance. 2.3 Catalyst enables composable, symbolic modeling of CRNs Catalyst’s primary feature is that its models are represented using a CAS, enabling them to be algebraically manipulated. Examples of how this is utilised include automatic computation of system Jacobians, calculation and elimination of conservation laws, and simplification of generated symbolic DAE models via ModelingToolkit’s symbolic analysis tooling. These techniques can help speed up numeric simulations, while also facilitating higher level analysis. One example is enabling users to generate ODE models with non-singular Jacobians via the elimination of conservation laws, which can aid steady-state analysis tooling. Catalyst also provides a network analysis API, enabling the calculation of a variety of network properties beyond conservation laws, including linkage classes, weak reversibility, and deficiency indices. Catalyst’s symbolic representation permits model internals to be freely extracted, investigated, and manipulated, giving the user full control over their models (Fig 4). This enables various forms of programmatic model creation, extension and composition. Model structures that occur repetitively can be duplicated, and disjoint models can be connected together. For example, such functionality can be used to model a population of cells, each with defined neighbours, where each cell can be assigned a duplicate of the same simple CRN. The CRNs within each cell can then be connected to those of its neighbours, enabling models with spatial structures. Similarly, one could define a collection of genetic modules, and then compose such modules together into a larger gene regulatory network. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. The simulation pipeline of a Catalyst model, with internal intermediates displayed. Code as written by the user (yellow background), and as generated internally by Catalyst and ModelingToolkit (blue and grey backgrounds respectively) are shown, in addition to the generated structures and their fields (blue background, some of the internal fields are omitted in all displayed structures). (a) A symbolic ReactionSystem for a reversible dimerisation reaction is created using either the DSL, or programmatically using the Symbolics computer algebraic system. (b) The ReactionSystem can be converted into a ModelingToolkit ODESystem structure, corresponding to a symbolic RRE ODE model. (c) By providing initial conditions, parameter values, and a time span, the ODESystem can be simulated, generating an output solution. The generated (internal) Julia code for evaluating the derivatives defining the ODEs, which gets compiled and is input to the ODE solver, is displayed in grey. At each step, the user has the ability to investigate and manipulate the generated structures. https://doi.org/10.1371/journal.pcbi.1011530.g004 Catalyst is highly flexible in the allowed Julia functions that can be used in defining rates, rate laws, or stoichiometry coefficients. This means that while reaction rates and rate laws are typically constants, parameters, or simple functions, e.g. Hill functions, they may also include other terms, such as neural networks or data-driven, empirically defined, Julia functions. Likewise, stoichiometric coefficients can be random variables by defining them as a symbolic variable, and setting that variable equal to a Julia function sampling the appropriate probability distribution. Such functionality can be utilized, for example, to model transcriptional bursting [53], where the produced mRNA copy-numbers are random variables. Finally, standard Catalyst-generated ODE and SDE models are differentiable, in that the generated codes can be used in higher-level packages that rely on automatic differentiation [34]. In this way Catalyst-generated models can be used in machine-learning based analyses. That Catalyst gives full access to its model internals, combined with its composability, allows other packages to easily integrate into, and build upon, it. Indeed, this is already being utilised by independent package developers. The MomentClosure.jl Julia package, which implements several techniques for moment closure approximations, is built to be deployed on Catalyst models [54]. It can generate symbolic finite-dimensional ODE system approximations to the full, infinite system of moment equations associated with the chemical master equation. These symbolic approximations can then be compiled and solved via ModelingToolkit in a similar manner to how Catalyst’s generated RRE ODE models are handled. Similarly, FiniteStateProjection.jl [55] builds upon Catalyst and ModelingToolkit to enable the numerical solution of the chemical master equation, while DelaySSAToolkit.jl [56] can accept Catalyst models as input to its SSAs that handle stochastic chemical kinetics models with delays. Another example of how Catalyst’s flexibility enables its integration into the Julia ecosystem is that CRNs with polynomial ODEs (a condition that holds for pure mass action systems) can be easily converted to symbolic steady-state systems of polynomial equations. This enables polynomial methods, such as homotopy continuation, to be employed on Catalyst models. Here, homotopy continuation (implemented by the HomotopyContinuation.jl Julia package) can be used to reliably compute all roots of a polynomial system [57]. This is an effective approach for finding multiple steady states of a system. When the CRN contains Hill functions (with integer exponents), by multiplying by the denominators, one generates a polynomial system with identical roots to the original, on which homotopy continuation can still be used. 2.4 Catalyst models are compatible with a wide range of ancillary tools and methods The Julia SciML, and broader Julia, ecosystem offers a wide range of techniques for working with models and data based around the IR that Catalyst produces (Fig 5). While the reactions that constitute a CRN are often known in developing a model, system parameters (these typically correspond to reaction rates) rarely are. A first step in analyzing a model is identifiability analysis, where we determine whether the parameters can be uniquely identified from the data [58]. This is enabled through the StructuralIdentifiability.jl package. In the next step, parameters can be fitted to data. This can be done using DiffEqParamEstim.jl, which provides simple functions that are easy to use. Alternatively, more powerful packages, like Optimization.jl and the Turing.jl Julia library for Bayesian analysis, offer increased flexibility for experienced users [59]. Furthermore, unknown CRN structures (such as a species’s production rate) can be approximated using neural networks and then fitted to data. This functionality is enabled by the SciMLSensitivity package [60]. More broadly, system steady states can be computed using the NLSolve.jl or HomotopyContinuation.jl Julia packages [57]; bifurcation structures can be calculated, and bifurcation diagrams generated, with the BifurcationKit.jl library [61]; and SciMLSensitivity.jl and GlobalSensitivity.jl can be used to investigate the sensitivity and uncertainty of model solutions with regard to parameters [62]. Finally, options for displaying CRNs, either as network graphs (via Graphviz) or Latex formatted equations (via Latexify.jl), also exist. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. A wide range of features are available for Catalyst model analysis. A CRN model can be created either through the DSL, by manually declaring the reaction events, or by loading it from a file. The model is stored in the ReactionSystem IR, which can be used as input to a wide range of methods. Purple boxes indicate code written by the user, and green boxes the corresponding output. For some methods, either one, or both, boxes are omitted. https://doi.org/10.1371/journal.pcbi.1011530.g005 2.1 The Catalyst DSL enables models to be created using chemical reaction notation Catalyst offers several ways to define a CRN model, with the most effortless option being the @reaction_network DSL. This feature extends the natural Julia syntax via a macro, allowing users to declare CRN models using classic chemical reaction notation (as opposed to declaring models using equations, or by declaring reactions implicitly or through functions). This alternative notation makes scripts more human-legible, and greatly reduces code length (simplifying both script writing and debugging). Using the DSL, the CRN’s chemical reactions are listed, each preceded by its reaction rate (Fig 1). From this, the system’s species and parameters are automatically extracted and a ReactionSystem IR structure is created (which can be used as input to e.g. numerical simulators). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Catalyst connects an intuitive domain-specific language with a well-supported intermediate representation. The extracellular signal-regulated kinase (ERK) network is important to the regulation of many cellular functions, and its disruption has been implicated in cancer [35]. (a) A CRN representation of the ERK network. (b) A model of the ERK CRN can be implemented in Julia through the Catalyst DSL, using code very similar to the actual CRN representation. (c) From this code, the DSL generates a ReactionSystem intermediate representation (IR) that is the target structure for a range of supported simulation and analysis methods. https://doi.org/10.1371/journal.pcbi.1011530.g001 To facilitate a more concise notation, similar reactions (e.g. several degradation events) can be bundled together. Each reaction rate can either be a constant, a parameter, or a function. Predefined Michaelis–Menten and Hill functions are provided by Catalyst, but any user-defined Julia function can be used to define a rate. Both parametric and non-integer stoichiometric coefficients are possible. There are also several non-DSL methods for model creation. They include loading networks from files via SBMLToolkit.jl [36] (for SBML files) and ReactionNetworkImporters.jl [37] (for BioNetGen generated .net files). CRNs can also be created via defining symbolic variables via the combined Catalyst/ModelingToolkit API, and directly building ReactionSystems from collections of Reaction structures. This enables programmatic definition of CRNs, making it possible to create large models by iterating through a relatively small number of rules within standard Julia scripts. 2.2 Catalyst models can be simulated using a wide range of high-performance methods Numerical simulations of Catalyst models are generally carried out using the DifferentialEquations.jl package [31]. It contains a large number of numerical solvers and a wide range of additional features (such as event handling, support for GPUs and threading, flexibility in choice of linear solvers for stiff integrators, and more). The package is highly competitive, often outperforming packages written in C and Fortran [31]. Simulation syntax is straightforward, and output solutions can be plotted using the Plots.jl package [38] via a recipe that allows users to easily select the species and times to display. CRNs can be translated and simulated using the ODE-based RREs, the SDE-based CLE, and through discrete SSAs (Fig 2). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Catalyst models can be simulated using both deterministic and stochastic interpretations. (a) The Brusselator network contains two species (X and Y) and two parameters (A and B, in practical implementation these are species present in excess, but they can in practice be considered parameters) [39, 40]. Here, we show the four reactions of the Brusselator CRN, and its implementation using the Catalyst DSL. (b-d) Simulations of models for the Brusselator at the three physical scales supported by Catalyst (RRE, CLE, SSA). Post-processing has been carried out on the plots to improve their visualization in this article’s format. (b) While B > 1 + A2, the deterministic model exhibits a limit cycle. This is confirmed using ODE RRE simulations. (c) The model can also be simulated using the stochastic CLE interpretation. (d) Finally, the discrete, stochastic, jump process interpretation is simulated via Gillespie’s direct method. The system displays a limit cycle even though B < 1 + A2, confirming the well known phenomenon of noise induced oscillations [41]. https://doi.org/10.1371/journal.pcbi.1011530.g002 To demonstrate the performance of these solvers, we benchmarked simulations of CRN models using a range of CRN modeling tools (BioNetGen, Catalyst, COPASI, GillesPy2, and Matlab’s SimBiology toolbox). These tools were selected as they are popular and highly cited, well documented, scriptable for running benchmark studies, and actively maintained. The Matlab SimBiology toolbox was selected due to the enduring popularity of the Matlab language. Overall, they provide a representative sample of the broader chemical reaction network modeling software ecosystem. We used both ODE simulations and discrete SSAs. Fewer packages permit SDE simulations, hence such simulations were not benchmarked. We note, however, that DifferentialEquations’ SDE solvers are highly performative [42]. When comparing a range of models, from small to large, we see that Catalyst typically outperforms the other packages, often by at least an order of magnitude (Fig 3). For the ODE benchmarks, to try to provide as fair a comparison as possible, identical absolute and relative tolerances were used for all simulations. Furthermore, in Fig C in S1 Text we demonstrate the relation between simulation time and actual error across the Julia solvers, lsoda, and CVODE (with the native Julia solvers typically having smaller errors as compared to lsoda and CVODE for any given tolerance). All SSA methods tested generate exact realizations, in the sense that they should each give statistics consistent with the underlying Chemical Master Equation of the model [43], and their simulation times are hence directly comparable. Here, the wide range of methods provided by the JumpProcesses.jl package [44], a component of DifferentialEquations, enables Catalyst to outperform the other packages (most of which only provide Gillespie’s direct method or its sorting direct variant [45]). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Simulations of Catalyst models outperform those of other modeling packages. Benchmarks of simulation runtimes for Catalyst and four other modeling packages (BioNetGen, COPASI, GillesPy2, and Matlab SimBiology). The benchmarks were run on the multi-state (Multistate, 9 species and 18 reactions [47]), multi-site (Multisite 2, 66 species and 288 reactions [48]), epidermal growth factor receptor signalling (Egfr_net, 356 species and 3749 reactions [49]), B-cell receptor (1122 species and 24388 reactions [50]), and high-affinity human IgE receptor signalling (Fceri_gamma2, 3744 species and 58276 reactions [51]) models. (a-e) Benchmarks of deterministic RRE ODE simulations of the five models. Each bar shows, for a given method, the runtime to simulate the model (to steady-state for those that approach a steady-state). For Catalyst, we show the three best-performing native Julia methods, as well as the performance of lsoda and CVODE. For each of the other tools, we show its best-performing method. Identical values for absolute and relative tolerance are used across all packages and methods. For each benchmark, the method options used can be found in Section 4.1, the exact benchmark times in Table A in S1 Text, and further details on the solver options for each tool in Section B in S1 Text. While this figure only contains the most performant methods, a full list of methods investigated can be found in Section B in S1 Text, with their results described in Figs A and B in S1 Text. (f-j) Benchmarks of stochastic chemical kinetics SSA simulations of the five models. Via JumpProcesses.jl, Catalyst can use several different algorithms (e.g. Direct, Sorting Direct, RSSA, and RSSACR above) for exact Gillespie simulations. Here, the simulation runtime is plotted against the (physical) final time of the simulation. Due to their long runtimes, some tools were not benchmarked for the largest models. We note that, in [52], it was remarked that BioNetGen (dashed orange lines) use a pseudo-random number generator in SSAs that, while fast, is of lower quality than many (slower) modern generators such as Mersenne Twister. For full details on benchmarks, see Section 4.1. https://doi.org/10.1371/journal.pcbi.1011530.g003 In contrast to the exact SSA methods, timestep-based ODE integrators typically provide a variety of numerical parameters, such as error tolerances and configuration options for implicit solvers (i.e. how to calculate Jacobians, how to solve linear and nonlinear systems, etc). ODE simulation performance then depends on which combinations of options are used with a given solver. Here, we limit ourselves to trying combinations of numeric solvers (Julia-native solvers for comparing performance of Catalyst-generated models, and lsoda and/or CVODE for comparisons between tools), methods for Jacobian computation and representation (automatic differentiation, finite differences, or symbolic computation, and dense vs. sparse representations), linear solvers (LU, GMRES, or KLU), and whether to use a preconditioner or not when using GMRES. The non-Catalyst simulators generally provide limited ability to change these options, in which case only the default was used in benchmarking. In contrast, the DifferentialEquations.jl solvers that Catalyst utilise, while they do not require the user to set these options, do give them full control to do so. Full documentation is available at [46]. The details of the most performant options we used for each tool and model are provided in Section 4.1. A list of all benchmarks we carried out (for various combinations of tool, method, and options) is provided in Section B in S1 Text, with their results described in Figs A and B in S1 Text. Finally, the benchmarking process is described in more detail in Section 4.1. The observed performance results for Catalyst-generated models arise from a variety of factors. For example, Catalyst inlines all mass action reaction terms for ODE models within a single generated function that evaluates the ODE derivative. This provides opportunities for the compiler to optimise expression evaluation, and avoids the overhead of repeatedly calling non-inlined functions to evaluate such terms. For the largest ODE models, Catalyst and ModelingToolkit’s support for generating explicit sparse Jacobians led to significant performance improvements when using the CVODE solver, see Section 4.1 and S1 Text. For jump process SSA simulations, Catalyst uses a sparse reaction specification that automatically analyses each reaction, and then classifies the reaction into the most performant but physically valid representation supported by JumpProcesses.jl (corresponding to jumps with general time-varying intensities, jumps with general rate expressions but for which the intensity is constant between the occurrence of two jumps, and jumps for which the intensity is a mass action type rate law). This enables JumpProcesses.jl to avoid the overhead of calling a large collection of user-provided functions via pointers, by using a single pre-defined and inlined function to evaluate individual mass action reaction intensities, while still supporting calling general user rate functions via pointers (for non-mass action rate laws). These Catalyst-specific features, when coupled to the large variety of solvers in DifferentialEquations.jl and broad flexibility in tuning solver components (i.e. different Jacobian and jump representations, flexibility in choice of linear solvers, etc.), help enable Catalyst’s observed performance. 2.3 Catalyst enables composable, symbolic modeling of CRNs Catalyst’s primary feature is that its models are represented using a CAS, enabling them to be algebraically manipulated. Examples of how this is utilised include automatic computation of system Jacobians, calculation and elimination of conservation laws, and simplification of generated symbolic DAE models via ModelingToolkit’s symbolic analysis tooling. These techniques can help speed up numeric simulations, while also facilitating higher level analysis. One example is enabling users to generate ODE models with non-singular Jacobians via the elimination of conservation laws, which can aid steady-state analysis tooling. Catalyst also provides a network analysis API, enabling the calculation of a variety of network properties beyond conservation laws, including linkage classes, weak reversibility, and deficiency indices. Catalyst’s symbolic representation permits model internals to be freely extracted, investigated, and manipulated, giving the user full control over their models (Fig 4). This enables various forms of programmatic model creation, extension and composition. Model structures that occur repetitively can be duplicated, and disjoint models can be connected together. For example, such functionality can be used to model a population of cells, each with defined neighbours, where each cell can be assigned a duplicate of the same simple CRN. The CRNs within each cell can then be connected to those of its neighbours, enabling models with spatial structures. Similarly, one could define a collection of genetic modules, and then compose such modules together into a larger gene regulatory network. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. The simulation pipeline of a Catalyst model, with internal intermediates displayed. Code as written by the user (yellow background), and as generated internally by Catalyst and ModelingToolkit (blue and grey backgrounds respectively) are shown, in addition to the generated structures and their fields (blue background, some of the internal fields are omitted in all displayed structures). (a) A symbolic ReactionSystem for a reversible dimerisation reaction is created using either the DSL, or programmatically using the Symbolics computer algebraic system. (b) The ReactionSystem can be converted into a ModelingToolkit ODESystem structure, corresponding to a symbolic RRE ODE model. (c) By providing initial conditions, parameter values, and a time span, the ODESystem can be simulated, generating an output solution. The generated (internal) Julia code for evaluating the derivatives defining the ODEs, which gets compiled and is input to the ODE solver, is displayed in grey. At each step, the user has the ability to investigate and manipulate the generated structures. https://doi.org/10.1371/journal.pcbi.1011530.g004 Catalyst is highly flexible in the allowed Julia functions that can be used in defining rates, rate laws, or stoichiometry coefficients. This means that while reaction rates and rate laws are typically constants, parameters, or simple functions, e.g. Hill functions, they may also include other terms, such as neural networks or data-driven, empirically defined, Julia functions. Likewise, stoichiometric coefficients can be random variables by defining them as a symbolic variable, and setting that variable equal to a Julia function sampling the appropriate probability distribution. Such functionality can be utilized, for example, to model transcriptional bursting [53], where the produced mRNA copy-numbers are random variables. Finally, standard Catalyst-generated ODE and SDE models are differentiable, in that the generated codes can be used in higher-level packages that rely on automatic differentiation [34]. In this way Catalyst-generated models can be used in machine-learning based analyses. That Catalyst gives full access to its model internals, combined with its composability, allows other packages to easily integrate into, and build upon, it. Indeed, this is already being utilised by independent package developers. The MomentClosure.jl Julia package, which implements several techniques for moment closure approximations, is built to be deployed on Catalyst models [54]. It can generate symbolic finite-dimensional ODE system approximations to the full, infinite system of moment equations associated with the chemical master equation. These symbolic approximations can then be compiled and solved via ModelingToolkit in a similar manner to how Catalyst’s generated RRE ODE models are handled. Similarly, FiniteStateProjection.jl [55] builds upon Catalyst and ModelingToolkit to enable the numerical solution of the chemical master equation, while DelaySSAToolkit.jl [56] can accept Catalyst models as input to its SSAs that handle stochastic chemical kinetics models with delays. Another example of how Catalyst’s flexibility enables its integration into the Julia ecosystem is that CRNs with polynomial ODEs (a condition that holds for pure mass action systems) can be easily converted to symbolic steady-state systems of polynomial equations. This enables polynomial methods, such as homotopy continuation, to be employed on Catalyst models. Here, homotopy continuation (implemented by the HomotopyContinuation.jl Julia package) can be used to reliably compute all roots of a polynomial system [57]. This is an effective approach for finding multiple steady states of a system. When the CRN contains Hill functions (with integer exponents), by multiplying by the denominators, one generates a polynomial system with identical roots to the original, on which homotopy continuation can still be used. 2.4 Catalyst models are compatible with a wide range of ancillary tools and methods The Julia SciML, and broader Julia, ecosystem offers a wide range of techniques for working with models and data based around the IR that Catalyst produces (Fig 5). While the reactions that constitute a CRN are often known in developing a model, system parameters (these typically correspond to reaction rates) rarely are. A first step in analyzing a model is identifiability analysis, where we determine whether the parameters can be uniquely identified from the data [58]. This is enabled through the StructuralIdentifiability.jl package. In the next step, parameters can be fitted to data. This can be done using DiffEqParamEstim.jl, which provides simple functions that are easy to use. Alternatively, more powerful packages, like Optimization.jl and the Turing.jl Julia library for Bayesian analysis, offer increased flexibility for experienced users [59]. Furthermore, unknown CRN structures (such as a species’s production rate) can be approximated using neural networks and then fitted to data. This functionality is enabled by the SciMLSensitivity package [60]. More broadly, system steady states can be computed using the NLSolve.jl or HomotopyContinuation.jl Julia packages [57]; bifurcation structures can be calculated, and bifurcation diagrams generated, with the BifurcationKit.jl library [61]; and SciMLSensitivity.jl and GlobalSensitivity.jl can be used to investigate the sensitivity and uncertainty of model solutions with regard to parameters [62]. Finally, options for displaying CRNs, either as network graphs (via Graphviz) or Latex formatted equations (via Latexify.jl), also exist. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. A wide range of features are available for Catalyst model analysis. A CRN model can be created either through the DSL, by manually declaring the reaction events, or by loading it from a file. The model is stored in the ReactionSystem IR, which can be used as input to a wide range of methods. Purple boxes indicate code written by the user, and green boxes the corresponding output. For some methods, either one, or both, boxes are omitted. https://doi.org/10.1371/journal.pcbi.1011530.g005 3 Discussion In this article, we have introduced the Catalyst library for modeling of CRNs. It represents models through the ModelingToolkit.jl IR, which is used across the SciML organization and Julia ecosystem libraries, and can be automatically translated into optimized inputs for numerical simulations (RRE ODE, CLE SDE, and stochastic chemical kinetics jump process models). Our benchmarks demonstrate that Catalyst often outperforms other tools by an order of magnitude or more. Moreover, it can compose with a variety of other Julia packages, including data-driven modeling tooling (parameter fitting and model inference), and other functionality (identifiable analysis, sensitivity analysis, steady state analysis, etc). The IR is based on the Symbolics.jl CAS, enabling algebraic manipulation and simplification of Catalyst models. This can both be harnessed by the user (e.g. to create models programmatically) and by software (e.g. for automated Jacobian computations). Finally, this also enables easy connection to other Julia packages for symbolic analysis, such as enabling polynomial methods (e.g homotopy continuations) to act on CRN ODEs that have a polynomial form. In addition to the wide range of powerful tools enabled by the combination of the ModelingToolkit IR and the Symbolics CAS, Catalyst also provides a DSL that simplifies the declaration of smaller models. Of a finalized pipeline that evaluates a model with respect to a specific scientific problem, the model declaration is typically only a minor part. However, reaching a final model often requires the production and analysis of several alternative network topologies. If the barrier to create, or modify, a model can be reduced, more topologies can be explored in a shorter time. Thus, an intuitive interface can greatly simplify the model exploration portion of a research project. By providing a DSL that reads CRN models in their most natural form, Catalyst helps to facilitate model construction. In addition, this form of declaration makes code easier to debug, as well as making it easier to understand for non-experts. While several previous tools for CRN modeling have been primarily designed around their own interface, we have instead designed Catalyst to be called from within standard Julia programs and scripts. This is advantageous, since it allows the flexibility of analysing a model with custom code, without having to save and load simulation results to and from files. Furthermore, by integrating our tool into a larger context (SciML), support for a large number of higher-order features is provided, without requiring any separate implementation within Catalyst. This strategy, with modeling software targeting an IR (here provided by ModelingToolkit) enables modelers across widely different domains to collaborate in the development and maintenance of tools. We believe this is the ideal setting for a package like Catalyst. Development of Catalyst is still active, with several types of additional functionality planned. This includes specialised support for spatial models, including spatial SSA solvers for the reaction-diffusion master equation, and general support for reaction models with transport on graphs at both the ODE and jump process level. A longer-term goal is to enable the specification of continuous-space reaction models with transport, and interface with Julia partial differential equation libraries to seamlessly generate such spatially-discrete ODE and jump process models. Furthermore, unlike BioNetGen, COPASI, and GillesPy2, Catalyst does not currently support hybrid methods. These allow model components to be defined at different physical scales (such as resolving some reactions via ODEs and others via jump processes) [63, 64]. This, as well as τ-leaping-based solvers [65, 66], are planned for future updates. Such hybrid approaches can help to overcome the potential negativity of solutions that can arise in τ-leaping and CLE-based models [67]. In the CLE case, Catalyst currently wraps rate laws within the coefficients of noise terms in absolute values to avoid square roots of negative numbers, allowing SDE solvers to continue time-stepping even when solutions become negative (following the approach in [68]). We hope to also integrate alternative modelling approaches, such as the constrained CLE [67], which avoid negativity of solutions via modification of the dynamics at the positive-negative population boundary. Finally, given Catalyst’s support for units we hope to implement functionality for automatically converting between concentration and “number of” units within system specifications by allowing users to specify compartments with associated size units. Catalyst is available for free under the permissive MIT License. The source code can be found at https://github.com/SciML/Catalyst.jl. It is also a registered package within the Julia ecosystem and can be installed from within a Julia environment using the commands using Pkg; Pkg.add("Catalyst"). Full documentation, including tutorials and an API, can be found at https://catalyst.sciml.ai/stable/. Issues and help requests can be raised either at the Catalyst GitHub page, on the Julia discourse forum (https://discourse.julialang.org/), or at the SciML organization’s Julia language Slack channels (#diffeq-bridged and #sciml-bridged). The library is open to pull requests from anyone who wishes to contribute to its development. Users are encouraged to engage in the project. 4 Materials and methods 4.1 Benchmarks Benchmarks were carried out using the five CRN models used in [52]. The .bngl files provided in [52] were used as input to BioNetGen, while COPASI, GillesPy2, and Matlab used the corresponding (BioNetGen generated) .xml files. Catalyst used the corresponding (by BioNetGen generated) .net files. The exception was the BCR model, for which we used the .bngl file from [50], rather than the one from [52]. Throughout the simulations, no observable values were saved. Where options were available to reduce solution time point save frequency, and these improved performances, these were used (Section C in S1 Text). BioNetGen, COPASI, and GillesPy2 simulations were performed using their corresponding Python interfaces. To ensure the correctness of the solvers, for each combination of model, tool, method, and options, ODE and SSA simulations were carried out and the results were plotted. The plots were inspected to ensure consistency across all simulations (Figs D-M in S1 Text). Runtimes were measured using timeit (in Python), BenchmarkTools.jl (in Julia, [69]), and timeit (in Matlab). For each benchmark, the median runtime over several simulations was used (the number of simulations carried out for each benchmark, over which we took the median, is described in Table 1). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Number of simulations used to calculate median simulation times. https://doi.org/10.1371/journal.pcbi.1011530.t001 For ODE benchmarks, simulation run times were measured from the initial conditions used in [52] to the time for the model to reach its (approximate) steady state (Table 2). The exception was the BCR model, which exhibited a pulsing limit cycle behaviour. For this, we simulated it over 20,000 time units, allowing it to complete three pulse events (Fig G in S1 Text). For ODE simulations, for all tools, the absolute tolerance was set to 10−9 and the relative tolerance 10−6. Primarily tests were carried out using the lsoda and CVODE solvers [70, 71]. However, Catalyst has access to additional ODE solvers via DifferentialEquations.jl (more specifically OrdinaryDiffEq.jl). Some of these (such as QNDF and TRBDF2) are competitive with lsoda and CVODE, hence these additional solvers were also benchmarked [72, 73]. All benchmarks were carried out on the MIT supercloud HPC [74]. We used its Intel Xeon Platinum 8260 units (each node has access to 192 GB RAM and contains 48 cores, of which only a single one was used). Each benchmark was carried out on a single, exclusive, node, to ensure they were not affected by the presence of other jobs. Julia, Matlab, and Python all were set to use only a single thread, ensuring multi-threading did not affect performance (e.g. Julia solvers will automatically utilise additional available threads to speed up the linear solvers of implicit simulators). Finally, work-precision diagrams were investigated to determine the relationship between simulation time and error in the native Julia solvers (Fig C in S1 Text). All benchmarking code is avaiable at (Code availability) under a permissive MIT license. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Final (physical) time for model steady states in ODE benchmarks. https://doi.org/10.1371/journal.pcbi.1011530.t002 When using CVODE or implicit solvers, Catalyst permits a range of simulation options. By default, Jacobians are computed through automatic differentiation [34]. This option can either be disabled (with the Jacobian then being automatically computed through finite differences), or an option can be set to automatically compute, and use, a symbolic Jacobian from Catalyst models. Another option enables a sparse representation of the Jacobian matrix. Furthermore, the underlying linear solver for all implicit methods can be specified. We tried both the default option (which automatically selects one), but also specified either the LapackDense (using LU), GMRES, or KLU linear solvers. When the GMRES linear solver is used, a preconditioner can be set. Here we investigated both using no preconditioner, and using an incomplete LU preconditioner (described further in Section B in S1 Text). Jacobians were generated using either automatic differentiation (when either the Multistate, Multisite2, or Egfr_net models were simulated using Julia solvers) or finite differences. The exception was for the KLU linear solver, for which we used a symbolically computed Jacobian. When we used either the KLU linear solver, or preconditioned GMRES, a sparse Jacobian representation was used. Generally, the non-Catalyst tools have fewer available solvers (typically depending on CVODE) and options, however, we tried those we found available. We also note that Catalyst CVODE simulations without any options specified still compare favourably to the other tools (Fig A in S1 Text). The methods and options used for the benchmarks in Fig 3 are described in Table 3. Their performance is also described in Table A in S1 Text (this contains the same information as Fig 3, but as numbers rather than a bar chart). For a full list of benchmarks carried out, and the options used, see Section B in S1 Text. Furthermore, Fig A in S1 Text shows the performance of all trialed combinations of methods and options, with Fig B in S1 Text showing the performance when the simulations are carried out for increasing final model (physical) times. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Options used for the benchmarked ODE methods displayed in the main text figure. https://doi.org/10.1371/journal.pcbi.1011530.t003 Stochastic chemical kinetics simulations of Catalyst models used SSAs defined in JumpProcesses.jl [44], a component of DifferentialEquations.jl. In Fig 3, Direct refers to Gillespie’s direct method [9], SortingDirect to the sorting direct method of [45], RSSA and RSSACR to the rejection and composition-rejection SSA methods of [75–77]. Dependency graphs needed for the different methods are automatically generated via Catalyst and ModelingToolkit as input to the JumpProcesses.jl solvers. Due to supercloud not permitting single runs longer than 4 days, for the largest models, the slowest tools and methods were not benchmarked. The BCR model exhibits pulses, to ensure that at least some pulses were included in each SSA simulation, this model was simulated over very long timespans (> 10, 000 seconds). For a full list of SSA benchmarks and their options, please see Section C in S1 Text. The benchmarks were carried out on Julia version 1.8.5, using Catalyst version 13.1.0, JumpProcesses version 9.5.1, and OrdinaryDiffEq version 6.49.0. Note that JumpProcesses and OrdinaryDiffEq are both components in the meta DifferentialEquations.jl package. We used Python version 3.9.15, the version 0.7.9 python interface for BioNetGen, the basico version 4.47 python interface for COPASI, GillesPy2 version 1.8.1, and Matlab version 9.8 with SimBiology version 5.10. 4.1 Benchmarks Benchmarks were carried out using the five CRN models used in [52]. The .bngl files provided in [52] were used as input to BioNetGen, while COPASI, GillesPy2, and Matlab used the corresponding (BioNetGen generated) .xml files. Catalyst used the corresponding (by BioNetGen generated) .net files. The exception was the BCR model, for which we used the .bngl file from [50], rather than the one from [52]. Throughout the simulations, no observable values were saved. Where options were available to reduce solution time point save frequency, and these improved performances, these were used (Section C in S1 Text). BioNetGen, COPASI, and GillesPy2 simulations were performed using their corresponding Python interfaces. To ensure the correctness of the solvers, for each combination of model, tool, method, and options, ODE and SSA simulations were carried out and the results were plotted. The plots were inspected to ensure consistency across all simulations (Figs D-M in S1 Text). Runtimes were measured using timeit (in Python), BenchmarkTools.jl (in Julia, [69]), and timeit (in Matlab). For each benchmark, the median runtime over several simulations was used (the number of simulations carried out for each benchmark, over which we took the median, is described in Table 1). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Number of simulations used to calculate median simulation times. https://doi.org/10.1371/journal.pcbi.1011530.t001 For ODE benchmarks, simulation run times were measured from the initial conditions used in [52] to the time for the model to reach its (approximate) steady state (Table 2). The exception was the BCR model, which exhibited a pulsing limit cycle behaviour. For this, we simulated it over 20,000 time units, allowing it to complete three pulse events (Fig G in S1 Text). For ODE simulations, for all tools, the absolute tolerance was set to 10−9 and the relative tolerance 10−6. Primarily tests were carried out using the lsoda and CVODE solvers [70, 71]. However, Catalyst has access to additional ODE solvers via DifferentialEquations.jl (more specifically OrdinaryDiffEq.jl). Some of these (such as QNDF and TRBDF2) are competitive with lsoda and CVODE, hence these additional solvers were also benchmarked [72, 73]. All benchmarks were carried out on the MIT supercloud HPC [74]. We used its Intel Xeon Platinum 8260 units (each node has access to 192 GB RAM and contains 48 cores, of which only a single one was used). Each benchmark was carried out on a single, exclusive, node, to ensure they were not affected by the presence of other jobs. Julia, Matlab, and Python all were set to use only a single thread, ensuring multi-threading did not affect performance (e.g. Julia solvers will automatically utilise additional available threads to speed up the linear solvers of implicit simulators). Finally, work-precision diagrams were investigated to determine the relationship between simulation time and error in the native Julia solvers (Fig C in S1 Text). All benchmarking code is avaiable at (Code availability) under a permissive MIT license. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Final (physical) time for model steady states in ODE benchmarks. https://doi.org/10.1371/journal.pcbi.1011530.t002 When using CVODE or implicit solvers, Catalyst permits a range of simulation options. By default, Jacobians are computed through automatic differentiation [34]. This option can either be disabled (with the Jacobian then being automatically computed through finite differences), or an option can be set to automatically compute, and use, a symbolic Jacobian from Catalyst models. Another option enables a sparse representation of the Jacobian matrix. Furthermore, the underlying linear solver for all implicit methods can be specified. We tried both the default option (which automatically selects one), but also specified either the LapackDense (using LU), GMRES, or KLU linear solvers. When the GMRES linear solver is used, a preconditioner can be set. Here we investigated both using no preconditioner, and using an incomplete LU preconditioner (described further in Section B in S1 Text). Jacobians were generated using either automatic differentiation (when either the Multistate, Multisite2, or Egfr_net models were simulated using Julia solvers) or finite differences. The exception was for the KLU linear solver, for which we used a symbolically computed Jacobian. When we used either the KLU linear solver, or preconditioned GMRES, a sparse Jacobian representation was used. Generally, the non-Catalyst tools have fewer available solvers (typically depending on CVODE) and options, however, we tried those we found available. We also note that Catalyst CVODE simulations without any options specified still compare favourably to the other tools (Fig A in S1 Text). The methods and options used for the benchmarks in Fig 3 are described in Table 3. Their performance is also described in Table A in S1 Text (this contains the same information as Fig 3, but as numbers rather than a bar chart). For a full list of benchmarks carried out, and the options used, see Section B in S1 Text. Furthermore, Fig A in S1 Text shows the performance of all trialed combinations of methods and options, with Fig B in S1 Text showing the performance when the simulations are carried out for increasing final model (physical) times. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Options used for the benchmarked ODE methods displayed in the main text figure. https://doi.org/10.1371/journal.pcbi.1011530.t003 Stochastic chemical kinetics simulations of Catalyst models used SSAs defined in JumpProcesses.jl [44], a component of DifferentialEquations.jl. In Fig 3, Direct refers to Gillespie’s direct method [9], SortingDirect to the sorting direct method of [45], RSSA and RSSACR to the rejection and composition-rejection SSA methods of [75–77]. Dependency graphs needed for the different methods are automatically generated via Catalyst and ModelingToolkit as input to the JumpProcesses.jl solvers. Due to supercloud not permitting single runs longer than 4 days, for the largest models, the slowest tools and methods were not benchmarked. The BCR model exhibits pulses, to ensure that at least some pulses were included in each SSA simulation, this model was simulated over very long timespans (> 10, 000 seconds). For a full list of SSA benchmarks and their options, please see Section C in S1 Text. The benchmarks were carried out on Julia version 1.8.5, using Catalyst version 13.1.0, JumpProcesses version 9.5.1, and OrdinaryDiffEq version 6.49.0. Note that JumpProcesses and OrdinaryDiffEq are both components in the meta DifferentialEquations.jl package. We used Python version 3.9.15, the version 0.7.9 python interface for BioNetGen, the basico version 4.47 python interface for COPASI, GillesPy2 version 1.8.1, and Matlab version 9.8 with SimBiology version 5.10. Supporting information S1 Text. Additional benchmarks and benchmark information. https://doi.org/10.1371/journal.pcbi.1011530.s001 (PDF) Acknowledgments The authors thank the 26 other individuals who contributed commits to Catalyst, the Catalyst tutorials, and the Catalyst documentation, along with the many users who have offered suggestions and opened issues. The authors acknowledge the MIT SuperCloud and Lincoln Laboratory Supercomputing Center for providing (HPC, database, consultation) resources that have contributed to the research results reported within this paper/report.

LaBelle, Steven A.;IV, A. Marsh Poulson;Maas, Steve A.;Rauff, Adam;Ateshian, Gerard A.;Weiss, Jeffrey A.

doi: 10.1371/journal.pcbi.1011553pmid: 37871113

Introduction Structural properties of the extracellular matrix (ECM)–namely collagen fibril anisotropy and collagen density–regulate cellular proliferation and directional guidance (taxis) [1–3]. During neovascularization, for instance, both homeostatic (wound healing, implant inosculation) and pathologic (tumorigenesis, chronic inflammation, arthritis) angiogenesis are associated with inhomogeneous matrix structures that can encourage or inhibit vascular growth [4–7]. At the cellular level, tissue structure directly influences cellular proliferation and cytokine signaling kinetics [1, 8, 9]. Experimental methods to delineate the role of individual ECM structural features on cellular proliferation and directional guidance remain impractical or impossible. Thus, computational simulation is emerging as a tool to investigate the influence of continuum-based measures of ECM collagen microstructure on cell guidance. The mass density of collagen fibrils and their anisotropy in the ECM play unique roles in modulating growth rate during angiogenesis and guiding growing neovessels [2, 8]. Collagen density regulates ECM stiffness and thus contributes to durotaxis–the guidance of cells by rigidity gradients [10]. Fibril anisotropy can be described by the fractional density of collagen fibrils oriented along all spatial directions emanating from a material point. This anisotropy contributes to durotaxis via guidance of cells along variously-tensed fibrils, as well as contact guidance and haptotaxis–the guidance of cells by matrix binding site gradients (e.g., integrins) [11, 12]. In a recent study, we demonstrated that both microvessel rate of extension and guidance (reorientation and taxis) increased with matrix anisotropy during in vitro microvascular angiogenesis [1]. Interestingly, collagen density attenuated the effects of anisotropy so that stronger alignment was required for significant proangiogenic effects to occur in dense scaffolds. The specific configuration of collagen density gradients and fibril alignment are thought to contribute to the development of new vasculatures during pathologic and homeostatic angiogenesis [13, 14]. However, it remains difficult (or impossible) to fabricate scaffolds that adequately and exhaustively represent complex collagen matrix architectures such as granulation tissue or along tissue-tissue interfaces. In vivo studies present additional challenges due to the number of confounding immunoregulatory, patient-specific, and etiology-specific factors [15–17]. Thus, simulation can be used to study the isolated effects and coordination of multiple stimuli in biomedical problems. Further, simulation can identify specific hypotheses and narrow down experimental parameter spaces. Toward this end, we present a computational simulation framework that predicts the effects of different configurations of collagen density and fibril anisotropy during angiogenesis. Several approaches have been employed to simulate fibril guidance during angiogenesis and other cell migration phenomena. Vector fields have long been used to represent spatial variations in collagen alignment at the sub-millimeter scale [18–22]. Here, the predominant orientation of the collagen matrix at each spatial discretization is represented by a unit vector. In a similar approach, discrete fibril networks have been superimposed on the simulation domain so that cells encounter multiple fibrils of differing orientation [5]. In both cases, cells have been modeled as discrete agents that migrate along the direction of the locally interpolated vector or embedded fibrils. However, cells are substantially larger than fibrils, and they interact with many fibrils of varying orientation. Additionally, these simple approaches are insufficient to determine the degree of anisotropy. Recently, orientation distribution functions (ODFs) have become a popular alternative for describing fibril anisotropy since they summarize the orientations of countless fibrils at a point in a continuum [11, 23–27]. More generally, ODFs characterize diverse types of oriented data including fibril alignment, trajectories of migrating cells, and entire microvascular networks [7, 8, 13, 20]. The values of the ODF correspond to the relative probability of observations (fibrils, cell trajectories, vessels) oriented along each direction. Mathematically, the ODF can be defined as , where is the space of all unit vector orientations u∈ℝn originating from the (n-1)-sphere, and is the space of all real, positive scalars [28]. The ODF is often formulated as a probability density function with the constraint that integration over the ODF sums to 1: [1] For most practical cases, integration takes place along the circumference of a circle (2D distributed data, n = 2) or over the surface a sphere (3D distributed data, n = 3). ODFs are commonly parameterized as ellipsoidal fibril distributions (EFDs) or periodic von-Mises distributions based on the shape of their boundary [11, 27]. Matrix-valued tensors (structure tensors) can be formulated that encode the shape, orientation, and anisotropy of the parameterized ODF. Unlike vector field models, ODF-based models typically model cell populations as continuous concentrations whose migration (modeled as anisotropic diffusion) is influenced by the ODF shape and orientation. Such models, however, have generally neglected the degree of anisotropy as a model input, despite evidence that anisotropy significantly affects neovessel extension rate, proliferation, guidance, and persistence [1, 8, 29]. Furthermore, the assumption that cell populations move as a continuum is ill-suited for phenomena characterized by stochastic behaviors of individual migrating cells (e.g., neovessel recruitment by solid tumors, circulating tumor cell extrusion/extravasation) [30, 31]. This is particularly relevant for simulations of microvascular angiogenesis, which model the formation and extension of discrete endothelial cell networks that interact with each other and the ECM. In the case of discrete cell modeling, vector fields have been preferred over ODFs because vectors can easily be interpolated and they identify a single direction for each cell agent to grow. In contrast, ODFs require advanced calculations to interpolate, and the determination of which direction each cell agent migrates is non-intuitive. The objectives of this study were 1) to develop an enhanced simulation framework to predict microvascular growth and guidance based on 3D EFD representations of matrix collagen anisotropy and density, and 2) to apply the framework to predict experimental observations of microvascular growth in the contexts of wound healing and tumorigenesis. To do this, we extended our AngioFE simulation framework to integrate prior experimental findings that the degree of matrix anisotropy and density simultaneously affect microvascular growth [1]. First, we simulated anisotropic fibril guidance during sprouting angiogenesis where matrix anisotropy was described using a field of either vectors or EFDs. We parameterized collagen fibril ODFs as EFDs due to their simplicity, as well as their agreement with experimental ODFs extracted from images of collagen hydrogels. Simulation results were compared to experimental measures of microvascular growth and guidance in anisotropic matrices of differing densities. Relationships between the local matrix structure (collagen density, anisotropy) and the growth behavior (rate, orientation) were updated in AngioFE to reflect experimental findings. Additionally, we evaluated the accuracy of pseudo-deformed structure tensors using techniques from differential geometry. Second, we investigated the role of physiologically relevant configurations of anisotropy and density such as anisotropy gradients in soft tissues during wound healing as well as structural alterations of the stroma during cancer tumorigenesis. The presented 3D simulation framework is the first to include matrix density and the degree of anisotropy simultaneously as model inputs. Our results suggest that spatially patterned structural cues can determine the success or failure of neovascularization in physiological and pathological contexts. Results AngioFE model formulation To address current shortcomings in modeling anisotropic guidance of angiogenesis, we extended our AngioFE simulation framework that models sprouting angiogenesis from parent microvessel fragments within the FEBio finite element software suite (Finite Elements for Biomechanics, FEBioStudio, www.febio.org) [21, 32]. Briefly, microvessel networks were explicitly represented as connected line segments whose tips grew through and deformed a finite element mesh representing the surrounding tissue. Previously, AngioFE modeled fibril guidance via vector fields that described the average collagen orientation at any position in the simulation. However, this approach failed to account for the local degree of ECM anisotropy and thus tended to over-constrain vessel reorientation. Thus, we modified AngioFE to represent fibril guidance via EFD fields. Notably, each EFD encoded the degree of anisotropy. This section briefly summarizes the formulation of FEBio + AngioFE as shown in the graphical summary (Fig 1). The full algorithmic details are described in the Methods and Supporting Methods A-M in S1 Text. All code and data are available on the AngioFE GitHub repository at github.com/febiosoftware/AngioFE. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. FEBio + AngioFE Graphical Summary. Presented here is a graphical summary of the simulation framework. The physical parameters of the model were generated and prescribed in FEBio/FEBioStudio. Other user-defined parameters associated with vessel growth and traction behaviors were included to configure AngioFE. Model initialization: Parent microvessel fragments were superimposed on the finite element mesh. Degrees of freedom (DOF) such as collagen EFDs and density were mapped to the finite element mesh. Vessel orientation: The direction each vessel grew, ψnew, was determined by 1) interpolating EFDs from the finite element integration points to the vessel tip, 2) pseudo-deforming the local EFD, 3) sampling the EFD for a single contact guidance direction θ, and 4) determining the balance between persistence along the previous vessel orientation ψ and contact guidance by θ. Vessel extension: The function ν(ρ,FA) scaled the vessel extension rate. This function decreased inversely with collagen density and increased directly with collagen anisotropy. After new vessels grew, branches were added along the existing vessels. Finally, cellular tractions were applied at the tips of the new vessels. Mechanics & kinematics: Cell tractions were sent to FEBio, which then solved the equations of motion. Vessel positions were then updated by AngioFE. Model I/O: The updated vessel positions and finite element degrees of freedom were saved after each time step. https://doi.org/10.1371/journal.pcbi.1011553.g001 Model inputs. FEBioStudio, the graphical user interface for FEBio, was used to generate the geometry and finite element discretizations, assign the constitutive model and material coefficients, and prescribe boundary conditions. AngioFE parameters were provided to dictate the distribution of parent microvessel fragments and to define the rules that controlled the growth rate, growth direction, and tip cell contraction. Model initialization. We initialized our models to mimic the starting density of parent microvessels in collagen hydrogel microvessel culture experiments [1]. Spatial maps of model physical parameters (e.g., EFD and density maps) were then initialized. Vessel orientation. The new direction each vessel tip grew, ψnew, was assumed to emerge from competition between persistence (growth along the vessel’s current orientation, ψ) and growth along the direction of contact guidance (θ) [2, 4]. To determine θ, we first interpolated the local ODF from the integration points of the finite element containing the tip [33]. Next, we determined the pseudo-deformed EFD configuration. Pseudo-deformation is a process that identifies an EFD that closely approximates the true ODF, which is important since deformed EFDs may morph into ODFs that are not ellipsoidal [11]. To mimic filopodial probing of the ECM prior to cell elongation, we introduced a method to sample orientation vectors θ from EFDs via Monte-Carlo methods. The direction that each vessel grew, ψnew, was determined by partially rotating ψ towards θ about their shared orthogonal axis (See Methods: “FEBio + AngioFE computational model of microvascular growth”, Supporting Methods A in S1 Text and S1 Fig). Vessel extension rate and tractions. The length each vessel grew during a simulation time step was determined from the vessel extension rate. The total vessel length was assumed to increase sigmoidally with respect to time based on time-series morphometric measurements of angiogenic microvessels in culture [20]. The time-dependent growth was scaled by ν(ρ,FA), a Lorentzian function which decreased with matrix density and increased with anisotropy. New vessel tips were added at the position determined by ψnew and the vessel extension rate. Afterwards, branches were stochastically introduced as new vessel tips emerging from older vessel segments (Supporting Methods J in S1 Text). Finally, traction fields were imposed around new vessel tips, which allowed them to deform the surrounding ECM [19, 34]. Updating mechanics and kinematics. FEBio solved the equations of motion in response to cell tractions, geometric boundary constraints, viscoelasticity, and interstitial fluid motion. The deformed matrix was returned to AngioFE, which updated vessel positions via kinematics. Model I/O. The solution was saved after each time step. Output data included vessel positions, vessel lengths, branch counts, finite element nodal displacements, and finite element integration point values for vessel tractions, EFDs, and matrix density. Model results were visualized in FEBioStudio and data were analyzed and post-processed in MATLAB (MathWorks, Natick, MA). EFD fields enhanced predictions of cell guidance and migration To highlight the crucial role that the degree of anisotropy plays during cell guidance, we simulated microvessel growth using either vector fields or EFD fields. Simulations were generated to correspond to experiments from our recent in vitro study on the roles of matrix density and anisotropy during angiogenesis [1]. Microvessel culture was performed or simulated in low, medium, or high degrees of collagen alignment as well as low or high collagen density (Figs 2, 3A, and S3A). ODFs of microvascular orientation were extracted either from confocal images or from model outputs. Extracted ODFs were then projected onto the XY (Figs 3B and S3B) and XZ planes. We primarily evaluated the XY plane projections since there was minimal growth in the Z direction (as previously observed [3, 35]). There was good agreement between the experimental (confocal) microvascular ODFs and those generated by EFD simulations. ODFs associated with vector field simulations were less polarized and under-predicted microvascular reorientation for the cases of medium and high anisotropy. Further, vector field approaches under-predicted microvascular alignment in sensitivity studies even when growth was simulated to occur solely along the local collagen orientation (i.e., when the internal mechanism representing persistence was disabled; Supporting Methods A-C in S1 Text and S1 and S2 Figs). Similar trends were observed regardless of matrix density (S2 and S3 Figs). Both approaches were insensitive to finite element mesh refinement (Supporting Methods D in S1 Text and S4 Fig). Furthermore, both approaches were insensitive to experimentally established levels of vessel traction (Supporting Methods M in S1 Text and S15 and S16 Figs) [3, 19, 34]. However, we found that stronger tractions (10X experimental observations) initiated a positive feedback loop, resulting in heightened polarization of emerging vascular networks along the horizontal orientation. This feedback loop was more pronounced in vector field simulations compared to EFD simulations. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Generation of collagen gels with varying anisotropy. Collagen gels were aligned to low, moderate, and high anisotropy and imaged via second harmonic generation (SHG) in a prior study [1]. Image data was used to extract ODFs and fit EFDs for each level of alignment. https://doi.org/10.1371/journal.pcbi.1011553.g002 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Comparison of experimental and simulated angiogenesis. A. Z projections of experimental (confocal) and simulated microvascular networks grown in low, medium, or high anisotropy collagen from our previous study [1]. Qualitative agreement was visible for all cases of low and medium anisotropy but vector field simulations visually differed from experiments and EFD simulations at high anisotropy. Depth of field = 200 μm. B. Averaged microvessel ODFs for each experimental or simulated case were projected onto the XY plane to simplify comparison since there was little growth in the Z direction. Microvessel orientations from EFD simulations were in good agreement with experimental microvascular ODFs at all three levels of anisotropy. In contrast, microvascular ODFs from vector field simulations diverged from the experimental data for the cases of medium and high anisotropy. https://doi.org/10.1371/journal.pcbi.1011553.g003 Pseudo-deformed EFDs closely approximated 3D fiber distribution deformation The accuracy of pseudo-deformed EFDs has not previously been calculated, despite growing adaptation of the method [11, 25, 27]. Thus, we validated and verified pseudo-deformation 1) qualitatively by assessing the pseudo-deformation of local EFDs during tissue-level strains and 2) quantitatively by using differential geometry measures to compare ODFs after non-affine deformation (Ω) to ODFs after pseudo-deformation (Ωp). Notably, pseudo-deformation assumes there is no interaction between fibrils (i.e., fibrils deform due to tissue-level strains and cell tractions, but not due to bonds between fibrils). For qualitative analysis, the uniaxial tension test of a collagen gel was simulated to 100% elongation using FEBio (Fig 4A). Visual observation demonstrated that pseudo-deformed EFDs experienced rotation and stretch in regions characterized by symmetric or non-affine local deformations that occur at the gel edges, center, and through the thickness. We quantitatively verified our approach by calculating the difference in generalized fractional anisotropy (GFA, no units, range [0, 1]) between deformed and pseudo-deformed distributions after undergoing tension, compression, simple shear, or pure shear (Supporting Methods E in S1 Text). Differences in GFA were less than 1x10-3 for all cases (Figs 4B and 4C, and S5–S8). Similarly, we measured the Fisher-Rao distance between the deformed and pseudo-deformed ODFs. The Fisher-Rao distance is a measure of dissimilarity between distributions, where the distance between identical distributions is 0° and the maximum possible distance between distributions is 90°. In our numerical simulations, the distance for all cases was below 6°, a threshold that we previously identified was explainable by random sampling [28]. Pseudo-deformed EFDs were close to the true ODFs for up to 50% applied tension/compression and 45% shear in numerical experiments regardless of the initial matrix anisotropy. These results demonstrate that pseudo-deformed EFDs (Ωp) adequately approximate non-affine deformations Ω at physiologically relevant levels of strain. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Visualizations and verification of EFD pseudo-deformation. A. Visualization of pseudo-deformed collagen fibril ODFs (glyphs colored by GFA) during large-scale uniaxial tension in biphasic materials. Cutouts highlight scaling and rotation at the top-right model corner and along the center row of elements (quarter-symmetry view from the edge to the center). B-C. Comparison between ODFs undergoing “true” deformation (Ω) and ODFs undergoing pseudo-deformation (Ωp) for tension/compression (stretch ratio λ ∈ [0.5, 1.5]) and simple shear (shear ratio κ ∈ [0, 0.5]) of a single element. Differences in GFA were less than 1e-3 for all cases, which indicated good agreement between Ω and Ωp. Heat maps of the 3D ODFs were generated for the test cases with the highest strain to demonstrate the agreement in ODF magnitude and orientation between Ω and Ωp. https://doi.org/10.1371/journal.pcbi.1011553.g004 Anisotropy gradients recruit angiogenic neovessels Recent experimental efforts have established that endothelial cell migration and guidance are affected by the degree of tissue anisotropy. This led us to the question: “how might spatial variations in anisotropy (i.e., anisotropy gradients) affect neovessel guidance and vascular recruitment?” Thus, we generalized physiologically relevant anisotropy gradients in AngioFE. A rectilinear simulation domain was created and classified into three regions: the proximal (left), middle, and distal (right) regions (Fig 5). Parent microvessels were generated in the proximal region of the domain then three cases were developed: 1) baseline case—all regions were isotropic (Fig 5A, glyphs), 2) positive gradient case—horizontal anisotropy increased across the domain (Fig 5B, glyphs), and 3) negative gradient case–anisotropy decreased across the domain (Fig 5C, glyphs). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Predictions of microvascular growth in response to spatial anisotropy gradients. Simulations were performed with microvessels seeded on the proximal (left) end of a rectilinear domain. Growth was simulated across an isotropic matrix (baseline) or a matrix characterized by a positive or negative anisotropy gradient. Local matrix alignment is indicated by ellipsoidal glyphs above each representative image with the color indicating the anisotropy (blue: low; red: high). Microvessels in the baseline model failed to reach the distal region after 12 days. Anisotropy gradients resulted in increased vascularization of the middle and distal regions. A negative anisotropy gradient resulted in the most vascularization in the middle region, although there was no difference in vascularization of the distal region between gradient cases (1 way ANOVA with Sidakholm post hoc. *: p<0.05 w.r.t baseline; @: p<0.05 post hoc pairwise comparison). https://doi.org/10.1371/journal.pcbi.1011553.g005 In the baseline case, microvessels vascularized the middle region but failed to grow into the distal region (Fig 5A–5D, and 5E). In contrast, positive and negative anisotropy gradients guided microvessels to the distal end of the simulation domain (Fig 5B–5E; 1 Way ANOVA, Sidakholm post hoc, F [2,29] = 170.12, p<0.001). Interestingly, long-range vascularization of the distal region did not differ between positive and negative anisotropy gradients (Fig 5E; post hoc, p = 0.74). Short-range vascularization of the middle region also significantly increased due to anisotropy gradients (Fig 5D; 1 Way ANOVA, Sidakholm post hoc, F [2,27] = 426.05, p<0.001). Notably, the negative anisotropy gradient case had a significantly greater impact on vascularization of the middle region than positive gradients as well (Fig 5B–5D, post hoc, p<0.001). We also performed these simulations using the vector field method to compare predictions with those from EFD simulations (Supporting Methods F in S1 Text and S9A–S9C Fig). Vessels failed to reach the distal region over a period of 12 days for all vector field simulations (S9D–S9F Fig; 1 Way ANOVA, Sidakholm post hoc, F [2,27] = 62.07, p<0.001). Further, only a negative anisotropy gradient was predicted to enhance vascularization of the middle region (S9G Fig; post hoc, p<0.001). Tumor-associated structural interfaces passively recruit microvessels Tumor-associated collagen signatures (TACS) are unique, heterogeneous configurations of matrix density and anisotropy found in the stroma/desmoplasia near some solid tumors [13, 31, 36]. Certain TACS are associated with an enhanced ability to recruit cells from the stroma, resulting in aggressive cancers and poor prognoses. Thus, we used AngioFE to explore how 5 different TACS affect tumor vascular recruitment during cancer tumorigenesis. Similar to before, a rectilinear simulation domain was generated with three regions: 1) the peritumoral stroma containing the parent microvessels, 2) a thin interface, and 3) a tumor (Fig 6). The density and anisotropy of the interface was modified to reflect clinically observed TACS. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. Tumor associated collagen signatures (TACS) differentially facilitate neovessel recruitment. Microvessels were simulated to originate in the stroma of a tumor periphery (inset, orange). Microvessels grew within the periphery toward the tumor (inset, magenta), separated by a structural interface (inset, yellow). Alignment and density of the interface was varied to mimic various TACS. The interface comprised an isotropic collagen ODF (circle), or aligned ODFs (ellipses) running perpendicular or along interface. The interface density was either low or high. Representative Z projections of the interface and tumor region are presented at the bottom. High interface density reduced the length of microvessels that crossed into the field (TACS-1 interfaces). Interface alignment along the tumor (TACS-2) deflected vessels or trapped them within the interface, while alignment radiating from the tumor facilitated vascular invasion (TACS-3). Fibril alignment in TACS-3 nullified the effects of increased matrix density. 1 way ANOVA. ***: p<0.001 w.r.t. baseline. https://doi.org/10.1371/journal.pcbi.1011553.g006 We discovered a significant relationship between TACS presentations and the degree of tumor vascularization (1 Way ANOVA, Sidakholm post hoc; F [5,54] = 52.05, p<0.001). In the baseline case (no difference in structure between regions), microvessels grew across the interface from the periphery into the tumor (Fig 6). A high-density interface (TACS-1) greatly reduced tumor vascularization (post hoc, p<0.001). Fibril alignment oriented along the interface (TACS-2) led to microvascular growth around the interface, but there was no change in tumor vascularization when compared to the baseline case (post hoc, p = 0.1). In contrast, fibril alignment across the interface (TACS-3) facilitated the greatest tumor vascularization overall (post hoc, p<0.001). The combined presence of elevated density and fibril alignment along the interface (TACS-1+2) resulted in tumor vascularization comparable to TACS-1 (post hoc, p = 0.39). However, in this case, the vessels continued to grow along the interface rather than stop in their tracks as was seen with TACS-1. Finally, the combined presence of elevated density and fibril alignment perpendicular to the interface (TACS-1+3) resulted in a similar degree of tumor vascularization as the baseline case (post hoc, p = 0.044, testwise α = 0.0127). We repeated these simulations using a vector field approach (Supporting Methods F in S1 Text). Trends in tumor vascularization were similar aside from the TACS-1+3 case. The vector field simulations predicted that TACS-1+3 would result in less tumor vascularization than the baseline case (S10 Fig). AngioFE model formulation To address current shortcomings in modeling anisotropic guidance of angiogenesis, we extended our AngioFE simulation framework that models sprouting angiogenesis from parent microvessel fragments within the FEBio finite element software suite (Finite Elements for Biomechanics, FEBioStudio, www.febio.org) [21, 32]. Briefly, microvessel networks were explicitly represented as connected line segments whose tips grew through and deformed a finite element mesh representing the surrounding tissue. Previously, AngioFE modeled fibril guidance via vector fields that described the average collagen orientation at any position in the simulation. However, this approach failed to account for the local degree of ECM anisotropy and thus tended to over-constrain vessel reorientation. Thus, we modified AngioFE to represent fibril guidance via EFD fields. Notably, each EFD encoded the degree of anisotropy. This section briefly summarizes the formulation of FEBio + AngioFE as shown in the graphical summary (Fig 1). The full algorithmic details are described in the Methods and Supporting Methods A-M in S1 Text. All code and data are available on the AngioFE GitHub repository at github.com/febiosoftware/AngioFE. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. FEBio + AngioFE Graphical Summary. Presented here is a graphical summary of the simulation framework. The physical parameters of the model were generated and prescribed in FEBio/FEBioStudio. Other user-defined parameters associated with vessel growth and traction behaviors were included to configure AngioFE. Model initialization: Parent microvessel fragments were superimposed on the finite element mesh. Degrees of freedom (DOF) such as collagen EFDs and density were mapped to the finite element mesh. Vessel orientation: The direction each vessel grew, ψnew, was determined by 1) interpolating EFDs from the finite element integration points to the vessel tip, 2) pseudo-deforming the local EFD, 3) sampling the EFD for a single contact guidance direction θ, and 4) determining the balance between persistence along the previous vessel orientation ψ and contact guidance by θ. Vessel extension: The function ν(ρ,FA) scaled the vessel extension rate. This function decreased inversely with collagen density and increased directly with collagen anisotropy. After new vessels grew, branches were added along the existing vessels. Finally, cellular tractions were applied at the tips of the new vessels. Mechanics & kinematics: Cell tractions were sent to FEBio, which then solved the equations of motion. Vessel positions were then updated by AngioFE. Model I/O: The updated vessel positions and finite element degrees of freedom were saved after each time step. https://doi.org/10.1371/journal.pcbi.1011553.g001 Model inputs. FEBioStudio, the graphical user interface for FEBio, was used to generate the geometry and finite element discretizations, assign the constitutive model and material coefficients, and prescribe boundary conditions. AngioFE parameters were provided to dictate the distribution of parent microvessel fragments and to define the rules that controlled the growth rate, growth direction, and tip cell contraction. Model initialization. We initialized our models to mimic the starting density of parent microvessels in collagen hydrogel microvessel culture experiments [1]. Spatial maps of model physical parameters (e.g., EFD and density maps) were then initialized. Vessel orientation. The new direction each vessel tip grew, ψnew, was assumed to emerge from competition between persistence (growth along the vessel’s current orientation, ψ) and growth along the direction of contact guidance (θ) [2, 4]. To determine θ, we first interpolated the local ODF from the integration points of the finite element containing the tip [33]. Next, we determined the pseudo-deformed EFD configuration. Pseudo-deformation is a process that identifies an EFD that closely approximates the true ODF, which is important since deformed EFDs may morph into ODFs that are not ellipsoidal [11]. To mimic filopodial probing of the ECM prior to cell elongation, we introduced a method to sample orientation vectors θ from EFDs via Monte-Carlo methods. The direction that each vessel grew, ψnew, was determined by partially rotating ψ towards θ about their shared orthogonal axis (See Methods: “FEBio + AngioFE computational model of microvascular growth”, Supporting Methods A in S1 Text and S1 Fig). Vessel extension rate and tractions. The length each vessel grew during a simulation time step was determined from the vessel extension rate. The total vessel length was assumed to increase sigmoidally with respect to time based on time-series morphometric measurements of angiogenic microvessels in culture [20]. The time-dependent growth was scaled by ν(ρ,FA), a Lorentzian function which decreased with matrix density and increased with anisotropy. New vessel tips were added at the position determined by ψnew and the vessel extension rate. Afterwards, branches were stochastically introduced as new vessel tips emerging from older vessel segments (Supporting Methods J in S1 Text). Finally, traction fields were imposed around new vessel tips, which allowed them to deform the surrounding ECM [19, 34]. Updating mechanics and kinematics. FEBio solved the equations of motion in response to cell tractions, geometric boundary constraints, viscoelasticity, and interstitial fluid motion. The deformed matrix was returned to AngioFE, which updated vessel positions via kinematics. Model I/O. The solution was saved after each time step. Output data included vessel positions, vessel lengths, branch counts, finite element nodal displacements, and finite element integration point values for vessel tractions, EFDs, and matrix density. Model results were visualized in FEBioStudio and data were analyzed and post-processed in MATLAB (MathWorks, Natick, MA). Model inputs. FEBioStudio, the graphical user interface for FEBio, was used to generate the geometry and finite element discretizations, assign the constitutive model and material coefficients, and prescribe boundary conditions. AngioFE parameters were provided to dictate the distribution of parent microvessel fragments and to define the rules that controlled the growth rate, growth direction, and tip cell contraction. Model initialization. We initialized our models to mimic the starting density of parent microvessels in collagen hydrogel microvessel culture experiments [1]. Spatial maps of model physical parameters (e.g., EFD and density maps) were then initialized. Vessel orientation. The new direction each vessel tip grew, ψnew, was assumed to emerge from competition between persistence (growth along the vessel’s current orientation, ψ) and growth along the direction of contact guidance (θ) [2, 4]. To determine θ, we first interpolated the local ODF from the integration points of the finite element containing the tip [33]. Next, we determined the pseudo-deformed EFD configuration. Pseudo-deformation is a process that identifies an EFD that closely approximates the true ODF, which is important since deformed EFDs may morph into ODFs that are not ellipsoidal [11]. To mimic filopodial probing of the ECM prior to cell elongation, we introduced a method to sample orientation vectors θ from EFDs via Monte-Carlo methods. The direction that each vessel grew, ψnew, was determined by partially rotating ψ towards θ about their shared orthogonal axis (See Methods: “FEBio + AngioFE computational model of microvascular growth”, Supporting Methods A in S1 Text and S1 Fig). Vessel extension rate and tractions. The length each vessel grew during a simulation time step was determined from the vessel extension rate. The total vessel length was assumed to increase sigmoidally with respect to time based on time-series morphometric measurements of angiogenic microvessels in culture [20]. The time-dependent growth was scaled by ν(ρ,FA), a Lorentzian function which decreased with matrix density and increased with anisotropy. New vessel tips were added at the position determined by ψnew and the vessel extension rate. Afterwards, branches were stochastically introduced as new vessel tips emerging from older vessel segments (Supporting Methods J in S1 Text). Finally, traction fields were imposed around new vessel tips, which allowed them to deform the surrounding ECM [19, 34]. Updating mechanics and kinematics. FEBio solved the equations of motion in response to cell tractions, geometric boundary constraints, viscoelasticity, and interstitial fluid motion. The deformed matrix was returned to AngioFE, which updated vessel positions via kinematics. Model I/O. The solution was saved after each time step. Output data included vessel positions, vessel lengths, branch counts, finite element nodal displacements, and finite element integration point values for vessel tractions, EFDs, and matrix density. Model results were visualized in FEBioStudio and data were analyzed and post-processed in MATLAB (MathWorks, Natick, MA). EFD fields enhanced predictions of cell guidance and migration To highlight the crucial role that the degree of anisotropy plays during cell guidance, we simulated microvessel growth using either vector fields or EFD fields. Simulations were generated to correspond to experiments from our recent in vitro study on the roles of matrix density and anisotropy during angiogenesis [1]. Microvessel culture was performed or simulated in low, medium, or high degrees of collagen alignment as well as low or high collagen density (Figs 2, 3A, and S3A). ODFs of microvascular orientation were extracted either from confocal images or from model outputs. Extracted ODFs were then projected onto the XY (Figs 3B and S3B) and XZ planes. We primarily evaluated the XY plane projections since there was minimal growth in the Z direction (as previously observed [3, 35]). There was good agreement between the experimental (confocal) microvascular ODFs and those generated by EFD simulations. ODFs associated with vector field simulations were less polarized and under-predicted microvascular reorientation for the cases of medium and high anisotropy. Further, vector field approaches under-predicted microvascular alignment in sensitivity studies even when growth was simulated to occur solely along the local collagen orientation (i.e., when the internal mechanism representing persistence was disabled; Supporting Methods A-C in S1 Text and S1 and S2 Figs). Similar trends were observed regardless of matrix density (S2 and S3 Figs). Both approaches were insensitive to finite element mesh refinement (Supporting Methods D in S1 Text and S4 Fig). Furthermore, both approaches were insensitive to experimentally established levels of vessel traction (Supporting Methods M in S1 Text and S15 and S16 Figs) [3, 19, 34]. However, we found that stronger tractions (10X experimental observations) initiated a positive feedback loop, resulting in heightened polarization of emerging vascular networks along the horizontal orientation. This feedback loop was more pronounced in vector field simulations compared to EFD simulations. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Generation of collagen gels with varying anisotropy. Collagen gels were aligned to low, moderate, and high anisotropy and imaged via second harmonic generation (SHG) in a prior study [1]. Image data was used to extract ODFs and fit EFDs for each level of alignment. https://doi.org/10.1371/journal.pcbi.1011553.g002 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Comparison of experimental and simulated angiogenesis. A. Z projections of experimental (confocal) and simulated microvascular networks grown in low, medium, or high anisotropy collagen from our previous study [1]. Qualitative agreement was visible for all cases of low and medium anisotropy but vector field simulations visually differed from experiments and EFD simulations at high anisotropy. Depth of field = 200 μm. B. Averaged microvessel ODFs for each experimental or simulated case were projected onto the XY plane to simplify comparison since there was little growth in the Z direction. Microvessel orientations from EFD simulations were in good agreement with experimental microvascular ODFs at all three levels of anisotropy. In contrast, microvascular ODFs from vector field simulations diverged from the experimental data for the cases of medium and high anisotropy. https://doi.org/10.1371/journal.pcbi.1011553.g003 Pseudo-deformed EFDs closely approximated 3D fiber distribution deformation The accuracy of pseudo-deformed EFDs has not previously been calculated, despite growing adaptation of the method [11, 25, 27]. Thus, we validated and verified pseudo-deformation 1) qualitatively by assessing the pseudo-deformation of local EFDs during tissue-level strains and 2) quantitatively by using differential geometry measures to compare ODFs after non-affine deformation (Ω) to ODFs after pseudo-deformation (Ωp). Notably, pseudo-deformation assumes there is no interaction between fibrils (i.e., fibrils deform due to tissue-level strains and cell tractions, but not due to bonds between fibrils). For qualitative analysis, the uniaxial tension test of a collagen gel was simulated to 100% elongation using FEBio (Fig 4A). Visual observation demonstrated that pseudo-deformed EFDs experienced rotation and stretch in regions characterized by symmetric or non-affine local deformations that occur at the gel edges, center, and through the thickness. We quantitatively verified our approach by calculating the difference in generalized fractional anisotropy (GFA, no units, range [0, 1]) between deformed and pseudo-deformed distributions after undergoing tension, compression, simple shear, or pure shear (Supporting Methods E in S1 Text). Differences in GFA were less than 1x10-3 for all cases (Figs 4B and 4C, and S5–S8). Similarly, we measured the Fisher-Rao distance between the deformed and pseudo-deformed ODFs. The Fisher-Rao distance is a measure of dissimilarity between distributions, where the distance between identical distributions is 0° and the maximum possible distance between distributions is 90°. In our numerical simulations, the distance for all cases was below 6°, a threshold that we previously identified was explainable by random sampling [28]. Pseudo-deformed EFDs were close to the true ODFs for up to 50% applied tension/compression and 45% shear in numerical experiments regardless of the initial matrix anisotropy. These results demonstrate that pseudo-deformed EFDs (Ωp) adequately approximate non-affine deformations Ω at physiologically relevant levels of strain. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Visualizations and verification of EFD pseudo-deformation. A. Visualization of pseudo-deformed collagen fibril ODFs (glyphs colored by GFA) during large-scale uniaxial tension in biphasic materials. Cutouts highlight scaling and rotation at the top-right model corner and along the center row of elements (quarter-symmetry view from the edge to the center). B-C. Comparison between ODFs undergoing “true” deformation (Ω) and ODFs undergoing pseudo-deformation (Ωp) for tension/compression (stretch ratio λ ∈ [0.5, 1.5]) and simple shear (shear ratio κ ∈ [0, 0.5]) of a single element. Differences in GFA were less than 1e-3 for all cases, which indicated good agreement between Ω and Ωp. Heat maps of the 3D ODFs were generated for the test cases with the highest strain to demonstrate the agreement in ODF magnitude and orientation between Ω and Ωp. https://doi.org/10.1371/journal.pcbi.1011553.g004 Anisotropy gradients recruit angiogenic neovessels Recent experimental efforts have established that endothelial cell migration and guidance are affected by the degree of tissue anisotropy. This led us to the question: “how might spatial variations in anisotropy (i.e., anisotropy gradients) affect neovessel guidance and vascular recruitment?” Thus, we generalized physiologically relevant anisotropy gradients in AngioFE. A rectilinear simulation domain was created and classified into three regions: the proximal (left), middle, and distal (right) regions (Fig 5). Parent microvessels were generated in the proximal region of the domain then three cases were developed: 1) baseline case—all regions were isotropic (Fig 5A, glyphs), 2) positive gradient case—horizontal anisotropy increased across the domain (Fig 5B, glyphs), and 3) negative gradient case–anisotropy decreased across the domain (Fig 5C, glyphs). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Predictions of microvascular growth in response to spatial anisotropy gradients. Simulations were performed with microvessels seeded on the proximal (left) end of a rectilinear domain. Growth was simulated across an isotropic matrix (baseline) or a matrix characterized by a positive or negative anisotropy gradient. Local matrix alignment is indicated by ellipsoidal glyphs above each representative image with the color indicating the anisotropy (blue: low; red: high). Microvessels in the baseline model failed to reach the distal region after 12 days. Anisotropy gradients resulted in increased vascularization of the middle and distal regions. A negative anisotropy gradient resulted in the most vascularization in the middle region, although there was no difference in vascularization of the distal region between gradient cases (1 way ANOVA with Sidakholm post hoc. *: p<0.05 w.r.t baseline; @: p<0.05 post hoc pairwise comparison). https://doi.org/10.1371/journal.pcbi.1011553.g005 In the baseline case, microvessels vascularized the middle region but failed to grow into the distal region (Fig 5A–5D, and 5E). In contrast, positive and negative anisotropy gradients guided microvessels to the distal end of the simulation domain (Fig 5B–5E; 1 Way ANOVA, Sidakholm post hoc, F [2,29] = 170.12, p<0.001). Interestingly, long-range vascularization of the distal region did not differ between positive and negative anisotropy gradients (Fig 5E; post hoc, p = 0.74). Short-range vascularization of the middle region also significantly increased due to anisotropy gradients (Fig 5D; 1 Way ANOVA, Sidakholm post hoc, F [2,27] = 426.05, p<0.001). Notably, the negative anisotropy gradient case had a significantly greater impact on vascularization of the middle region than positive gradients as well (Fig 5B–5D, post hoc, p<0.001). We also performed these simulations using the vector field method to compare predictions with those from EFD simulations (Supporting Methods F in S1 Text and S9A–S9C Fig). Vessels failed to reach the distal region over a period of 12 days for all vector field simulations (S9D–S9F Fig; 1 Way ANOVA, Sidakholm post hoc, F [2,27] = 62.07, p<0.001). Further, only a negative anisotropy gradient was predicted to enhance vascularization of the middle region (S9G Fig; post hoc, p<0.001). Tumor-associated structural interfaces passively recruit microvessels Tumor-associated collagen signatures (TACS) are unique, heterogeneous configurations of matrix density and anisotropy found in the stroma/desmoplasia near some solid tumors [13, 31, 36]. Certain TACS are associated with an enhanced ability to recruit cells from the stroma, resulting in aggressive cancers and poor prognoses. Thus, we used AngioFE to explore how 5 different TACS affect tumor vascular recruitment during cancer tumorigenesis. Similar to before, a rectilinear simulation domain was generated with three regions: 1) the peritumoral stroma containing the parent microvessels, 2) a thin interface, and 3) a tumor (Fig 6). The density and anisotropy of the interface was modified to reflect clinically observed TACS. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. Tumor associated collagen signatures (TACS) differentially facilitate neovessel recruitment. Microvessels were simulated to originate in the stroma of a tumor periphery (inset, orange). Microvessels grew within the periphery toward the tumor (inset, magenta), separated by a structural interface (inset, yellow). Alignment and density of the interface was varied to mimic various TACS. The interface comprised an isotropic collagen ODF (circle), or aligned ODFs (ellipses) running perpendicular or along interface. The interface density was either low or high. Representative Z projections of the interface and tumor region are presented at the bottom. High interface density reduced the length of microvessels that crossed into the field (TACS-1 interfaces). Interface alignment along the tumor (TACS-2) deflected vessels or trapped them within the interface, while alignment radiating from the tumor facilitated vascular invasion (TACS-3). Fibril alignment in TACS-3 nullified the effects of increased matrix density. 1 way ANOVA. ***: p<0.001 w.r.t. baseline. https://doi.org/10.1371/journal.pcbi.1011553.g006 We discovered a significant relationship between TACS presentations and the degree of tumor vascularization (1 Way ANOVA, Sidakholm post hoc; F [5,54] = 52.05, p<0.001). In the baseline case (no difference in structure between regions), microvessels grew across the interface from the periphery into the tumor (Fig 6). A high-density interface (TACS-1) greatly reduced tumor vascularization (post hoc, p<0.001). Fibril alignment oriented along the interface (TACS-2) led to microvascular growth around the interface, but there was no change in tumor vascularization when compared to the baseline case (post hoc, p = 0.1). In contrast, fibril alignment across the interface (TACS-3) facilitated the greatest tumor vascularization overall (post hoc, p<0.001). The combined presence of elevated density and fibril alignment along the interface (TACS-1+2) resulted in tumor vascularization comparable to TACS-1 (post hoc, p = 0.39). However, in this case, the vessels continued to grow along the interface rather than stop in their tracks as was seen with TACS-1. Finally, the combined presence of elevated density and fibril alignment perpendicular to the interface (TACS-1+3) resulted in a similar degree of tumor vascularization as the baseline case (post hoc, p = 0.044, testwise α = 0.0127). We repeated these simulations using a vector field approach (Supporting Methods F in S1 Text). Trends in tumor vascularization were similar aside from the TACS-1+3 case. The vector field simulations predicted that TACS-1+3 would result in less tumor vascularization than the baseline case (S10 Fig). Discussion We developed a new approach to predict the guidance of neovessels during angiogenesis that accounts for the simultaneous influence of the ECM collagen orientation, degree of anisotropy, and density. Cells dynamically interact with the matrix during migration; microvessel sprouts constantly probe the matrix in time-series images of in vitro angiogenesis [37]. In our simulations, vessel tips grown using the EFD field approach continuously sampled the local collagen orientations, emulating filopodia probing during angiogenesis. As a result, these simulations accurately predicted anisotropic guidance in agreement with prior experimental measures. In contrast, vector field simulations under-predicted anisotropic guidance (e.g., less vessels were oriented along the X direction in Fig 3). There are some reasons why these differences emerged. First, vector fields were only generated during model initialization (i.e., the collagen orientation is determined by sampling the ODF once). Thus, individual vectors could be randomly assigned orientations that differed greatly from the primary collagen direction. Furthermore, isotropic and near-isotropic ODFs are incompatible with vector fields, which distilled a 3D distribution into a single orientation. In contrast, EFDs were sampled at the beginning of each growth step, which better captured the underlying distribution and was compatible with isotropic ODFs. The final disadvantage of the vector field approach is that neovessels growing through the same finite element will be guided by the same vectors, resulting in similar trajectories despite isotropic underlying collagen networks. In contrast, EFDs allowed neovessels to sample unique directions from the same local EFD, resulting in different trajectories. We theorized that mesh refinement could improve the ability of vessels in the vector field simulations to “probe” their local environment; however, this was not the case as mesh refinement had little effect on vessel guidance. Thus, differences between approaches likely occurred due to their differing interpolation methods. The vector field approach interpolated the collagen fibril direction from the finite element nodes via linear weighted averaging of vectors. This method failed to incorporate the properties of the underlying ODF [33]. In contrast, EFD interpolation was accurate because it calculated a geodesic average, which preserved the properties of the underlying orientation distributions. We can extend this reasoning to the observation that discrete vector field simulations displayed higher sensitivity to elevated sprout tractions than EFD simulations. As cellular tractions intensified, nodal vectors rapidly reoriented towards tractions, which skewed the arithmetically averaged fibril orientation. Thus, additional care must be taken when studying pathologies characterized by elevated cell tractions (due to biochemical signaling, pharmaceuticals, mutations, etc.), or when the initial vascular density (cell concentration) is much greater. The degree of anisotropy has not been widely used as a model input in studies of cell migration, despite evidence that the degree of anisotropy affects cell behaviors and guidance [1, 2, 5, 8, 20, 25, 34]. Some contributions to the literature have implicitly modeled this phenomenon, such as 2D simulations of angiogenesis by Sun et al. They incorporated a “matrix conductivity tensor” as an analogue for anisotropy to study how heterogeneously oriented matrices affect vascular growth [24]. Enhanced vessel growth was observed along directions of higher conductivity. However, they concluded that matrix anisotropy decreased vessel growth and increased network tortuosity, contradicting new experimental evidence. This discrepancy can be explained by three factors. First, anisotropy was modeled at a different scale and there was only local alignment (i.e., anisotropy was high but each ODF’s orientation was globally random). Their mesh resolution was about the width of a microvessel, at which point the parameters of a continuous fibril distribution becomes sensitive to individual fibril bundles. Additionally, vessels respond to long-range matrix stiffness and anisotropy gradients, which could not be identified by an over-refined mesh [3, 38]. Second, vessel proliferation/extension rates were insensitive to the degree of anisotropy. Third, vascular persistence was not modeled. This last factor piqued our interest, as we had similarly noticed vessel tortuosity when persistence was diminished in our sensitivity studies (Supporting Methods C in S1 Text and S14 Fig). In this case, both modeling frameworks support the theory that pathological vessel tortuosity emerges due to upregulated mechanosensitivity and downregulated persistence [39]. The framework presented here extends methods to simulate anisotropic guidance of cell continua to discretely modeled cells. First, we modified the pseudo-deformation approach of Barocas et al., which approximated EFD deformations as the ellipsoid associated with B, the left Cauchy-Green deformation tensor [11, 40]. They appropriately identified the orientation of the deformed EFD but scaled the axial lengths by the square of the eigenvalues of the left stretch tensor V. Although this was inconsequential for prior applications (which only used the orientation of the pseudo-deformed EFD as a model input), it was necessary for us to make modifications since our growth model relied on the degree of anisotropy, which depends on physical levels of stretch. We verified the accuracy of our modified pseudo-deformation in the context of various non-affine deformations. The second novelty of this study is the use of stochastic methods for discrete cells to “probe” the matrix by sampling 3D ODFs. In prior modeling approaches, cell migration primarily followed the collagen principal direction and stochastic variation was introduced via random or Brownian processes [20, 25, 41]. Here, we sampled EFDs, which introduced a stochastic component that was directly related to the underlying matrix structure. Further, we avoided tedious calculations of ellipsoidal integrals by using Monte-Carlo methods to sample EFDs [11]. We simulated microvascular growth along a positive or negative gradient of matrix anisotropy to determine how different spatial configurations of matrix anisotropy are involved in neovessel guidance. Short-range and long-range vascularization of the middle and distal regions were increased by positive and negative anisotropy gradients, suggesting that anisotropy gradients can passively recruit neovessels across various length scales. Unexpectedly, vascularization of the distal region was comparable between the positive and negative gradient cases. However, vascularization of the middle region was greater for the negative anisotropy gradient. In this scenario, positive gradients were more efficient at vascularizing the far region. These differences emerged due to the dependence of the vessel elongation rate on the local anisotropy. Collectively, these results highlight the ability of anisotropy gradients passively recruited microvessels. Anisotropy gradients are found in both healthy and diseased tissues. For instance, connective tissues prominently feature anisotropy gradients that provide mechanical function and are distributed at the level of individual tissues (e.g., skin, long bones, articular cartilage, etc.) as well as boundaries between adjacent tissues (e.g., attachments to fascia, joints, synovia, entheses, etc.) [42–47]. Wounds may disrupt the native tissue anisotropy (e.g., tendon & ligament), and scars may amplify tissue alignment [48, 49]. Deviations in collagen organization also arise from disease (chronic inflammation, arthritis) and old age. In extreme cases, microdamage accumulation can erode tissue boundaries, allowing vessels to invade traditionally avascular niches and induce inflammation [47, 50–53]. Previously, we demonstrated that the proangiogenic effects of matrix anisotropy are attenuated by elevated matrix density in vitro [1]. Changes in both matrix anisotropy and density are common at structural interfaces between adjacent tissues. For instance, a number of cancers are characterized by elevated density and/or fibril alignment in the region surrounding the tumor. These hallmarks were termed tumor-associated collagen signatures (TACS) by Provenzano et al. [13]. To test how various TACSs affect tumor vascularization, we simulated angiogenesis near the tumor-stroma interface. The density and anisotropy of the interface were modified to resemble either a single TACS or a clinically relevant combination of multiple TACSs [13, 36, 54]. TACS-1 is characterized by dense collagen around the tumor periphery that forms due to hypertrophic cell growth and myofibroblast collagen deposition [55]. As expected, this type of interface prevented most neovessels from vascularizing the tumor in our simulations. TACS-2 presents as fibril alignment along the surface of the tumor, which can form as the tumor expands and stretches collagen fibrils surrounding it [56]. In our simulations, vessels grew into this interface and began to reorient along it, but still invaded and vascularized the tumor to a similar degree as the baseline case. The combined TACS-1+2 resulted in the least tumor vascularization. Our findings mostly agree with prior simulations of angiogenesis based on an in vitro model of angiogenesis across tissue barriers [4]. The present simulations differ in that the interface was more vascularized by TACS-2 and TACS-1+2 because vessel growth increased with matrix anisotropy. These vessels would be susceptible to recruitment by tumor cells, suggesting that TACS-2 increases the odds of vascular invasion by recruiting vessels to the tumor interface. We also investigated the effects of TACS-3, which is characterized by spiculated collagen fibrils radiating outward from the tumor and poor prognoses [57]. TACS-3 is presumed to emerge due to outward cell migration from the tumor or intense radial contractions by the tumor [13, 56]. Our simulations predicted similar outcomes associated with TACS-3 prognoses, as evident by the large increase tumor vascularization. Further, a TACS-1+3 interface was predicted to facilitate baseline levels of tumor vascularization since anisotropy emboldened vessels to persist in spite of tissue density. In summary, we demonstrated that clinical relationships between tissue structure and solid tumorigenesis could be predicted by simultaneously accounting for matrix density, orientation, and the degree of anisotropy. Our findings support the growing interest in stromal-based cancer diagnostics and therapeutics [58]. In the future, we will expand AngioFE to include biochemical signaling to explore the coordination of biophysical and biochemical stimuli during microvessel guidance across tissue structures. In our predictive simulations, microvessel velocity increased with matrix anisotropy and decreased with matrix density based on experimental observations [1]. This approach can be extended to simulate the roles of other anisotropic biophysical and biochemical stimuli. For instance, there is evidence that microvessels grow along the primary direction of stretch (1st principal strain) and avoid compression (3rd principal strain) [3]. Additionally, the mechanical model can accommodate anisotropic fibril stiffness that emerges from differing fibril properties (as modeled by ξ which varies with fibril diameter, crosslink types, etc.) or variations in fibril stretch along various orientations (e.g., fibrils assume a kinked conformation when relaxed and a linear conformation when engaged in tension). Thus, the presented approach could easily be adapted to evaluate hypotheses of mechanoregulation based on tissue deformation. Similarly, diffusion and permeability tensors can be used to evaluate the role of anisotropic diffusion and matrix porosity on cell growth and guidance. Methods Ethics statement The animal study was reviewed and approved by University of Utah Institutional Animal Care and Use Committee (IACUC). In-vitro pre-alignment of collagen hydrogels and microvessel culture Experimental data for this study were derived from previously published studies in which microvessels were cultured at 3 levels of ECM anisotropy (low, medium, and high anisotropy) as well as 2 levels of ECM density (low or high) [1]. The methods used to characterize ECM architecture and the morphology of microvascular networks after 10 days of culture are summarized below. Type-I collagen hydrogels were cast at low (3 mg/mL) or high (4 mg/mL) density in rectangular chambers flanked with steel mesh anchors at the ends before incubation at 37°C (rat-tail tendon, Corning Inc., Corning, NY). Steel mesh anchors were then stretched between 0 and 20% along the long axis at 25 minutes after casting in order to pre-align the ECM collagen fibrils. Gels were cut free from mesh anchors after 1 day then transferred to phosphate buffered saline or cell-culture media depending on the experiment. The ECM collagen microstructure was visualized via second harmonic generation (SHG) imaging (Fig 2). SHG images were then processed and orientation distribution functions (ODFs) were extracted via fast Fourier transforms using a custom MATLAB script (MathWorks, Natick, MA). Ellipsoids were fit to the ODFs, which allowed the calculation of the anisotropy and ellipsoid semiprincipal axes associated with each level of collagen fibril alignment and density. Intact microvessel fragments were isolated from male Sprague Dawley retired breeder rats using an established protocol [59]. Microvessel fragments were suspended in liquid collagen at low or high density before casting gels in rectangular chambers. Vascularized gels were pre-strained to achieve low, moderate, or high anisotropy, resulting in 6 unique combinations of matrix anisotropy and density. These gels were similarly cut free from the steel mesh anchors after 1 day. Gels were fixed with 2% paraformaldehyde (VWR, Radnor, PA) after 10 days of culture, stained for endothelial cells (Isolectin; ThermoFisher), and then imaged via confocal microscopy. Confocal volumes were processed in AMIRA (ThermoFisher, Waltham, MA) which allowed the extraction of microvessel network ODFs and vascularity (mm vessels / mm3 image volume). A set of vascularized gels were also fixed, stained, and imaged on Day 1 to provide baseline measures of vascularity and vessel polarization that would later be used to generate the initial simulation conditions. Pseudo-deformation and interpolation of ellipsoidal fibril distribution structure tensors We assumed that EFDs approximate collagen ODFs for spatial scales on the order of 100 μm based on prior collagen structural imaging [3, 20]. Further, we assumed the individual fibrils in an ODF deform independent of each other (i.e., no interaction during deformation). These assumptions allowed us to approximate 3D collagen distributions with 6-term symmetric positive-definite tensors (SPDs) which are inherently ellipsoidal. In general, a fiber distribution that is ellipsoidal in the reference (unloaded) configuration of a material does not remain ellipsoidal upon finite deformation, since the semi-principal axes of the ellipsoid may not remain orthogonal. In order to maintain the symmetry properties of the EFDs after deformation, we assumed that EFDs remain ellipsoidal but experience affine deformation (finite rotation and stretching) of their principal radii of curvature. This was necessary to conveniently update and sample EFDs, as well as to spatially interpolate local matrix ODFs and visualize the deformed ODFs. The collagen fibril matrix undergoes deformation due to cellular tractions, boundary constraints, external forces, and fluid flow [11, 35]. Under the action of material deformations an SPD does not necessarily map to an SPD; i.e., non-affine mappings will deform initial EFDs Ω0 into ODFs Ω which may lack symmetry and/or orthogonality (Fig 7A). In general, deformed collagen ODFs are ellipsoidal when observing neighborhoods at physical scales of ~100 μm. Thus, we approximated a symmetric, orthogonal ODF Ωp that approximated the stretch, orientation, and anisotropy of Ω. This was accomplished by modifying the methods previously outlined by Barocas et al. [11, 40]. First, we considered the SPD tensor P0, which describes the initial collagen EFD. The components of P0 are given by the semiprincipal axis magnitudes βi and orientations ni0 which are related via the spectral decomposition [2] Further, P0 can be used to map radii from a unit sphere ΩI to radii of the undeformed EFD Ω0 (Fig 7A). We next mapped the radii of Ω0 to Ω via the deformation gradient F = ∂x/∂x0. The net transformation from ΩI to Ω was given by the mapping [3] Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. Pseudo-deformation concept map & Monte-Carlo EFD sampling visualization. A. EFD deformation & pseudo-deformation concept map. Vectors originating from the center of a unit sphere ΩI were mapped by P0 to the undeformed EFD Ω0. After deformation by F, the fibrils of Ω0 were stretched and rotated, yielding the deformed ODF Ω. The net mapping from ΩI to Ω was given by Fn. Polar decomposition of Fn yielded Fn = VnRn. An EFD Ωp that closely approximated Ω was found by applying the pseudo-deformation Fp = VnI to ΩI. B. Visualizations of EFD Monte-Carlo sampling are provided for the cases of transverse isotropy (semiprincipal axis lengths β1 = 5, β2 = β3 = 1) and orthotropy (semiprincipal axis lengths β1 = 5, β2 = 3, β3 = 1). Sampled directions were mapped onto the surface of a unit sphere. The ODF was generated by summing points on the surface of the unit sphere that corresponded to faces on an icosahedron. https://doi.org/10.1371/journal.pcbi.1011553.g007 The polar decomposition of Eq [3] gives the SPD Vn and the orthogonal rotational tensor Rn. Previously, Barocas et al. [40] approximated the pseudo-deformed EFD via an analog of the Left-Cauchy deformation tensor given by [4] Here, differs from the Left-Cauchy deformation tensor B since transforms points from a unit sphere to the final state (points xi on the unit sphere surface to x) while B transforms points between the initial and final states (points x0 on the surface of an undeformed EFD to x). However, we were concerned with the physical levels of stretch, whereas was related to the squared stretch of Vn. Thus, we approximated Fn with the SPD Fp that replaced Rn with the identity I in order to remove non-affine deformations: [5] Fp allowed us to map radii from ΩI to the pseudo-deformed EFD Ωp that approximated the true Ω (Fig 7A). The semiprincipal axis magnitudes βi and directions ni of Ωp were calculated from the spectral decomposition of Fp as previously done in Eq [2]. In a finite element analysis, the deformation gradient is evaluated at the element integration points within the finite element domain. However, directed guidance of angiogenesis relies on knowledge of the fiber distribution throughout the element domain. Therefore, EFDs were interpolated across the spatial domain using a geodesic weighted average [33]. The local EFD at a growing vessel tip was interpolated from EFDs stored at the integration points of the element containing the tip where weights of each EFD were determined by the finite element shape functions. The fractional anisotropy of an EFD was calculated from its principal radii of curvature as [6] where std is the standard deviation of the ellipsoid radii βi and rms is the root mean square [60]. Importantly, the fractional anisotropy is only applicable to SPDs since the metric assumes an underlying ellipsoidal distribution. Validation of pseudo-deformed EFDs against true orientation distribution function deformation EFDs were subjected to either uniaxial tension/compression with applied stretch ratios λ ∈ [0.5, 1.5] or shear with applied shear ratios of κ ∈ [0, 0.5] (Supporting Methods E in S1 Text and S5–S8 Figs). The initial EFDs, Ω0, were specified as isotropic, transversely isotropic (1 preferred direction), or planarly anisotropic (2 orthogonal preferred directions of equal magnitude) to evaluate how pseudo-deformation accuracy depended on the underlying fibril distribution. Each point, s0, of an icosahedron mesh containing 10,242 nodes was mapped to the surface of either the true deformed ODF Ω (derived from Fn), or the pseudo-deformed ODF Ωp (derived from Fp) using a custom MATLAB script [61]. Deformed positions were mapped back to the surface of a unit sphere so that they could be assigned to faces of an icosahedron. Points s on the surface of Ω were calculated and mapped to the unit sphere via: [7] Here, the values of the probability density function of Ω were computed by determining the fraction of points s that mapped onto each triangular facet of an icosahedron. Similarly, the values of the probability density function of Ωp were calculated by mapping points sp on the surface of Ωp to the unit sphere icosahedron via: [8] Two quantitative methods were used to assess similarity between deformed and pseudo-deformed ODFs. First, the generalized fractional anisotropy (GFA) for the true and pseudo-deformed distributions via: [9] Notably, Eq [9] simplifies to Eq [6] for ellipsoidal distributions. The second quantitative method involved calculation of the Fisher-Rao distance, a Riemannian metric quantifying the difference in probability distributions between the pseudo-deformed EFD and the true deformation [62, 63]. This metric ranges between 0 and 90°, where 0° indicates a complete overlap in distributions, and 90° indicates no overlap between distributions. The MATLAB scripts used for this analysis are available at github.com/febiosoftware/AngioFE under the documentation section. Random sampling of EFDs The direction that a vessel tip grew in AngioFE depended on the prior direction ψ and the predominant local collagen fibril orientation θ. Previously, the collagen fibril direction was represented as a field of vectors stored at the finite element nodes. However, vessels in vitro and in vivo probe fibrils oriented in many different directions and ultimately decide to grow along a single one of those directions. To simulate this stochastic phenomenon, we introduced a method that randomly samples a single fibril orientation from the local collagen EFD. In our new approach, θ was sampled from the collagen EFDs using a Monte-Carlo approach. For an EFD with an arbitrary basis (i.e., 3D rotation) this was performed by sampling a direction in the global basis before rotating it into the EFD basis. First, the EFD was split via Eigen-decomposition to yield βi, the semiprincipal axes in the global basis. Random direction vectors r were generated inside of a cube bounding the EFD such that each component of r was constrained by ri ∈ [-max(βi), max(βi)]. Points were accepted if they fell within the ellipsoid as determined by the inequality: [10] Visualizations of the 3D sampling method for transverse isotropy and anisotropy are presented in Fig 7B. Interpolation of fibril directions and EFDs Vessel tips can occupy any position within the simulation domain, but data about the local fibril distributions are stored at the local finite element integration points. Thus, we developed a method to interpolate EFDs from the integration points to the position of a vessel tip. Simulations were performed using either a vector-field or an EFD approach for representing collagen fibril ODFs. Vector fields were spatially interpolated via a linear weighted sum of the orientations stored at each integration point in the finite element mesh [20]. The weights were calculated using the finite element shape functions based on the current position of the vessel tip in the element. EFDs were defined at the finite element integration points as SPDs. The local EFD was calculated via a geodesic weighted sum of the integration points to ensure interpolated tensors maintained orthogonality and positive-definiteness [33]. This was accomplished using algorithms 1, 2, and 5 from [33]. FEBio + AngioFE computational model of microvascular growth AngioFE is developed as a plugin for FEBio to simulate microvascular growth and cell-matrix interactions within the FEBio framework [21]. Briefly, microvessel parent fragments are represented as line segments seeded within a finite element mesh. The end of each line is a vessel tip that is capable of growing and contracting the surrounding matrix. Textual and graphic summaries of the FEBio + AngioFE modeling framework are presented in the results (“AngioFE model formulation”, Fig 1). AngioFE version 3.0 and FEBio 3.7 were used for all simulations. AngioFE version 3.0 was developed to address prior limitations in the growth model and to introduce new mechanisms of directional guidance by ECM fibril orientation (Supporting Methods H-K in S1 Text). This software is freely available at github.com/febiosoftware/AngioFE. The builds of FEBio and AngioFE that were used as well as all input files are available in the documentation section of the AngioFE GitHub page. Model generation & inputs. Model geometries, materials, and boundary conditions were prescribed in FEBioStudio. Linear hexahedral elements (Hex8) were used for all simulations. Material behavior in the finite element models was represented with a biphasic (i.e., fluid-solid mixture) viscoelastic constitutive model comprising a mix of a neo-Hookean ground matrix, a continuous collagen fiber distribution, and suspended neo-Hookean neovessels [19, 64]. The moduli of the collagen fibrils varied with strain due to stretch and rotation. Collagen mechanical parameters were derived from prior stress-relaxation experiments of collagen gels [65]. All exposed edges on models were prescribed free-draining boundaries to allow fluid exudation. Model initialization. Model initialization comprised of generating the initial parent microvessels as well as maps of material properties relevant to angiogenesis (ODFs, density). To generate the initial parent microvessels, we first examined confocal images of 1-day-old collagen gel cultures of microvessel fragments prior to sprouting. Probability distributions of the initial vessel lengths and XY plane orientations were extracted from images. Vessels were randomly added to the simulation domain in a manner that recapitulated the image-derived distributions (Supporting Methods H in S1 Text and S12 Fig). Finally, data maps were generated to project the density and collagen ODF to the finite element integration points. Vessel orientation. The new direction each vessel tip grew, ψnew, depended on persistence along the current orientation of the vessel, ψ, and contact guidance along θ [2, 4]. The contact guidance direction was determined by 1) interpolating the local EFD from the integration points, 2) pseudo-deforming the local EFD, and 3) Monte-Carlo sampling of the pseudo-deformed EFD [33]. The direction that each vessel grew, ψnew, was determined by the equation [11] The function R(θ, ψ,α) was formulated as a rotation matrix about the shared orthogonal axis of ψ and θ (Supporting Methods A in S1 Text and S1 Fig). The fibril weight parameter α ∈ [0, 1] scaled the rotation from ψ towards θ, such that α = 0 → ψnew = ψ, and α = 1 → ψnew = θ. Vessel extension rate. The length each tip grew per time step was determined by scaling a time-dependent sigmoidal growth function. Unless otherwise noted, the time-dependent growth function was modeled using the time-derivative of a sigmoidal curve, with parameters derived from prior experiments (Supporting Methods G in S1 Text) [1, 4, 20]. The time-dependent growth function was scaled by the 3D Lorentzian function ν(ρ,FA) (S11 Fig) to introduce a dependence on both the density (ρ) and anisotropy (FA): [12] The total length the tip grew was determined by the product of the time-dependent function and the scalar function. New vessel tips were then grown along ψnew. Parameters for Eq [12] were identified from prior microvessel cultures in collagen hydrogels of varying density and anisotropy [1]. After culture, the microvessel network lengths were measured and the growth corresponding to each matrix configuration was normalized relative to measurements from unaligned 3 mg/mL collagen matrices. These relative growth values were then used to fit parameters aν, bν, cν, ρ0,ν and FA0,ν, in the MATLAB curve fitting toolbox. The parameter dν was added to sustain minimal, non-negative growth. Branching and cell tractions Vessel branches and cell tractions were added after the main vessel growth step. Branches were generated to appear along recently grown vessel segments based on prior implementations in AngioFE (Supporting Methods J in S1 Text). Traction fields were imposed around growing vessel tips that allowed them to deform their surrounding ECM [19, 34]. The traction magnitude was assumed to increase over time as the vascular network grew and sprouts matured. The traction magnitude was also assumed to decrease with increasing matrix density. Additional information is presented in Supporting Methods L-M in S1 Text. Simulations and data collection All simulations were performed 10 times using unique random engine seeds. The random engine generated the initial vessel positions, initial vessel lengths, the length a vessel grew before branching, and random vectors used to sample directions from EFDs. Results from the 10 simulations were averaged. Simulation parameters and justifications are presented in S1 Table. Comparison of discrete and continuous fibril distribution approaches A parametric study was performed on the collagen fibril orientation weight (α) to determine appropriate values to simulate microvessel guidance in a range of matrix anisotropies and densities. This study was performed using either discrete fibril vector fields or continuous EFDs to represent collagen ODFs. The results for each approach were compared to prior experiments of microvascular growth in low, moderate, and high anisotropy matrices at low (3.0 mg/mL) or high (4.0 mg/mL) density [1]. Branching was not simulated for this study to simplify analysis. Vectors for the vector field models were stored at finite element integration points and were randomly sampled from the initial EFDs during the initialization step. First, the initial parent fragments were generated by AngioFE’s initialization step. Initial microvessel orientations were prescribed so that the orientations and lengths of fragments matched experimental measurements prior to sprouting [1]. To do this, vessel segments from confocal images acquired on day 0 of culture were projected onto the XY and XZ planes and the ratio of the resulting ODF semiprincipal axes was calculated. We found that microvessel segments were initially isotropic in the XY plane; thus, the semiprincipal axes for the X and Y direction during initialization are set to β1p = β2p = 1.0. The semiprincipal axis for the Z direction was varied between 0.1–0.9 until the vessel XZ distribution matched the experimental values. The collagen fibril weight during initialization was set to 0.3. The collagen fibril semiprincipal axes and collagen fibril orientation weight, α, were determined in a similar manner as done for the initialization step. Microvascular growth velocity for each condition was prescribed based on prior experimental measures. The EFD semiprincipal axes β1 and β2 for each simulation are derived from second harmonic generation (SHG) images of collagen gels aligned by stretch during polymerization [1]. The ratio of β1 and β2 was determined by fitting ellipsoids to the XY ODF in the images. Next, the weight α was varied between 0 and 1. The resulting microvessel network was projected onto the XY planes to calculate ODFs for guidance in the horizontal plane. Values of α were chosen for each condition and approach and then the parameter β3 was varied between 0.1–1.0 to approximate the collagen orientation in the vertical direction. All simulations for this study were performed using a 1 x 1 x 1 mm3 cube. Trilinear hexahedral finite elements were used to discretize the domain with a size of 0.1 x 0.1 x 0.1 mm3 each. Nodes along the X, Y, and Z-axes were fixed in lateral directions to keep the model centered. Visualizations with similar depths of field as confocal Z projections from prior experiments were obtained by using XY plane cuts. In the model, fibril orientation and growing neovessels partake in a 2-way feedback loop known as dynamic reciprocity [66]. Neovessel orientation is determined by the fibril orientation, and then the fibril orientation is displaced by tractions of growing neovessels. To determine the influence of vessel traction on this feedback system, the above simulations were repeated with 3 values of traction amplitude (Supporting Methods M in S1 Text). Mesh sensitivity study A mesh sensitivity study was performed to determine how mesh size affected directional guidance. Simulations from the parametric study were repeated with 8, 64, 512, 1000, or 8000 total elements, with element side lengths of 500, 250, 125, 100, and 50 μm respectively. We evaluated the effect of mesh refinement on the ratio of the semiprincipal axes of the ellipsoids fit to ODFs of microvessel growth in the XY plane. Simulation of anisotropy gradients during healing We examined how microvessels respond to gradients of matrix anisotropy. The simulation domain contained a middle region that was relatively isotropic for a baseline simulation or contained a gradient of increasing anisotropy (positive) or decreasing anisotropy (negative) across the simulation domain. The magnitude of the semiprincipal axis in the X direction β1 increased linearly across the middle region from 1.3–5.0. The magnitudes in the Y and Z directions were fixed at β2 = 1.0 and β3 = 0.5 respectively. Microvessels were seeded on the proximal (left) end of the simulation domain. The flanking end geometries were 0.4 x 2.0 x 0.5 mm3 and the center region geometry was 1.0 x 2.0 x 0.5 mm3. All elements were specified as 0.1 x 0.1 x 0.1 mm3 FEBio hex8 elements. Microvessel cultures were simulated for 12 days using a growth velocity rule that supported linear growth after the first few days of culture until the local vessel volume fraction exceeded a threshold wthresh (Supporting Methods K in S1 Text). Simulations of neovascular tumor invasion The second set of predictive simulations examined the role of fibril alignment and density in preventing or facilitating vessel crossing at a structural tissue interface. The properties of the structural interface varied based on tumor associated collagen signatures (TACS) and our prior in vitro interface models [4, 13]. We examined how different TACSs affect the ability of microvessels to cross a structural interface between a tumor and its peripheral stroma. The simulation domain was divided into the tumor (1 x 0.8 x 0.6 mm3), the periphery (1 x 0.8 x 0.6 mm3), and an 80 μm thick interface separating them (1 x 0.08 x 0.6 mm3). The interface geometry was based on prior simulations of microvessel crossing at a tissue interface [4]. Elements were specified as 0.1 x 0.1 x 0.1 mm3 FEBio hex8 elements in the stroma and the tumor, and as 0.1 x 0.04 x 0.01 mm3 FEBio hex8 elements in the interface. Six scenarios were simulated with modifications to the structure of the interface: a baseline simulation (no change in density or anisotropy), TACS-1 (high-density interface), TACS-2 (fibril alignment along interface), TACS-3 (fibril alignment across interface), TACS-1+2 in combination, and TACS-1+3 in combination. The resulting length of microvessels in the tumor after 10 days was quantified to compare how each TACS facilitated microvascular invasion. Statistical analysis Statistical analyses were performed in Origin 2020b (OriginLab, Northampton, MA). One-way analysis of variance (ANOVA) with a Sidakholm means comparison post hoc test was used for assessing statistical significance. Box plot data are centered at the median with edges at 25% and 75% of the data. Whiskers are at 1.5 x IQR. Significance was measured with a family-wise α set to 0.05. Ethics statement The animal study was reviewed and approved by University of Utah Institutional Animal Care and Use Committee (IACUC). In-vitro pre-alignment of collagen hydrogels and microvessel culture Experimental data for this study were derived from previously published studies in which microvessels were cultured at 3 levels of ECM anisotropy (low, medium, and high anisotropy) as well as 2 levels of ECM density (low or high) [1]. The methods used to characterize ECM architecture and the morphology of microvascular networks after 10 days of culture are summarized below. Type-I collagen hydrogels were cast at low (3 mg/mL) or high (4 mg/mL) density in rectangular chambers flanked with steel mesh anchors at the ends before incubation at 37°C (rat-tail tendon, Corning Inc., Corning, NY). Steel mesh anchors were then stretched between 0 and 20% along the long axis at 25 minutes after casting in order to pre-align the ECM collagen fibrils. Gels were cut free from mesh anchors after 1 day then transferred to phosphate buffered saline or cell-culture media depending on the experiment. The ECM collagen microstructure was visualized via second harmonic generation (SHG) imaging (Fig 2). SHG images were then processed and orientation distribution functions (ODFs) were extracted via fast Fourier transforms using a custom MATLAB script (MathWorks, Natick, MA). Ellipsoids were fit to the ODFs, which allowed the calculation of the anisotropy and ellipsoid semiprincipal axes associated with each level of collagen fibril alignment and density. Intact microvessel fragments were isolated from male Sprague Dawley retired breeder rats using an established protocol [59]. Microvessel fragments were suspended in liquid collagen at low or high density before casting gels in rectangular chambers. Vascularized gels were pre-strained to achieve low, moderate, or high anisotropy, resulting in 6 unique combinations of matrix anisotropy and density. These gels were similarly cut free from the steel mesh anchors after 1 day. Gels were fixed with 2% paraformaldehyde (VWR, Radnor, PA) after 10 days of culture, stained for endothelial cells (Isolectin; ThermoFisher), and then imaged via confocal microscopy. Confocal volumes were processed in AMIRA (ThermoFisher, Waltham, MA) which allowed the extraction of microvessel network ODFs and vascularity (mm vessels / mm3 image volume). A set of vascularized gels were also fixed, stained, and imaged on Day 1 to provide baseline measures of vascularity and vessel polarization that would later be used to generate the initial simulation conditions. Pseudo-deformation and interpolation of ellipsoidal fibril distribution structure tensors We assumed that EFDs approximate collagen ODFs for spatial scales on the order of 100 μm based on prior collagen structural imaging [3, 20]. Further, we assumed the individual fibrils in an ODF deform independent of each other (i.e., no interaction during deformation). These assumptions allowed us to approximate 3D collagen distributions with 6-term symmetric positive-definite tensors (SPDs) which are inherently ellipsoidal. In general, a fiber distribution that is ellipsoidal in the reference (unloaded) configuration of a material does not remain ellipsoidal upon finite deformation, since the semi-principal axes of the ellipsoid may not remain orthogonal. In order to maintain the symmetry properties of the EFDs after deformation, we assumed that EFDs remain ellipsoidal but experience affine deformation (finite rotation and stretching) of their principal radii of curvature. This was necessary to conveniently update and sample EFDs, as well as to spatially interpolate local matrix ODFs and visualize the deformed ODFs. The collagen fibril matrix undergoes deformation due to cellular tractions, boundary constraints, external forces, and fluid flow [11, 35]. Under the action of material deformations an SPD does not necessarily map to an SPD; i.e., non-affine mappings will deform initial EFDs Ω0 into ODFs Ω which may lack symmetry and/or orthogonality (Fig 7A). In general, deformed collagen ODFs are ellipsoidal when observing neighborhoods at physical scales of ~100 μm. Thus, we approximated a symmetric, orthogonal ODF Ωp that approximated the stretch, orientation, and anisotropy of Ω. This was accomplished by modifying the methods previously outlined by Barocas et al. [11, 40]. First, we considered the SPD tensor P0, which describes the initial collagen EFD. The components of P0 are given by the semiprincipal axis magnitudes βi and orientations ni0 which are related via the spectral decomposition [2] Further, P0 can be used to map radii from a unit sphere ΩI to radii of the undeformed EFD Ω0 (Fig 7A). We next mapped the radii of Ω0 to Ω via the deformation gradient F = ∂x/∂x0. The net transformation from ΩI to Ω was given by the mapping [3] Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. Pseudo-deformation concept map & Monte-Carlo EFD sampling visualization. A. EFD deformation & pseudo-deformation concept map. Vectors originating from the center of a unit sphere ΩI were mapped by P0 to the undeformed EFD Ω0. After deformation by F, the fibrils of Ω0 were stretched and rotated, yielding the deformed ODF Ω. The net mapping from ΩI to Ω was given by Fn. Polar decomposition of Fn yielded Fn = VnRn. An EFD Ωp that closely approximated Ω was found by applying the pseudo-deformation Fp = VnI to ΩI. B. Visualizations of EFD Monte-Carlo sampling are provided for the cases of transverse isotropy (semiprincipal axis lengths β1 = 5, β2 = β3 = 1) and orthotropy (semiprincipal axis lengths β1 = 5, β2 = 3, β3 = 1). Sampled directions were mapped onto the surface of a unit sphere. The ODF was generated by summing points on the surface of the unit sphere that corresponded to faces on an icosahedron. https://doi.org/10.1371/journal.pcbi.1011553.g007 The polar decomposition of Eq [3] gives the SPD Vn and the orthogonal rotational tensor Rn. Previously, Barocas et al. [40] approximated the pseudo-deformed EFD via an analog of the Left-Cauchy deformation tensor given by [4] Here, differs from the Left-Cauchy deformation tensor B since transforms points from a unit sphere to the final state (points xi on the unit sphere surface to x) while B transforms points between the initial and final states (points x0 on the surface of an undeformed EFD to x). However, we were concerned with the physical levels of stretch, whereas was related to the squared stretch of Vn. Thus, we approximated Fn with the SPD Fp that replaced Rn with the identity I in order to remove non-affine deformations: [5] Fp allowed us to map radii from ΩI to the pseudo-deformed EFD Ωp that approximated the true Ω (Fig 7A). The semiprincipal axis magnitudes βi and directions ni of Ωp were calculated from the spectral decomposition of Fp as previously done in Eq [2]. In a finite element analysis, the deformation gradient is evaluated at the element integration points within the finite element domain. However, directed guidance of angiogenesis relies on knowledge of the fiber distribution throughout the element domain. Therefore, EFDs were interpolated across the spatial domain using a geodesic weighted average [33]. The local EFD at a growing vessel tip was interpolated from EFDs stored at the integration points of the element containing the tip where weights of each EFD were determined by the finite element shape functions. The fractional anisotropy of an EFD was calculated from its principal radii of curvature as [6] where std is the standard deviation of the ellipsoid radii βi and rms is the root mean square [60]. Importantly, the fractional anisotropy is only applicable to SPDs since the metric assumes an underlying ellipsoidal distribution. Validation of pseudo-deformed EFDs against true orientation distribution function deformation EFDs were subjected to either uniaxial tension/compression with applied stretch ratios λ ∈ [0.5, 1.5] or shear with applied shear ratios of κ ∈ [0, 0.5] (Supporting Methods E in S1 Text and S5–S8 Figs). The initial EFDs, Ω0, were specified as isotropic, transversely isotropic (1 preferred direction), or planarly anisotropic (2 orthogonal preferred directions of equal magnitude) to evaluate how pseudo-deformation accuracy depended on the underlying fibril distribution. Each point, s0, of an icosahedron mesh containing 10,242 nodes was mapped to the surface of either the true deformed ODF Ω (derived from Fn), or the pseudo-deformed ODF Ωp (derived from Fp) using a custom MATLAB script [61]. Deformed positions were mapped back to the surface of a unit sphere so that they could be assigned to faces of an icosahedron. Points s on the surface of Ω were calculated and mapped to the unit sphere via: [7] Here, the values of the probability density function of Ω were computed by determining the fraction of points s that mapped onto each triangular facet of an icosahedron. Similarly, the values of the probability density function of Ωp were calculated by mapping points sp on the surface of Ωp to the unit sphere icosahedron via: [8] Two quantitative methods were used to assess similarity between deformed and pseudo-deformed ODFs. First, the generalized fractional anisotropy (GFA) for the true and pseudo-deformed distributions via: [9] Notably, Eq [9] simplifies to Eq [6] for ellipsoidal distributions. The second quantitative method involved calculation of the Fisher-Rao distance, a Riemannian metric quantifying the difference in probability distributions between the pseudo-deformed EFD and the true deformation [62, 63]. This metric ranges between 0 and 90°, where 0° indicates a complete overlap in distributions, and 90° indicates no overlap between distributions. The MATLAB scripts used for this analysis are available at github.com/febiosoftware/AngioFE under the documentation section. Random sampling of EFDs The direction that a vessel tip grew in AngioFE depended on the prior direction ψ and the predominant local collagen fibril orientation θ. Previously, the collagen fibril direction was represented as a field of vectors stored at the finite element nodes. However, vessels in vitro and in vivo probe fibrils oriented in many different directions and ultimately decide to grow along a single one of those directions. To simulate this stochastic phenomenon, we introduced a method that randomly samples a single fibril orientation from the local collagen EFD. In our new approach, θ was sampled from the collagen EFDs using a Monte-Carlo approach. For an EFD with an arbitrary basis (i.e., 3D rotation) this was performed by sampling a direction in the global basis before rotating it into the EFD basis. First, the EFD was split via Eigen-decomposition to yield βi, the semiprincipal axes in the global basis. Random direction vectors r were generated inside of a cube bounding the EFD such that each component of r was constrained by ri ∈ [-max(βi), max(βi)]. Points were accepted if they fell within the ellipsoid as determined by the inequality: [10] Visualizations of the 3D sampling method for transverse isotropy and anisotropy are presented in Fig 7B. Interpolation of fibril directions and EFDs Vessel tips can occupy any position within the simulation domain, but data about the local fibril distributions are stored at the local finite element integration points. Thus, we developed a method to interpolate EFDs from the integration points to the position of a vessel tip. Simulations were performed using either a vector-field or an EFD approach for representing collagen fibril ODFs. Vector fields were spatially interpolated via a linear weighted sum of the orientations stored at each integration point in the finite element mesh [20]. The weights were calculated using the finite element shape functions based on the current position of the vessel tip in the element. EFDs were defined at the finite element integration points as SPDs. The local EFD was calculated via a geodesic weighted sum of the integration points to ensure interpolated tensors maintained orthogonality and positive-definiteness [33]. This was accomplished using algorithms 1, 2, and 5 from [33]. FEBio + AngioFE computational model of microvascular growth AngioFE is developed as a plugin for FEBio to simulate microvascular growth and cell-matrix interactions within the FEBio framework [21]. Briefly, microvessel parent fragments are represented as line segments seeded within a finite element mesh. The end of each line is a vessel tip that is capable of growing and contracting the surrounding matrix. Textual and graphic summaries of the FEBio + AngioFE modeling framework are presented in the results (“AngioFE model formulation”, Fig 1). AngioFE version 3.0 and FEBio 3.7 were used for all simulations. AngioFE version 3.0 was developed to address prior limitations in the growth model and to introduce new mechanisms of directional guidance by ECM fibril orientation (Supporting Methods H-K in S1 Text). This software is freely available at github.com/febiosoftware/AngioFE. The builds of FEBio and AngioFE that were used as well as all input files are available in the documentation section of the AngioFE GitHub page. Model generation & inputs. Model geometries, materials, and boundary conditions were prescribed in FEBioStudio. Linear hexahedral elements (Hex8) were used for all simulations. Material behavior in the finite element models was represented with a biphasic (i.e., fluid-solid mixture) viscoelastic constitutive model comprising a mix of a neo-Hookean ground matrix, a continuous collagen fiber distribution, and suspended neo-Hookean neovessels [19, 64]. The moduli of the collagen fibrils varied with strain due to stretch and rotation. Collagen mechanical parameters were derived from prior stress-relaxation experiments of collagen gels [65]. All exposed edges on models were prescribed free-draining boundaries to allow fluid exudation. Model initialization. Model initialization comprised of generating the initial parent microvessels as well as maps of material properties relevant to angiogenesis (ODFs, density). To generate the initial parent microvessels, we first examined confocal images of 1-day-old collagen gel cultures of microvessel fragments prior to sprouting. Probability distributions of the initial vessel lengths and XY plane orientations were extracted from images. Vessels were randomly added to the simulation domain in a manner that recapitulated the image-derived distributions (Supporting Methods H in S1 Text and S12 Fig). Finally, data maps were generated to project the density and collagen ODF to the finite element integration points. Vessel orientation. The new direction each vessel tip grew, ψnew, depended on persistence along the current orientation of the vessel, ψ, and contact guidance along θ [2, 4]. The contact guidance direction was determined by 1) interpolating the local EFD from the integration points, 2) pseudo-deforming the local EFD, and 3) Monte-Carlo sampling of the pseudo-deformed EFD [33]. The direction that each vessel grew, ψnew, was determined by the equation [11] The function R(θ, ψ,α) was formulated as a rotation matrix about the shared orthogonal axis of ψ and θ (Supporting Methods A in S1 Text and S1 Fig). The fibril weight parameter α ∈ [0, 1] scaled the rotation from ψ towards θ, such that α = 0 → ψnew = ψ, and α = 1 → ψnew = θ. Vessel extension rate. The length each tip grew per time step was determined by scaling a time-dependent sigmoidal growth function. Unless otherwise noted, the time-dependent growth function was modeled using the time-derivative of a sigmoidal curve, with parameters derived from prior experiments (Supporting Methods G in S1 Text) [1, 4, 20]. The time-dependent growth function was scaled by the 3D Lorentzian function ν(ρ,FA) (S11 Fig) to introduce a dependence on both the density (ρ) and anisotropy (FA): [12] The total length the tip grew was determined by the product of the time-dependent function and the scalar function. New vessel tips were then grown along ψnew. Parameters for Eq [12] were identified from prior microvessel cultures in collagen hydrogels of varying density and anisotropy [1]. After culture, the microvessel network lengths were measured and the growth corresponding to each matrix configuration was normalized relative to measurements from unaligned 3 mg/mL collagen matrices. These relative growth values were then used to fit parameters aν, bν, cν, ρ0,ν and FA0,ν, in the MATLAB curve fitting toolbox. The parameter dν was added to sustain minimal, non-negative growth. Model generation & inputs. Model geometries, materials, and boundary conditions were prescribed in FEBioStudio. Linear hexahedral elements (Hex8) were used for all simulations. Material behavior in the finite element models was represented with a biphasic (i.e., fluid-solid mixture) viscoelastic constitutive model comprising a mix of a neo-Hookean ground matrix, a continuous collagen fiber distribution, and suspended neo-Hookean neovessels [19, 64]. The moduli of the collagen fibrils varied with strain due to stretch and rotation. Collagen mechanical parameters were derived from prior stress-relaxation experiments of collagen gels [65]. All exposed edges on models were prescribed free-draining boundaries to allow fluid exudation. Model initialization. Model initialization comprised of generating the initial parent microvessels as well as maps of material properties relevant to angiogenesis (ODFs, density). To generate the initial parent microvessels, we first examined confocal images of 1-day-old collagen gel cultures of microvessel fragments prior to sprouting. Probability distributions of the initial vessel lengths and XY plane orientations were extracted from images. Vessels were randomly added to the simulation domain in a manner that recapitulated the image-derived distributions (Supporting Methods H in S1 Text and S12 Fig). Finally, data maps were generated to project the density and collagen ODF to the finite element integration points. Vessel orientation. The new direction each vessel tip grew, ψnew, depended on persistence along the current orientation of the vessel, ψ, and contact guidance along θ [2, 4]. The contact guidance direction was determined by 1) interpolating the local EFD from the integration points, 2) pseudo-deforming the local EFD, and 3) Monte-Carlo sampling of the pseudo-deformed EFD [33]. The direction that each vessel grew, ψnew, was determined by the equation [11] The function R(θ, ψ,α) was formulated as a rotation matrix about the shared orthogonal axis of ψ and θ (Supporting Methods A in S1 Text and S1 Fig). The fibril weight parameter α ∈ [0, 1] scaled the rotation from ψ towards θ, such that α = 0 → ψnew = ψ, and α = 1 → ψnew = θ. Vessel extension rate. The length each tip grew per time step was determined by scaling a time-dependent sigmoidal growth function. Unless otherwise noted, the time-dependent growth function was modeled using the time-derivative of a sigmoidal curve, with parameters derived from prior experiments (Supporting Methods G in S1 Text) [1, 4, 20]. The time-dependent growth function was scaled by the 3D Lorentzian function ν(ρ,FA) (S11 Fig) to introduce a dependence on both the density (ρ) and anisotropy (FA): [12] The total length the tip grew was determined by the product of the time-dependent function and the scalar function. New vessel tips were then grown along ψnew. Parameters for Eq [12] were identified from prior microvessel cultures in collagen hydrogels of varying density and anisotropy [1]. After culture, the microvessel network lengths were measured and the growth corresponding to each matrix configuration was normalized relative to measurements from unaligned 3 mg/mL collagen matrices. These relative growth values were then used to fit parameters aν, bν, cν, ρ0,ν and FA0,ν, in the MATLAB curve fitting toolbox. The parameter dν was added to sustain minimal, non-negative growth. Branching and cell tractions Vessel branches and cell tractions were added after the main vessel growth step. Branches were generated to appear along recently grown vessel segments based on prior implementations in AngioFE (Supporting Methods J in S1 Text). Traction fields were imposed around growing vessel tips that allowed them to deform their surrounding ECM [19, 34]. The traction magnitude was assumed to increase over time as the vascular network grew and sprouts matured. The traction magnitude was also assumed to decrease with increasing matrix density. Additional information is presented in Supporting Methods L-M in S1 Text. Simulations and data collection All simulations were performed 10 times using unique random engine seeds. The random engine generated the initial vessel positions, initial vessel lengths, the length a vessel grew before branching, and random vectors used to sample directions from EFDs. Results from the 10 simulations were averaged. Simulation parameters and justifications are presented in S1 Table. Comparison of discrete and continuous fibril distribution approaches A parametric study was performed on the collagen fibril orientation weight (α) to determine appropriate values to simulate microvessel guidance in a range of matrix anisotropies and densities. This study was performed using either discrete fibril vector fields or continuous EFDs to represent collagen ODFs. The results for each approach were compared to prior experiments of microvascular growth in low, moderate, and high anisotropy matrices at low (3.0 mg/mL) or high (4.0 mg/mL) density [1]. Branching was not simulated for this study to simplify analysis. Vectors for the vector field models were stored at finite element integration points and were randomly sampled from the initial EFDs during the initialization step. First, the initial parent fragments were generated by AngioFE’s initialization step. Initial microvessel orientations were prescribed so that the orientations and lengths of fragments matched experimental measurements prior to sprouting [1]. To do this, vessel segments from confocal images acquired on day 0 of culture were projected onto the XY and XZ planes and the ratio of the resulting ODF semiprincipal axes was calculated. We found that microvessel segments were initially isotropic in the XY plane; thus, the semiprincipal axes for the X and Y direction during initialization are set to β1p = β2p = 1.0. The semiprincipal axis for the Z direction was varied between 0.1–0.9 until the vessel XZ distribution matched the experimental values. The collagen fibril weight during initialization was set to 0.3. The collagen fibril semiprincipal axes and collagen fibril orientation weight, α, were determined in a similar manner as done for the initialization step. Microvascular growth velocity for each condition was prescribed based on prior experimental measures. The EFD semiprincipal axes β1 and β2 for each simulation are derived from second harmonic generation (SHG) images of collagen gels aligned by stretch during polymerization [1]. The ratio of β1 and β2 was determined by fitting ellipsoids to the XY ODF in the images. Next, the weight α was varied between 0 and 1. The resulting microvessel network was projected onto the XY planes to calculate ODFs for guidance in the horizontal plane. Values of α were chosen for each condition and approach and then the parameter β3 was varied between 0.1–1.0 to approximate the collagen orientation in the vertical direction. All simulations for this study were performed using a 1 x 1 x 1 mm3 cube. Trilinear hexahedral finite elements were used to discretize the domain with a size of 0.1 x 0.1 x 0.1 mm3 each. Nodes along the X, Y, and Z-axes were fixed in lateral directions to keep the model centered. Visualizations with similar depths of field as confocal Z projections from prior experiments were obtained by using XY plane cuts. In the model, fibril orientation and growing neovessels partake in a 2-way feedback loop known as dynamic reciprocity [66]. Neovessel orientation is determined by the fibril orientation, and then the fibril orientation is displaced by tractions of growing neovessels. To determine the influence of vessel traction on this feedback system, the above simulations were repeated with 3 values of traction amplitude (Supporting Methods M in S1 Text). Mesh sensitivity study A mesh sensitivity study was performed to determine how mesh size affected directional guidance. Simulations from the parametric study were repeated with 8, 64, 512, 1000, or 8000 total elements, with element side lengths of 500, 250, 125, 100, and 50 μm respectively. We evaluated the effect of mesh refinement on the ratio of the semiprincipal axes of the ellipsoids fit to ODFs of microvessel growth in the XY plane. Simulation of anisotropy gradients during healing We examined how microvessels respond to gradients of matrix anisotropy. The simulation domain contained a middle region that was relatively isotropic for a baseline simulation or contained a gradient of increasing anisotropy (positive) or decreasing anisotropy (negative) across the simulation domain. The magnitude of the semiprincipal axis in the X direction β1 increased linearly across the middle region from 1.3–5.0. The magnitudes in the Y and Z directions were fixed at β2 = 1.0 and β3 = 0.5 respectively. Microvessels were seeded on the proximal (left) end of the simulation domain. The flanking end geometries were 0.4 x 2.0 x 0.5 mm3 and the center region geometry was 1.0 x 2.0 x 0.5 mm3. All elements were specified as 0.1 x 0.1 x 0.1 mm3 FEBio hex8 elements. Microvessel cultures were simulated for 12 days using a growth velocity rule that supported linear growth after the first few days of culture until the local vessel volume fraction exceeded a threshold wthresh (Supporting Methods K in S1 Text). Simulations of neovascular tumor invasion The second set of predictive simulations examined the role of fibril alignment and density in preventing or facilitating vessel crossing at a structural tissue interface. The properties of the structural interface varied based on tumor associated collagen signatures (TACS) and our prior in vitro interface models [4, 13]. We examined how different TACSs affect the ability of microvessels to cross a structural interface between a tumor and its peripheral stroma. The simulation domain was divided into the tumor (1 x 0.8 x 0.6 mm3), the periphery (1 x 0.8 x 0.6 mm3), and an 80 μm thick interface separating them (1 x 0.08 x 0.6 mm3). The interface geometry was based on prior simulations of microvessel crossing at a tissue interface [4]. Elements were specified as 0.1 x 0.1 x 0.1 mm3 FEBio hex8 elements in the stroma and the tumor, and as 0.1 x 0.04 x 0.01 mm3 FEBio hex8 elements in the interface. Six scenarios were simulated with modifications to the structure of the interface: a baseline simulation (no change in density or anisotropy), TACS-1 (high-density interface), TACS-2 (fibril alignment along interface), TACS-3 (fibril alignment across interface), TACS-1+2 in combination, and TACS-1+3 in combination. The resulting length of microvessels in the tumor after 10 days was quantified to compare how each TACS facilitated microvascular invasion. Statistical analysis Statistical analyses were performed in Origin 2020b (OriginLab, Northampton, MA). One-way analysis of variance (ANOVA) with a Sidakholm means comparison post hoc test was used for assessing statistical significance. Box plot data are centered at the median with edges at 25% and 75% of the data. Whiskers are at 1.5 x IQR. Significance was measured with a family-wise α set to 0.05. Supporting information S1 Text. Supporting methods including algorithmic details and additional studies. The supporting information contains additional explanations and details of algorithmic updates to AngioFE. We have also included parameter and sensitivity studies that we used to guide finite element meshing and parameter selection. Finally, a table is provided with the coefficients used for various parameters in our simulations. https://doi.org/10.1371/journal.pcbi.1011553.s001 (DOCX) S1 Table. Model coefficients and justification. The coefficients used for various parameters in our simulations are presented in this table. Justifications and any additional assumptions are presented as well. https://doi.org/10.1371/journal.pcbi.1011553.s002 (DOCX) S1 Fig. Visualization of Equation S2. A vessel tip is centered at the origin. The new direction that a vessel grows depends on the previous direction (ψ) and the local collagen direction (θ). A coplanar rotation from ψ to θ occurs about the axis u = ψ × θ. The new direction ψnew is found by partially rotating ψ towards θ. The fraction of the whole rotation is determined by α ∈ [0, 1] where α = 0 performs no rotation, and α = 1 performs the entire rotation. https://doi.org/10.1371/journal.pcbi.1011553.s003 (TIF) S2 Fig. Sensitivity study–Collagen fibril orientation weight (α). Top: Low-density collagen results for simulations where the fibril weight (α) was varied from 0.0–1.0. Here, 0 indicates vessel grow along the persistence direction while 1 indicates that vessels grow entirely along the sampled fibril direction. Plotted is the ratio of the major and minor semiprincipal axes for ODFs of microvascular orientation. Discrete fibril approaches and continuous EFD approaches to representing matrix fibril distributions were compared to prior experimental measures (target) for simulations of growth in low, medium, or high anisotropy collagen. Bottom: Similar results were generated from simulations with high-density matrix collagen. https://doi.org/10.1371/journal.pcbi.1011553.s004 (TIF) S3 Fig. Angiogenesis in high-density collagen of varying anisotropy. A. Z projections of experimental (confocal) and simulated microvascular networks grown in low, medium, or high anisotropy collagen. Depth of field = 200 μm. B. XY-plane ODFs for each experimental or simulated case. Overlap is seen between the confocal and EFD field ODFs for all cases. The presented results are for 4 mg/mL collagen. https://doi.org/10.1371/journal.pcbi.1011553.s005 (TIF) S4 Fig. Mesh refinement study. Directional guidance was generally unaffected by mesh refinement. A 1x1x1 mm3 cube was meshed between 8 and 8,000 FEBio hex8 elements. https://doi.org/10.1371/journal.pcbi.1011553.s006 (TIF) S5 Fig. Verification of pseudo-deformation–uniaxial tension. Three different initial EFDs were tested with the EFD designated by P0 (Supporting Methods E in S1 Text). P0 is an EFD representing a uniform, uniaxially aligned, or planarly isotropic (transversely anisotropic) EFD. These EFDs are rotated by R, a rotation matrix of 45° about the X, Y, and Z-axes. The rotation ensures that deformations do not occur along the EFD principal directions (in which cases the pseudo-deformation and “true deformation” are identical during tension and some shear cases). The resulting GFA for each are compared (deformation solid, pseudo-deformation dashed). The Fisher-Rao distance between ODFs was also calculated for each. The resulting ODFs are visualized via heat maps. Heat maps correspond to the highest level of stretch tested. https://doi.org/10.1371/journal.pcbi.1011553.s007 (TIF) S6 Fig. Verification of pseudo-deformation–biaxial tension. Three different initial EFDs were tested with the EFD designated by P0 (Supporting Methods E in S1 Text). P0 is an EFD representing a uniform, uniaxially aligned, or planarly isotropic (transversely anisotropic) EFD. These EFDs are rotated by R, a rotation matrix of 45° about the X, Y, and Z-axes. The rotation ensures that deformations do not occur along the EFD principal directions (in which cases the pseudo-deformation and “true deformation” are identical during tension and some shear cases). The resulting GFA for each are compared (deformation solid, pseudo-deformation dashed). The Fisher-Rao distance between ODFs was also calculated for each. The resulting ODFs are visualized via heat maps. Heat maps correspond to the highest level of stretch tested. https://doi.org/10.1371/journal.pcbi.1011553.s008 (TIF) S7 Fig. Verification of pseudo-deformation–simple shear. Three different initial EFDs were tested with the EFD designated by P0 (Supporting Methods E in S1 Text). P0 is an EFD representing a uniform, uniaxially aligned, or planarly isotropic (transversely anisotropic) EFD. These EFDs are rotated by R, a rotation matrix of 45° about the X, Y, and Z-axes. The rotation ensures that deformations do not occur along the EFD principal directions (in which cases the pseudo-deformation and “true deformation” are identical during tension and some shear cases). The resulting GFA for each are compared (deformation solid, pseudo-deformation dashed). The Fisher-Rao distance between ODFs was also calculated for each. The resulting ODFs are visualized via heat maps. Heat maps correspond to the highest level of stretch tested. https://doi.org/10.1371/journal.pcbi.1011553.s009 (TIF) S8 Fig. Verification of pseudo-deformation–pure shear. Three different initial EFDs were tested with the EFD designated by P0 (Supporting Methods E in S1 Text). P0 is an EFD representing a uniform, uniaxially aligned, or planarly isotropic (transversely anisotropic) EFD. These EFDs are rotated by R, a rotation matrix of 45° about the X, Y, and Z-axes. The rotation ensures that deformations do not occur along the EFD principal directions (in which cases the pseudo-deformation and “true deformation” are identical during tension and some shear cases). The resulting GFA for each are compared (deformation solid, pseudo-deformation dashed). The Fisher-Rao distance between ODFs was also calculated for each. The resulting ODFs are visualized via heat maps. Heat maps correspond to the highest level of stretch tested. https://doi.org/10.1371/journal.pcbi.1011553.s010 (TIF) S9 Fig. Vector field simulations of anisotropy gradients. A-C. Top-down view of the initial fibril directions for each element. Directions were randomly sampled from the element’s collagen EFD. Local matrix alignment is indicated by ellipsoidal glyphs above each representative image with the color indicating the anisotropy (blue: low; red: high). D-F. Microvessels were seeded on the left end of a rectilinear domain. Growth was either simulated in a relatively isotropic matrix (baseline) or a matrix characterized by a positive or negative anisotropy gradient. Microvessels for all models failed to reach the far region on the right. EFD simulations, on the other hand, predicted vascularization of the far region for both gradient models. G. Differences were also observed for growth in the middle region. Here, only a negative gradient resulted in increased vascularization, differing from the EFD simulations where both gradients resulted in increased growth in the middle region. (*: p<0.001 w.r.t baseline; @: p<0.001 pairwise comparison, 1 way ANOVA with Sidakholm post hoc). https://doi.org/10.1371/journal.pcbi.1011553.s011 (TIF) S10 Fig. Vector field simulations of TACS and neovessel invasion. Microvessels were simulated to originate in a tumor periphery (inset, orange). Microvessels grow towards the tumor (inset, magenta) but must cross a tissue interface (inset, yellow). Alignment and density of the interface is varied to mimic various tumor associated collagen signatures (TACS). The interface structure contained either an isotropic fibril ODF (circle) or aligned ODFs (ellipses) running either perpendicular to the tumor or towards the tumor. The interface density was either 3 mg/mL or 5 mg/mL. Representative Z projections of the interface and tumor region are presented at the bottom. High interface density generally reduced the length of microvessels that crossed into the tumor. Fibril alignment towards the tumor facilitated crossing while alignment perpendicular to the tumor deflected vessels or trapped them within the interface. Unlike the EFD simulations, fibril alignment towards the tumor did not nullify the effects of increased matrix density for TACS-1+3. ***: p<0.001. 1 Way ANOVA with Sidakholm post hoc test. https://doi.org/10.1371/journal.pcbi.1011553.s012 (TIF) S11 Fig. Visualization of ν(ρ,FA) (Eq 12). Eq [12] is a function designed to scale vessel growth based on matrix density and anisotropy. Scaling decreases velocity with increased matrix density (ρ) and increases with anisotropy (FA). Left: contour level plot with 0.2 level increments. Right: 3D contour plot with experimental points overlaid. https://doi.org/10.1371/journal.pcbi.1011553.s013 (TIF) S12 Fig. Sampling of initial vessel lengths based on confocal images. Top: The initial length of each discrete microvessel was sampled from a rational function P(L0) that had been fit to experimental measures of initial microvessel length. Bottom: Left: Representative confocal Z projection of microvessels at the beginning of culture. Right: Representative Z projection of initial microvessels in simulations. Depth of field for both images is 200 μm. https://doi.org/10.1371/journal.pcbi.1011553.s014 (TIF) S13 Fig. Schematic representation of simulated branching. The direction of a new branch is determined by the direction of the parent microvessel, the zenith angle, and the azimuth angle. The zenith angle is the angle describing the elevation of the branch direction from the parent direction. The azimuth angle represents a rotation about the axis of the parent direction. https://doi.org/10.1371/journal.pcbi.1011553.s015 (TIF) S14 Fig. Emergent vessel tortuosity. Z projections of microvascular networks grown in low (left) and high (right) anisotropy collagen using a continuous EFD method with the fibril weight α set to 0.9 (minimal persistence). Depth of field = 200 μm. High fibril weight leads to high curvature that resembles pathological angiogenesis. https://doi.org/10.1371/journal.pcbi.1011553.s016 (TIF) S15 Fig. Sensitivity study–Collagen fibril orientation weight (α) and stress magnitude (aamp). Low-density results: The sensitivity of the model predictions to the neovessel sprout stress magnitude (aamp) was studied across the full range of possible collagen orientation weights (α). The ratio of the 2 greatest ODF semiprincipal axes was plotted for each condition on the y-axis. The results for 3 mg/mL simulations are displayed with results from discrete fibril vector simulations on the left and results from EFD simulations on the right. No differences are observable between the stress-free (aamp = 0.0 μPa) and baseline stress (aamp = 3.72 μPa). In contrast, the high-stress case led to increased polarization of the vascular network for a number of cases, as indicated by increased values on the y-axis. These effects are visibly more pronounced for the discrete simulations. https://doi.org/10.1371/journal.pcbi.1011553.s017 (TIF) S16 Fig. Sensitivity study–Collagen fibril orientation weight (α) and stress magnitude (aamp). High-density results: The sensitivity of the model predictions to the neovessel sprout stress magnitude (aamp) was studied across the full range of possible collagen orientation weights (α). The ratio of the 2 greatest ODF semiprincipal axes was plotted for each condition on the y-axis. The results for 4 mg/mL simulations are displayed with results from discrete fibril vector simulations on the left and results from EFD simulations on the right. No differences are observable between the stress-free (aamp = 0.0 μPa) and baseline stress (aamp = 3.72 μPa). In contrast, the high-stress case led to increased polarization of the vascular network for a number of cases, as indicated by increased values on the y-axis. These effects are visibly more pronounced for the discrete simulations. https://doi.org/10.1371/journal.pcbi.1011553.s018 (TIF) Acknowledgments We thank James B. Hoying and Hannah A. Strobel from Advanced Solutions Life Sciences LLC for their input and guidance for the predictive simulations.

Honasoge, Kailas Shankar;Karagöz, Zeynep;Goult, Benjamin T.;Wolfenson, Haguy;LaPointe, Vanessa L. S.;Carlier, Aurélie

doi: 10.1371/journal.pcbi.1011500pmid: 37801464

Introduction Direct contact between cells and the extracellular matrix (ECM) through adhesions is a crucial component of multicellular organisms [1]. Integrins are transmembrane ECM receptor proteins that assemble as non-covalently bonded heterodimers with α and β subunits [2]. The integrin ectodomain binds ECM ligands while the cytoplasmic tail is indirectly linked to the actomyosin cytoskeleton of the cell forming a supramolecular assembly or ‘clutch’ [3–6]. This indirect link consists of a dynamic network of over 200 proteins, collectively termed the ‘integrin adhesome’ [7, 8]. Central to integrin function are the dynamics and balance of extra- and intracellular forces [9] which drive the force-dependent evolution of the integrin adhesion complexes (IACs) [7] leading to changes in their size and composition. In vitro studies have shown that adhesion assembly is a multi-step process where integrins are first activated by binding to intracellular adaptor protein molecules such as talin [10–12] and/or to an ECM ligand [13]. Once activated, integrins cluster at the site of adhesion, independent of force and substrate rigidity, to form nascent adhesions (NAs) [14–16]. Then, NAs either undergo disassembly or force-dependent maturation by the recruitment of other adaptor proteins such as vinculin, to form focal adhesions (FA) (Fig 1A) [17, 18]. These three major steps of adhesion assembly also overlap in time and are not strictly sequential. Understanding interactions between key proteins of the integrin adhesome and force generation will provide valuable insight into cell-ECM interactions, with consequences for developmental biology. A better understanding of cellular responses and signalling can potentially highlight new therapeutic targets, and improve engineered substrates to better mimic biological tissue, thus advancing regenerative medicine. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Overview of the processes modelled in this study. (A)—Integrins bind to talin and vinculin in a precomplexation step, then form a small cluster, termed ‘seed’. Seeds can dimerise to form larger clusters, termed ‘clusts’. Actin filaments pull on talin and vinculin causing cryptic vinculin-binding sites on talin to be exposed, promoting more vinculin recruitment. This chain can then break at the integrin–ligand catch-slip bond or the talin–actin slip bond (black boxes). See Table 1 for a detailed description of the terminology. (B)—Overview of reactions in the model. Int, tal, and vinc refer to concentrations of integrins, talin and vinculin respectively. Black rectangle encloses the reinforcement reactions (expanded further in Fig 1C). Grey arrows represent clust formation reactions. Red arrows represent actin binding reactions. Dotted arrows represent force-dependent reactions—blue dotted: reinforcement, black dotted: actin unbinding. Dashed arrows represent adhesion disassembly reactions, black dashed: talin refolding, purple dashed: cluster breakdown. Yellow lightning bolts indicate rates that undergo signal-dependent rate modification (SDRM), dark green solid hourglasses represent rates that undergo time-dependent rate modification (TDRM). The rate constants undergoing signal-dependent modifications are driven to zero after ∼158 s leaving active only the lower part of the model, enclosed in the blue box, representing adhesions that will undergo further maturation. (C)—Talin and vinculin are modelled as Hookean springs (also see Fig A in S1 Appendix). In this model, to capture the process of reinforcement, a maximum of three vinculin binding events occur sequentially (blue arrows) at different points along the talin rod, thereby increasing the stiffness of individual integrin–talin–vinculin spring systems. Clustering is modelled as an increase in the number of integrin–talin–vinculin spring systems in parallel (grey arrows). This figure was created using BioRender.com. https://doi.org/10.1371/journal.pcbi.1011500.g001 In addition to questions pertaining to adhesion (dis)assembly, adhesion maturation is also a complex process influenced by the mechanical properties of the substrate [19, 20], force-dependent conformational changes [21–25], different catch and slip bond strengths [5, 26, 27] and intracellular forces [20, 28, 29]. How changes in these mechanical factors affect the biochemical composition of adhesions, and which factors determine the decision to mature a particular NA remains unclear. Given the constraints and challenges of experimental studies, computational modelling can be a valuable resource. Many computational models of cell–ECM interactions have been developed since the first molecular-clutch model by Chan and Odde (2008) [4] that explained filopodial traction dynamics on compliant substrates [30–34]. Elosegui-Artola and colleagues have extended the Chan and Odde model to include adhesion reinforcement through increases in integrin density [35] and multiple integrin types [36]. Integrin-based Rho signalling [34] and reversible cross-links in the actin filament network [37] have also been included in previous studies by other groups. More recently, Venturini and Saez (2023) [38] have developed an extensive multi-scale model of molecular clutch-driven adhesion mechanics. All these models explore adhesion formation, growth, and the influence of substrate stiffness and actomyosin forces on traction forces, but they are discrete models that simulate a relatively small number of individual particles. They also do not account for the increase in clutch stiffness after the recruitment of vinculin and do not consider the disassembly processes to be dynamic and active. In addition, these models give little information about the changes to the overall biochemical composition of adhesions in the cell during the process of maturation of NAs to FAs. In this study, we developed a new model using ordinary differential equations (ODEs) to describe the biochemical composition of cell–ECM adhesions over time based on mechanical properties like substrate stiffness, adaptor protein stiffness, actomyosin-generated forces, and bond characteristics. Using our model, we studied the fraction of NAs that have the potential to become mature FAs under different mechanical circumstances. Overall, the results from this study shed light on the mechanotransduction mechanisms underlying adhesion maturation and disassembly. This model also provides a reliable starting point to model the larger focal adhesome with over 200 identified proteins [7]. Methods Differential equation model We developed an ODE-based model that captures changes in the biochemical composition of cell–ECM adhesions based on the mechanical properties of the environment and intracellular proteins. Below we shortly describe the particular phases of the adhesion maturation process—formation of integrin-talin-vinculin precomplexes, formation and growth of precomplex clusters, actin binding and unbinding, adhesion reinforcement with vinculin, and adhesion breakdown—and how they are modelled (Fig 1B provides an overview of all the species in the system and their interactions). For each subprocess (in bold), we give a brief explanation and also label the corresponding terms in the differential equations (Eqs 1–17). Unless otherwise mentioned, reactions are reversible with forward rate constants having ‘f’ in the subscript and reverse rate constants having ‘r’. Reactions follow mass-action kinetics unless mentioned otherwise. When referring to concentrations in the text, they are written between square brackets (e.g., [S3a]), and when referred to as a species they are written as is (e.g., S3a). A detailed explanation of the reactions and parameters can be found in Text A and Table A in S1 Appendix. Table 1 provides an overview of the terminology used throughout the manuscript. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Terminology used in this manuscript. https://doi.org/10.1371/journal.pcbi.1011500.t001 Adhesion assembly starts with integrin activation. In this study, we model α5β1 integrins and assume they are activated. We also assume that the ligand spacing on the substrate is sufficiently close for integrin clusters to form. Pcomp formation and dissociation. Activated integrins, [Int], bind to talin, [tal], and vinculin, [vinc], forming pre-complexes [Pcomp], a necessary step for adhesion maturation [18] (Text A in S1 Appendix). Seed formation and dimerisation. Up to 50 IAPCs cluster independent of substrate rigidity and tension to form NAs [16]. Here, as a simplification, the growth of clusters to the maximal size (50 IAPCs) occurs in two stages—first, a small cluster of 25 IAPCs, termed ‘seed’ (denoted by ‘Sx’ (Fig 1C, Table 1) is formed, and a second stage where seeds dimerise, forming a large cluster with 50 IAPCs, termed ‘clust’ (denoted by ‘Cx’ (Fig 1C, Table 1, Text A in S1 Appendix)). Here, x ∈ {1,2,3} denotes the number of vinculin molecules in the individual IAPCs. When a clutch is AUB, the stretched talin is likely to refold [24, 39]. We assume this makes it very unlikely for AUB seeds of mid- and high- order (S2, S3) to dimerise. AB seeds of all orders (S1a, S2a, S3a) can dimerise to form AB clusts (C1a, C2a, and C3a) (grey arrows in Fig 1B). Actin binding/unbinding. Actin-unbound (AUB) seeds and clusts can bind to actin through the actin-binding sites on talin and vinculin, giving AB seeds and clusts (denoted by ‘Sxa’ and ‘Cxa’ respectively, (Text A in S1 Appendix)) that can stretch to different extents (based on the value of x) and hence transmit varying magnitudes of force (Fig 1C). While the baseline actin-binding rate is kact for all actin-binding reactions, signalling molecules such as focal adhesion kinase (FAK), Src and ERK play a role in adhesion turnover, and their inhibition leads to more maturation [40]. To implement a similar mechanism to stop indefinite adhesion formation and maturation, we introduce in the model a signal-dependent rate modification (SDRM) (see ‘Signal dependent rate modification (SDRM)’ for details). Depending on the force on the integrin-ligand (catch-slip) bonds [41] and the talin-actin (slip) bonds [5], the force-chain between the cell and the substrate can break at either of these bonds. We capture these phenomena through force-dependent (and consequently substrate rigidity-dependent) actin unbinding rates (black dotted arrows in Fig 1B, also see Text A in S1 Appendix for details). These bonds may also rupture due to random thermodynamic fluctuations before the clutches can reach their maximum force-carrying capacity, reducing the total force exerted by the clutches. To account for the spontaneous clutch unbinding in a continuous framework, we introduce a time-dependent rate modification (TDRM) (see ‘Force-dependent actin-unbinding and time-dependent rate modification (TDRM)’ for details). Reinforcement. The AB clutches form a mechanical link between the substrate and the actin cytoskeleton, enabling force transmission and protein unfolding. Up to eleven cryptic vinculin-binding sites (VBS) are uncovered when talin is stretched and unfolded [42, 43], leading to reinforcement by vinculin recruitment. The rate of talin unfolding and reinforcement occurring depends on the force experienced by the AB clutch, and increases with increasing force, similar to the Bell model [24, 44, 45] (for more details see section ‘Adhesion reinforcement rates’). In this simplified model, two vinculin-reinforcement events are considered, one from low- to mid-order clutches (S1a to S2a and C1a to C2a), followed by one from mid- to high-order clutches (S2a to S3a and C2a to C3a) (see Text A in S1 Appendix for a detailed explanation). Thus, in this model, the talin rod can be bound to at least 1 and at most 3 vinculin molecules. With this framework, we consider low-order clutches (S1, S1a, C1, C1a (Fig 1C)) to represent NAs, and mid- and high-order clutches (S2, S2a, C2, C2a, S3, S3a, C3, C3a, (Fig 1C)) to represent more mature stages of adhesions, indicative of the fraction of NAs that mature into FAs (see Section ‘NA formation is rigidity- and force-independent’ for reasoning). Reinforcement is modelled as a single-step reaction where simultaneous recruitment of 25 (for seeds) and 50 (for clusts) vinculin molecules respectively occurs (indicated by blue dotted arrows in Fig 1B). The order of these reactions with respect to vinculin, however, was chosen to be 2 (Eqs 3, 8, 9 and 10), to account for the effects of possible intermediate stages in the reactions. Refolding and breakdown. In the absence of sufficient force, adhesions disassemble because of mechanical and chemical signals [46]. Here, we model two parallel processes of disassembly, namely 1) talin refolding (black dashed arrows in Fig 1B) leading to the loss of vinculin from, and weakening of, clutches, and 2) breakdown of AUB clusters into seeds and Pcomp (purple dashed arrows in Fig 1B) leading to reduced force carrying capacity of the adhesions (Text A in S1 Appendix). These are irreversible reactions. To account for the mechanical aspects of adhesion assembly, the substrate–integrin–adaptor protein system was formulated as a system of Hookean springs. When clutches bind to the actin filaments, they provide resistance to the motion of actin filaments until bond rupture, caused either randomly or because the catch/slip bond force threshold is reached. We assume that the force exerted by myosin II motors on actin filaments is balanced by the drag force arising due to the viscosity of the cytoplasm. Thus, in the absence of integrin-mediated forces on actin filaments, they move with a constant retrograde velocity (see Text A in S1 Appendix). As a continuous ODE framework is used, we consider the same actin retrograde velocity for all clutches. The force on a clutch depends on its stiffness and extension (according to Hooke’s law). The stiffness of a clutch depends on the number of constituent IAPCs and the number of vinculin molecules in each IAPC (equivalent spring constants are calculated, see Text A in S1 Appendix). The total force exerted on the actin filament network thus depends on the number of AB clutches of each type and their stiffnesses. Since we use a continuum approach to account for the abundance of each species, we discretise concentrations of AB clutches by assuming a volume of 1 μm3 to calculate the total force (see Text A in S1 Appendix). Together, the above-described processes result in the following set of differential equations: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) Baseline parameter values and rate constants can be found in Table A in S1 Appendix. We refer the reader to Text A in S1 Appendix for detailed descriptions of all reactions in the model and the underlying reasoning. Below, we highlight the novel methodological approaches (signal- and time-dependent rate modification (SDRM and TDRM)), and provide brief explanations of a few mathematical formulations and assumptions that are used in this model. Signal dependent rate modification (SDRM) Nascent adhesions (NAs) form in large numbers and most are disassembled within a time scale of a few minutes [17, 47]. As the cell protrudes, the distance between the cell membrane and the NAs increases, and actin depolymerization rates are higher away from the cell membrane [48]. Thus, numerous NAs may be supported near the cell membrane but in the absence of this scaffold, many NAs disassemble. Various signal cascades also regulate adhesion disassembly. Signalling molecules such as focal adhesion kinase (FAK), Src and ERK kinases play a role in adhesion turnover, and their inhibition leads to more maturation [40]. However, most studies have investigated the effects of signalling molecules on the turnover of FAs and not NAs [49, 50], and the exact mechanical or chemical triggers for NA disassembly remain elusive [51]. NA assembly at the cell front, maturation, and disassembly away from the leading edge occur constantly due to above-mentioned mechanisms. In this study, we focus on one cycle of NA formation and investigated the differences in NA maturation on different substrate stiffnesses. To implement a mechanism to stop indefinite adhesion formation and maturation, it is hypothesized that there exists a signal molecule of which a minimum concentration, signalthresh, is required for new NA formation and low-order AUB clutches (S1, C1) to bind actin for maturation. The concentration [signal] of this molecule is initially high and decreases at an arbitrary rate following Michaelis-Menten kinetics given by: (17) Initial estimates for values of the maximum velocity and the Michaelis constant were based on those reported in the literature for FAK Tyr-397 phosphorylation in the presence of ATP [52] but were adjusted such that the concentration of [signal] reaches the signalthresh in 58 s, which is the approximate duration of the NA assembly phase as measured in experiments [4]. When the [signal] falls below signalthresh, it is analogous to a signalling pathway being activated, and particular reaction rate constants (k1f, k1r, k2f, k2r, k3f, k3r, k4f, k9f, k18f, k20f, k21f, k22f, indicated by yellow lighting bolts in Fig 1B) are modified as detailed in Text A in S1 Appendix. Thus, the rate of decay of [signal] determines the amount of time available before NA disassembly starts in which adhesion maturation can occur. Note that the model behaviour does not change if we assume the opposite i.e., [signal] increases over time and there is an upper limit for its concentration beyond which actin binding does not occur (Fig C.B in S1 Appendix). Force-dependent actin-unbinding and time-dependent rate modification (TDRM) A cell–ECM force chain is broken if either the integrin–ligand (catch-slip) bond [41] or the talin–actin (slip) bond [5] ruptures as a result of reaching the respective force thresholds or due to random thermodynamic fluctuations. This is described in detail in Text A in S1 Appendix. To capture the combined dynamics of the catch-slip and slip bonds, as well as the effect of random bond ruptures, the actin unbinding rates (dark green hourglasses in Fig 1B) are defined as follows: (18) where Fclutch is the force on an individual IAPC in the clutch (see Text A in S1 Appendix where Fclutch is described in detail). The first and second terms, where A and C are scaling factors and b and d control the force-dependency, describe the integrin–ligand catch-slip bond, and the third term describes the talin-actin slip bond where is bond rupture threshold for a given clutch type (i ∈ {1, 2, 3}, refer Table A in S1 Appendix) and is the unloaded dissociation rate. kTDRM is the time-dependent rate modification (TDRM) factor that is required to qualitatively account for the reduction in the total force caused by spontaneous clutch unbinding events. kTDRM is given by: (19) where tclutch is the number of simulated time-steps that a clutch remains actin-bound, dt is the time-step and ksens is a parameter that determines the magnitude of the influence. In short, this definition captures the decreasing likelihood of an AB clutch remaining AB for long periods of time. It has been shown that integrin-ligand bonds undergo cyclic mechanical reinforcement (CMR) leading to longer lifetimes [53]. This implies that on soft substrates where the force-loading rate is low, integrin-ligand bonds experience fewer force cycles in a given time period compared to stiff substrates and consequently are more likely to break on soft substrates. Previous studies model CMR with an increased bond-dissociation rate at low forces [35, 38]. Here, TDRM can capture the effects of these phenomena. TDRM affects softer substrates, where clutches take longer to reach their force thresholds, more than stiff substrates. When an AB clutch experiences a force equal to its force threshold, it unbinds from actin and becomes an AUB clutch. Thus, the concentration of the AB clutch is set to 0, and the concentration of its AUB counterpart is increased by the same amount. Adhesion reinforcement rates The rate at which the talin rod unfolds increases with applied force and has been described in previous studies by the Bell model [24, 44, 45]. Here, the Bell formulation was adapted such that the rate increases exponentially with force until the vinculin binding force threshold Fvb is reached, beyond which it remains constant. The rate is given by: (20) where is the rate of unloaded talin unfolding, kUF is a parameter controlling force-dependence, Fclutch is the force experienced by an individual IAPC in the clutch, and is the vinculin binding force threshold, with i ∈ {1, 2} corresponding to the first and second vinculin binding steps. Here, and [24, 42]. Vinculin binding is assumed to occur instantaneously once the VBS is uncovered [45, 54], and hence the rates of reinforcement were determined based on the force-dependent unfolding kinetics of talin as observed in single-molecule experiments using magnetic tweezers [42]. A detailed description of the method used for curve-fitting can be found in Text A in S1 Appendix. Range of substrate rigidities The stiffness of the ECM, generally measured in terms of Young’s modulus, varies across three orders of magnitude, between 1–3 kPa in the brain, 23–42 kPa in muscular tissue, 1000 kPa-860 MPa in blood vessels, to 15–40 GPa in bone [55]. In this study, however, we consider spring constants, which can be converted to Young’s moduli based on a few assumptions as detailed in [34, 36] and summarized in Text A in S1 Appendix. Accordingly, we restrict the scope of the main investigations and predictions to a stiffness range equivalent to 0.1–130 kPa and only qualitatively discuss the results up to 1000 kPa. This is in line with previous computational and experimental studies which have also primarily studied adhesion mechanobiology in similar ranges of substrate rigidities [4, 36]. Local sensitivity analysis As the model included many parameters whose values were either estimated or adapted to fit experimental data, a local sensitivity analysis was performed. The range of values tested for each parameter was the baseline value (Table A in S1 Appendix) ±10% and ±20%. To quantify the influence, two different metrics were used as outcomes, namely 1) maturation fraction: the concentration of integrins in mid- and high-order AB and AUB clutches ([S2]+[S2a]+[S3]+[S3a]+[C2]+[C2a]+[C3]+[C3a]) at equilibrium (the last time point), and 2) the optimal stiffness: the substrate stiffness with the lowest mean actin retrograde velocity. Outcome 1 represents the total fraction of integrins in the system that made it beyond the initial force-independent stage of adhesion formation, indicative of the fraction of NAs that mature into FAs. Outcome 2 represents an overall influence on the system as it quantifies the mean force exerted during the length of the simulation for a range of substrate stiffnesses. In addition, cells are known to be able to tune their mechanosensitive ranges to adapt to their environments, an aspect on which outcome 2 can shed light. As different parameters may have different levels of influence based on the substrate stiffness, the sensitivity of outcome 1 to each parameter was evaluated for four substrate stiffnesses (ksub = 0.1, 1, 10, 100 pN/nm). Parameter sensitivity analysis was performed on 21 parameters (Text B, Fig I and Fig J in S1 Appendix), and the ones with the highest influence or of key importance are presented in the main text. The parameter sensitivity for a parameter p for an outcome i was calculated as follows: (21) where Outcomei(p + Δp) represents the value of the outcome metric with the changed parameter value, Outcomei(p) is the value of the outcome metric with the baseline parameter value, and Δp and p are the change in the parameter and the baseline parameter value respectively. Initial conditions The initial concentrations of integrins, talin and vinculin were assumed to be equal and set to 1 μM for simplicity, and that of all other species were set to 0. Vinculin was assumed to be abundantly available within the cytoplasm and thus modelled at a constant concentration of 1 μM throughout the simulation. Simulation steps All simulations were run for 600 s. Euler’s forward integration method was used to solve the ODEs with a time step dt of 5 ms as used in previous computational studies [4, 35]. For mass conservation steps, see Text A in S1 Appendix. The steps of integration and the order of updates (Fig 2) of the different aspects of the model are as follows: Force-dependent rate constants are calculated. In particular, the following rates are evaluated at the current force: First reinforcement rates: (k7f, k12f) using Eq 20. Second reinforcement rates: (k8f, k13f) using Eq 20. Signal-dependent rate modification: (k1f, k1r, k2f, k2r, k3f, k3r, k4f, k9f, k18f, k20f, k21f, k22f) are updated as detailed in Text A in S1 Appendix. Catch-slip bond rates with time-dependent rate modification: (k4r, k5r, k6r, k9r, k10r, k11r) using Eqs 18 and 19. Concentrations are updated based on current rate constants by solving the differential equations listed above. The slip bond threshold is checked for each clutch type If the slip bond threshold is reached, the force on the clutch is reset to 0. The concentration of the actin-bound form of the clutch is converted to the actin-unbound form. The total force exerted by actin-bound clutches is calculated based on discretised concentrations using eq. S53. Retrograde velocity, vretro, is updated based on the current total force in the system using eq. S49. All substrate-clutch spring systems are extended by an amount vretro ⋅ dt. Force on each clutch is updated using eq. S51. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Schematic showing the flowchart for simulation and force quantification. https://doi.org/10.1371/journal.pcbi.1011500.g002 Differential equation model We developed an ODE-based model that captures changes in the biochemical composition of cell–ECM adhesions based on the mechanical properties of the environment and intracellular proteins. Below we shortly describe the particular phases of the adhesion maturation process—formation of integrin-talin-vinculin precomplexes, formation and growth of precomplex clusters, actin binding and unbinding, adhesion reinforcement with vinculin, and adhesion breakdown—and how they are modelled (Fig 1B provides an overview of all the species in the system and their interactions). For each subprocess (in bold), we give a brief explanation and also label the corresponding terms in the differential equations (Eqs 1–17). Unless otherwise mentioned, reactions are reversible with forward rate constants having ‘f’ in the subscript and reverse rate constants having ‘r’. Reactions follow mass-action kinetics unless mentioned otherwise. When referring to concentrations in the text, they are written between square brackets (e.g., [S3a]), and when referred to as a species they are written as is (e.g., S3a). A detailed explanation of the reactions and parameters can be found in Text A and Table A in S1 Appendix. Table 1 provides an overview of the terminology used throughout the manuscript. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Terminology used in this manuscript. https://doi.org/10.1371/journal.pcbi.1011500.t001 Adhesion assembly starts with integrin activation. In this study, we model α5β1 integrins and assume they are activated. We also assume that the ligand spacing on the substrate is sufficiently close for integrin clusters to form. Pcomp formation and dissociation. Activated integrins, [Int], bind to talin, [tal], and vinculin, [vinc], forming pre-complexes [Pcomp], a necessary step for adhesion maturation [18] (Text A in S1 Appendix). Seed formation and dimerisation. Up to 50 IAPCs cluster independent of substrate rigidity and tension to form NAs [16]. Here, as a simplification, the growth of clusters to the maximal size (50 IAPCs) occurs in two stages—first, a small cluster of 25 IAPCs, termed ‘seed’ (denoted by ‘Sx’ (Fig 1C, Table 1) is formed, and a second stage where seeds dimerise, forming a large cluster with 50 IAPCs, termed ‘clust’ (denoted by ‘Cx’ (Fig 1C, Table 1, Text A in S1 Appendix)). Here, x ∈ {1,2,3} denotes the number of vinculin molecules in the individual IAPCs. When a clutch is AUB, the stretched talin is likely to refold [24, 39]. We assume this makes it very unlikely for AUB seeds of mid- and high- order (S2, S3) to dimerise. AB seeds of all orders (S1a, S2a, S3a) can dimerise to form AB clusts (C1a, C2a, and C3a) (grey arrows in Fig 1B). Actin binding/unbinding. Actin-unbound (AUB) seeds and clusts can bind to actin through the actin-binding sites on talin and vinculin, giving AB seeds and clusts (denoted by ‘Sxa’ and ‘Cxa’ respectively, (Text A in S1 Appendix)) that can stretch to different extents (based on the value of x) and hence transmit varying magnitudes of force (Fig 1C). While the baseline actin-binding rate is kact for all actin-binding reactions, signalling molecules such as focal adhesion kinase (FAK), Src and ERK play a role in adhesion turnover, and their inhibition leads to more maturation [40]. To implement a similar mechanism to stop indefinite adhesion formation and maturation, we introduce in the model a signal-dependent rate modification (SDRM) (see ‘Signal dependent rate modification (SDRM)’ for details). Depending on the force on the integrin-ligand (catch-slip) bonds [41] and the talin-actin (slip) bonds [5], the force-chain between the cell and the substrate can break at either of these bonds. We capture these phenomena through force-dependent (and consequently substrate rigidity-dependent) actin unbinding rates (black dotted arrows in Fig 1B, also see Text A in S1 Appendix for details). These bonds may also rupture due to random thermodynamic fluctuations before the clutches can reach their maximum force-carrying capacity, reducing the total force exerted by the clutches. To account for the spontaneous clutch unbinding in a continuous framework, we introduce a time-dependent rate modification (TDRM) (see ‘Force-dependent actin-unbinding and time-dependent rate modification (TDRM)’ for details). Reinforcement. The AB clutches form a mechanical link between the substrate and the actin cytoskeleton, enabling force transmission and protein unfolding. Up to eleven cryptic vinculin-binding sites (VBS) are uncovered when talin is stretched and unfolded [42, 43], leading to reinforcement by vinculin recruitment. The rate of talin unfolding and reinforcement occurring depends on the force experienced by the AB clutch, and increases with increasing force, similar to the Bell model [24, 44, 45] (for more details see section ‘Adhesion reinforcement rates’). In this simplified model, two vinculin-reinforcement events are considered, one from low- to mid-order clutches (S1a to S2a and C1a to C2a), followed by one from mid- to high-order clutches (S2a to S3a and C2a to C3a) (see Text A in S1 Appendix for a detailed explanation). Thus, in this model, the talin rod can be bound to at least 1 and at most 3 vinculin molecules. With this framework, we consider low-order clutches (S1, S1a, C1, C1a (Fig 1C)) to represent NAs, and mid- and high-order clutches (S2, S2a, C2, C2a, S3, S3a, C3, C3a, (Fig 1C)) to represent more mature stages of adhesions, indicative of the fraction of NAs that mature into FAs (see Section ‘NA formation is rigidity- and force-independent’ for reasoning). Reinforcement is modelled as a single-step reaction where simultaneous recruitment of 25 (for seeds) and 50 (for clusts) vinculin molecules respectively occurs (indicated by blue dotted arrows in Fig 1B). The order of these reactions with respect to vinculin, however, was chosen to be 2 (Eqs 3, 8, 9 and 10), to account for the effects of possible intermediate stages in the reactions. Refolding and breakdown. In the absence of sufficient force, adhesions disassemble because of mechanical and chemical signals [46]. Here, we model two parallel processes of disassembly, namely 1) talin refolding (black dashed arrows in Fig 1B) leading to the loss of vinculin from, and weakening of, clutches, and 2) breakdown of AUB clusters into seeds and Pcomp (purple dashed arrows in Fig 1B) leading to reduced force carrying capacity of the adhesions (Text A in S1 Appendix). These are irreversible reactions. To account for the mechanical aspects of adhesion assembly, the substrate–integrin–adaptor protein system was formulated as a system of Hookean springs. When clutches bind to the actin filaments, they provide resistance to the motion of actin filaments until bond rupture, caused either randomly or because the catch/slip bond force threshold is reached. We assume that the force exerted by myosin II motors on actin filaments is balanced by the drag force arising due to the viscosity of the cytoplasm. Thus, in the absence of integrin-mediated forces on actin filaments, they move with a constant retrograde velocity (see Text A in S1 Appendix). As a continuous ODE framework is used, we consider the same actin retrograde velocity for all clutches. The force on a clutch depends on its stiffness and extension (according to Hooke’s law). The stiffness of a clutch depends on the number of constituent IAPCs and the number of vinculin molecules in each IAPC (equivalent spring constants are calculated, see Text A in S1 Appendix). The total force exerted on the actin filament network thus depends on the number of AB clutches of each type and their stiffnesses. Since we use a continuum approach to account for the abundance of each species, we discretise concentrations of AB clutches by assuming a volume of 1 μm3 to calculate the total force (see Text A in S1 Appendix). Together, the above-described processes result in the following set of differential equations: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) Baseline parameter values and rate constants can be found in Table A in S1 Appendix. We refer the reader to Text A in S1 Appendix for detailed descriptions of all reactions in the model and the underlying reasoning. Below, we highlight the novel methodological approaches (signal- and time-dependent rate modification (SDRM and TDRM)), and provide brief explanations of a few mathematical formulations and assumptions that are used in this model. Pcomp formation and dissociation. Activated integrins, [Int], bind to talin, [tal], and vinculin, [vinc], forming pre-complexes [Pcomp], a necessary step for adhesion maturation [18] (Text A in S1 Appendix). Seed formation and dimerisation. Up to 50 IAPCs cluster independent of substrate rigidity and tension to form NAs [16]. Here, as a simplification, the growth of clusters to the maximal size (50 IAPCs) occurs in two stages—first, a small cluster of 25 IAPCs, termed ‘seed’ (denoted by ‘Sx’ (Fig 1C, Table 1) is formed, and a second stage where seeds dimerise, forming a large cluster with 50 IAPCs, termed ‘clust’ (denoted by ‘Cx’ (Fig 1C, Table 1, Text A in S1 Appendix)). Here, x ∈ {1,2,3} denotes the number of vinculin molecules in the individual IAPCs. When a clutch is AUB, the stretched talin is likely to refold [24, 39]. We assume this makes it very unlikely for AUB seeds of mid- and high- order (S2, S3) to dimerise. AB seeds of all orders (S1a, S2a, S3a) can dimerise to form AB clusts (C1a, C2a, and C3a) (grey arrows in Fig 1B). Actin binding/unbinding. Actin-unbound (AUB) seeds and clusts can bind to actin through the actin-binding sites on talin and vinculin, giving AB seeds and clusts (denoted by ‘Sxa’ and ‘Cxa’ respectively, (Text A in S1 Appendix)) that can stretch to different extents (based on the value of x) and hence transmit varying magnitudes of force (Fig 1C). While the baseline actin-binding rate is kact for all actin-binding reactions, signalling molecules such as focal adhesion kinase (FAK), Src and ERK play a role in adhesion turnover, and their inhibition leads to more maturation [40]. To implement a similar mechanism to stop indefinite adhesion formation and maturation, we introduce in the model a signal-dependent rate modification (SDRM) (see ‘Signal dependent rate modification (SDRM)’ for details). Depending on the force on the integrin-ligand (catch-slip) bonds [41] and the talin-actin (slip) bonds [5], the force-chain between the cell and the substrate can break at either of these bonds. We capture these phenomena through force-dependent (and consequently substrate rigidity-dependent) actin unbinding rates (black dotted arrows in Fig 1B, also see Text A in S1 Appendix for details). These bonds may also rupture due to random thermodynamic fluctuations before the clutches can reach their maximum force-carrying capacity, reducing the total force exerted by the clutches. To account for the spontaneous clutch unbinding in a continuous framework, we introduce a time-dependent rate modification (TDRM) (see ‘Force-dependent actin-unbinding and time-dependent rate modification (TDRM)’ for details). Reinforcement. The AB clutches form a mechanical link between the substrate and the actin cytoskeleton, enabling force transmission and protein unfolding. Up to eleven cryptic vinculin-binding sites (VBS) are uncovered when talin is stretched and unfolded [42, 43], leading to reinforcement by vinculin recruitment. The rate of talin unfolding and reinforcement occurring depends on the force experienced by the AB clutch, and increases with increasing force, similar to the Bell model [24, 44, 45] (for more details see section ‘Adhesion reinforcement rates’). In this simplified model, two vinculin-reinforcement events are considered, one from low- to mid-order clutches (S1a to S2a and C1a to C2a), followed by one from mid- to high-order clutches (S2a to S3a and C2a to C3a) (see Text A in S1 Appendix for a detailed explanation). Thus, in this model, the talin rod can be bound to at least 1 and at most 3 vinculin molecules. With this framework, we consider low-order clutches (S1, S1a, C1, C1a (Fig 1C)) to represent NAs, and mid- and high-order clutches (S2, S2a, C2, C2a, S3, S3a, C3, C3a, (Fig 1C)) to represent more mature stages of adhesions, indicative of the fraction of NAs that mature into FAs (see Section ‘NA formation is rigidity- and force-independent’ for reasoning). Reinforcement is modelled as a single-step reaction where simultaneous recruitment of 25 (for seeds) and 50 (for clusts) vinculin molecules respectively occurs (indicated by blue dotted arrows in Fig 1B). The order of these reactions with respect to vinculin, however, was chosen to be 2 (Eqs 3, 8, 9 and 10), to account for the effects of possible intermediate stages in the reactions. Refolding and breakdown. In the absence of sufficient force, adhesions disassemble because of mechanical and chemical signals [46]. Here, we model two parallel processes of disassembly, namely 1) talin refolding (black dashed arrows in Fig 1B) leading to the loss of vinculin from, and weakening of, clutches, and 2) breakdown of AUB clusters into seeds and Pcomp (purple dashed arrows in Fig 1B) leading to reduced force carrying capacity of the adhesions (Text A in S1 Appendix). These are irreversible reactions. To account for the mechanical aspects of adhesion assembly, the substrate–integrin–adaptor protein system was formulated as a system of Hookean springs. When clutches bind to the actin filaments, they provide resistance to the motion of actin filaments until bond rupture, caused either randomly or because the catch/slip bond force threshold is reached. We assume that the force exerted by myosin II motors on actin filaments is balanced by the drag force arising due to the viscosity of the cytoplasm. Thus, in the absence of integrin-mediated forces on actin filaments, they move with a constant retrograde velocity (see Text A in S1 Appendix). As a continuous ODE framework is used, we consider the same actin retrograde velocity for all clutches. The force on a clutch depends on its stiffness and extension (according to Hooke’s law). The stiffness of a clutch depends on the number of constituent IAPCs and the number of vinculin molecules in each IAPC (equivalent spring constants are calculated, see Text A in S1 Appendix). The total force exerted on the actin filament network thus depends on the number of AB clutches of each type and their stiffnesses. Since we use a continuum approach to account for the abundance of each species, we discretise concentrations of AB clutches by assuming a volume of 1 μm3 to calculate the total force (see Text A in S1 Appendix). Together, the above-described processes result in the following set of differential equations: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) Baseline parameter values and rate constants can be found in Table A in S1 Appendix. We refer the reader to Text A in S1 Appendix for detailed descriptions of all reactions in the model and the underlying reasoning. Below, we highlight the novel methodological approaches (signal- and time-dependent rate modification (SDRM and TDRM)), and provide brief explanations of a few mathematical formulations and assumptions that are used in this model. Signal dependent rate modification (SDRM) Nascent adhesions (NAs) form in large numbers and most are disassembled within a time scale of a few minutes [17, 47]. As the cell protrudes, the distance between the cell membrane and the NAs increases, and actin depolymerization rates are higher away from the cell membrane [48]. Thus, numerous NAs may be supported near the cell membrane but in the absence of this scaffold, many NAs disassemble. Various signal cascades also regulate adhesion disassembly. Signalling molecules such as focal adhesion kinase (FAK), Src and ERK kinases play a role in adhesion turnover, and their inhibition leads to more maturation [40]. However, most studies have investigated the effects of signalling molecules on the turnover of FAs and not NAs [49, 50], and the exact mechanical or chemical triggers for NA disassembly remain elusive [51]. NA assembly at the cell front, maturation, and disassembly away from the leading edge occur constantly due to above-mentioned mechanisms. In this study, we focus on one cycle of NA formation and investigated the differences in NA maturation on different substrate stiffnesses. To implement a mechanism to stop indefinite adhesion formation and maturation, it is hypothesized that there exists a signal molecule of which a minimum concentration, signalthresh, is required for new NA formation and low-order AUB clutches (S1, C1) to bind actin for maturation. The concentration [signal] of this molecule is initially high and decreases at an arbitrary rate following Michaelis-Menten kinetics given by: (17) Initial estimates for values of the maximum velocity and the Michaelis constant were based on those reported in the literature for FAK Tyr-397 phosphorylation in the presence of ATP [52] but were adjusted such that the concentration of [signal] reaches the signalthresh in 58 s, which is the approximate duration of the NA assembly phase as measured in experiments [4]. When the [signal] falls below signalthresh, it is analogous to a signalling pathway being activated, and particular reaction rate constants (k1f, k1r, k2f, k2r, k3f, k3r, k4f, k9f, k18f, k20f, k21f, k22f, indicated by yellow lighting bolts in Fig 1B) are modified as detailed in Text A in S1 Appendix. Thus, the rate of decay of [signal] determines the amount of time available before NA disassembly starts in which adhesion maturation can occur. Note that the model behaviour does not change if we assume the opposite i.e., [signal] increases over time and there is an upper limit for its concentration beyond which actin binding does not occur (Fig C.B in S1 Appendix). Force-dependent actin-unbinding and time-dependent rate modification (TDRM) A cell–ECM force chain is broken if either the integrin–ligand (catch-slip) bond [41] or the talin–actin (slip) bond [5] ruptures as a result of reaching the respective force thresholds or due to random thermodynamic fluctuations. This is described in detail in Text A in S1 Appendix. To capture the combined dynamics of the catch-slip and slip bonds, as well as the effect of random bond ruptures, the actin unbinding rates (dark green hourglasses in Fig 1B) are defined as follows: (18) where Fclutch is the force on an individual IAPC in the clutch (see Text A in S1 Appendix where Fclutch is described in detail). The first and second terms, where A and C are scaling factors and b and d control the force-dependency, describe the integrin–ligand catch-slip bond, and the third term describes the talin-actin slip bond where is bond rupture threshold for a given clutch type (i ∈ {1, 2, 3}, refer Table A in S1 Appendix) and is the unloaded dissociation rate. kTDRM is the time-dependent rate modification (TDRM) factor that is required to qualitatively account for the reduction in the total force caused by spontaneous clutch unbinding events. kTDRM is given by: (19) where tclutch is the number of simulated time-steps that a clutch remains actin-bound, dt is the time-step and ksens is a parameter that determines the magnitude of the influence. In short, this definition captures the decreasing likelihood of an AB clutch remaining AB for long periods of time. It has been shown that integrin-ligand bonds undergo cyclic mechanical reinforcement (CMR) leading to longer lifetimes [53]. This implies that on soft substrates where the force-loading rate is low, integrin-ligand bonds experience fewer force cycles in a given time period compared to stiff substrates and consequently are more likely to break on soft substrates. Previous studies model CMR with an increased bond-dissociation rate at low forces [35, 38]. Here, TDRM can capture the effects of these phenomena. TDRM affects softer substrates, where clutches take longer to reach their force thresholds, more than stiff substrates. When an AB clutch experiences a force equal to its force threshold, it unbinds from actin and becomes an AUB clutch. Thus, the concentration of the AB clutch is set to 0, and the concentration of its AUB counterpart is increased by the same amount. Adhesion reinforcement rates The rate at which the talin rod unfolds increases with applied force and has been described in previous studies by the Bell model [24, 44, 45]. Here, the Bell formulation was adapted such that the rate increases exponentially with force until the vinculin binding force threshold Fvb is reached, beyond which it remains constant. The rate is given by: (20) where is the rate of unloaded talin unfolding, kUF is a parameter controlling force-dependence, Fclutch is the force experienced by an individual IAPC in the clutch, and is the vinculin binding force threshold, with i ∈ {1, 2} corresponding to the first and second vinculin binding steps. Here, and [24, 42]. Vinculin binding is assumed to occur instantaneously once the VBS is uncovered [45, 54], and hence the rates of reinforcement were determined based on the force-dependent unfolding kinetics of talin as observed in single-molecule experiments using magnetic tweezers [42]. A detailed description of the method used for curve-fitting can be found in Text A in S1 Appendix. Range of substrate rigidities The stiffness of the ECM, generally measured in terms of Young’s modulus, varies across three orders of magnitude, between 1–3 kPa in the brain, 23–42 kPa in muscular tissue, 1000 kPa-860 MPa in blood vessels, to 15–40 GPa in bone [55]. In this study, however, we consider spring constants, which can be converted to Young’s moduli based on a few assumptions as detailed in [34, 36] and summarized in Text A in S1 Appendix. Accordingly, we restrict the scope of the main investigations and predictions to a stiffness range equivalent to 0.1–130 kPa and only qualitatively discuss the results up to 1000 kPa. This is in line with previous computational and experimental studies which have also primarily studied adhesion mechanobiology in similar ranges of substrate rigidities [4, 36]. Local sensitivity analysis As the model included many parameters whose values were either estimated or adapted to fit experimental data, a local sensitivity analysis was performed. The range of values tested for each parameter was the baseline value (Table A in S1 Appendix) ±10% and ±20%. To quantify the influence, two different metrics were used as outcomes, namely 1) maturation fraction: the concentration of integrins in mid- and high-order AB and AUB clutches ([S2]+[S2a]+[S3]+[S3a]+[C2]+[C2a]+[C3]+[C3a]) at equilibrium (the last time point), and 2) the optimal stiffness: the substrate stiffness with the lowest mean actin retrograde velocity. Outcome 1 represents the total fraction of integrins in the system that made it beyond the initial force-independent stage of adhesion formation, indicative of the fraction of NAs that mature into FAs. Outcome 2 represents an overall influence on the system as it quantifies the mean force exerted during the length of the simulation for a range of substrate stiffnesses. In addition, cells are known to be able to tune their mechanosensitive ranges to adapt to their environments, an aspect on which outcome 2 can shed light. As different parameters may have different levels of influence based on the substrate stiffness, the sensitivity of outcome 1 to each parameter was evaluated for four substrate stiffnesses (ksub = 0.1, 1, 10, 100 pN/nm). Parameter sensitivity analysis was performed on 21 parameters (Text B, Fig I and Fig J in S1 Appendix), and the ones with the highest influence or of key importance are presented in the main text. The parameter sensitivity for a parameter p for an outcome i was calculated as follows: (21) where Outcomei(p + Δp) represents the value of the outcome metric with the changed parameter value, Outcomei(p) is the value of the outcome metric with the baseline parameter value, and Δp and p are the change in the parameter and the baseline parameter value respectively. Initial conditions The initial concentrations of integrins, talin and vinculin were assumed to be equal and set to 1 μM for simplicity, and that of all other species were set to 0. Vinculin was assumed to be abundantly available within the cytoplasm and thus modelled at a constant concentration of 1 μM throughout the simulation. Simulation steps All simulations were run for 600 s. Euler’s forward integration method was used to solve the ODEs with a time step dt of 5 ms as used in previous computational studies [4, 35]. For mass conservation steps, see Text A in S1 Appendix. The steps of integration and the order of updates (Fig 2) of the different aspects of the model are as follows: Force-dependent rate constants are calculated. In particular, the following rates are evaluated at the current force: First reinforcement rates: (k7f, k12f) using Eq 20. Second reinforcement rates: (k8f, k13f) using Eq 20. Signal-dependent rate modification: (k1f, k1r, k2f, k2r, k3f, k3r, k4f, k9f, k18f, k20f, k21f, k22f) are updated as detailed in Text A in S1 Appendix. Catch-slip bond rates with time-dependent rate modification: (k4r, k5r, k6r, k9r, k10r, k11r) using Eqs 18 and 19. Concentrations are updated based on current rate constants by solving the differential equations listed above. The slip bond threshold is checked for each clutch type If the slip bond threshold is reached, the force on the clutch is reset to 0. The concentration of the actin-bound form of the clutch is converted to the actin-unbound form. The total force exerted by actin-bound clutches is calculated based on discretised concentrations using eq. S53. Retrograde velocity, vretro, is updated based on the current total force in the system using eq. S49. All substrate-clutch spring systems are extended by an amount vretro ⋅ dt. Force on each clutch is updated using eq. S51. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Schematic showing the flowchart for simulation and force quantification. https://doi.org/10.1371/journal.pcbi.1011500.g002 Results To explore the influence of mechanical properties like substrate stiffness, adaptor protein stiffness, actomyosin-generated forces and bond characteristics on adhesion maturation, we developed a computational model that captures the overall changes to the IAC compositions as adhesions form and mature. The model, based on ODEs, consists of a single compartment that represents a patch of the cell where adhesions form, and considers three core components: integrins, talin and vinculin, from which 14 other species are made. The dynamic NA/FA maturation is modelled by a total of 22 reactions (explained in Methods and Text A, Tables A and B in S1 Appendix) that largely represent three distinct processes (Fig 1): (i) adhesion formation, (ii) reinforcement and growth, and (iii) adhesion disassembly. Using our mechanochemical computational model, we find that dynamic rates of assembly and disassembly, which are likely regulated by biochemical signalling events, are essential to determine the subset of NAs that mature. The model was found to satisfy mass conservation (Text B, Fig B in S1 Appendix). NA formation is rigidity- and force-independent When only pre-complex, initial seed and clust formation reactions (Rx1, Rx2 and Rx3) were active (see Text B, Table B in S1 Appendix) the concentration of seeds and clusts for all substrate stiffnesses tested were equal, in line with previous experimental evidence showing that NA formation is substrate rigidity-independent [16] (Fig 3A). This is the result of the rigidity- and force-independent rate constants (k1f, k1r, k2f, k2r, k3f, k3r) for reactions Rx1, Rx2 and Rx3. Thus, the concentration of seeds and clusts formed only depends on the initial concentrations of [int], [tal], and [vinc], which were all set to 1 μM, with [vinc] being constant throughout the simulation (see Methods). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. While NA formation is substrate stiffness independent, maturation is influenced considerably by stiffness. (A)—Concentration over time of species in the model that represent NAs (S1 and C1). The curves for all tested substrate rigidities overlap and hence appear as a single (blue) line. The vertical dotted line marks the time point when the signal threshold is crossed and hence new NA formation reduces. (B)—Concentrations over time of all the actin-bound species. Species representing NAs (S1a, C1a) increase initially before being driven to 0 after the signal concentration drops below the threshold. The highest levels of maturation occur on substrate of moderate stiffness (ksub = 1 pN/nm). https://doi.org/10.1371/journal.pcbi.1011500.g003 The baseline signal decay parameters (Table A in S1 Appendix) were set to match experimentally measured time periods for the assembly phase of NAs [17], leading to the concentration of signal crossing the signalthresh at tsig = 58.04 s (Fig 3A, Fig C and Text B in S1 Appendix). Additionally, when maturation (actin-binding) reactions were disabled, the predicted concentration of integrins in seeds and clusts was ≥ 0.1 μM (Fig 3A) for approximately 158 s, a duration indicative of the lifetime of nascent adhesions and is in line with experimentally measured average lifetimes of NAs of 135–180 s [16, 17, 47]. Additionally, the concentrations of [S1] and [C1] also matched experimentally observed trends in abundance of early NAs [17] (Fig 3A) When actin-binding reactions were allowed however, the concentration of [S1a] and [C1a] reached a peak at 58 s (Fig 3B), followed by a sharp fall to 0. This decrease is because the signal-dependent reduction in actin-binding rates reduced the formation of these species, but the rate constants (k7f and k12f) of reinforcement reactions, Rx7 and Rx12, that transform S1a and C1a to S2a and C2a, respectively, remained unchanged. Thus, S1a and C1a were almost completely consumed after approximately 158 s. The predicted concentrations of [S1a] and [S3a] are highest on ksub = 100 pN/nm and ksub = 1 pN/nm substrates respectively (Fig 3B), whereas [C1a] and [C3a] are always higher on ksub = 1 pN/nm. These differences arise from the stiffness-dependent reinforcement rates, and the reversible mass-action kinetics considered here. This is explained in more detail in Text B in S1 Appendix. As such, the model accounts for one cycle of NA assembly, followed by either maturation or disassembly. When maturation was enabled, the concentrations of [S2a], [C2a], [S3a] and[C3a] reach a steady state after 600 s, in that they oscillate between 0 and an almost constant peak concentration (Fig 3B). In summary, the balance between the reversible mass-action kinetics, stiffness-dependent reinforcement rates, and signal concentration decay results in the formation and subsequent disassembly of low-order species (S1, C1, S1a, C1a) representing NAs, while mid- and high-order species (S2, C2, S2a, C2a, S3, C3, S3a, C3a) represent stable adhesions that may further mature to become FAs. Adhesion maturation is highest on moderate substrate stiffness The concentrations of [S3a], [C2a] and [C3a] are highest on a moderate substrate stiffness (1 pN/nm), and lower on stiffer or softer substrates (Fig 3B) in accordance with experimental findings [4, 34, 56, 57]. This is a result of the time taken for the force on a clutch to reach the bond-rupture threshold being roughly equal to the lifetime of an unloaded AB-clutch that spontaneously dissociates from actin (or the substrate) due to thermodynamic fluctuations [4, 58]. Thus, the lifetime of a complete ECM-integrin-adaptor protein actin chain is maximized, resulting in more maturation. In addition, in the early periods of the simulations (0 to 70 s, Fig D in S1 Appendix), the concentrations [C3a], [C2a] and [S3a] increase most rapidly on ksub=1 pN/nm. While these results are for simulations with a constant vinculin concentration, similar results were obtained for limited vinculin conditions (see Text B and Fig E in S1 Appendix). Although the concentration plots in Fig 3B are oscillatory due to the repeated bond-rupture events that transform AB clutches to AUB clutches, pushing the concentration of AB clutches to 0 and causing a spike in the concentration of AUB clutches, the peaks approach a steady state. We observed generally shorter periods of oscillations for AB clutches on stiffer substrates (Table 2), which is also reported by Venturini and Saez (2023) [38]. The periods predicted in our simulations were in good agreement with previous studies (Table 2) [4, 38, 58, 59]. Note that the periods for C2a and C3a on ksub = 0.1 pN/nm, are of the order of the lifetime of NAs (∼60 s) or higher. Thus, these results suggest that C2a and C3a can represent (partially) mature adhesions and not NAs, and that adhesions are likely to disassemble before sufficient reinforcement can occur on very soft substrates. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Mean periods in s of different actin-bound clutches for different substrate stiffnesses. https://doi.org/10.1371/journal.pcbi.1011500.t002 In essence, after the initial NA formation phase, the number of mature adhesions grows, most rapidly on a moderate substrate stiffness of ksub = 1 pN/nm, and approaches a steady state. While the number of adhesions actively bound to the actin network constantly fluctuates due to force thresholds being reached or stochastic bond-rupture events, on average the number approaches a stable steady state. Traction force is highest on substrates of moderate stiffness An optimal substrate stiffness is one at which the highest traction force is generated at the adhesions [60]. As explained in the previous section, this arises due to a balance between the lifetime of an unloaded AB clutch and the time taken for a clutch to reach the force threshold. In our simulations, the highest traction force was reached at ksub = 1 pN/nm, which also corresponded to the point where the lowest retrograde velocity was recorded (Fig 4A). The frequency of oscillations was higher for stiffer substrates (Table 2) given that the force on clutches increases faster on stiff substrates, thus reaching the force thresholds earlier, causing bond-rupture, and subsequent rebinding of the AUB-clutches to actin. While on softer substrates the force build-up is much slower as the substrate is more compliant (hence the lower oscillation frequency), actin unbinding due to thermodynamic fluctuations dominates as bonds break spontaneously much before the force thresholds are reached, giving rise to a higher actin unbinding rate (Fig F in S1 Appendix). These results suggest that ksub = 1 pN/nm gives rise to a ‘load-and-fail’ regime where clutches are loaded at a moderate rate, reach their force thresholds and subsequently break, ‘frictional slippage’ occurs on stiffer substrates where rapid loading causes clutches to disengage too quickly, resulting in lower average AB-clutch concentration [4, 60]. Altogether, these observations show that the optimal stiffness for NA maturation in our model is at ksub = 1 pN/nm. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Model predictions of mean actin retrograde velocity and maturation fraction for the baseline model. (A) shows the predicted velocity vs substrate stiffness compared to previous studies, (B) shows the predicted velocity (blue) and mean force exerted by all adhesions (red) in this model, and (C) shows the NA maturation fraction vs substrate stiffness. https://doi.org/10.1371/journal.pcbi.1011500.g004 Fig 4A shows the agreement between the mean retrograde velocity in our simulations and other computational [36, 38] and experimental [4] studies. A recent computational study also reports a similar biphasic behaviour with an optimal stiffness of around 10 pN/nm [38]. However, this is only observed when the weakest link in the force chain in their model is simulated as a catch bond. Additionally, our model predicts a linear increase in the mean velocity between 100 pN/nm and 102 pN/nm (Fig 4B), which is the stiffness range where the cell is mechanosensitive—a change in stiffness translates linearly into a change in actin retrograde velocity. This is in good agreement with previous studies which report ranges of 100–101 to 100–102 pN/nm [4, 57, 61]. The decrease in the predicted retrograde velocity for stiffnesses > 101 pN/nm in the computational study of Elosegui-Artola et al. [36] (Fig 4A) arises because of reinforcement which they model as an increase in integrin density beyond a certain threshold force on a clutch. While the range of velocities reported varies, in our model, the lowest velocity depends on the concentration of myosin motors concmyo, which is a free parameter that was adjusted such that the lowest velocity was within 10% of that reported by Chan and Odde (2008) [4]. It is also important to note that the optimal stiffness of 1 pN/nm is reached only when the TDRM of actin-unbinding rates was applied (see subsection ‘Force-dependent actin-unbinding and time-dependent rate modification (TDRM)’ in Methods and Text A in S1 Appendix). For high substrate stiffnesses, since the time taken to reach the force thresholds is short, maturation is limited. For soft stiffnesses, however, this is not the case. Hence, in the absence of TDRM, a method we use to account for the force-independent spontaneous bond ruptures, the equilibrium concentrations of mid- and high-order AB clutches ([S2a], [S3a], [C2a], [C3a]) are highest on ksub = 0.1 pN/nm and decrease monotonically with increasing substrate stiffness (Fig G in S1 Appendix). Importantly, the decrease in equilibrium concentrations of mid- and high-order AB clutches caused by TDRM is the largest on ksub = 0.1 pN/nm and least on ksub = 100 pN/nm (Figs G.A and G.B in S1 Appendix). Thus, TDRM of disassembly rates is essential for obtaining an optimal stiffness through mechanosensing. As adhesion assembly and disassembly are tightly regulated processes, these results suggest altering factors affecting adhesion disassembly allows for more robustness and resilience in the mechanosensing and adhesion maturation processes. Predicted NA maturation fraction is most sensitive to talin stiffness and vinculin availability After identifying species in the model that represent NAs (S1, C1, S1a, C1a) and adhesions that mature to FAs (S2, C2, S2a, C2a, S3, C3, S3a, C3a) based on comparisons of their concentrations, bond formation and rupture times to values reported in literature, we used our model to predict the fraction of NAs that may mature into FAs on a range of substrate stiffnesses. Notably, our model also predicts a biphasic trend in maturation fraction (MF) (Fig 4C). More specifically, the MF ranges from approximately 18% on very soft substrates (10-2 pN/nm) to around 34.3% on the optimal substrate stiffness of 100 pN/nm, which lies within experimentally determined ranges of MFs under different conditions [18, 62]. To ensure that both the optimal stiffness and MF predictions were not heavily influenced by the choice of parameter values, we performed a local sensitivity analysis on 21 parameters (Text B, Figs I and J in S1 Appendix), and address the most important and representative ones here. The optimal substrate stiffness, the stiffness at which the lowest mean retrograde velocity is observed (Fig 5B), was most sensitive to changes in ktal, the stiffness of talin. Increasing ktal shifts the optimum stiffness to softer substrates and reduces the MF (Fig 5B). Talin is the most abundant mechanosensitive component in the model and majorly contributes to determining the stiffness of clutches, which effectively determines the optimum substrate stiffness. Increasing the stiffness of talin results in stiffer clutches that reach the bond force thresholds sooner, leaving less time for maturation reactions and consequently lower MF. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Sensitivity analysis results. (A)—Maturation fraction vs stiffness (outcome 1) and (B)—Mean retrograde velocity vs stiffness for a local variation in parameter values of different parameters. Black arrows in (B) point in the direction of increasing parameter value and track the optimal stiffness (outcome 2). https://doi.org/10.1371/journal.pcbi.1011500.g005 Increases in initial vinculin concentration Initialvinc leads to large increases in MF (Fig 5A) and small increases in the optimal stiffness (Fig 5B). A higher vinculin concentration increases the likelihood of maturation leading to increased force-carrying capacity and consequently a shift of the optimal stiffness to stiffer regimes. In contrast, a lower vinculin availability leads to decreased maturation fractions and traction force and a higher mean retrograde velocity (Figs 5 and 6). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. Vinculin concentration can influence maturation fraction. (A) and (B) are cell types or biological contexts where vinculin availability is low and high respectively. In A, the likelihood of vinculin binding to the exposed vinculin-binding sites on talin is low leading to a low maturation fraction. However, in B, due to a relatively higher vinculin availability, the integrin-actin link is highly likely to be reinforced by vinculin, increasing the maturation fraction. This figure was created using BioRender.com. https://doi.org/10.1371/journal.pcbi.1011500.g006 Changes in the talin refolding rate, talrf, affect the mean retrograde velocities more than the optimum stiffness (Fig 5). A higher talrf leads to lower maturation fractions but has negligible effects on the optimum stiffness. This is because a higher talin refolding rate causes AUB clutches unbinding from vinculin more likely and hence lowers the concentration of more mature, higher force-carrying-capacity clutches on all substrate stiffnesses. This, however, does not cause any changes in the relative proportions of concentrations of different clutches across substrate stiffnesses and hence does not change the optimum stiffness considerably. A similar reasoning is valid for a lower talrf. The talin refolding factor, , controls the rate of vinculin-dissociation and hence relative additional stability of higher order species in our model (see Text A in S1 Appendix) and its baseline value is set to 0.5 in our simulations (talin in higher order species is half as likely to refold compared to mid-order species). Increasing results in lower MFs and shifts the optimal stiffness to softer substrates due to the decreased ‘additional’ stability of higher-order species. Similar reasoning can be used to explain the effects of decreasing . As vinculin is known to ‘lock’ talin in the unfolded conformation [39], we further reduced the value of to 0.2 to investigate the effects of having highly stable high-order species. This resulted in the same trends of maturation across different stiffnesses but with slightly higher maturation fractions (Fig H in S1 Appendix). An increase in vu, the unloaded actin retrograde velocity, pushed the optimal substrate stiffness to softer substrates in line with previous computational studies [57, 60] and leads to lower MF (Fig 5). A higher retrograde velocity causes faster force build-up resulting in frictional slippage on softer substrates. Similar to ktal, it also results in lower MF. On the contrary, increases in kact pushed the optimal substrate stiffness towards stiffer substrates and increases MF (Fig 5), which is due to the ‘strengthening’ of clutches as they are more likely to bind actin, get stretched and recruit vinculin, and on average there are more AB clutches resulting in higher forces on softer substrates [57, 60]. Out of all the parameters, the stiffness of talin ktal, initial vinculin concentration initialvinc, talin refolding rate talrf, and the actin-binding rate kact had the greatest influence on MF, similar for both an increase and decrease in the parameter values (Fig I in S1 Appendix). Importantly, the TDRM factor ksens, and the cluster formation (k14f, k15f, k16f) and disassembly (k21f, k22f) rates had negligible influences on the MF and optimal stiffness (Figs I and J in S1 Appendix) for the tested range of values (±20%). NA formation is rigidity- and force-independent When only pre-complex, initial seed and clust formation reactions (Rx1, Rx2 and Rx3) were active (see Text B, Table B in S1 Appendix) the concentration of seeds and clusts for all substrate stiffnesses tested were equal, in line with previous experimental evidence showing that NA formation is substrate rigidity-independent [16] (Fig 3A). This is the result of the rigidity- and force-independent rate constants (k1f, k1r, k2f, k2r, k3f, k3r) for reactions Rx1, Rx2 and Rx3. Thus, the concentration of seeds and clusts formed only depends on the initial concentrations of [int], [tal], and [vinc], which were all set to 1 μM, with [vinc] being constant throughout the simulation (see Methods). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. While NA formation is substrate stiffness independent, maturation is influenced considerably by stiffness. (A)—Concentration over time of species in the model that represent NAs (S1 and C1). The curves for all tested substrate rigidities overlap and hence appear as a single (blue) line. The vertical dotted line marks the time point when the signal threshold is crossed and hence new NA formation reduces. (B)—Concentrations over time of all the actin-bound species. Species representing NAs (S1a, C1a) increase initially before being driven to 0 after the signal concentration drops below the threshold. The highest levels of maturation occur on substrate of moderate stiffness (ksub = 1 pN/nm). https://doi.org/10.1371/journal.pcbi.1011500.g003 The baseline signal decay parameters (Table A in S1 Appendix) were set to match experimentally measured time periods for the assembly phase of NAs [17], leading to the concentration of signal crossing the signalthresh at tsig = 58.04 s (Fig 3A, Fig C and Text B in S1 Appendix). Additionally, when maturation (actin-binding) reactions were disabled, the predicted concentration of integrins in seeds and clusts was ≥ 0.1 μM (Fig 3A) for approximately 158 s, a duration indicative of the lifetime of nascent adhesions and is in line with experimentally measured average lifetimes of NAs of 135–180 s [16, 17, 47]. Additionally, the concentrations of [S1] and [C1] also matched experimentally observed trends in abundance of early NAs [17] (Fig 3A) When actin-binding reactions were allowed however, the concentration of [S1a] and [C1a] reached a peak at 58 s (Fig 3B), followed by a sharp fall to 0. This decrease is because the signal-dependent reduction in actin-binding rates reduced the formation of these species, but the rate constants (k7f and k12f) of reinforcement reactions, Rx7 and Rx12, that transform S1a and C1a to S2a and C2a, respectively, remained unchanged. Thus, S1a and C1a were almost completely consumed after approximately 158 s. The predicted concentrations of [S1a] and [S3a] are highest on ksub = 100 pN/nm and ksub = 1 pN/nm substrates respectively (Fig 3B), whereas [C1a] and [C3a] are always higher on ksub = 1 pN/nm. These differences arise from the stiffness-dependent reinforcement rates, and the reversible mass-action kinetics considered here. This is explained in more detail in Text B in S1 Appendix. As such, the model accounts for one cycle of NA assembly, followed by either maturation or disassembly. When maturation was enabled, the concentrations of [S2a], [C2a], [S3a] and[C3a] reach a steady state after 600 s, in that they oscillate between 0 and an almost constant peak concentration (Fig 3B). In summary, the balance between the reversible mass-action kinetics, stiffness-dependent reinforcement rates, and signal concentration decay results in the formation and subsequent disassembly of low-order species (S1, C1, S1a, C1a) representing NAs, while mid- and high-order species (S2, C2, S2a, C2a, S3, C3, S3a, C3a) represent stable adhesions that may further mature to become FAs. Adhesion maturation is highest on moderate substrate stiffness The concentrations of [S3a], [C2a] and [C3a] are highest on a moderate substrate stiffness (1 pN/nm), and lower on stiffer or softer substrates (Fig 3B) in accordance with experimental findings [4, 34, 56, 57]. This is a result of the time taken for the force on a clutch to reach the bond-rupture threshold being roughly equal to the lifetime of an unloaded AB-clutch that spontaneously dissociates from actin (or the substrate) due to thermodynamic fluctuations [4, 58]. Thus, the lifetime of a complete ECM-integrin-adaptor protein actin chain is maximized, resulting in more maturation. In addition, in the early periods of the simulations (0 to 70 s, Fig D in S1 Appendix), the concentrations [C3a], [C2a] and [S3a] increase most rapidly on ksub=1 pN/nm. While these results are for simulations with a constant vinculin concentration, similar results were obtained for limited vinculin conditions (see Text B and Fig E in S1 Appendix). Although the concentration plots in Fig 3B are oscillatory due to the repeated bond-rupture events that transform AB clutches to AUB clutches, pushing the concentration of AB clutches to 0 and causing a spike in the concentration of AUB clutches, the peaks approach a steady state. We observed generally shorter periods of oscillations for AB clutches on stiffer substrates (Table 2), which is also reported by Venturini and Saez (2023) [38]. The periods predicted in our simulations were in good agreement with previous studies (Table 2) [4, 38, 58, 59]. Note that the periods for C2a and C3a on ksub = 0.1 pN/nm, are of the order of the lifetime of NAs (∼60 s) or higher. Thus, these results suggest that C2a and C3a can represent (partially) mature adhesions and not NAs, and that adhesions are likely to disassemble before sufficient reinforcement can occur on very soft substrates. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Mean periods in s of different actin-bound clutches for different substrate stiffnesses. https://doi.org/10.1371/journal.pcbi.1011500.t002 In essence, after the initial NA formation phase, the number of mature adhesions grows, most rapidly on a moderate substrate stiffness of ksub = 1 pN/nm, and approaches a steady state. While the number of adhesions actively bound to the actin network constantly fluctuates due to force thresholds being reached or stochastic bond-rupture events, on average the number approaches a stable steady state. Traction force is highest on substrates of moderate stiffness An optimal substrate stiffness is one at which the highest traction force is generated at the adhesions [60]. As explained in the previous section, this arises due to a balance between the lifetime of an unloaded AB clutch and the time taken for a clutch to reach the force threshold. In our simulations, the highest traction force was reached at ksub = 1 pN/nm, which also corresponded to the point where the lowest retrograde velocity was recorded (Fig 4A). The frequency of oscillations was higher for stiffer substrates (Table 2) given that the force on clutches increases faster on stiff substrates, thus reaching the force thresholds earlier, causing bond-rupture, and subsequent rebinding of the AUB-clutches to actin. While on softer substrates the force build-up is much slower as the substrate is more compliant (hence the lower oscillation frequency), actin unbinding due to thermodynamic fluctuations dominates as bonds break spontaneously much before the force thresholds are reached, giving rise to a higher actin unbinding rate (Fig F in S1 Appendix). These results suggest that ksub = 1 pN/nm gives rise to a ‘load-and-fail’ regime where clutches are loaded at a moderate rate, reach their force thresholds and subsequently break, ‘frictional slippage’ occurs on stiffer substrates where rapid loading causes clutches to disengage too quickly, resulting in lower average AB-clutch concentration [4, 60]. Altogether, these observations show that the optimal stiffness for NA maturation in our model is at ksub = 1 pN/nm. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Model predictions of mean actin retrograde velocity and maturation fraction for the baseline model. (A) shows the predicted velocity vs substrate stiffness compared to previous studies, (B) shows the predicted velocity (blue) and mean force exerted by all adhesions (red) in this model, and (C) shows the NA maturation fraction vs substrate stiffness. https://doi.org/10.1371/journal.pcbi.1011500.g004 Fig 4A shows the agreement between the mean retrograde velocity in our simulations and other computational [36, 38] and experimental [4] studies. A recent computational study also reports a similar biphasic behaviour with an optimal stiffness of around 10 pN/nm [38]. However, this is only observed when the weakest link in the force chain in their model is simulated as a catch bond. Additionally, our model predicts a linear increase in the mean velocity between 100 pN/nm and 102 pN/nm (Fig 4B), which is the stiffness range where the cell is mechanosensitive—a change in stiffness translates linearly into a change in actin retrograde velocity. This is in good agreement with previous studies which report ranges of 100–101 to 100–102 pN/nm [4, 57, 61]. The decrease in the predicted retrograde velocity for stiffnesses > 101 pN/nm in the computational study of Elosegui-Artola et al. [36] (Fig 4A) arises because of reinforcement which they model as an increase in integrin density beyond a certain threshold force on a clutch. While the range of velocities reported varies, in our model, the lowest velocity depends on the concentration of myosin motors concmyo, which is a free parameter that was adjusted such that the lowest velocity was within 10% of that reported by Chan and Odde (2008) [4]. It is also important to note that the optimal stiffness of 1 pN/nm is reached only when the TDRM of actin-unbinding rates was applied (see subsection ‘Force-dependent actin-unbinding and time-dependent rate modification (TDRM)’ in Methods and Text A in S1 Appendix). For high substrate stiffnesses, since the time taken to reach the force thresholds is short, maturation is limited. For soft stiffnesses, however, this is not the case. Hence, in the absence of TDRM, a method we use to account for the force-independent spontaneous bond ruptures, the equilibrium concentrations of mid- and high-order AB clutches ([S2a], [S3a], [C2a], [C3a]) are highest on ksub = 0.1 pN/nm and decrease monotonically with increasing substrate stiffness (Fig G in S1 Appendix). Importantly, the decrease in equilibrium concentrations of mid- and high-order AB clutches caused by TDRM is the largest on ksub = 0.1 pN/nm and least on ksub = 100 pN/nm (Figs G.A and G.B in S1 Appendix). Thus, TDRM of disassembly rates is essential for obtaining an optimal stiffness through mechanosensing. As adhesion assembly and disassembly are tightly regulated processes, these results suggest altering factors affecting adhesion disassembly allows for more robustness and resilience in the mechanosensing and adhesion maturation processes. Predicted NA maturation fraction is most sensitive to talin stiffness and vinculin availability After identifying species in the model that represent NAs (S1, C1, S1a, C1a) and adhesions that mature to FAs (S2, C2, S2a, C2a, S3, C3, S3a, C3a) based on comparisons of their concentrations, bond formation and rupture times to values reported in literature, we used our model to predict the fraction of NAs that may mature into FAs on a range of substrate stiffnesses. Notably, our model also predicts a biphasic trend in maturation fraction (MF) (Fig 4C). More specifically, the MF ranges from approximately 18% on very soft substrates (10-2 pN/nm) to around 34.3% on the optimal substrate stiffness of 100 pN/nm, which lies within experimentally determined ranges of MFs under different conditions [18, 62]. To ensure that both the optimal stiffness and MF predictions were not heavily influenced by the choice of parameter values, we performed a local sensitivity analysis on 21 parameters (Text B, Figs I and J in S1 Appendix), and address the most important and representative ones here. The optimal substrate stiffness, the stiffness at which the lowest mean retrograde velocity is observed (Fig 5B), was most sensitive to changes in ktal, the stiffness of talin. Increasing ktal shifts the optimum stiffness to softer substrates and reduces the MF (Fig 5B). Talin is the most abundant mechanosensitive component in the model and majorly contributes to determining the stiffness of clutches, which effectively determines the optimum substrate stiffness. Increasing the stiffness of talin results in stiffer clutches that reach the bond force thresholds sooner, leaving less time for maturation reactions and consequently lower MF. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Sensitivity analysis results. (A)—Maturation fraction vs stiffness (outcome 1) and (B)—Mean retrograde velocity vs stiffness for a local variation in parameter values of different parameters. Black arrows in (B) point in the direction of increasing parameter value and track the optimal stiffness (outcome 2). https://doi.org/10.1371/journal.pcbi.1011500.g005 Increases in initial vinculin concentration Initialvinc leads to large increases in MF (Fig 5A) and small increases in the optimal stiffness (Fig 5B). A higher vinculin concentration increases the likelihood of maturation leading to increased force-carrying capacity and consequently a shift of the optimal stiffness to stiffer regimes. In contrast, a lower vinculin availability leads to decreased maturation fractions and traction force and a higher mean retrograde velocity (Figs 5 and 6). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. Vinculin concentration can influence maturation fraction. (A) and (B) are cell types or biological contexts where vinculin availability is low and high respectively. In A, the likelihood of vinculin binding to the exposed vinculin-binding sites on talin is low leading to a low maturation fraction. However, in B, due to a relatively higher vinculin availability, the integrin-actin link is highly likely to be reinforced by vinculin, increasing the maturation fraction. This figure was created using BioRender.com. https://doi.org/10.1371/journal.pcbi.1011500.g006 Changes in the talin refolding rate, talrf, affect the mean retrograde velocities more than the optimum stiffness (Fig 5). A higher talrf leads to lower maturation fractions but has negligible effects on the optimum stiffness. This is because a higher talin refolding rate causes AUB clutches unbinding from vinculin more likely and hence lowers the concentration of more mature, higher force-carrying-capacity clutches on all substrate stiffnesses. This, however, does not cause any changes in the relative proportions of concentrations of different clutches across substrate stiffnesses and hence does not change the optimum stiffness considerably. A similar reasoning is valid for a lower talrf. The talin refolding factor, , controls the rate of vinculin-dissociation and hence relative additional stability of higher order species in our model (see Text A in S1 Appendix) and its baseline value is set to 0.5 in our simulations (talin in higher order species is half as likely to refold compared to mid-order species). Increasing results in lower MFs and shifts the optimal stiffness to softer substrates due to the decreased ‘additional’ stability of higher-order species. Similar reasoning can be used to explain the effects of decreasing . As vinculin is known to ‘lock’ talin in the unfolded conformation [39], we further reduced the value of to 0.2 to investigate the effects of having highly stable high-order species. This resulted in the same trends of maturation across different stiffnesses but with slightly higher maturation fractions (Fig H in S1 Appendix). An increase in vu, the unloaded actin retrograde velocity, pushed the optimal substrate stiffness to softer substrates in line with previous computational studies [57, 60] and leads to lower MF (Fig 5). A higher retrograde velocity causes faster force build-up resulting in frictional slippage on softer substrates. Similar to ktal, it also results in lower MF. On the contrary, increases in kact pushed the optimal substrate stiffness towards stiffer substrates and increases MF (Fig 5), which is due to the ‘strengthening’ of clutches as they are more likely to bind actin, get stretched and recruit vinculin, and on average there are more AB clutches resulting in higher forces on softer substrates [57, 60]. Out of all the parameters, the stiffness of talin ktal, initial vinculin concentration initialvinc, talin refolding rate talrf, and the actin-binding rate kact had the greatest influence on MF, similar for both an increase and decrease in the parameter values (Fig I in S1 Appendix). Importantly, the TDRM factor ksens, and the cluster formation (k14f, k15f, k16f) and disassembly (k21f, k22f) rates had negligible influences on the MF and optimal stiffness (Figs I and J in S1 Appendix) for the tested range of values (±20%). Discussion Although cell-ECM adhesions are extensively studied, the effects of mechanical properties of the ECM and intracellular proteins on the early processes of adhesion assembly, maturation and traction force generation remain unclear. Here, we present a computational model that innovatively bridges the discrete mechanical and continuous biochemical aspects of adhesion formation. Our model captures key trends in the maturation fraction (MF) of NAs, actin retrograde velocity, and the periods of bond formation-rupture cycles, all in agreement with experimental evidence [4, 18, 59, 62]. The predicted optimal substrate stiffness [4] and stiffness sensitivity range [57, 61] also lie within experimentally determined ranges. While the predicted mean actin retrograde velocity across the stiffness range tested is in agreement with an experimental study using embryonic chick forebrain neurons [4], the agreement with the study of Elosegui-Artola and colleagues (2014) [36] is limited to the softer regimes where the biphasic trend is also seen (compare yellow and blue lines in Fig 4A). This discrepancy arises from the way reinforcement of adhesions is modelled. In particular, reinforcement in that study is modelled as an increase in the integrin density that occurs if a clutch experiences a force ≥87 pN, leading to an increase in integrin–ECM binding events and a larger number of bound clutches. In a recent computational study where reinforcement is also modelled similarly, a biphasic behaviour is observed just as in our model but with the optimal stiffness being around 10 pN/nm [38]. In our model, while there is an increase in the cluster size of clutches and additional vinculin recruitment leading to larger force-carrying capacities, there is no change in the number of available integrins or the adhesion formation rates. In this study, we assume relatively fast kinetics for the signal molecule to keep the NA assembly and disassembly phases in line with experimental data [17]. It is important to note that there may be considerable differences in experimental results based on the cell types used, resulting in different time scales. However, since the model is relatively insensitive to changes in signalthresh (Figs I and J in S1 Appendix), and consequently changes in tsig, the overall behaviour of the model is unlikely to change drastically when these parameters are tuned to represent specific cell types or, for instance, signal molecule kinetics. Thus, the generic signal molecule in the model can potentially represent the level of unphosphorylated FAK or similar molecules whose change in (phosphorylation) state can set off signalling cascades leading to adhesion disassembly. Future work should aim to determine the underlying factors that induce and influence adhesion disassembly so that the generic signal molecule can be replaced with more accurate formulations and interactions. In particular, the identification of such concrete factors could help in determining the settings of signalthresh. Based on our results, the factors affecting the NA disassembly dynamics play a more important role than those affecting assembly dynamics. We applied TDRM, an innovative method to account for spontaneous bond-rupture events in NA formation in an ODE framework. TDRM was necessary to establish the optimal stiffness because in its absence, maturation is highest on soft substrates as forces on the clutches build up slowly, giving long durations for maturation reactions to occur. TDRM counters this by increasing the rate of clutch-actin bond rupture and hence prevents maturation. With the baseline value of the TDRM factor ksens, the effect of TDRM on the bond-rupture rate is highest on soft substrates and negligible on stiffer substrates due to the short clutch lifetimes. Walcott et al. (2011) [63] predicted and experimentally verified that disassembly processes begin earlier for soft substrates, and this arises from the force- and strain-dependent bond formation and rupture probabilities. In addition, cyclic mechanical reinforcement (CMR) of integrin-ligand bonds strengthens them, increasing the lifetimes, implying that on soft substrates where force-loading is relatively slow and force remains low for longer durations, these bonds are less reinforced and are more likely to break [53]. In previous studies, CMR has been modelled as an increase in bond-dissociation rates at low forces [35, 38]. TDRM can be considered as a method to coarsely account for these processes. However, while the outcomes of TDRM are similar to the effects of CMR as modelled in [35, 38], the differences between the two methods need to be investigated further. Surprisingly, the optimal stiffness was insensitive to changes in the parameter that controls the magnitude of TDRM (ksens, (Fig 5B). This was unexpected since TDRM was essential for establishing an optimal stiffness implying a major role of this parameter in determining model behaviour (Figs F and G in S1 Appendix). It is likely that the explored sensitivity range (±20%) was too narrow to considerably change the behaviour of the model, which should be investigated more in-depth in the future. Note as well that in this study we performed a local sensitivity analysis focusing on those parameters that can directly be traced to a biological phenomenon (for instance kact, Initialvinc), those that we introduced as part of SDRM and TDRM (for instance ksens, signalthresh), or assumed (for instance, ). Considering the non-linear nature of the model, it would be interesting in future studies, to conduct a more rigorous, global sensitivity analysis (i.e. using Bayesian Optimization) to further identify the most significant parameters of the model. Another benefit of our model is that it allows the prediction of the maturation fraction of adhesions for a range of substrate stiffnesses. While there are no studies to the best of our knowledge that explicitly investigate the MFs for different substrate stiffnesses, the predicted range of MFs for the stiffness range tested in this study was within the range of experimentally determined fractions [18, 62]. Our sensitivity analysis results show that the MF is highly influenced by the concentration of integrins and vinculin available, the actin-binding rate, and the talin-refolding rate (Fig 5A). These factors can possibly be experimentally controlled, by introducing mutations in the proteins, allowing the predictions to be tested. Additionally, the MF and optimal stiffness were found to be insensitive to variations in the TDRM factor ksens and the cluster formation and disassembly rates, suggesting that the model is locally robust to these factors and the parameters can be tuned to be specific to experimental conditions or cell lines. This suggests that the model can be used to predict the MFs for a variety of conditions by varying molecular stiffnesses, initial concentrations of talin, integrin, and vinculin, different clustering, maturation, and disassembly rates among many other parameters. This can potentially shed light on how traction force exerted by the cell is affected by biochemical alterations within the cell. For instance, vinculin plays an important role in both cell-ECM adhesions and cell-cell adhesions through cadherins. Numerous studies indicate interdependence and cooperativity of these two processes, mediated through signal cascades or proteins that are essential in both types of adhesions, to varying degrees in different cell types [64–69]. While vinculin knockout studies have shown that traction force generation is impaired, with some studies reporting a decrease of nearly 50% in the absence of vinculin, overexpression of vinculin results in extremely strong adhesions that suppresses cell motility [70–75]. However, this leaves unanswered questions about how relatively less drastic changes in vinculin availability arising from cross-talk between integrins and cadherins adhesion complexes affect traction force generation and adhesion maturation. Our model predicts that a 20% decrease in the vinculin concentration results in a ∼9% increase in the actin retrograde velocity (or equivalently a 9% decrease in the traction force exerted due to lower MFs) at the optimal substrate stiffness (Fig 5). Thus, our model can be especially valuable to make predictions and generate hypotheses about how (local) adhesion protein concentrations influence the early processes of adhesion assembly, maturation, and traction force generation. Overall, our model improves on previous studies in several aspects. Firstly, the process of maturation is more accurately captured by accounting for multiple vinculin recruitment events that progressively increase the clutch stiffness in a continuous ODE framework. Previous studies either did not account for this or at most accounted for recruitment of one vinculin [4, 30–35]. Secondly, this model couples changes in discrete mechanical factors of adhesion maturation such as clutch stiffness with the continuous framework of biochemical reactions underlying adhesion maturation. This is particularly important because the continuous biochemical models do not explicitly account for force on the clutches, and discrete mechanical models of adhesion formation do not capture the resulting experimentally measurable biochemical changes that occur. While it is clear from our results that the adhesion assembly and disassembly rates must be dynamic and dependent on a signal to achieve the maturation of only a fraction of the NAs that are initially formed, we acknowledge several limitations to this study. First, we do not model numerous proteins involved in the process of maturation or the continuous increase in the area of the adhesion [76]. Second, we simplified vinculin recruitment and growth of cluster size to occur in two discrete steps, and no spatial effects (e.g. proximity to an actin fibre, the distance of adhesion from the cell membrane) are accounted for. And third, we assume that integrins, talin and vinculin are available in roughly equal proportions near the adhesions (initial concentrations are the same), which may not necessarily be true. Despite these limitations, our model reproduced experimentally observed trends with respect to force, substrate stiffness, and time periods of oscillation in concentrations of the different seeds and clusts [4, 59, 60]. Furthermore, our results are reasonably close to discrete, stochastic computational studies as mentioned earlier even though our model bridges discrete and continuous aspects. The model thus provides a reliable foundation for further investigations. What remains to be explored, perhaps by building on our model, is the interaction between the various signal cascades that regulate NA maturation. The ubiquitous signaling molecule FAK is also force-activated adding a further layer of interactions and complexity [77, 78]. In addition, the KANK family of proteins are known to impair the actin-binding capacity of talin, thereby weakening the integrin-actin linkage, and affecting the catch and slip bond dynamics [79]. They also play a role in targeting microtubules to focal adhesions which aids in their disassembly through multiple signal cascades [80]. By expanding the current model framework to include these interactions, it has the potential to robustly simulate the mechanochemical processes underlying mechanotransduction and provide valuable insight into cell signalling, communication and organization, hence contributing to advances in developmental biology and regenerative medicine. Implementation The model and the simulations were implemented using MATLAB R2020a [81]. All code and scripts used in this study are publicly available via GitHub at https://github.com/CarlierComputationalLab/force-dependent-adhesion-composition.git. Implementation The model and the simulations were implemented using MATLAB R2020a [81]. All code and scripts used in this study are publicly available via GitHub at https://github.com/CarlierComputationalLab/force-dependent-adhesion-composition.git. Supporting information S1 Appendix. Detailed explanations of the methods (Text A), additional results (Text B), figures (Fig A—Fig N) and tables (Table A and Table B). [83–104] are cited in this file. https://doi.org/10.1371/journal.pcbi.1011500.s001 (ZIP) Acknowledgments We thank Dr. José Manuel Garcia-Aznar for providing key inputs and ideas, and Hang Nguyen for providing guidance in scientific writing. For the colours in Figs I and J in S1 Appendix, a publicly available user-created MATLAB package was used [82].

Winner, Taniel S.;Rosenberg, Michael C.;Jain, Kanishk;Kesar, Trisha M.;Ting, Lena H.;Berman, Gordon J.

doi: 10.1371/journal.pcbi.1011556pmid: 37889927

Introduction Locomotion is a ubiquitous, complex, and dynamic behavior that is essential for survival. Using cyclic patterns of joint angles, inter-limb and inter-joint coordination, animals effectively move through their environments: walking, running, trotting, swimming, flying, and crawling. Even within species and types of locomotion, variations in locomotor patterns often occur across behavioral contexts, groups, and individuals. Thus, although locomotor patterns can appear highly stereotyped, considerable inter- and intra-individual variability exists. Studies of locomotor behaviors have shown systematic differences in movement patterns based on a wide range of neural [1–4] and biomechanical perturbations [5–8] environmental challenges [9,10], psychological state [11,12], social status [13,14], injury [15–17], and disease [4,18–23]. Furthermore, locomotor impairments can arise from a wide range of physiological and neurological changes, from the subtle changes that may be indicators of progressive disorders (e.g., aging, cognitive impairments) to profound impairments with brain injury (e.g., stroke, spinal cord injury) that can severely limit locomotor function. Although locomotor deficits are often subjectively visible to a human observer, objectively characterizing and understanding sometimes subtle yet important differences in locomotion from a scientific and mechanistic standpoint has been challenging [24–26]. For example, kinematic movement patterns (the continuous motion of joint angles over time) have been collected across a wide range of locomotor modes and species but revealing individual-specific differences in kinematics remains difficult. One barrier to progress is that interpreting individual differences in kinematics without an underlying dynamical model is challenging, as kinematics are the result of the complex neuromechanical dynamics that drive the spatiotemporal dependencies of joint kinematics over time. Thus, capturing these underlying gait dynamics is likely essential for interpreting differences in gait and movement across conditions and individuals. Traditionally, gait dynamics are modeled using physiologically detailed neuromechanical equations, however making predictive models using this approach has often proved challenging [27–29]. Partially, this difficulty arises because in order to understand the dynamics underlying gait, we also need to understand how neural feedback and control shape these dynamics. While many models (e.g., musculoskeletal models) that use principles like optimal control can generate simulations of unimpaired gait, as well as changes in gait due to altered biomechanical or neural constraints, they often fail to predict changes in gait kinematics following neurological injury [28] or more subtle perturbations [30,31]. Progress in the physiological modeling of locomotor circuitry in the spinal cord and brainstem demonstrates the role of neural circuits in gait dynamics. However, these models typically rely on simplified [32,33] biomechanical properties and cannot yet predict the deficits in gait specific to an individual [25,34–36]. More importantly, if a hyper-realistic model of the neural and biomechanical system did exist, the relationships between the high-dimensional parameters and actual movement patterns would not likely be unique, as many parameters would not be identifiable, even given massive amounts of data, as many different parameter choices could lead to the same biomechanical output [37,38]. This non-identifiability limits the predictive power and generalizability of these models to other interventions and conditions outside of limited contexts, suggesting a need for a more holistic approach. Despite these challenges, rich individual-specific information exists in gait data. For instance, through observation of movement, the human brain can perceive many socially salient features of an individual’s gait, suggesting that it should be possible to infer aspects of gait dynamics from kinematic data. As an example, humans can derive a host of information about individuals from movement patterns, including gender [39], body size [40], sexual orientation [41], emotion [42], individual differences in dancing [43], perceived affective states [44] and underlying intention [45]. Furthermore, judgements based on how individuals move can drive decisions such as partner desirability or attractiveness [46] diagnosis [47,48], and treatment planning [49,50]. Despite the rapid advent of technologies providing kinematic measurements through a wide range of techniques, from videos to wearable sensors, we are still limited in how kinematic data can help interpret individual differences in gait [51,52]. Current approaches to comparing biomechanical features or kinematic trajectories quantify between-group differences or inter-individual similarity but lack sufficient sensitivity to reveal interpretable differences in individuals’ gaits [53–55]. Inter-joint coordination differs across individuals, as muscular coordination patterns vary across a variety of motor skills and deficits in individual-specific ways. Indeed, metrics of muscle coordination in children with cerebral palsy are consistent with clinician judgements of motor control complexity that predict intervention outcomes [56]. Recently, supervised machine learning methods have been used to classify differences in a large sets of gait kinematics that were labeled by groups or individuals [55,57]. However, these approaches have not modeled the underlying gait dynamics, nor can they discover subtle differences in gait that are not labeled a priori. Here we develop a data-driven framework for modeling gait dynamics that represents multiple individuals in the same latent space. This latent space reveals individual- and group-level differences in the neuromechanical dynamics of gait. We used kinematic data from multiple healthy and neurologically impaired individuals, each walking at six different speeds, to train a recurrent neural network (RNN) that learns gait dynamics. This phenomenological approach infers complex spatiotemporal dynamics and enables future kinematic predictions to be made based on current and prior kinematic postures. Once trained, differences in gait dynamics across groups, individuals, and walking speed were projected onto a common, low-dimensional latent space of the model parameters. The stride-averaged representation of gait dynamics in the latent space constitutes a “gait signature” that we use to characterize differences across individuals, groups, gait speed, and impairment severity. To demonstrate the generalizability of gait dynamics, we show that interpolating gait signatures to predict gait kinematics at new walking speeds is more accurate than interpolating the kinematics themselves in healthy individuals. Further, we show that the low-dimensional basis functions we discovered have biomechanical interpretability in terms of the inter- and intra-limb coordination patterns that they generate. The dynamical projections onto each basis function for each trial can be independently driven through the trained gait dynamics model to reconstruct the kinematics associated with that specific basis function. We generated illustrations of the reconstructed joint angle kinematics to visualize and infer what aspects of gait coordination each subcomponent influences. These subcomponents of gait coordination can be manipulated independently (i.e., gait sculpting) to infer the relationships between specific underlying dynamical components and their corresponding kinematic phenotypes and to identify what specific gait rehabilitation strategies are likely required for individuals. Finally, our gait dynamics model is generative; it can predict individual-specific time evolution of kinematics from an initial arbitrary posture (self-driving) once the network is primed with several gait cycles of the individual’s kinematic data. This study establishes a new data-driven framework to quantitatively interpret individual-specific differences in gait dynamics with the potential to enable discovery in a wide range of gait coordination deficits, contexts and interventions in humans and other animals. Results Gait signatures: A low-dimensional representation of gait dynamics We used motion capture to collect sagittal-plane kinematic data that consisted of 15 seconds of continuous gait kinematics from bilateral, hip, knee, and ankle joints from 5 able-bodied (AB) participants and 7 stroke survivors (> 6 months post-stroke, gait speeds 0.1 to 0.8 m/s) walking on a treadmill at a range of six different speeds each. Taking inspiration from neural network models that capture neural dynamics [58–60] and biological systems, we implemented a recurrent neural network (RNN) model to capture the dynamical properties of gait. Our model input parameters only include kinematic data and do not include anthropometric information or clinical characteristics and do not account for differences in joint kinematics due to neural versus biomechanical constraints. Developing the recurrent neural network (RNN) architecture and training the model The gait dynamics model was developed in Python using common Python libraries, including TensorFlow, Keras, Pandas, and NumPy. We developed our code in Google Colab to facilitate open-source sharing of our dynamic framework, which can be found here: https://github.com/bermanlabemory/gait_signatures. The model architecture was selected based on two criteria: 1) minimizing model training and validation loss during model fitting, and 2) maximizing the similarity of short-time (single stride) and long-time (multiple strides) self-driven model predictions (termed: gait signature alignment) post model training (S1 Fig). By implementing these two model selection criteria we ensure 1) a high goodness-of-fit (model that best represents the underlying dynamics across all participants and gait speeds) and 2) the model is capable of predicting the time-evolution of gait (encode gait dynamics). We evaluated these criteria against alternative models by varying 2 hyperparameters (number of LSTM units and the lookback time, see Methods). The selected model architecture is a sequence-to-sequence RNN [61] consisting of an input layer, a hidden layer of 512 LSTM units, and an output layer. The RNN learns a map from time-series kinematic input data (0 to T-1) to kinematics one time-step in the future (1 to T) for all training trials (Fig 1A). The model was trained using the ‘mean squared error’ (mse) loss function until training and validation error converged and stabilized around the same point (< 0.03 degrees2). Thus, the model successfully learns the underlying dynamics of gait (S2 Fig). The model’s internal states capture trial-specific dynamics predicting the time evolution of joint kinematics; activation coefficients (H) and memory cell states (C) and are tuned based on kinematic inputs. Kinematic data was input in multivariate format, not concatenated [62,63]. In brief, our RNN model was designed to capture short and long-term gait dependencies in time [64,65] as well as inter-and intra-limb coordination over time, uncovering features of gait that were not previously targeted or used in gait analysis. To verify whether our model was generalizable, we conducted leave-one-out cross validation, where 12 different models were trained leaving a single individual’s 6 trials on each model run (S3 Fig). Stroke-survivors are known for having neurological impairments that result in heterogeneous gait dysfunction that are not fully understood; thus, we anticipate that our gait dynamics model will capture and shed light on these individual-specific deficits in gait coordination, identify similar coordination strategies or deficits amongst our stroke cohort, and allow us to compare these different gait dysfunctions to the able-bodied ‘normative’ gait (controls). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Pipeline figure outlining the steps to generating individual-specific gait signatures. Continuous, multi-joint kinematics from multiple individuals are fed into the RNN model as input data and the model is trained sequence-to-sequence to predict one-step time shifted output kinematics. High dimensional internal parameter (H and C) time traces per individual are extracted and principal component analysis was applied to reduce the dimensionality of the data to form individual gait signatures (A). 3D time trace visualizations of 3 representative individuals (able-bodied (blue), high-functioning (red), low-functioning stroke (orange)) of the 1st 3 dominant principal component contributions (B, left). 3D projections of the 6-D gait signatures using multi-dimensional scaling (MDS) reveal different gait dynamics amongst the three gait groups: able-bodied (blue), high-functioning (red) and low-functioning (orange) stroke survivors (B, right). The size of the circles represents the individual’s trial speed (i.e., the smallest circles represent an individual’s slowest gait speed, and the size of the circles increase with gait speed). https://doi.org/10.1371/journal.pcbi.1011556.g001 Generating gait signatures To generate gait signatures, kinematic trajectories from each walking speed trial across participants were fed as input into the trained neural network and the corresponding internal states (H and C parameters, see above) were extracted (Fig 1A). The internal activations prescribe the spatial and temporal dependencies generating the input kinematics. The resulting time-series of 1024 internal states (512 H, 512 C parameters) were reduced in dimension using Principal Components Analysis (PCA) and phase averaged [66]. Phase averaging is applicable here, as the underlying gait dynamics are periodic, and the translation from time to a phase between 0 and 2π allows us to describe all internal state dynamics in a speed-independent manner. The first 6 Principal Components (PCs) explain ~72% of the variance in gait dynamics (S4 Fig), allowing us to focus on these modes for our visualization and analysis. The time-varying contributions of the first 3 dominant PCs were plotted in 3D for 3 representative individuals—able-bodied adults, high-functioning stroke (self-selected (SS) walking speed > 0.4m/s) and low-functioning stroke (SS speed < 0.4m/s)—highlighting that the gait dynamics between all 3 individuals are different (Fig 1B, left). The gait dynamics of the high-functioning stroke survivor (red), while spatially closer to the able-bodied individual (blue) than the low-functioning stroke survivor (orange), show observable differences in its dynamical trajectory between to the two individuals. To determine whether some structure exists amongst the three different subject groups, all the 6-dimensional gait signatures were projected onto a 3D map using Multidimensional scaling (MDS) [67] to visualize relative distances between all gait signatures (Fig 1B, right). The locations of the 3 MDS projections of the 3 representative individuals are not arbitrary, as they belong to clusters of gait signatures of the same gait group. Thus, gait signatures preserve key clinically relevant features of the underlying gait dynamics, independent of the individual or speed. Gait signatures reveal that individual-specific differences in dynamics are favored in the gait representation, over differences in gait speed Gait signatures of individuals’ 6 speed trials within both cohorts (healthy and stroke) are tightly grouped together. Gait signatures represent individual-specific dynamics; the unimpaired cohort exhibit a stereotyped low-dimensional structure across individuals in the able-bodied cohort (Fig 2Ai, left) vs. the impaired cohort, which display much more variable (i.e., highly individualized) low-dimensional representations (Fig 2Ai, right). Because the data are phase averaged over the gait cycle, we demonstrate that gait signature trajectories are well-aligned with the four gait phases (leg 1 swing, leg 1 stance, leg 2 swing, leg 2 stance), enabling phase-specific comparisons of differences in gait dynamics. The unimpaired group showed similar structure across the four gait events (Fig 2Aii, left), whereas there was much more variability within the impaired group (Fig 2Aii, right), revealing individual-specific differences within and across distinct parts of the gait cycle. The similarity between gait signatures was computed and visualized in a dimensionally reduced gait map space using MDS and colored according to the different individuals in the dataset (Fig 2Bi). The unimpaired group form a cluster in the gait map, showing that individuals in the unimpaired group are distinct from the impaired group. Stroke-survivors occupy distinct positions from other impaired individuals’ sub-clusters in the gait space that highlight the well-established but poorly understood heterogeneity in gait deficits in the stroke cohort. Furthermore, individual-specific gait signatures change slightly as individuals walk faster than their self-selected pace (Fig 2Bi). However, these within-subject speed-induced changes are much smaller than between-individual difference in gait signatures. The gait signatures of the individuals belonging to the able-bodied, high-functioning, and low functioning stroke survivor cohorts show 3x, 4x and 7x larger distances between individuals in the group versus within each individual 6 speed trials, respectively. We calculated the Euclidean distance between individuals’ self-selected speed trial gait signature and the calculated able-bodied centroid (Fig 2Bi, black square) and the results shown on the plot to the right reveal that low-functioning stroke survivors (characterized based on the clinical definition of having a self-selected walking speed of <0.4 m/s) are further away from the able-bodied cluster than the high-functioning stroke survivors. Note that no information from a clustering algorithm was used to characterize low- vs high-functioning individuals. The encircled low-functioning stroke survivors (Fig 2Bi, orange enclosure) were labeled post-hoc to demonstrate the lack of a single cluster characterizing low- vs high-functioning stroke survivors. Showing the validity of our approach, low-functioning stroke survivors are less dynamically similar to AB than higher functioning stroke survivors. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Gait signatures reveal highly individualized low dimensional representations of gait dynamics irrespective of absolute gait speed. A) 3D unimpaired (left) and impaired (right) gait signatures colored by i) individual and ii) gait phase. Gait signatures are grouped together according to individuals within both cohorts (same hues of blue cluster together for unimpaired (i, left) and similarly the same hues of red cluster in the impaired cohort (i, right)). In our convention the right leg of all unimpaired individuals was assigned to be the paretic leg and left leg the non-paretic leg. Impaired individuals can have either left or right leg paresis. Unimpaired gait signatures reveal a similar looped structure across the four gait phases that occur during a gait cycle (leg 1 swing, leg 1 stance, leg 2 swing, leg 2 stance) (ii, left) whereas impaired signatures showed individual-specific differences across the four phases and were more variable (ii, right). B) 3D multidimensional scaling applied to all gait signatures shows the pronounced separation between unimpaired (blue hues in left section of map) and impaired (red hues in right section of map) gait dynamics (i). Impaired signatures (red hues) are located further away from the centroid of all unimpaired gait signatures (black square), indicating that they are less dynamically similar to the unimpaired individuals. The smallest circles represent an individual’s self-selected walking speed trial and larger circles correspond to the faster speed trials. Low-functioning stroke survivors (encapsulated in orange; based on self-selected gait speed < 0.4m/s) are located furthest away (largest Euclidean distances) from the unimpaired centroid (i). Gait speed does not appear to strongly influence the differences in dynamics between individuals as similar speed gait signatures are in different regions of the gait map (ii). Particularly, gait speed does not explain the heterogeneity in low-functioning stroke survivors’ gait dynamics. https://doi.org/10.1371/journal.pcbi.1011556.g002 Gait speed does not appear to strongly influence the differences in dynamics between individuals’ gaits (although the range of gait speeds for each participant may not have been wide enough to elicit major differences in their overall dynamics). It is worth noting that the walking speeds in our post-stroke cohort spanned the full speed range of each participant’s safe walking capacity, whereas this was not the case for our able-bodied cohort. Overall, as expected, the unimpaired group walked at faster speeds than the impaired group (Fig 2Bii). Individuals in the able-bodied cluster walk at a range of different speeds, but individual gait signatures still cluster tightly together. Despite the able-bodied dynamics being similar, there still exists inherent variability in their gait dynamics that may be explained by factors such as prior exercise and sports-training history, injury, disease, etc. Post-stroke individuals who walk at similar slower speeds, however, maintain their own distinct individualized groupings. Thus, individuals’ characteristic gait signatures were preserved across their range of walking speeds and were not grouped based on absolute walking speed. For example, several clinically similar post-stroke individuals (similar overground walking speed and Fugl-Meyer score [68]) have very different gait signatures that remain recognizable across a range of gait speeds (Fig 2). Although the low-functioning individuals in our sample are more dispersed than high-functioning individuals, we expect that the spaces between individuals represent a continuum of gait dynamics that would be filled given a larger sample size. Furthermore, when used to distinguish between gait groups and identify individuals, gait signatures perform similarly to using a set of 26 commonly used discrete variables (S5 Fig). Discrete variables are already sufficient to classify between able-bodied and stroke gaits, with numerous studies identifying key variables that map to function/impairment [69–71]. With Gait signatures, we achieve the same level of classification without needing to hand-pick discrete variables or to use force plates or inverse-dynamics analyses that would require more equipment, computation, and subject-specific anthropometry for each observation. Gait signatures also perform better than continuous kinematics and joint velocities at these same discrimination tasks (S5 Fig). These results serve as a positive control, as researchers previously could distinguish gait groups by building a classifier based on important subjectively selected discrete variables. Here, we have created a dynamical representation that can distinguish groups with similar accuracy. It is not surprising that the continuous kinematics performed worse than the RNN gait signatures (which were developed from these very same data), as the RNN model used the data to encode important time-varying changes in the kinematics, allowing for more information to be extracted. Thus, parameterizing the evolution of individuals’ walking patterns into a common subspace allows for a more holistic, less biased, and straightforward analysis of primarily their overall differences in gait dynamics, inter- and intra-limb coordination over any differences attributed to absolute gait speed. Gait signatures can allow gait researchers to study or analyze the dynamical differences underlying impairment independently from gait speed, facilitating analysis of dynamics between individuals who may not be capable of walking at the same speeds and allowing investigation of changes in the underlying mechanism of gait changes under different conditions (walking speed, gait rehabilitation intervention, age etc.) Low-functioning stroke-survivors are less dynamically analogous to able-bodied and more dynamically variable compared to high-functioning stroke-survivors Clinically, gait rehabilitation researchers use gait speed as a primary quantitative indicator of gait dysfunction [19,72,73]. While this coarse metric gives an overall value or number to one’s overall gait function, it does not identify the specific impairments underlying the individuals’ gait. To derive more precise measures or indicators of gait impairment, we anticipated that utilizing this gait signatures framework, we would be able to capture both subtle and obvious differences in kinematic patterns underlying impaired gait. In the clinic, stroke survivors are typically segmented into subgroups according to their self-selected walking speeds: high-functioning stroke survivors with a self-selected (SS) walking speed above 0.4m/s and low-functioning stroke survivors who adopt SS walking speeds less than 0.4m/s [74]. It is assumed that low-functioning stroke survivors are more impaired and thus adopt slower walking speeds to be able to navigate the environment safely. However, gait deficits of stroke survivors within either sub-group are heterogeneous across individuals and include different impairments such as foot drop, reduced paretic push-off during late stance, limited initial heel contact during early stance, as well as compensatory gait strategies such as hip circumduction and hip hiking. We expected that higher functioning individuals would have less severe impairments and would be more dynamically analogous to able-bodied individuals, whereas low-functioning stroke survivors would exhibit highly variable impairments from each other and be even less dynamically analogous to able-bodied dynamics compared to higher functioning stroke survivors. To better visualize all developed individuals’ gait signatures across their 6 different speed trials in our dataset, we again used MDS to project the 6D gait signatures to 3D. This mapping allows us to visualize the relative locations of individuals in comparison to all the other gait signatures to gain insights on how dynamically similar they are from one another. A 3D MDS gait map of all gait signatures reveals that able-bodied and high-functioning stroke survivors are located near each other, whereas low-functioning stroke survivors are farther and more dispersed and form distinct clusters in different regions of the map (Fig 3A). Sub-group level analysis reveals significant differences in the Euclidean distance metric (distance between each gait signature and the able-bodied centroid) between the able-bodied group and the low- and high-functioning stroke survivor groups, respectively (Fig 3B). Able-bodied gait signatures are located closest to the centroid, followed by high-functioning and low-functioning stroke survivors (Fig 3B). The within-group dispersion of gait dynamics for the low- and high- functioning stroke survivors was calculated based on the radius of a hypersphere enclosing 95% of the groups’ gait signatures. Using a leave-one-out sample with replacement method, multiple within-group dispersion calculations were conducted for each group and the average within-group dispersion was expressed alongside the standard error in Fig 3C. The 95th percent radius was significantly higher in the low-functioning stroke-survivors gait signatures compared to the high-functioning, highlighting that low-functioning gait signatures were more dispersed from each other (higher inter-individual variability) and the RNN model can capture these individual-specific gait deficits in individuals with more severe gait impairment. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Comparison of gait signatures across three gait subgroups: able-bodied (AB), high functioning (HF) and low functioning (LF). A) 3D gait map using multidimensional scaling highlights the relative distances between AB (blue), HF (red) and LF (orange) stroke survivors. LF stroke survivors are less clustered and occupy distinct regions of the map away from the able-bodied centroid (black square). B) Gait dynamics similarity based on Euclidean distance between AB centroid and each participant, showing larger distances within the low-versus high-functioning groups. C) Within-group dispersion of gait signatures based on the radius of a hypersphere enclosing 95% of each group’s gait signature reveals more dispersed gait signatures in low- versus high-functioning stroke survivors, highlighting the potential of gait signatures to capture individual differences in more severe gait impairments. https://doi.org/10.1371/journal.pcbi.1011556.g003 Gait signatures are biomechanically interpretable While Principal Component trajectories and low-dimensional maps provide one way to compare the overall dynamics between individuals and groups, it remains to be seen what information the independent components of the 6D gait signature represent biomechanically. The contributions of each principal component (PC) to a gait signature fluctuates over the gait cycle, shown for an exemplar able-bodied, one high-functioning stroke survivor, and one low-functioning stroke survivor in Fig 4A. Superimposed individual stride-averaged PC projections from these 3 individuals (Fig 4B) highlight the specific differences in each PC. For PC1, both able-bodied and high-functioning stroke survivor traces are within the able-bodied 95% confidence interval, whereas the low-functioning stroke survivor is outside of these bounds around the middle of the gait cycle. For PC2, some regions of the low and high-functioning stroke survivor can be found outside of the confidence interval, however the entirety of the PC3 projection of the low-functioning stroke survivor is found outside of interval (vertically shifted). Given the generative nature of our RNN-based model, a specified number of the loadings on the PCs can be driven through the trained RNN model to reconstruct the corresponding kinematics. Thus, to interpret the individual PC components, the internal parameters corresponding to each isolated PC were driven through the gait dynamics model, generating gait predictions, i.e., a multi-joint coordination pattern and their temporal evolution over the gait cycle that can be visualized in an animation or gait movie. Stick figure snapshots (7 equally spaced samples of 100 frames) show that PC1 encodes dynamics driving hip flexion and extension, PC2 encodes dynamics driving knee flexion and extension and PC3 encodes dynamics driving primarily postural coordination (trunk location relative to joints) (S1–S4 Videos). This framework can potentially allow for the identification and targeting of individual-specific gait deficits, informing the tailoring of precision rehabilitation strategies. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Biomechanical interpretation of gait signatures. A) Gait signatures reveal different gait dynamics between exemplar AB, low-and high-functioning stroke survivors. B) The loadings on each principal component (PC), e.g., the contributions of each PC vary over the gait cycle and can be compared to the AB 95% confidence interval (gray). C) Each PC generates specific multi-joint gait coordination patterns when used to drive the gait model, enabling biomechanical interpretation of gait deficits and effects of treatment. https://doi.org/10.1371/journal.pcbi.1011556.g004 The gait dynamics model generalizes to unmeasured speeds Our gait signature model can capture and predict nonlinear changes in dynamics in response to speed in cases where interpolation of kinematics may fail. We trained a different gait dynamics model using only 15s of data of the 2 fastest and 2 slowest walking speeds of each subject. Weighted averages of gait signatures from an individual walking at these four different gait speeds can be used to generate multi-joint kinematic trajectories that predict data from a gait speed that was not used to train the model (Fig 5). Predicted kinematics from interpolation of gait signatures across the four speeds resemble the measured kinematic reference more accurately than do the kinematics generated from interpolating gait kinematics, shown for an exemplary AB individual (Fig 5A) and low-functioning stroke survivor (Fig 5B). Kinematic prediction from interpolation of dynamics did considerably better than interpolating kinematics directly for the exemplary low-functioning stroke survivor shown in Fig 5B, indicating that interpolating gait signatures capture nonlinear (non-monotonic) changes in kinematics between speeds. The kinematic output of the interpolated kinematics follows that of the fast speed in the paretic hip closely but does not resemble the measured kinematic reference waveforms for the paretic knee or ankle angles. In some cases where interpolation of kinematics fails, the averaged dynamics do a better job at predicting kinematic trajectories at unseen speeds. Group level analyses show that the R2 values between the measured and predicted kinematics from interpolated gait dynamics are significantly higher (Wilcoxon paired signed rank test) than interpolating kinematics within the able-bodied cohort (Fig 5C), but not for stroke (Fig 5D). In general, averaging gait dynamics produced less variable R2 values and less R2 outliers than averaging kinematics in both the able-bodied (Fig 5C) and stroke survivors (Fig 5D). The range of R2 values in the able-bodied cohort for averaged dynamics was -0.20 to 1.00 compared to –1.30 to 0.98 in averaged kinematics whereas the range of R2 values in the stroke cohort for averaged dynamics was 0.46 to 1.00 compared to -0.50 to 1.00 in averaged kinematics. Two low-functioning stroke survivors show higher R2 values of their hip, knee and ankle kinematic traces when interpolating kinematics vs. dynamics. Post hoc analysis revealed that these two stroke survivors (ST4 and ST2) were furthest away from the able-bodied centroid (least dynamically similar to able-bodied) as shown in Fig 2Bi. These results suggest that the RNN largely captures more stereotyped able-bodied dynamics and has a harder time learning the dynamics from more variable stroke individuals, especially those that deviate furthest from able-bodied. We acknowledge that our model likely is not capable of generalizing to speeds beyond the ranges of the input data (extrapolating), as RNNs are highly dependent on the training data that it sees to learn patterns in the dataset. One benefit of this capability, however, is that any data that deviates from the walking speeds in the training set can still be analyzed, reducing the number of speeds required in the training set to achieve a model that is valid across a range of speeds. Additionally, our small sample size limits the amount of data the RNN sees for each diverse type of stroke dynamics, thus, with a larger sample size of stroke survivors and longer trials, the RNN may be able to make better kinematic predictions of lower-functioning stroke survivors. Moreover, this result highlights the utility in predicting kinematics in unseen conditions which in contrast cannot be made using discrete biomechanical or clinical metrics, nor with current biophysical models. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Data-driven gait dynamics model predicts non-linear changes in joint kinematics with gait speed. Gait predictions of joint kinematics (green) at intermediate gait speeds not used in model training were generated by interpolating gait signatures between slow (dashed grey) and fast speeds (dashed black) lines and using them to drive the gait model. Interpolated kinematics from gait dynamics (green) and interpolated directly from kinematics (blue) were compared to the measured reference kinematics (black solid). A) Predictions in an exemplar AB participant are more accurate when interpolating gait signatures compared to interpolating gait kinematics across speeds. B) In an exemplar low-functioning stroke survivor, interpolated gait signatures predict nonlinear changes in kinematics better at intermediate speeds than interpolated gait kinematics. Averaging the kinematics fail in this case where there are larger differences between the slow and fast speed paretic kinematics; the averaged kinematics (blue) follow the fast speed paretic hip kinematics whereas the other angles do not reflect waveforms that resemble either the fast or slow speed. The gait model can therefore predict movement reasonably well when interpolating between tested speeds. There is a statistically significant difference between group level R2 comparisons (kinematics generated from interpolated dynamics vs interpolated kinematics) in the able-bodied (C) but not in stroke (D) cohorts. However, the range of R2 values are larger in both able-bodied and stroke kinematic predictions resulting from interpolated kinematics (-1.30–0.98, -0.50–1.00 respectively) vs. predicted from interpolated gait dynamics (-0.20–1.00,0.46–1.00 respectively). Thus, while the R2 values may not improve on average for the stroke survivors, the model’s performance is more robust overall. https://doi.org/10.1371/journal.pcbi.1011556.g005 Gait sculpting: Manipulating the PC components of an individual’s gait signature identifies specific coordination deficits in stroke survivors Previously, we showed that we can leverage our model to reconstruct the kinematics of healthy PC projections of the gait signature to gain insight into their independent biomechanical interpretations. However, identifying and interpreting the biomechanics related to impaired PC dynamics of stroke-survivors’ gait would prove to be even more beneficial, as these dynamics can potentially serve as rehabilitation targets when designing tailored gait intervention/strategies for individuals. Here we present an example of how we use gait signatures to identify specific biomechanical or coordination targets in specific stroke survivors. Specifically, we utilize our finding that the phase-varying contributions of the 6 principal projections of the gait signature differ in individual-specific manners (Fig 6A). For example, AB2’s 6 PC contributions all lie within the 95% confidence interval of all able-bodied individuals. ST4 primarily shows major deviation from AB in PC 3 (located entirely above the AB confidence interval), impaired dynamics during paretic swing in PC 4 and overall irregular shapes in PC 5 and 6. ST2’s PC3 is largely within the AB confidence interval, however PC 4’s paretic swing shows major deviation, their PC 5 contribution is shifted below the AB confidence interval and PC 6 shows an irregular shape. ST3’s PC5 projection lies below the AB confidence interval and PC 6 projection is irregularly shaped compared to AB. To validate our finding that suggested that PC 3 primarily influences hip flexion or extension, we exchanged AB2’s healthy PC1 with that of ST4 (Fig 6A, orange boxes and arrow) and we observed if and how AB2’s original hip joint kinematics (Fig 6B, orange box, black trace) deviated (Fig 6B, orange box, red dashed trace and S5 Video). To gain further insight into how the other PC deviations manifest in movement, we manipulated the PC3 projection of AB2 by replacing it with that of ST4 (Fig 6A, brown boxes and arrow). The kinematic reconstruction from this manipulation (Fig 6B, brown box, red dashed trace and S6 Video) shows a vertical shift downwards for bilateral hip angles and the non-paretic knee. The vertical shifts in the hip flexion/extension angles suggest a major difference in this individual’s posture (perhaps stroke individual leaned forward more during gait) compared to able-bodied. We manipulated AB2’s PC4 projection by replacing it with that of ST2 (Fig 6A, green boxes and arrow). This manipulation affected specifically the paretic and non-paretic ankle angles and both knee joints primarily during the period between non-paretic stance and paretic swing (Fig 6B, green box, red dashed trace and S7 Video). This result highlights a coordination deficit between these specific joint angles and, if targeted accurately, may allow for corrected gait patterns of this stroke survivor. Conversely, we tested the effects of replacing an impaired PC projection with a healthy one to observe how gait impairments can potentially be improved. We replaced ST3’s PC5 with that of AB2 (Fig 6A, purple boxes and arrow) and observed a substantial change in the magnitude and shape of bilateral ankle angle trajectories and slight increase in non-paretic knee magnitude (Fig 6B, purple box, red dashed trace and S8 Video). We can infer that to make improvements to ST3’s PC5 towards able-bodied or normative kinematics, rehabilitation focusing on these specific knee and ankle strategies may prove useful. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. Gait sculpting: interpolating between components of able-bodied and stroke gait dynamics to visualize anticipated gait improvement. The components of individuals’ gait signatures can be manipulated (gait sculpting) to understand the relationship between specific underlying dynamics and their corresponding kinematic phenotype. A) The projection on each of the 1st 6 principal components (PCs) can be observed for a representative able-bodied (AB2), two low functioning stroke-survivors each having similar self-selected (SS) speeds and Fugl-Meyer (FM) scores (ST2 & ST4, as denoted in Fig 2) and another low functioning stroke survivor (ST3) who has a higher FM score and faster SS walking speed. The PC projections are colored according to the 4 gait phases (non-paretic swing, non-paretic stance, paretic swing, paretic stance). The right leg of the unimpaired individuals was arbitrarily assigned to be paretic and the left leg, non-paretic for consistency. Colored boxes and arrows (orange, brown, green, purple) show specific, single PC manipulations, for example, the orange boxes and arrow illustrate that the PC 1 projection of AB2 was replaced with the impaired PC 1 projection from ST4. B) The AB2:ST4 manipulation (orange) shows how AB2’s original phase averaged kinematics (black trace) was manipulated by ST4’s impaired PC 1 projection (red dashed traced). ST4’s impaired PC 1 manifests in AB2’s healthy kinematics showing deviation primarily in the hip kinematics (as suggested in Fig 4 where healthy PC 1 encodes a kinematic subcomponent corresponding to hip flexion/extension) and some deviation in the ankle angles, especially the paretic ankle. The AB2:ST4 manipulation (brown) shows how ST4’s impaired PC3 manifests in AB2’s healthy kinematics; we observe a vertical shift downwards (red trace) of the bilateral hip angles as well as the non-paretic knee. This change in hip flexion highlights that this impaired PC3 encodes a reduction in the hip flexion angles; pointing to a more crouched gait (trunk is leaning forward more). The AB2:ST2 manipulation (green) shows replacing AB2’s PC4 projection with ST2’s impaired PC4 dynamics shows deviation in the knee joints especially during paretic swing, a vertical shift upwards in the paretic ankle angle kinematics and deviations around the middle of the gait cycle (transition between non-paretic stance and paretic swing) in the non-paretic ankle kinematics. Alternatively, the AB2:ST3 manipulation (purple) the impaired PC5 in ST3 is replaced with the healthy PC5 projection from AB2 resulting in slight increase in non-paretic knee magnitude and reduced amplitude of paretic and non-paretic ankle flexion. The result of this manipulation points to potential predicted improvements (or deviations) that can occur when aiming to mimic PC5 healthy dynamics in this stroke survivor allowing offline in-silico testing of potential avenues for gait rehabilitation for this stroke survivor. https://doi.org/10.1371/journal.pcbi.1011556.g006 Self-driven signatures: Our gait dynamics model revealed robustness of gait predictions establishing the utility of gait signatures in precision medicine The ability to predict future kinematics based on measured data is key to rapid, virtual design of personalized interventions. We demonstrate that the recurrent neural network model of gait dynamics, once primed with several gait cycles of data from either able-bodied or stroke participants, can predict future joint angle trajectories (Fig 7). Once the network is primed, an initial posture is presented (initial condition, denoted by blue vertical bar) after which the model self-drives i.e., predicts the general shape of future kinematics in a feedforward manner (without referencing previous measured data points) in an able-bodied (Fig 7Ai, left) and stroke individual (Fig 7Bii, left). A smooth transition is seen between the previously measured gait cycle (green) and the self-driven cycle (red trace) for both AB and stroke (Fig 7Ai, right and 7Aii, right respectively).To verify that the model was not generating a gait cycle prediction entirely by chance, we calculated the Euclidean distance between the kinematics of the predicted (self-driven) gait cycle and the kinematics from each of the measured gait cycles. We computed the distribution of distances between each predicted gait cycle with all other gait cycles from the same individual (Fig 7B, purple bars). We then compared the predicted gait cycle to the target gait cycle (Fig 7B, red bars). In the able-bodied individual, the predicted gait cycles are more similar to the target gait cycle (Fig 7Bi, red bar) than 79% of all gait cycles. However, in the stroke survivor, 60% of other gait cycles were more similar to the predicted gait cycle than the target gait cycle (Fig 7Bii, red bar). This suggests that the model is less able to accurately predict future kinematics in stroke gait. Note, to calculate the Euclidean distances between the gait cycles, we need to normalize the period of each gait cycle to the period of the self-driven cycle. To avoid the potential of bias due to this normalization, we also performed a comparison using a metric that was not manipulated in time–gait cycle duration. After priming the model, we presented the model with the first posture of the trial and ran the network forward in self-driving mode for the remainder of the trial length (15 seconds). Able-bodied self-driven predicted kinematics resembled the reference kinematics closely (Fig 7Ci, top plot) whereas stroke self-driven predicted kinematics matched the first gait cycle closely but soon converged to patterns reflecting able-bodied kinematics (Fig 7Cii, top plot). The gait cycle duration of the first few cycles of the self-driven kinematics match those of the measured kinematics (blue dots located close to the y = x line) in both the exemplary able-bodied (Fig 7Ci, bottom plot) and stroke individual shown (Fig 7Cii, bottom plot), however kinematics soon diverged to shorter and relatively consistent gait cycle durations (blue dots appearing almost horizontal in the plots) result in both cases. The model may preferentially predict able-bodied kinematics, which were less variable between individuals than were post-stroke kinematics. These results highlight that the model encodes gait dynamics that can predict kinematics over short timescales, but the variability and amount of training data may influence predictive power over long timescales. Our model provides a foundation for a sample-specific gait dynamics model to predict the effects of environmental perturbations, assistive devices, and treatments without extensive experimental sessions. In-silico predictions using the gait dynamics model may thus reduce the experimental time, cost, and participant burden required for personalized gait characterization, treatment personalization, and device design. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. Our trained RNN model can predict the time evolution of kinematics from an initial posture. The trained gait dynamics model can predict individual-specific time-evolution of gait kinematics from an arbitrary initial posture (self-driving) in able-bodied (A, i) and stroke (A, ii) once the network is primed with several gait cycles of an individual’s data (gait cyclex-n(measured), black solid). This predictive ability shows that the model encodes the gait dynamics underlying movement. Despite inter-cycle kinematic variability, the gait dynamics model can predict the general shape of the next gait cycle of kinematics (gait cyclex(predicted), red) in an able-bodied individual (A, i) and stroke survivor (A, ii), however, predicted kinematics (red) show larger deviation from the measured reference gait cycle (gait cyclex(measured), black dashed) in the stroke survivor. A smooth transition exists between the measured kinematics from the gait cycle preceding (gait cyclex-1(measured), green) the self-driven predicted cycle (red). For the representative able-bodied individual (B, i), the Euclidean distance (deviation) between the predicted gait cycle of kinematics and its respective measured kinematics (reference) is ~79% lower than the distance between the other gait cycles in the trial; ruling out that the kinematic predictions are attributed to chance. The deviation (Euclidean distance) of the predicted gait cycle of stroke (B, ii) kinematics to its reference gait cycle is ~40% lower than the distance between the other gait cycles in the trial. This suggests that the dynamical model is less able to accurately predict stroke kinematics better than chance. The dynamical model was first initialized with all the trial’s kinematics data (15 seconds) (black trace) after which the trial’s initial posture was presented to the model to self-drive kinematics (red trace) in feedforward mode for 15 seconds (C, i, top plot). The duration of each gait cycle from the measured kinematics is not well encoded by the dynamical model; gait cycle durations of the predicted kinematics are typically underestimated in both able-bodied (C, i, bottom plot) and stroke (C, ii, bottom plot) (to a larger degree) in self-driving mode and as such deviate from the y = x reference line (black). https://doi.org/10.1371/journal.pcbi.1011556.g007 Gait signatures: A low-dimensional representation of gait dynamics We used motion capture to collect sagittal-plane kinematic data that consisted of 15 seconds of continuous gait kinematics from bilateral, hip, knee, and ankle joints from 5 able-bodied (AB) participants and 7 stroke survivors (> 6 months post-stroke, gait speeds 0.1 to 0.8 m/s) walking on a treadmill at a range of six different speeds each. Taking inspiration from neural network models that capture neural dynamics [58–60] and biological systems, we implemented a recurrent neural network (RNN) model to capture the dynamical properties of gait. Our model input parameters only include kinematic data and do not include anthropometric information or clinical characteristics and do not account for differences in joint kinematics due to neural versus biomechanical constraints. Developing the recurrent neural network (RNN) architecture and training the model The gait dynamics model was developed in Python using common Python libraries, including TensorFlow, Keras, Pandas, and NumPy. We developed our code in Google Colab to facilitate open-source sharing of our dynamic framework, which can be found here: https://github.com/bermanlabemory/gait_signatures. The model architecture was selected based on two criteria: 1) minimizing model training and validation loss during model fitting, and 2) maximizing the similarity of short-time (single stride) and long-time (multiple strides) self-driven model predictions (termed: gait signature alignment) post model training (S1 Fig). By implementing these two model selection criteria we ensure 1) a high goodness-of-fit (model that best represents the underlying dynamics across all participants and gait speeds) and 2) the model is capable of predicting the time-evolution of gait (encode gait dynamics). We evaluated these criteria against alternative models by varying 2 hyperparameters (number of LSTM units and the lookback time, see Methods). The selected model architecture is a sequence-to-sequence RNN [61] consisting of an input layer, a hidden layer of 512 LSTM units, and an output layer. The RNN learns a map from time-series kinematic input data (0 to T-1) to kinematics one time-step in the future (1 to T) for all training trials (Fig 1A). The model was trained using the ‘mean squared error’ (mse) loss function until training and validation error converged and stabilized around the same point (< 0.03 degrees2). Thus, the model successfully learns the underlying dynamics of gait (S2 Fig). The model’s internal states capture trial-specific dynamics predicting the time evolution of joint kinematics; activation coefficients (H) and memory cell states (C) and are tuned based on kinematic inputs. Kinematic data was input in multivariate format, not concatenated [62,63]. In brief, our RNN model was designed to capture short and long-term gait dependencies in time [64,65] as well as inter-and intra-limb coordination over time, uncovering features of gait that were not previously targeted or used in gait analysis. To verify whether our model was generalizable, we conducted leave-one-out cross validation, where 12 different models were trained leaving a single individual’s 6 trials on each model run (S3 Fig). Stroke-survivors are known for having neurological impairments that result in heterogeneous gait dysfunction that are not fully understood; thus, we anticipate that our gait dynamics model will capture and shed light on these individual-specific deficits in gait coordination, identify similar coordination strategies or deficits amongst our stroke cohort, and allow us to compare these different gait dysfunctions to the able-bodied ‘normative’ gait (controls). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Pipeline figure outlining the steps to generating individual-specific gait signatures. Continuous, multi-joint kinematics from multiple individuals are fed into the RNN model as input data and the model is trained sequence-to-sequence to predict one-step time shifted output kinematics. High dimensional internal parameter (H and C) time traces per individual are extracted and principal component analysis was applied to reduce the dimensionality of the data to form individual gait signatures (A). 3D time trace visualizations of 3 representative individuals (able-bodied (blue), high-functioning (red), low-functioning stroke (orange)) of the 1st 3 dominant principal component contributions (B, left). 3D projections of the 6-D gait signatures using multi-dimensional scaling (MDS) reveal different gait dynamics amongst the three gait groups: able-bodied (blue), high-functioning (red) and low-functioning (orange) stroke survivors (B, right). The size of the circles represents the individual’s trial speed (i.e., the smallest circles represent an individual’s slowest gait speed, and the size of the circles increase with gait speed). https://doi.org/10.1371/journal.pcbi.1011556.g001 Generating gait signatures To generate gait signatures, kinematic trajectories from each walking speed trial across participants were fed as input into the trained neural network and the corresponding internal states (H and C parameters, see above) were extracted (Fig 1A). The internal activations prescribe the spatial and temporal dependencies generating the input kinematics. The resulting time-series of 1024 internal states (512 H, 512 C parameters) were reduced in dimension using Principal Components Analysis (PCA) and phase averaged [66]. Phase averaging is applicable here, as the underlying gait dynamics are periodic, and the translation from time to a phase between 0 and 2π allows us to describe all internal state dynamics in a speed-independent manner. The first 6 Principal Components (PCs) explain ~72% of the variance in gait dynamics (S4 Fig), allowing us to focus on these modes for our visualization and analysis. The time-varying contributions of the first 3 dominant PCs were plotted in 3D for 3 representative individuals—able-bodied adults, high-functioning stroke (self-selected (SS) walking speed > 0.4m/s) and low-functioning stroke (SS speed < 0.4m/s)—highlighting that the gait dynamics between all 3 individuals are different (Fig 1B, left). The gait dynamics of the high-functioning stroke survivor (red), while spatially closer to the able-bodied individual (blue) than the low-functioning stroke survivor (orange), show observable differences in its dynamical trajectory between to the two individuals. To determine whether some structure exists amongst the three different subject groups, all the 6-dimensional gait signatures were projected onto a 3D map using Multidimensional scaling (MDS) [67] to visualize relative distances between all gait signatures (Fig 1B, right). The locations of the 3 MDS projections of the 3 representative individuals are not arbitrary, as they belong to clusters of gait signatures of the same gait group. Thus, gait signatures preserve key clinically relevant features of the underlying gait dynamics, independent of the individual or speed. Gait signatures reveal that individual-specific differences in dynamics are favored in the gait representation, over differences in gait speed Gait signatures of individuals’ 6 speed trials within both cohorts (healthy and stroke) are tightly grouped together. Gait signatures represent individual-specific dynamics; the unimpaired cohort exhibit a stereotyped low-dimensional structure across individuals in the able-bodied cohort (Fig 2Ai, left) vs. the impaired cohort, which display much more variable (i.e., highly individualized) low-dimensional representations (Fig 2Ai, right). Because the data are phase averaged over the gait cycle, we demonstrate that gait signature trajectories are well-aligned with the four gait phases (leg 1 swing, leg 1 stance, leg 2 swing, leg 2 stance), enabling phase-specific comparisons of differences in gait dynamics. The unimpaired group showed similar structure across the four gait events (Fig 2Aii, left), whereas there was much more variability within the impaired group (Fig 2Aii, right), revealing individual-specific differences within and across distinct parts of the gait cycle. The similarity between gait signatures was computed and visualized in a dimensionally reduced gait map space using MDS and colored according to the different individuals in the dataset (Fig 2Bi). The unimpaired group form a cluster in the gait map, showing that individuals in the unimpaired group are distinct from the impaired group. Stroke-survivors occupy distinct positions from other impaired individuals’ sub-clusters in the gait space that highlight the well-established but poorly understood heterogeneity in gait deficits in the stroke cohort. Furthermore, individual-specific gait signatures change slightly as individuals walk faster than their self-selected pace (Fig 2Bi). However, these within-subject speed-induced changes are much smaller than between-individual difference in gait signatures. The gait signatures of the individuals belonging to the able-bodied, high-functioning, and low functioning stroke survivor cohorts show 3x, 4x and 7x larger distances between individuals in the group versus within each individual 6 speed trials, respectively. We calculated the Euclidean distance between individuals’ self-selected speed trial gait signature and the calculated able-bodied centroid (Fig 2Bi, black square) and the results shown on the plot to the right reveal that low-functioning stroke survivors (characterized based on the clinical definition of having a self-selected walking speed of <0.4 m/s) are further away from the able-bodied cluster than the high-functioning stroke survivors. Note that no information from a clustering algorithm was used to characterize low- vs high-functioning individuals. The encircled low-functioning stroke survivors (Fig 2Bi, orange enclosure) were labeled post-hoc to demonstrate the lack of a single cluster characterizing low- vs high-functioning stroke survivors. Showing the validity of our approach, low-functioning stroke survivors are less dynamically similar to AB than higher functioning stroke survivors. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Gait signatures reveal highly individualized low dimensional representations of gait dynamics irrespective of absolute gait speed. A) 3D unimpaired (left) and impaired (right) gait signatures colored by i) individual and ii) gait phase. Gait signatures are grouped together according to individuals within both cohorts (same hues of blue cluster together for unimpaired (i, left) and similarly the same hues of red cluster in the impaired cohort (i, right)). In our convention the right leg of all unimpaired individuals was assigned to be the paretic leg and left leg the non-paretic leg. Impaired individuals can have either left or right leg paresis. Unimpaired gait signatures reveal a similar looped structure across the four gait phases that occur during a gait cycle (leg 1 swing, leg 1 stance, leg 2 swing, leg 2 stance) (ii, left) whereas impaired signatures showed individual-specific differences across the four phases and were more variable (ii, right). B) 3D multidimensional scaling applied to all gait signatures shows the pronounced separation between unimpaired (blue hues in left section of map) and impaired (red hues in right section of map) gait dynamics (i). Impaired signatures (red hues) are located further away from the centroid of all unimpaired gait signatures (black square), indicating that they are less dynamically similar to the unimpaired individuals. The smallest circles represent an individual’s self-selected walking speed trial and larger circles correspond to the faster speed trials. Low-functioning stroke survivors (encapsulated in orange; based on self-selected gait speed < 0.4m/s) are located furthest away (largest Euclidean distances) from the unimpaired centroid (i). Gait speed does not appear to strongly influence the differences in dynamics between individuals as similar speed gait signatures are in different regions of the gait map (ii). Particularly, gait speed does not explain the heterogeneity in low-functioning stroke survivors’ gait dynamics. https://doi.org/10.1371/journal.pcbi.1011556.g002 Gait speed does not appear to strongly influence the differences in dynamics between individuals’ gaits (although the range of gait speeds for each participant may not have been wide enough to elicit major differences in their overall dynamics). It is worth noting that the walking speeds in our post-stroke cohort spanned the full speed range of each participant’s safe walking capacity, whereas this was not the case for our able-bodied cohort. Overall, as expected, the unimpaired group walked at faster speeds than the impaired group (Fig 2Bii). Individuals in the able-bodied cluster walk at a range of different speeds, but individual gait signatures still cluster tightly together. Despite the able-bodied dynamics being similar, there still exists inherent variability in their gait dynamics that may be explained by factors such as prior exercise and sports-training history, injury, disease, etc. Post-stroke individuals who walk at similar slower speeds, however, maintain their own distinct individualized groupings. Thus, individuals’ characteristic gait signatures were preserved across their range of walking speeds and were not grouped based on absolute walking speed. For example, several clinically similar post-stroke individuals (similar overground walking speed and Fugl-Meyer score [68]) have very different gait signatures that remain recognizable across a range of gait speeds (Fig 2). Although the low-functioning individuals in our sample are more dispersed than high-functioning individuals, we expect that the spaces between individuals represent a continuum of gait dynamics that would be filled given a larger sample size. Furthermore, when used to distinguish between gait groups and identify individuals, gait signatures perform similarly to using a set of 26 commonly used discrete variables (S5 Fig). Discrete variables are already sufficient to classify between able-bodied and stroke gaits, with numerous studies identifying key variables that map to function/impairment [69–71]. With Gait signatures, we achieve the same level of classification without needing to hand-pick discrete variables or to use force plates or inverse-dynamics analyses that would require more equipment, computation, and subject-specific anthropometry for each observation. Gait signatures also perform better than continuous kinematics and joint velocities at these same discrimination tasks (S5 Fig). These results serve as a positive control, as researchers previously could distinguish gait groups by building a classifier based on important subjectively selected discrete variables. Here, we have created a dynamical representation that can distinguish groups with similar accuracy. It is not surprising that the continuous kinematics performed worse than the RNN gait signatures (which were developed from these very same data), as the RNN model used the data to encode important time-varying changes in the kinematics, allowing for more information to be extracted. Thus, parameterizing the evolution of individuals’ walking patterns into a common subspace allows for a more holistic, less biased, and straightforward analysis of primarily their overall differences in gait dynamics, inter- and intra-limb coordination over any differences attributed to absolute gait speed. Gait signatures can allow gait researchers to study or analyze the dynamical differences underlying impairment independently from gait speed, facilitating analysis of dynamics between individuals who may not be capable of walking at the same speeds and allowing investigation of changes in the underlying mechanism of gait changes under different conditions (walking speed, gait rehabilitation intervention, age etc.) Low-functioning stroke-survivors are less dynamically analogous to able-bodied and more dynamically variable compared to high-functioning stroke-survivors Clinically, gait rehabilitation researchers use gait speed as a primary quantitative indicator of gait dysfunction [19,72,73]. While this coarse metric gives an overall value or number to one’s overall gait function, it does not identify the specific impairments underlying the individuals’ gait. To derive more precise measures or indicators of gait impairment, we anticipated that utilizing this gait signatures framework, we would be able to capture both subtle and obvious differences in kinematic patterns underlying impaired gait. In the clinic, stroke survivors are typically segmented into subgroups according to their self-selected walking speeds: high-functioning stroke survivors with a self-selected (SS) walking speed above 0.4m/s and low-functioning stroke survivors who adopt SS walking speeds less than 0.4m/s [74]. It is assumed that low-functioning stroke survivors are more impaired and thus adopt slower walking speeds to be able to navigate the environment safely. However, gait deficits of stroke survivors within either sub-group are heterogeneous across individuals and include different impairments such as foot drop, reduced paretic push-off during late stance, limited initial heel contact during early stance, as well as compensatory gait strategies such as hip circumduction and hip hiking. We expected that higher functioning individuals would have less severe impairments and would be more dynamically analogous to able-bodied individuals, whereas low-functioning stroke survivors would exhibit highly variable impairments from each other and be even less dynamically analogous to able-bodied dynamics compared to higher functioning stroke survivors. To better visualize all developed individuals’ gait signatures across their 6 different speed trials in our dataset, we again used MDS to project the 6D gait signatures to 3D. This mapping allows us to visualize the relative locations of individuals in comparison to all the other gait signatures to gain insights on how dynamically similar they are from one another. A 3D MDS gait map of all gait signatures reveals that able-bodied and high-functioning stroke survivors are located near each other, whereas low-functioning stroke survivors are farther and more dispersed and form distinct clusters in different regions of the map (Fig 3A). Sub-group level analysis reveals significant differences in the Euclidean distance metric (distance between each gait signature and the able-bodied centroid) between the able-bodied group and the low- and high-functioning stroke survivor groups, respectively (Fig 3B). Able-bodied gait signatures are located closest to the centroid, followed by high-functioning and low-functioning stroke survivors (Fig 3B). The within-group dispersion of gait dynamics for the low- and high- functioning stroke survivors was calculated based on the radius of a hypersphere enclosing 95% of the groups’ gait signatures. Using a leave-one-out sample with replacement method, multiple within-group dispersion calculations were conducted for each group and the average within-group dispersion was expressed alongside the standard error in Fig 3C. The 95th percent radius was significantly higher in the low-functioning stroke-survivors gait signatures compared to the high-functioning, highlighting that low-functioning gait signatures were more dispersed from each other (higher inter-individual variability) and the RNN model can capture these individual-specific gait deficits in individuals with more severe gait impairment. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Comparison of gait signatures across three gait subgroups: able-bodied (AB), high functioning (HF) and low functioning (LF). A) 3D gait map using multidimensional scaling highlights the relative distances between AB (blue), HF (red) and LF (orange) stroke survivors. LF stroke survivors are less clustered and occupy distinct regions of the map away from the able-bodied centroid (black square). B) Gait dynamics similarity based on Euclidean distance between AB centroid and each participant, showing larger distances within the low-versus high-functioning groups. C) Within-group dispersion of gait signatures based on the radius of a hypersphere enclosing 95% of each group’s gait signature reveals more dispersed gait signatures in low- versus high-functioning stroke survivors, highlighting the potential of gait signatures to capture individual differences in more severe gait impairments. https://doi.org/10.1371/journal.pcbi.1011556.g003 Gait signatures are biomechanically interpretable While Principal Component trajectories and low-dimensional maps provide one way to compare the overall dynamics between individuals and groups, it remains to be seen what information the independent components of the 6D gait signature represent biomechanically. The contributions of each principal component (PC) to a gait signature fluctuates over the gait cycle, shown for an exemplar able-bodied, one high-functioning stroke survivor, and one low-functioning stroke survivor in Fig 4A. Superimposed individual stride-averaged PC projections from these 3 individuals (Fig 4B) highlight the specific differences in each PC. For PC1, both able-bodied and high-functioning stroke survivor traces are within the able-bodied 95% confidence interval, whereas the low-functioning stroke survivor is outside of these bounds around the middle of the gait cycle. For PC2, some regions of the low and high-functioning stroke survivor can be found outside of the confidence interval, however the entirety of the PC3 projection of the low-functioning stroke survivor is found outside of interval (vertically shifted). Given the generative nature of our RNN-based model, a specified number of the loadings on the PCs can be driven through the trained RNN model to reconstruct the corresponding kinematics. Thus, to interpret the individual PC components, the internal parameters corresponding to each isolated PC were driven through the gait dynamics model, generating gait predictions, i.e., a multi-joint coordination pattern and their temporal evolution over the gait cycle that can be visualized in an animation or gait movie. Stick figure snapshots (7 equally spaced samples of 100 frames) show that PC1 encodes dynamics driving hip flexion and extension, PC2 encodes dynamics driving knee flexion and extension and PC3 encodes dynamics driving primarily postural coordination (trunk location relative to joints) (S1–S4 Videos). This framework can potentially allow for the identification and targeting of individual-specific gait deficits, informing the tailoring of precision rehabilitation strategies. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Biomechanical interpretation of gait signatures. A) Gait signatures reveal different gait dynamics between exemplar AB, low-and high-functioning stroke survivors. B) The loadings on each principal component (PC), e.g., the contributions of each PC vary over the gait cycle and can be compared to the AB 95% confidence interval (gray). C) Each PC generates specific multi-joint gait coordination patterns when used to drive the gait model, enabling biomechanical interpretation of gait deficits and effects of treatment. https://doi.org/10.1371/journal.pcbi.1011556.g004 The gait dynamics model generalizes to unmeasured speeds Our gait signature model can capture and predict nonlinear changes in dynamics in response to speed in cases where interpolation of kinematics may fail. We trained a different gait dynamics model using only 15s of data of the 2 fastest and 2 slowest walking speeds of each subject. Weighted averages of gait signatures from an individual walking at these four different gait speeds can be used to generate multi-joint kinematic trajectories that predict data from a gait speed that was not used to train the model (Fig 5). Predicted kinematics from interpolation of gait signatures across the four speeds resemble the measured kinematic reference more accurately than do the kinematics generated from interpolating gait kinematics, shown for an exemplary AB individual (Fig 5A) and low-functioning stroke survivor (Fig 5B). Kinematic prediction from interpolation of dynamics did considerably better than interpolating kinematics directly for the exemplary low-functioning stroke survivor shown in Fig 5B, indicating that interpolating gait signatures capture nonlinear (non-monotonic) changes in kinematics between speeds. The kinematic output of the interpolated kinematics follows that of the fast speed in the paretic hip closely but does not resemble the measured kinematic reference waveforms for the paretic knee or ankle angles. In some cases where interpolation of kinematics fails, the averaged dynamics do a better job at predicting kinematic trajectories at unseen speeds. Group level analyses show that the R2 values between the measured and predicted kinematics from interpolated gait dynamics are significantly higher (Wilcoxon paired signed rank test) than interpolating kinematics within the able-bodied cohort (Fig 5C), but not for stroke (Fig 5D). In general, averaging gait dynamics produced less variable R2 values and less R2 outliers than averaging kinematics in both the able-bodied (Fig 5C) and stroke survivors (Fig 5D). The range of R2 values in the able-bodied cohort for averaged dynamics was -0.20 to 1.00 compared to –1.30 to 0.98 in averaged kinematics whereas the range of R2 values in the stroke cohort for averaged dynamics was 0.46 to 1.00 compared to -0.50 to 1.00 in averaged kinematics. Two low-functioning stroke survivors show higher R2 values of their hip, knee and ankle kinematic traces when interpolating kinematics vs. dynamics. Post hoc analysis revealed that these two stroke survivors (ST4 and ST2) were furthest away from the able-bodied centroid (least dynamically similar to able-bodied) as shown in Fig 2Bi. These results suggest that the RNN largely captures more stereotyped able-bodied dynamics and has a harder time learning the dynamics from more variable stroke individuals, especially those that deviate furthest from able-bodied. We acknowledge that our model likely is not capable of generalizing to speeds beyond the ranges of the input data (extrapolating), as RNNs are highly dependent on the training data that it sees to learn patterns in the dataset. One benefit of this capability, however, is that any data that deviates from the walking speeds in the training set can still be analyzed, reducing the number of speeds required in the training set to achieve a model that is valid across a range of speeds. Additionally, our small sample size limits the amount of data the RNN sees for each diverse type of stroke dynamics, thus, with a larger sample size of stroke survivors and longer trials, the RNN may be able to make better kinematic predictions of lower-functioning stroke survivors. Moreover, this result highlights the utility in predicting kinematics in unseen conditions which in contrast cannot be made using discrete biomechanical or clinical metrics, nor with current biophysical models. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Data-driven gait dynamics model predicts non-linear changes in joint kinematics with gait speed. Gait predictions of joint kinematics (green) at intermediate gait speeds not used in model training were generated by interpolating gait signatures between slow (dashed grey) and fast speeds (dashed black) lines and using them to drive the gait model. Interpolated kinematics from gait dynamics (green) and interpolated directly from kinematics (blue) were compared to the measured reference kinematics (black solid). A) Predictions in an exemplar AB participant are more accurate when interpolating gait signatures compared to interpolating gait kinematics across speeds. B) In an exemplar low-functioning stroke survivor, interpolated gait signatures predict nonlinear changes in kinematics better at intermediate speeds than interpolated gait kinematics. Averaging the kinematics fail in this case where there are larger differences between the slow and fast speed paretic kinematics; the averaged kinematics (blue) follow the fast speed paretic hip kinematics whereas the other angles do not reflect waveforms that resemble either the fast or slow speed. The gait model can therefore predict movement reasonably well when interpolating between tested speeds. There is a statistically significant difference between group level R2 comparisons (kinematics generated from interpolated dynamics vs interpolated kinematics) in the able-bodied (C) but not in stroke (D) cohorts. However, the range of R2 values are larger in both able-bodied and stroke kinematic predictions resulting from interpolated kinematics (-1.30–0.98, -0.50–1.00 respectively) vs. predicted from interpolated gait dynamics (-0.20–1.00,0.46–1.00 respectively). Thus, while the R2 values may not improve on average for the stroke survivors, the model’s performance is more robust overall. https://doi.org/10.1371/journal.pcbi.1011556.g005 Gait sculpting: Manipulating the PC components of an individual’s gait signature identifies specific coordination deficits in stroke survivors Previously, we showed that we can leverage our model to reconstruct the kinematics of healthy PC projections of the gait signature to gain insight into their independent biomechanical interpretations. However, identifying and interpreting the biomechanics related to impaired PC dynamics of stroke-survivors’ gait would prove to be even more beneficial, as these dynamics can potentially serve as rehabilitation targets when designing tailored gait intervention/strategies for individuals. Here we present an example of how we use gait signatures to identify specific biomechanical or coordination targets in specific stroke survivors. Specifically, we utilize our finding that the phase-varying contributions of the 6 principal projections of the gait signature differ in individual-specific manners (Fig 6A). For example, AB2’s 6 PC contributions all lie within the 95% confidence interval of all able-bodied individuals. ST4 primarily shows major deviation from AB in PC 3 (located entirely above the AB confidence interval), impaired dynamics during paretic swing in PC 4 and overall irregular shapes in PC 5 and 6. ST2’s PC3 is largely within the AB confidence interval, however PC 4’s paretic swing shows major deviation, their PC 5 contribution is shifted below the AB confidence interval and PC 6 shows an irregular shape. ST3’s PC5 projection lies below the AB confidence interval and PC 6 projection is irregularly shaped compared to AB. To validate our finding that suggested that PC 3 primarily influences hip flexion or extension, we exchanged AB2’s healthy PC1 with that of ST4 (Fig 6A, orange boxes and arrow) and we observed if and how AB2’s original hip joint kinematics (Fig 6B, orange box, black trace) deviated (Fig 6B, orange box, red dashed trace and S5 Video). To gain further insight into how the other PC deviations manifest in movement, we manipulated the PC3 projection of AB2 by replacing it with that of ST4 (Fig 6A, brown boxes and arrow). The kinematic reconstruction from this manipulation (Fig 6B, brown box, red dashed trace and S6 Video) shows a vertical shift downwards for bilateral hip angles and the non-paretic knee. The vertical shifts in the hip flexion/extension angles suggest a major difference in this individual’s posture (perhaps stroke individual leaned forward more during gait) compared to able-bodied. We manipulated AB2’s PC4 projection by replacing it with that of ST2 (Fig 6A, green boxes and arrow). This manipulation affected specifically the paretic and non-paretic ankle angles and both knee joints primarily during the period between non-paretic stance and paretic swing (Fig 6B, green box, red dashed trace and S7 Video). This result highlights a coordination deficit between these specific joint angles and, if targeted accurately, may allow for corrected gait patterns of this stroke survivor. Conversely, we tested the effects of replacing an impaired PC projection with a healthy one to observe how gait impairments can potentially be improved. We replaced ST3’s PC5 with that of AB2 (Fig 6A, purple boxes and arrow) and observed a substantial change in the magnitude and shape of bilateral ankle angle trajectories and slight increase in non-paretic knee magnitude (Fig 6B, purple box, red dashed trace and S8 Video). We can infer that to make improvements to ST3’s PC5 towards able-bodied or normative kinematics, rehabilitation focusing on these specific knee and ankle strategies may prove useful. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. Gait sculpting: interpolating between components of able-bodied and stroke gait dynamics to visualize anticipated gait improvement. The components of individuals’ gait signatures can be manipulated (gait sculpting) to understand the relationship between specific underlying dynamics and their corresponding kinematic phenotype. A) The projection on each of the 1st 6 principal components (PCs) can be observed for a representative able-bodied (AB2), two low functioning stroke-survivors each having similar self-selected (SS) speeds and Fugl-Meyer (FM) scores (ST2 & ST4, as denoted in Fig 2) and another low functioning stroke survivor (ST3) who has a higher FM score and faster SS walking speed. The PC projections are colored according to the 4 gait phases (non-paretic swing, non-paretic stance, paretic swing, paretic stance). The right leg of the unimpaired individuals was arbitrarily assigned to be paretic and the left leg, non-paretic for consistency. Colored boxes and arrows (orange, brown, green, purple) show specific, single PC manipulations, for example, the orange boxes and arrow illustrate that the PC 1 projection of AB2 was replaced with the impaired PC 1 projection from ST4. B) The AB2:ST4 manipulation (orange) shows how AB2’s original phase averaged kinematics (black trace) was manipulated by ST4’s impaired PC 1 projection (red dashed traced). ST4’s impaired PC 1 manifests in AB2’s healthy kinematics showing deviation primarily in the hip kinematics (as suggested in Fig 4 where healthy PC 1 encodes a kinematic subcomponent corresponding to hip flexion/extension) and some deviation in the ankle angles, especially the paretic ankle. The AB2:ST4 manipulation (brown) shows how ST4’s impaired PC3 manifests in AB2’s healthy kinematics; we observe a vertical shift downwards (red trace) of the bilateral hip angles as well as the non-paretic knee. This change in hip flexion highlights that this impaired PC3 encodes a reduction in the hip flexion angles; pointing to a more crouched gait (trunk is leaning forward more). The AB2:ST2 manipulation (green) shows replacing AB2’s PC4 projection with ST2’s impaired PC4 dynamics shows deviation in the knee joints especially during paretic swing, a vertical shift upwards in the paretic ankle angle kinematics and deviations around the middle of the gait cycle (transition between non-paretic stance and paretic swing) in the non-paretic ankle kinematics. Alternatively, the AB2:ST3 manipulation (purple) the impaired PC5 in ST3 is replaced with the healthy PC5 projection from AB2 resulting in slight increase in non-paretic knee magnitude and reduced amplitude of paretic and non-paretic ankle flexion. The result of this manipulation points to potential predicted improvements (or deviations) that can occur when aiming to mimic PC5 healthy dynamics in this stroke survivor allowing offline in-silico testing of potential avenues for gait rehabilitation for this stroke survivor. https://doi.org/10.1371/journal.pcbi.1011556.g006 Self-driven signatures: Our gait dynamics model revealed robustness of gait predictions establishing the utility of gait signatures in precision medicine The ability to predict future kinematics based on measured data is key to rapid, virtual design of personalized interventions. We demonstrate that the recurrent neural network model of gait dynamics, once primed with several gait cycles of data from either able-bodied or stroke participants, can predict future joint angle trajectories (Fig 7). Once the network is primed, an initial posture is presented (initial condition, denoted by blue vertical bar) after which the model self-drives i.e., predicts the general shape of future kinematics in a feedforward manner (without referencing previous measured data points) in an able-bodied (Fig 7Ai, left) and stroke individual (Fig 7Bii, left). A smooth transition is seen between the previously measured gait cycle (green) and the self-driven cycle (red trace) for both AB and stroke (Fig 7Ai, right and 7Aii, right respectively).To verify that the model was not generating a gait cycle prediction entirely by chance, we calculated the Euclidean distance between the kinematics of the predicted (self-driven) gait cycle and the kinematics from each of the measured gait cycles. We computed the distribution of distances between each predicted gait cycle with all other gait cycles from the same individual (Fig 7B, purple bars). We then compared the predicted gait cycle to the target gait cycle (Fig 7B, red bars). In the able-bodied individual, the predicted gait cycles are more similar to the target gait cycle (Fig 7Bi, red bar) than 79% of all gait cycles. However, in the stroke survivor, 60% of other gait cycles were more similar to the predicted gait cycle than the target gait cycle (Fig 7Bii, red bar). This suggests that the model is less able to accurately predict future kinematics in stroke gait. Note, to calculate the Euclidean distances between the gait cycles, we need to normalize the period of each gait cycle to the period of the self-driven cycle. To avoid the potential of bias due to this normalization, we also performed a comparison using a metric that was not manipulated in time–gait cycle duration. After priming the model, we presented the model with the first posture of the trial and ran the network forward in self-driving mode for the remainder of the trial length (15 seconds). Able-bodied self-driven predicted kinematics resembled the reference kinematics closely (Fig 7Ci, top plot) whereas stroke self-driven predicted kinematics matched the first gait cycle closely but soon converged to patterns reflecting able-bodied kinematics (Fig 7Cii, top plot). The gait cycle duration of the first few cycles of the self-driven kinematics match those of the measured kinematics (blue dots located close to the y = x line) in both the exemplary able-bodied (Fig 7Ci, bottom plot) and stroke individual shown (Fig 7Cii, bottom plot), however kinematics soon diverged to shorter and relatively consistent gait cycle durations (blue dots appearing almost horizontal in the plots) result in both cases. The model may preferentially predict able-bodied kinematics, which were less variable between individuals than were post-stroke kinematics. These results highlight that the model encodes gait dynamics that can predict kinematics over short timescales, but the variability and amount of training data may influence predictive power over long timescales. Our model provides a foundation for a sample-specific gait dynamics model to predict the effects of environmental perturbations, assistive devices, and treatments without extensive experimental sessions. In-silico predictions using the gait dynamics model may thus reduce the experimental time, cost, and participant burden required for personalized gait characterization, treatment personalization, and device design. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. Our trained RNN model can predict the time evolution of kinematics from an initial posture. The trained gait dynamics model can predict individual-specific time-evolution of gait kinematics from an arbitrary initial posture (self-driving) in able-bodied (A, i) and stroke (A, ii) once the network is primed with several gait cycles of an individual’s data (gait cyclex-n(measured), black solid). This predictive ability shows that the model encodes the gait dynamics underlying movement. Despite inter-cycle kinematic variability, the gait dynamics model can predict the general shape of the next gait cycle of kinematics (gait cyclex(predicted), red) in an able-bodied individual (A, i) and stroke survivor (A, ii), however, predicted kinematics (red) show larger deviation from the measured reference gait cycle (gait cyclex(measured), black dashed) in the stroke survivor. A smooth transition exists between the measured kinematics from the gait cycle preceding (gait cyclex-1(measured), green) the self-driven predicted cycle (red). For the representative able-bodied individual (B, i), the Euclidean distance (deviation) between the predicted gait cycle of kinematics and its respective measured kinematics (reference) is ~79% lower than the distance between the other gait cycles in the trial; ruling out that the kinematic predictions are attributed to chance. The deviation (Euclidean distance) of the predicted gait cycle of stroke (B, ii) kinematics to its reference gait cycle is ~40% lower than the distance between the other gait cycles in the trial. This suggests that the dynamical model is less able to accurately predict stroke kinematics better than chance. The dynamical model was first initialized with all the trial’s kinematics data (15 seconds) (black trace) after which the trial’s initial posture was presented to the model to self-drive kinematics (red trace) in feedforward mode for 15 seconds (C, i, top plot). The duration of each gait cycle from the measured kinematics is not well encoded by the dynamical model; gait cycle durations of the predicted kinematics are typically underestimated in both able-bodied (C, i, bottom plot) and stroke (C, ii, bottom plot) (to a larger degree) in self-driving mode and as such deviate from the y = x reference line (black). https://doi.org/10.1371/journal.pcbi.1011556.g007 Discussion Summary Here we establish a data-driven framework for comparing and predicting individual-specific locomotor patterns without needing to construct physiologically based mechanistic models. As an initial proof of concept, complex neuromechanical gait dynamics were modeled using a relatively simple recurrent neural network that captures the rules by which joint kinematics during gait transition from one time point to the next. Because the network was trained on multiple healthy and impaired individuals walking at several speeds, its internal parameters provide a basis for comparing, interpreting, and predicting gait dynamics. Gait signatures further capture coordination between joints and limbs without the need for pre-selecting gait features that may introduce bias and ignore the continuous nature of gait. We show that individuals have little variance in gait dynamics across speeds, leading to the individual-specific “gait signature” concept and enabling comparisons between individuals moving at different speeds. Across stroke survivors, we found greater heterogeneity in low-functioning individuals who exhibited disparate gait dynamics despite similar clinical metrics, highlighting the potential utility of gait signatures in providing more sensitive diagnoses to personalize therapies. Gait signatures provide a predictive simulation framework for sculpting gait dynamics to understand coordination deficits and predict kinematics, potentially forecasting the effects of rehabilitative devices or treatments. Finally, the gait signatures methodology can be readily applied to other periodic motions across species and across conditions that alter movement and may be a powerful adjunct to modern experimental methods aimed at understanding the neural mechanisms underlying movement. Computational framework captures the neuromechanical dynamics of walking Using a data driven modeling approach enabled us to learn the underlying gait dynamics based on data rather than constructing a neuromechanical gait model based on first principles. Data-driven approaches in gait have not focused on gait dynamics but have solved tasks based on unique features in multi-dimensional gait data such as classifying gait based on pathologies [75] or conditions such as fatigue and non-fatigue [76], identifying gait events (e.g., initial contact, loading response [77–79]), and discriminating individuals [62,80]. Gait dynamics have typically been described though neuromusculoskeletal models based on physical principles focusing on musculoskeletal mechanics, [30,81] but they lack adequate representations of the neural systems that contribute to the resulting movement patterns, particularly in neurological impairments such as stroke [31]. Machine learning methods to capture dynamics have been used across physics, engineering, and neuroscience to learn the dynamics underlying complex systems when the governing equations are unknown [58,82,83]. Recently, machine learning models have been used in human gait to predict continuous kinetic variables such as ground-reaction forces [84] or joint torque [85,86] based on kinematic data. Dynamical machine learning models have also been used to encode gait dynamics, including responses to perturbations or assistive devices, but their model structure did not enable comparisons between individuals [87–91]. Here, our RNN-based gait dynamics model provides a means to capture the rules underlying continuous, multi-joint coordination between bilateral lower limb joints, and how they evolve over time. Accordingly, we do not explicitly capture mechanical dynamics (i.e., the relationship motion and force), but the effects of force interactions within the body and environment and implicitly represented in how multi-joint kinematics evolve over time, with the network parameters and the internal states at each time point determining the output kinematics. As gait arises from complex interactions between the nervous system and the musculoskeletal system that are not easily modeled from first principles, a data-driven approach provides a powerful framework for capturing and comparing neuromechanical constraints on gait dynamics. While biomechanical dynamics clearly play a role in movement, the activation of muscles by the nervous system enables the body to perform a variety of motor behaviors. However, the governing spatiotemporal dynamics of neuromuscular signals are poorly understood, especially in neuro-pathologies such as stroke. During behaviors such as locomotion, motor patterns can be characterized by the number and structure of motor modules, or muscle synergies, defining groups of co-activated muscles producing a biomechanical function for gait [92]. Similar motor modules are used within individuals across different task conditions [93–95], and are shaped by learning and disease [96,97]. Particularly in post-stroke gait, motor modules appear to constrain motor function. Fewer motor modules are observed post-stroke with the number of modules correlated to reduced walking speed [98,99]. Further, different patterns of motor module merging are seen in slower walking stroke survivors, differentially affecting gait biomechanics in a manner that may necessitate individualized rehabilitation approaches [100]. Adding neural constraints such as motor modules on muscle activations in musculoskeletal simulations improve predictions of key physiological variables such as joint loading in osteoarthritis [101]. However, relating motor modules to kinematic gait patterns post-stroke and in other neurological disorders has been challenging, likely because the neural constraints are underspecified [27,102–105]. Corroborating results from motor module analysis, there were greater differences in gait dynamics amongst the slowest walking stroke survivors. Since the gait signatures capture spatiotemporal constraints underlying gait dynamics, they provide a complementary approach to musculoskeletal simulations. Ultimately, gait signatures may play a complementary role to biophysical simulations, enabling the relationships between biomechanical principles, neural constraints, and the emergent gait dynamics to be revealed. Gait signatures enable holistic comparison of gait dynamics across individuals, speeds, and groups In contrast to other applications of dynamical machine learning models for gait, we capture multiple individuals within a single network, enabling comparisons of gait dynamics across groups, individuals, and gait conditions. Rather than using the network as a black box solely to generate predictions, we explicitly compare and interpret the model’s internal parameters to identify low-dimensional latent variables representing gait dynamics. To encourage a generalizable data-driven gait dynamics model, we omitted subject and trial condition (gait speed) labels as inputs to the neural network. Adding input labels might force the RNN to create separable gait models, whereas our goal was to have the network learn a structure that could be modified parametrically to represent individual differences in the neuromechanics of walking. Similarly, neuromusculoskeletal models assume common dynamic principles across individuals, using parameter variations to represent individual differences [27,28,103,104,106]. We intentionally designed a relatively simple RNN architecture (e.g., single layer, linear input/output) as a starting point to recover as much interpretability as possible, with the awareness that more complexity could be added to the model architecture (number of hidden layers, number of neurons, etc.) if required to fit a given data set robustly. The representations of gait dynamics that emerge from our model holistically capture the changes underlying measured kinematics, without being attributable to specific neural or biomechanical constraints. The loss of physiological interpretability is counterbalanced by the holistic approach to representing gait dynamics and explaining gait kinematics features. Analogous to written signatures, we find that features of individual-specific gait signatures are largely preserved across walking speeds. Recognizable qualitative features of handwriting are preserved even as the size of letters changes quantitatively, or if different limbs, or writing instruments are used [107]. Similarly, it is well known that individuals can be recognized based on how they move or walk [28,40,108–110], even if joint angle excursions are similar. We show that gait dynamics are more similar within individuals across speeds than between individuals, leading to the concept of the gait signature. In contrast, gait kinetics and kinematics vary characteristically across speeds, such that they cannot be directly compared across speeds [111]. The relatively small changes in gait signatures across speeds suggest that the signatures reflect changes in the spatiotemporal relationships between joint kinematics, rather than quantitative changes in their magnitude. As such, gait signatures appear to encode individual-specific constraints of walking, making it possible to compare gait either within or between individuals walking at different speeds. Gait signatures characterize the high inter-individual variability in gait impairment amongst stroke survivors beyond overall gait function explained by clinical gait metrics. This heterogeneity is a direct reflection of the wide range of impairments in stroke survivors, including muscle weakness, impaired coordination, spasticity, abnormal synergistic activation (muscles not independently coordinated), and compensatory motion [19,53,112]. We found that higher-functioning stroke survivors were more dynamically similar to each other, whereas lower functioning stroke survivors were more dispersed. In fact, two low-functioning stroke survivors with similar clinical metrics (Fugl-Meyer score and gait speed) had quite different gait signatures. As such, gait signatures have the potential to provide insights into individual differences in gait dynamics that are simply not captured by clinical metric such as gait speed. Moreover, in contrast to higher-functioning stroke survivors who share similar gait dynamics, lower-functioning stroke survivors may require more individuals individualized rehabilitation approaches targeting specific aspects of gait dysfunction. Further gait signatures do not require a priori selection of which gait variables to compare [113–116]. As such gait signatures provide a powerful, holistic approach to enhance the specificity and precision of gait diagnosis and treatment. Our study inclusion criteria exclude severe contractures or deformities that interfere with normal ambulation and in future work the gait signatures would need to be interpreted and correlated with clinical evaluation of strength, range of motion, sensorimotor impairment, and/or limb deformities. The demographic and clinical information of the stroke participants in our study are available in Supplementary Materials S1 Table. Gait signatures could be part of a set of multi-modal data to account for the diverse causal factors underlying each individual’s gait pattern (e.g., lesion neuroanatomy, medical confounding variables, musculoskeletal conditions, psychosocial variables, physiological contributors to gait and environmental factors). This framework can potentially extend to other diseases, disorders, injury, etc. to gain further insight into individuals’ specific impairments and uncover specific targets towards developing targeted therapies for individuals. Gait signatures enable biomechanical interpretation and manipulation Our gait dynamics model enables biomechanical interpretation of gait signatures and exploring “what if” scenarios to sculpt desirable gait dynamics. Gait signatures are based on principal components (PCs) of the gait model internal states, where the weightings on each PC vary over the gait cycle. The model parameters can be prescribed over the gait cycle, resulting in the predicted kinematic outputs (i.e., joint angles). The gait signature PCs and their time-varying weightings can be individually prescribed in the network as a method to reveal the specific inter- and intra-limb coordination patterns governed by each PC. Further, any combination of PC’s can be combined and reweighted to generate new kinematic output patterns. For example, we interchanged healthy and impaired PCs to gain deeper insight into how specific impaired PCs alter healthy gait and vice versa. Further, interpolating gait dynamics can predict gait kinematics at walking speeds that were not used in the training data. Especially when there was a nonlinear response in gait kinematics across speeds, interpolation of gait dynamics to predict gait kinematics performed better than interpolating gait kinematics directly. As such our data driven gait dynamics model can be used to show how changing select components of the gait signature alters gait kinematics, providing a potential framework to identify personalized therapeutic targets for gait rehabilitation. Gait signatures have potential to predict future kinematics Another powerful aspect of our gait signatures framework is its ability to generate future gait kinematics in the absence of new data. The model is self-driving for able-bodied individuals, predicting multiple cycles of gait kinematics in the future. However, the ability of the model to predict future stroke kinematics is limited to approximately one gait cycle in the future; rendering it promising in applications that provide control signals to rehabilitation devices (e.g., exoskeletons). Furthermore, our model is likely only capable of generalizing to speeds within the speed ranges of the input data. There was a moderate association between participants’ similarity to able-bodied gait signatures (distance to the able-bodied centroid) and the RNN’s ability to predict gait kinematics over one gait cycle (S6 Fig). This association is likely due to the RNN favoring able-bodied dynamics during model fitting, which were more homogeneous than those of high- or low-functioning stroke survivors. A larger post-stroke sample may improve the RNN’s ability to encode and predict pathological gait dynamics. Further, the reduced predictive power for the stroke participants can be attributed to our model architecture’s relative simplicity and short time-series (15 seconds/ 1500 sample points per trial). These factors should be addressed to improve the predictive capacity of the model for impaired gait in the future. Additionally, including more variables besides sagittal plane kinematics (e.g., frontal plane and coronal plane kinematics and joint forces, may improve learning of the underlying dynamics of gait and increase predictive capability of our model. Generalization to other species and rhythmic movements Because the input to this model are periodic sequences of behaviors, our gait dynamics framework should generalize to other species that display similar behavioral motions (e.g., flight, crawling, and walking). Physicists, computational biologists, and other scientists can benefit from this method by studying the dynamical behavior of species whose neuromechanical models and physics of complex terrains are difficult to model. This is the first study to our knowledge that uses a neural network to study the dynamics of gait in an interpretable manner. While much work is left to be done, we have provided a simplistic, unsupervised framework to discover individual-specific differences in walking in health and disease in humans. Despite being limited by a small dataset, we have shown that our model is generalizable to characterizing and predicting kinematics of one held-out subject using leave one out cross validation (S3 Fig). Here we focus on demonstrating the innovation, feasibility, and potential advantages of our RNN gait signature approach, justifying the need and potential for further development by scaling to larger-sample studies. Importantly, this methodology relies on having a periodic or quasi-periodic pattern, as non-periodic patterns would not be able to generate a phase and subsequent signature. We also limited our inputs to gait kinematics, anticipating applications to the proliferation of new measurement modalities for movement in humans and animals such as wearable sensors and markerless video-based motion capture [117–119]. However, the gait signatures framework could easily be extended to include other data types (e.g., force, muscle activity, joint loadings, center of mass dynamics) and experimental conditions (overground walking, biomechanical constraints, gait interventions, such as exoskeletons, functional electrical stimulation, or treatment e.g., drugs, optogenetics). In practice, a more comprehensive data set would be needed within each gait group to train a model capable of capturing the full range of variability in gait dynamics. Short of having a massive data set, it may also be possible to leverage synthetic gait data from simulations to span the full range of feasible gait dynamics variations. Overall, by modeling the dynamics of individual’s gait based on measured data, we uncovered individual-specific representations of individuals’ neuromechanical constraints that allows direct comparisons between individuals who do not walk at the same speed. The gait signatures framework has implications for the diagnosis of disease, development of future tailored gait therapies or interventions and tracking meaningful changes in the fundamental neuromechanical mechanism of walking. Summary Here we establish a data-driven framework for comparing and predicting individual-specific locomotor patterns without needing to construct physiologically based mechanistic models. As an initial proof of concept, complex neuromechanical gait dynamics were modeled using a relatively simple recurrent neural network that captures the rules by which joint kinematics during gait transition from one time point to the next. Because the network was trained on multiple healthy and impaired individuals walking at several speeds, its internal parameters provide a basis for comparing, interpreting, and predicting gait dynamics. Gait signatures further capture coordination between joints and limbs without the need for pre-selecting gait features that may introduce bias and ignore the continuous nature of gait. We show that individuals have little variance in gait dynamics across speeds, leading to the individual-specific “gait signature” concept and enabling comparisons between individuals moving at different speeds. Across stroke survivors, we found greater heterogeneity in low-functioning individuals who exhibited disparate gait dynamics despite similar clinical metrics, highlighting the potential utility of gait signatures in providing more sensitive diagnoses to personalize therapies. Gait signatures provide a predictive simulation framework for sculpting gait dynamics to understand coordination deficits and predict kinematics, potentially forecasting the effects of rehabilitative devices or treatments. Finally, the gait signatures methodology can be readily applied to other periodic motions across species and across conditions that alter movement and may be a powerful adjunct to modern experimental methods aimed at understanding the neural mechanisms underlying movement. Computational framework captures the neuromechanical dynamics of walking Using a data driven modeling approach enabled us to learn the underlying gait dynamics based on data rather than constructing a neuromechanical gait model based on first principles. Data-driven approaches in gait have not focused on gait dynamics but have solved tasks based on unique features in multi-dimensional gait data such as classifying gait based on pathologies [75] or conditions such as fatigue and non-fatigue [76], identifying gait events (e.g., initial contact, loading response [77–79]), and discriminating individuals [62,80]. Gait dynamics have typically been described though neuromusculoskeletal models based on physical principles focusing on musculoskeletal mechanics, [30,81] but they lack adequate representations of the neural systems that contribute to the resulting movement patterns, particularly in neurological impairments such as stroke [31]. Machine learning methods to capture dynamics have been used across physics, engineering, and neuroscience to learn the dynamics underlying complex systems when the governing equations are unknown [58,82,83]. Recently, machine learning models have been used in human gait to predict continuous kinetic variables such as ground-reaction forces [84] or joint torque [85,86] based on kinematic data. Dynamical machine learning models have also been used to encode gait dynamics, including responses to perturbations or assistive devices, but their model structure did not enable comparisons between individuals [87–91]. Here, our RNN-based gait dynamics model provides a means to capture the rules underlying continuous, multi-joint coordination between bilateral lower limb joints, and how they evolve over time. Accordingly, we do not explicitly capture mechanical dynamics (i.e., the relationship motion and force), but the effects of force interactions within the body and environment and implicitly represented in how multi-joint kinematics evolve over time, with the network parameters and the internal states at each time point determining the output kinematics. As gait arises from complex interactions between the nervous system and the musculoskeletal system that are not easily modeled from first principles, a data-driven approach provides a powerful framework for capturing and comparing neuromechanical constraints on gait dynamics. While biomechanical dynamics clearly play a role in movement, the activation of muscles by the nervous system enables the body to perform a variety of motor behaviors. However, the governing spatiotemporal dynamics of neuromuscular signals are poorly understood, especially in neuro-pathologies such as stroke. During behaviors such as locomotion, motor patterns can be characterized by the number and structure of motor modules, or muscle synergies, defining groups of co-activated muscles producing a biomechanical function for gait [92]. Similar motor modules are used within individuals across different task conditions [93–95], and are shaped by learning and disease [96,97]. Particularly in post-stroke gait, motor modules appear to constrain motor function. Fewer motor modules are observed post-stroke with the number of modules correlated to reduced walking speed [98,99]. Further, different patterns of motor module merging are seen in slower walking stroke survivors, differentially affecting gait biomechanics in a manner that may necessitate individualized rehabilitation approaches [100]. Adding neural constraints such as motor modules on muscle activations in musculoskeletal simulations improve predictions of key physiological variables such as joint loading in osteoarthritis [101]. However, relating motor modules to kinematic gait patterns post-stroke and in other neurological disorders has been challenging, likely because the neural constraints are underspecified [27,102–105]. Corroborating results from motor module analysis, there were greater differences in gait dynamics amongst the slowest walking stroke survivors. Since the gait signatures capture spatiotemporal constraints underlying gait dynamics, they provide a complementary approach to musculoskeletal simulations. Ultimately, gait signatures may play a complementary role to biophysical simulations, enabling the relationships between biomechanical principles, neural constraints, and the emergent gait dynamics to be revealed. Gait signatures enable holistic comparison of gait dynamics across individuals, speeds, and groups In contrast to other applications of dynamical machine learning models for gait, we capture multiple individuals within a single network, enabling comparisons of gait dynamics across groups, individuals, and gait conditions. Rather than using the network as a black box solely to generate predictions, we explicitly compare and interpret the model’s internal parameters to identify low-dimensional latent variables representing gait dynamics. To encourage a generalizable data-driven gait dynamics model, we omitted subject and trial condition (gait speed) labels as inputs to the neural network. Adding input labels might force the RNN to create separable gait models, whereas our goal was to have the network learn a structure that could be modified parametrically to represent individual differences in the neuromechanics of walking. Similarly, neuromusculoskeletal models assume common dynamic principles across individuals, using parameter variations to represent individual differences [27,28,103,104,106]. We intentionally designed a relatively simple RNN architecture (e.g., single layer, linear input/output) as a starting point to recover as much interpretability as possible, with the awareness that more complexity could be added to the model architecture (number of hidden layers, number of neurons, etc.) if required to fit a given data set robustly. The representations of gait dynamics that emerge from our model holistically capture the changes underlying measured kinematics, without being attributable to specific neural or biomechanical constraints. The loss of physiological interpretability is counterbalanced by the holistic approach to representing gait dynamics and explaining gait kinematics features. Analogous to written signatures, we find that features of individual-specific gait signatures are largely preserved across walking speeds. Recognizable qualitative features of handwriting are preserved even as the size of letters changes quantitatively, or if different limbs, or writing instruments are used [107]. Similarly, it is well known that individuals can be recognized based on how they move or walk [28,40,108–110], even if joint angle excursions are similar. We show that gait dynamics are more similar within individuals across speeds than between individuals, leading to the concept of the gait signature. In contrast, gait kinetics and kinematics vary characteristically across speeds, such that they cannot be directly compared across speeds [111]. The relatively small changes in gait signatures across speeds suggest that the signatures reflect changes in the spatiotemporal relationships between joint kinematics, rather than quantitative changes in their magnitude. As such, gait signatures appear to encode individual-specific constraints of walking, making it possible to compare gait either within or between individuals walking at different speeds. Gait signatures characterize the high inter-individual variability in gait impairment amongst stroke survivors beyond overall gait function explained by clinical gait metrics. This heterogeneity is a direct reflection of the wide range of impairments in stroke survivors, including muscle weakness, impaired coordination, spasticity, abnormal synergistic activation (muscles not independently coordinated), and compensatory motion [19,53,112]. We found that higher-functioning stroke survivors were more dynamically similar to each other, whereas lower functioning stroke survivors were more dispersed. In fact, two low-functioning stroke survivors with similar clinical metrics (Fugl-Meyer score and gait speed) had quite different gait signatures. As such, gait signatures have the potential to provide insights into individual differences in gait dynamics that are simply not captured by clinical metric such as gait speed. Moreover, in contrast to higher-functioning stroke survivors who share similar gait dynamics, lower-functioning stroke survivors may require more individuals individualized rehabilitation approaches targeting specific aspects of gait dysfunction. Further gait signatures do not require a priori selection of which gait variables to compare [113–116]. As such gait signatures provide a powerful, holistic approach to enhance the specificity and precision of gait diagnosis and treatment. Our study inclusion criteria exclude severe contractures or deformities that interfere with normal ambulation and in future work the gait signatures would need to be interpreted and correlated with clinical evaluation of strength, range of motion, sensorimotor impairment, and/or limb deformities. The demographic and clinical information of the stroke participants in our study are available in Supplementary Materials S1 Table. Gait signatures could be part of a set of multi-modal data to account for the diverse causal factors underlying each individual’s gait pattern (e.g., lesion neuroanatomy, medical confounding variables, musculoskeletal conditions, psychosocial variables, physiological contributors to gait and environmental factors). This framework can potentially extend to other diseases, disorders, injury, etc. to gain further insight into individuals’ specific impairments and uncover specific targets towards developing targeted therapies for individuals. Gait signatures enable biomechanical interpretation and manipulation Our gait dynamics model enables biomechanical interpretation of gait signatures and exploring “what if” scenarios to sculpt desirable gait dynamics. Gait signatures are based on principal components (PCs) of the gait model internal states, where the weightings on each PC vary over the gait cycle. The model parameters can be prescribed over the gait cycle, resulting in the predicted kinematic outputs (i.e., joint angles). The gait signature PCs and their time-varying weightings can be individually prescribed in the network as a method to reveal the specific inter- and intra-limb coordination patterns governed by each PC. Further, any combination of PC’s can be combined and reweighted to generate new kinematic output patterns. For example, we interchanged healthy and impaired PCs to gain deeper insight into how specific impaired PCs alter healthy gait and vice versa. Further, interpolating gait dynamics can predict gait kinematics at walking speeds that were not used in the training data. Especially when there was a nonlinear response in gait kinematics across speeds, interpolation of gait dynamics to predict gait kinematics performed better than interpolating gait kinematics directly. As such our data driven gait dynamics model can be used to show how changing select components of the gait signature alters gait kinematics, providing a potential framework to identify personalized therapeutic targets for gait rehabilitation. Gait signatures have potential to predict future kinematics Another powerful aspect of our gait signatures framework is its ability to generate future gait kinematics in the absence of new data. The model is self-driving for able-bodied individuals, predicting multiple cycles of gait kinematics in the future. However, the ability of the model to predict future stroke kinematics is limited to approximately one gait cycle in the future; rendering it promising in applications that provide control signals to rehabilitation devices (e.g., exoskeletons). Furthermore, our model is likely only capable of generalizing to speeds within the speed ranges of the input data. There was a moderate association between participants’ similarity to able-bodied gait signatures (distance to the able-bodied centroid) and the RNN’s ability to predict gait kinematics over one gait cycle (S6 Fig). This association is likely due to the RNN favoring able-bodied dynamics during model fitting, which were more homogeneous than those of high- or low-functioning stroke survivors. A larger post-stroke sample may improve the RNN’s ability to encode and predict pathological gait dynamics. Further, the reduced predictive power for the stroke participants can be attributed to our model architecture’s relative simplicity and short time-series (15 seconds/ 1500 sample points per trial). These factors should be addressed to improve the predictive capacity of the model for impaired gait in the future. Additionally, including more variables besides sagittal plane kinematics (e.g., frontal plane and coronal plane kinematics and joint forces, may improve learning of the underlying dynamics of gait and increase predictive capability of our model. Generalization to other species and rhythmic movements Because the input to this model are periodic sequences of behaviors, our gait dynamics framework should generalize to other species that display similar behavioral motions (e.g., flight, crawling, and walking). Physicists, computational biologists, and other scientists can benefit from this method by studying the dynamical behavior of species whose neuromechanical models and physics of complex terrains are difficult to model. This is the first study to our knowledge that uses a neural network to study the dynamics of gait in an interpretable manner. While much work is left to be done, we have provided a simplistic, unsupervised framework to discover individual-specific differences in walking in health and disease in humans. Despite being limited by a small dataset, we have shown that our model is generalizable to characterizing and predicting kinematics of one held-out subject using leave one out cross validation (S3 Fig). Here we focus on demonstrating the innovation, feasibility, and potential advantages of our RNN gait signature approach, justifying the need and potential for further development by scaling to larger-sample studies. Importantly, this methodology relies on having a periodic or quasi-periodic pattern, as non-periodic patterns would not be able to generate a phase and subsequent signature. We also limited our inputs to gait kinematics, anticipating applications to the proliferation of new measurement modalities for movement in humans and animals such as wearable sensors and markerless video-based motion capture [117–119]. However, the gait signatures framework could easily be extended to include other data types (e.g., force, muscle activity, joint loadings, center of mass dynamics) and experimental conditions (overground walking, biomechanical constraints, gait interventions, such as exoskeletons, functional electrical stimulation, or treatment e.g., drugs, optogenetics). In practice, a more comprehensive data set would be needed within each gait group to train a model capable of capturing the full range of variability in gait dynamics. Short of having a massive data set, it may also be possible to leverage synthetic gait data from simulations to span the full range of feasible gait dynamics variations. Overall, by modeling the dynamics of individual’s gait based on measured data, we uncovered individual-specific representations of individuals’ neuromechanical constraints that allows direct comparisons between individuals who do not walk at the same speed. The gait signatures framework has implications for the diagnosis of disease, development of future tailored gait therapies or interventions and tracking meaningful changes in the fundamental neuromechanical mechanism of walking. Materials and methods Ethics statement All participants provided written informed consent prior to study participation, and the study protocol was approved by the Emory University Institutional Review Board. Human subject participants To develop dynamical signatures of human gait, we collected data in seven post-stroke individuals (age = 56 ± 12 years; 2 females; 48 ± 25 months post-stroke; Lower Extremity Fugl-Meyer = 20 ± 4) and five able-bodied (AB) controls (age = 24 ± 4 years; 4 female). All post-stroke participants experienced a cortical or subcortical ischemic stroke, were able to walk on a treadmill for one minute without an orthotic device, and exhibited no signs of hemi-neglect, orthopedic conditions limiting walking, or cerebellar dysfunction. Experimental design Participants completed 15-second walking trials at six different speeds, distributed evenly between and ranging from each participant’s self-selected (SS) speed to the fastest safe and comfortable speed. Across stroke participants, gait speeds ranged from 0.3–1.6 m/s. Each participant’s fastest walking speed was determined by progressively increasing the treadmill speed from the SS speed until the participant could no longer comfortably or safely maintain the speed for 30 seconds. Participants rested for 1–2 minutes between consecutive gait trials. During data collection, speed increased from the participant’s SS to their fastest speed (i.e., not randomized). Data acquisition Reflective markers were attached to the trunk, pelvis, and bilateral shank, thigh, and foot segments [120]. We collected marker position data while participants walked on a split-belt instrumented treadmill (Bertec Corp., Ohio, USA) using a -7-camera motion analysis system (Vicon Motion Systems, Ltd., UK). Participants held onto a front handrail and wore an overhead safety harness that did not support body weight. Marker data were collected at 100 Hz, and synchronous ground reaction forces were recorded at 2000 Hz and were down sampled to 100Hz using previously established techniques [121–123]. Data processing Raw marker position data were labeled, gap-filled, and low-pass filtered in Vicon Nexus. Labeled marker trajectories and ground reaction force raw analog data were low-pass filtered In Visual 3D. Gait events (bilateral heel contact and toe-off) were determined using a 20-N vertical GRF cutoff, and sagittal-plane hip, knee, and ankle joint kinematics were calculated in Visual 3D (C-Motion Inc., Maryland, USA). RNN model development Our goal was to start with a simple RNN to reduce overfitting with too many parameters and deep layers. We wanted the simplest model capable of learning the dynamics underlying gait which also preserved interpretability. The simplest recurrent neural network (RNN) model architecture consisted of one input layer, one hidden layer and one output layer. The hidden layer was composed of long short-term memory (LSTM) units with a lookback parameter that spanned at least one gait cycle. Model hyperparameter selection is described in a later paragraph. RNN model training Model fitting on our selected dataset and architecture was executed on the order of minutes to tens of minutes, using Keras 3.7.13 and TensorFlow 2.8.2 on Google Colab’s standard GPU with high-RAM runtime (54.8 gigabytes). The RNN model was trained using bilateral, sagittal-plane, lower-limb joint angles from 5 able-bodied (AB) participants and 7 stroke survivors each walking on a treadmill at 6 steady speeds, ranging from each participant’s preferred speed to the fastest safe speed. Our training dataset was input to the RNN in multivariate format (not concatenated) [62,63]. We trained a sequence-to-sequence RNN with 512 long-short-term memory (LSTM) activations units in the single hidden layer, capable of using 15 seconds (sample rate of 100 Hz) time-series kinematic input data (0 to T-1) to predict kinematics one time-step in the future (1 to T) for all training data across individuals and speeds. Our data was batched according to the number of total trials (N = 72); thus, the LSTM maintains its internal state while a batch is being processed, after which the internal state can be maintained or cleared. Because our network retains its internal state from one time step to the next (i.e., the RNN is stateful), we have fine-grained control over when the internal state of the LSTM network is reset. The input data from all trials was ‘mini-batched’ into 2 training batches and 1 validation batch (499 samples each) that were used to update model weights on each model run (epoch). To format our data into equal length input and output mini batches for training and account for the output data being a one-time step shifted version of our input data, our lookback parameter must be one value less than a divisor of the trial length. For example, in our dataset (1500 sample length trials), a lookback parameter of 499 would result in the first mini batch input of samples [0:499] which will predict our reference output samples [1:500], our 2nd mini batch input data would include samples [501:999] and corresponding output [502:1000] and the last mini-batch input of samples [1001: 1499] predicts samples [1002: 1500]. This lookback parameter of 499 allows us to construct 3 mini-batches of shorter input and output data lengths which would be used to train and validate the RNN model (2:1 training: validation mini batch split). Similarly with the lookback parameter of 499 (2:1 training: validation mini-batch split) and 749 (1:1 training: validation split). Mean squared error was used as the LSTM loss function and ADAM as the optimization algorithm because it is fast, has a small memory footprint and is well suited for large-parameter deep learning models [124]. The model was trained for at least 5000 iterations or until training and validation error converged (< 0.75°). The training resulted in a sample-specific dynamical model structure defined by a single set of LSTM network weights (W). The model’s internal states capture trial-specific dynamics predicting the time evolution of joint kinematics; activation coefficients (H) and memory cell states (C) and are tuned based on kinematic input. Model hyperparameter selection We selected the hyperparameter values of 512 nodes in the LSTM layer and a 499-sample LSTM lookback length (number of samples preceding the current time point that is used to train the LSTM) were selected based on training and validation loss, as well as the ability to encode dynamics over short and long timescales. In two steps, we evaluated all pairs of the following hyperparameter values: 1024, 512, 256, and 128 LSTM nodes and 749, 499, and 249-sample lookback parameters. Because RNN performance can change with the parameters used to initialize the RNN, we fit an RNN gait dynamics model 20 times using random initial parameters, for each hyperparameter pair. First, we compared model training and validation loss for each hyperparameter pair: the ‘best’ hyperparameter pair would have low training and validation loss. The following [node-lookback] pairs were considered the best hyperparameter pairs: 512–499 (MSEtrain = 0.010 ± 0.001 deg2; MSEval = 0.018 ± 0.000 deg2), 256–749 (MSEtrain = 0.010 ± 0.002 deg2; MSEval = 0.015 ± 0.001 deg2), 256–499 (MSEtrain = 0.010 ± 0.001 deg2; MSEval = 0.017 ± 0.000 deg2). (S1A Fig). The training loss was not different between hyperparameter pairs (p > 0.235). The validation loss differed between all three models (p < 0.001), with the 256–749 model having the lowest validation loss. However, if the differences in validation loss of less than 0.003 deg2 corresponded to meaningful differences in performance was unclear. Our second analysis was, therefore, used to compare the three hyperparameter pairs deemed best in the prior analysis. Here, we evaluated the models’ abilities to encode the average dynamical behavior over long timescales (long-time) and the stride-to-stride behavior (short-time). We defined the best model as the one with the highest long- and short-time performance. The following analysis was performed for 10 of the 20 random initializations. For long- and short-time analyses, we created a single set of reference dynamics as done in the manuscript: we performed one time-step predictions over the full (1500-sample) time-series. This step provided best-case predictions of the gait dynamics (S1B Fig). Long-time performance. We generated long-time predictions of each trial’s gait signatures (RNN latent states) by simulating each participant’s gait dynamics forward in time, 1500 samples into the future. Each simulation was initialized by setting the RNNs’ latent states to those of the last sample of the trial’s reference dynamics and using the last sample of the trial’s kinematics. We then phase-averaged both the reference dynamics and the long-time predictions using the same technique as described in the main manuscript. Long-time performance was defined as the similarity of the phase-averaged latent states (i.e., the gait signatures) between the reference and the long-time predictions and was quantified using R2. Note that using R2 as a similarity metric, rather than the Euclidean distance metric used in the main manuscript, was needed to compare models with different numbers of nodes. Unlike R2, Euclidean distances are sensitive to the number of samples used to compare models, which would bias short- and long-time performance towards models with fewer nodes. Low R2 values between predictions indicates that the learned dynamics are sufficiently complex to capture instantaneous gait dynamics but can also accurately generate the time evolution of dynamics over the gait cycle—a major challenge in data-driven models of locomotion [89–90]. The 512-node model captured gait dynamics over long time scales significantly better (i.e., more accurate predictions of the time-varying dynamics) than the 249-node models (S1B Fig). For long-time predictions, the 512-node model predictions (R2 = 0.50 ± 0.46) were better than the 249-node 499-sample lookback model (ΔR2 = 0.27 ± 0.06; p < 0.001; independent-samples t-tests) and the 249-node 499-sample lookback model (ΔR2 = 0.31 ± 0.07; p < 0.001). Short-time performance. We generated short-time predictions by simulating single strides in each trial’s time-series, initialized from the first sample of each stride. Initialization used the latent RNN states and kinematics of the reference dynamics at the onset of a new stride (phase = 0 rad). For each initial condition, we integrated the dynamics forward in time, up to the onset of the next stride. For each stride, we then compare the similarity of the reference dynamics to the dynamics of the corresponding short-time prediction using R2. Short-time performance was quantified as the average R2 value across trials for a single model and initialization. The 512-node model captured gait dynamics over short time scales significantly better (i.e., more accurate predictions of the time-varying dynamics) than the 249-node models (S1B Fig). For short-time predictions, the 512-node model predictions (R2 = 0.11 ± 0.51) were more accurate than the 249-node 499-sample lookback model (ΔR2 = 0.51 ± 0.13; p = 0.055; independent-samples t-tests) and the 249-node 499-sample lookback model (ΔR2 = 0.34 ± 0.09; p < 0.001). Based on difference in short- and long-time prediction performance, we selected the 512-node, 499-sample lookback hyperparameters for the RNN model. Leave-one-out subject model evaluation for generalizability Using the selected hyperparameters, 12 different models were trained where one different subject (all 6 speed trials per subject) was held out for evaluation on each model run. The same model architecture, training and validation setup was used as the original model trained using the full dataset (12 subjects). The minimum training loss, validation loss, and overall evaluated test loss for each model were extracted and box plots of each generated. The Wilcoxon Rank-Sum Test statistic was used to compare the means. Each model was evaluated on the 6 held-out speed trials from training and an average loss was calculated for each model. The reference kinematics, externally driven and self-driven predictions of each of the 6 held-out trials per model were phase averaged and R2 between the phase averaged externally driven and long-time self-driven predictions (see Long-Time Performance section, above) were calculated. Box plots for each metric across the held-out trials were generated and the Wilcoxon Rank-Sum Test statistic used to compare the means. Computing gait signatures from RNN internal states To develop the gait signatures, we extract the activation and cell states from the LSTM (denoted “H” and “C” respectively) which evolve over time (the course of the gait cycle) as the kinematics of each trial are fed through the trained RNN. These H and C parameters represent how the model’s internal parameters change as it encodes the prediction of future kinematic trajectories. The selected 512-node LSTM layer had 512 H parameters and 512 C parameters. Time-varying gait signatures were computed by identifying dominant modes of variation in the internal states using principal components analysis (PCA). A single PCA operation was used to transform the internal states for all participants into a common basis. Consequently, inter-trial differences in the time-varying activations of the principal components (modes) reflect differences in the underlying dynamics of the individual(s). These activations constituted the time-varying (1500 sample) gait signatures, which had the same dimension as the RNN’s hidden layer (1024 units). However, the first six principal components accounted for ~72% of the variance in the internal states. To compare gait signatures within and between individuals, we phase averaged each trial’s signatures across strides. Rather than linearly interpolating the data between foot contact events before averaging, as is common in gait analysis [122,125–127] we computed a continuous phase using the first 3 gait signature modes for each trial using Von Mises interpolation [128]. Compared to averaging across linearly interpolated strides, phase averaging is expected to reduce the variance in the data at any point in the stride [66,90]. As the domain for interpolation, we estimated the time-varying phase for each trial separately using the Phaser algorithm, using the first 3 principal components as phase variables [66]. To align phase estimates across trials, we defined zero-phase as the maximum of the first principal component. Gait event estimation of phase averaged signatures The force plates embedded in the treadmill captured precise gait event timing information (left heel strike, right toe off, right heel strike, left toe off) across individuals’ trials which we represented as a vector of 1’s, 2’s, 3’s and 4’s, respectively (ground truth markings for the 4 gait events). We leveraged the Phaser algorithm again [66] to develop a phase estimator to transform these 4 gait events over time into gait events over phase. For each trial, we determined the mode phase that corresponded to each of the 4 gait events to gain a representation of where the 4 gait events occurred during phase averaged dynamics (0–2π) for each trial. Interpolation of unseen speed gait signatures to reconstruct kinematics To demonstrate the generalizability of gait dynamics, we show that linearly interpolating gait signatures to predict gait kinematics at new walking speeds is more accurate than linearly interpolating the kinematics themselves. We trained another RNN model with the same architecture and hyperparameters to the first model, however using only the 2 slowest and 2 fastest speeds from each participant (i.e., we held out the 2 middle speed trials from each participant). We then linearly interpolated the 2-middle speeds’ internal states and ran the data through the trained RNN to reconstruct or predict kinematics. We compared the original phase averaged kinematics to the predicted kinematics resulting from each of the two linear interpolations (dynamics and kinematics) using the coefficient of determination. Furthermore, even when we reduced the dimensionality of the model’s internal states from the full 1024 to the first 6 principal components (the selected dimension on the gait signatures), it still performed better than interpolating kinematics (also rank = 6). Biomechanical interpretation of principal components of the gait signature To reconstruct kinematics from the corresponding underlying dynamics (internal state representations), we restored our trained model’s weights to a new model using the ‘model.set’ and ‘model.get_weights’ Keras built in functions. The function ‘model.predict’ takes in the hidden state values (Hs) only (first 512 of the 1024 internal-state time trajectories) and predicts the corresponding kinematics for the provided internal states. Using this framework, we provided this new model with independent principal component representations of individuals’ hidden states and visualized the corresponding kinematics through stick figure movie representations of the resulting kinematics over the walking trials. Predicting time evolution of kinematics from an initial posture (self-driving) Our trained generative gait model can take in a single initial posture of size (6,1) corresponding to a single time point representation of each of the 6 joint angles to predict the next time step posture/kinematics using command ‘model.predict’. To make further predictions, the predicted value is used as the new initial condition (posture) and predictions are made on a one-time step basis in a similar fashion for a pre-specified prediction length (self-driving). However, it is important to note that even though our current framework only predicts one time-step in the future, the LSTM layer remains stateful through each gait cycle, allowing the model to learn much longer timescales as seen when self-driving the network (S7 Fig). Ethics statement All participants provided written informed consent prior to study participation, and the study protocol was approved by the Emory University Institutional Review Board. Human subject participants To develop dynamical signatures of human gait, we collected data in seven post-stroke individuals (age = 56 ± 12 years; 2 females; 48 ± 25 months post-stroke; Lower Extremity Fugl-Meyer = 20 ± 4) and five able-bodied (AB) controls (age = 24 ± 4 years; 4 female). All post-stroke participants experienced a cortical or subcortical ischemic stroke, were able to walk on a treadmill for one minute without an orthotic device, and exhibited no signs of hemi-neglect, orthopedic conditions limiting walking, or cerebellar dysfunction. Experimental design Participants completed 15-second walking trials at six different speeds, distributed evenly between and ranging from each participant’s self-selected (SS) speed to the fastest safe and comfortable speed. Across stroke participants, gait speeds ranged from 0.3–1.6 m/s. Each participant’s fastest walking speed was determined by progressively increasing the treadmill speed from the SS speed until the participant could no longer comfortably or safely maintain the speed for 30 seconds. Participants rested for 1–2 minutes between consecutive gait trials. During data collection, speed increased from the participant’s SS to their fastest speed (i.e., not randomized). Data acquisition Reflective markers were attached to the trunk, pelvis, and bilateral shank, thigh, and foot segments [120]. We collected marker position data while participants walked on a split-belt instrumented treadmill (Bertec Corp., Ohio, USA) using a -7-camera motion analysis system (Vicon Motion Systems, Ltd., UK). Participants held onto a front handrail and wore an overhead safety harness that did not support body weight. Marker data were collected at 100 Hz, and synchronous ground reaction forces were recorded at 2000 Hz and were down sampled to 100Hz using previously established techniques [121–123]. Data processing Raw marker position data were labeled, gap-filled, and low-pass filtered in Vicon Nexus. Labeled marker trajectories and ground reaction force raw analog data were low-pass filtered In Visual 3D. Gait events (bilateral heel contact and toe-off) were determined using a 20-N vertical GRF cutoff, and sagittal-plane hip, knee, and ankle joint kinematics were calculated in Visual 3D (C-Motion Inc., Maryland, USA). RNN model development Our goal was to start with a simple RNN to reduce overfitting with too many parameters and deep layers. We wanted the simplest model capable of learning the dynamics underlying gait which also preserved interpretability. The simplest recurrent neural network (RNN) model architecture consisted of one input layer, one hidden layer and one output layer. The hidden layer was composed of long short-term memory (LSTM) units with a lookback parameter that spanned at least one gait cycle. Model hyperparameter selection is described in a later paragraph. RNN model training Model fitting on our selected dataset and architecture was executed on the order of minutes to tens of minutes, using Keras 3.7.13 and TensorFlow 2.8.2 on Google Colab’s standard GPU with high-RAM runtime (54.8 gigabytes). The RNN model was trained using bilateral, sagittal-plane, lower-limb joint angles from 5 able-bodied (AB) participants and 7 stroke survivors each walking on a treadmill at 6 steady speeds, ranging from each participant’s preferred speed to the fastest safe speed. Our training dataset was input to the RNN in multivariate format (not concatenated) [62,63]. We trained a sequence-to-sequence RNN with 512 long-short-term memory (LSTM) activations units in the single hidden layer, capable of using 15 seconds (sample rate of 100 Hz) time-series kinematic input data (0 to T-1) to predict kinematics one time-step in the future (1 to T) for all training data across individuals and speeds. Our data was batched according to the number of total trials (N = 72); thus, the LSTM maintains its internal state while a batch is being processed, after which the internal state can be maintained or cleared. Because our network retains its internal state from one time step to the next (i.e., the RNN is stateful), we have fine-grained control over when the internal state of the LSTM network is reset. The input data from all trials was ‘mini-batched’ into 2 training batches and 1 validation batch (499 samples each) that were used to update model weights on each model run (epoch). To format our data into equal length input and output mini batches for training and account for the output data being a one-time step shifted version of our input data, our lookback parameter must be one value less than a divisor of the trial length. For example, in our dataset (1500 sample length trials), a lookback parameter of 499 would result in the first mini batch input of samples [0:499] which will predict our reference output samples [1:500], our 2nd mini batch input data would include samples [501:999] and corresponding output [502:1000] and the last mini-batch input of samples [1001: 1499] predicts samples [1002: 1500]. This lookback parameter of 499 allows us to construct 3 mini-batches of shorter input and output data lengths which would be used to train and validate the RNN model (2:1 training: validation mini batch split). Similarly with the lookback parameter of 499 (2:1 training: validation mini-batch split) and 749 (1:1 training: validation split). Mean squared error was used as the LSTM loss function and ADAM as the optimization algorithm because it is fast, has a small memory footprint and is well suited for large-parameter deep learning models [124]. The model was trained for at least 5000 iterations or until training and validation error converged (< 0.75°). The training resulted in a sample-specific dynamical model structure defined by a single set of LSTM network weights (W). The model’s internal states capture trial-specific dynamics predicting the time evolution of joint kinematics; activation coefficients (H) and memory cell states (C) and are tuned based on kinematic input. Model hyperparameter selection We selected the hyperparameter values of 512 nodes in the LSTM layer and a 499-sample LSTM lookback length (number of samples preceding the current time point that is used to train the LSTM) were selected based on training and validation loss, as well as the ability to encode dynamics over short and long timescales. In two steps, we evaluated all pairs of the following hyperparameter values: 1024, 512, 256, and 128 LSTM nodes and 749, 499, and 249-sample lookback parameters. Because RNN performance can change with the parameters used to initialize the RNN, we fit an RNN gait dynamics model 20 times using random initial parameters, for each hyperparameter pair. First, we compared model training and validation loss for each hyperparameter pair: the ‘best’ hyperparameter pair would have low training and validation loss. The following [node-lookback] pairs were considered the best hyperparameter pairs: 512–499 (MSEtrain = 0.010 ± 0.001 deg2; MSEval = 0.018 ± 0.000 deg2), 256–749 (MSEtrain = 0.010 ± 0.002 deg2; MSEval = 0.015 ± 0.001 deg2), 256–499 (MSEtrain = 0.010 ± 0.001 deg2; MSEval = 0.017 ± 0.000 deg2). (S1A Fig). The training loss was not different between hyperparameter pairs (p > 0.235). The validation loss differed between all three models (p < 0.001), with the 256–749 model having the lowest validation loss. However, if the differences in validation loss of less than 0.003 deg2 corresponded to meaningful differences in performance was unclear. Our second analysis was, therefore, used to compare the three hyperparameter pairs deemed best in the prior analysis. Here, we evaluated the models’ abilities to encode the average dynamical behavior over long timescales (long-time) and the stride-to-stride behavior (short-time). We defined the best model as the one with the highest long- and short-time performance. The following analysis was performed for 10 of the 20 random initializations. For long- and short-time analyses, we created a single set of reference dynamics as done in the manuscript: we performed one time-step predictions over the full (1500-sample) time-series. This step provided best-case predictions of the gait dynamics (S1B Fig). Long-time performance. We generated long-time predictions of each trial’s gait signatures (RNN latent states) by simulating each participant’s gait dynamics forward in time, 1500 samples into the future. Each simulation was initialized by setting the RNNs’ latent states to those of the last sample of the trial’s reference dynamics and using the last sample of the trial’s kinematics. We then phase-averaged both the reference dynamics and the long-time predictions using the same technique as described in the main manuscript. Long-time performance was defined as the similarity of the phase-averaged latent states (i.e., the gait signatures) between the reference and the long-time predictions and was quantified using R2. Note that using R2 as a similarity metric, rather than the Euclidean distance metric used in the main manuscript, was needed to compare models with different numbers of nodes. Unlike R2, Euclidean distances are sensitive to the number of samples used to compare models, which would bias short- and long-time performance towards models with fewer nodes. Low R2 values between predictions indicates that the learned dynamics are sufficiently complex to capture instantaneous gait dynamics but can also accurately generate the time evolution of dynamics over the gait cycle—a major challenge in data-driven models of locomotion [89–90]. The 512-node model captured gait dynamics over long time scales significantly better (i.e., more accurate predictions of the time-varying dynamics) than the 249-node models (S1B Fig). For long-time predictions, the 512-node model predictions (R2 = 0.50 ± 0.46) were better than the 249-node 499-sample lookback model (ΔR2 = 0.27 ± 0.06; p < 0.001; independent-samples t-tests) and the 249-node 499-sample lookback model (ΔR2 = 0.31 ± 0.07; p < 0.001). Short-time performance. We generated short-time predictions by simulating single strides in each trial’s time-series, initialized from the first sample of each stride. Initialization used the latent RNN states and kinematics of the reference dynamics at the onset of a new stride (phase = 0 rad). For each initial condition, we integrated the dynamics forward in time, up to the onset of the next stride. For each stride, we then compare the similarity of the reference dynamics to the dynamics of the corresponding short-time prediction using R2. Short-time performance was quantified as the average R2 value across trials for a single model and initialization. The 512-node model captured gait dynamics over short time scales significantly better (i.e., more accurate predictions of the time-varying dynamics) than the 249-node models (S1B Fig). For short-time predictions, the 512-node model predictions (R2 = 0.11 ± 0.51) were more accurate than the 249-node 499-sample lookback model (ΔR2 = 0.51 ± 0.13; p = 0.055; independent-samples t-tests) and the 249-node 499-sample lookback model (ΔR2 = 0.34 ± 0.09; p < 0.001). Based on difference in short- and long-time prediction performance, we selected the 512-node, 499-sample lookback hyperparameters for the RNN model. Long-time performance. We generated long-time predictions of each trial’s gait signatures (RNN latent states) by simulating each participant’s gait dynamics forward in time, 1500 samples into the future. Each simulation was initialized by setting the RNNs’ latent states to those of the last sample of the trial’s reference dynamics and using the last sample of the trial’s kinematics. We then phase-averaged both the reference dynamics and the long-time predictions using the same technique as described in the main manuscript. Long-time performance was defined as the similarity of the phase-averaged latent states (i.e., the gait signatures) between the reference and the long-time predictions and was quantified using R2. Note that using R2 as a similarity metric, rather than the Euclidean distance metric used in the main manuscript, was needed to compare models with different numbers of nodes. Unlike R2, Euclidean distances are sensitive to the number of samples used to compare models, which would bias short- and long-time performance towards models with fewer nodes. Low R2 values between predictions indicates that the learned dynamics are sufficiently complex to capture instantaneous gait dynamics but can also accurately generate the time evolution of dynamics over the gait cycle—a major challenge in data-driven models of locomotion [89–90]. The 512-node model captured gait dynamics over long time scales significantly better (i.e., more accurate predictions of the time-varying dynamics) than the 249-node models (S1B Fig). For long-time predictions, the 512-node model predictions (R2 = 0.50 ± 0.46) were better than the 249-node 499-sample lookback model (ΔR2 = 0.27 ± 0.06; p < 0.001; independent-samples t-tests) and the 249-node 499-sample lookback model (ΔR2 = 0.31 ± 0.07; p < 0.001). Short-time performance. We generated short-time predictions by simulating single strides in each trial’s time-series, initialized from the first sample of each stride. Initialization used the latent RNN states and kinematics of the reference dynamics at the onset of a new stride (phase = 0 rad). For each initial condition, we integrated the dynamics forward in time, up to the onset of the next stride. For each stride, we then compare the similarity of the reference dynamics to the dynamics of the corresponding short-time prediction using R2. Short-time performance was quantified as the average R2 value across trials for a single model and initialization. The 512-node model captured gait dynamics over short time scales significantly better (i.e., more accurate predictions of the time-varying dynamics) than the 249-node models (S1B Fig). For short-time predictions, the 512-node model predictions (R2 = 0.11 ± 0.51) were more accurate than the 249-node 499-sample lookback model (ΔR2 = 0.51 ± 0.13; p = 0.055; independent-samples t-tests) and the 249-node 499-sample lookback model (ΔR2 = 0.34 ± 0.09; p < 0.001). Based on difference in short- and long-time prediction performance, we selected the 512-node, 499-sample lookback hyperparameters for the RNN model. Leave-one-out subject model evaluation for generalizability Using the selected hyperparameters, 12 different models were trained where one different subject (all 6 speed trials per subject) was held out for evaluation on each model run. The same model architecture, training and validation setup was used as the original model trained using the full dataset (12 subjects). The minimum training loss, validation loss, and overall evaluated test loss for each model were extracted and box plots of each generated. The Wilcoxon Rank-Sum Test statistic was used to compare the means. Each model was evaluated on the 6 held-out speed trials from training and an average loss was calculated for each model. The reference kinematics, externally driven and self-driven predictions of each of the 6 held-out trials per model were phase averaged and R2 between the phase averaged externally driven and long-time self-driven predictions (see Long-Time Performance section, above) were calculated. Box plots for each metric across the held-out trials were generated and the Wilcoxon Rank-Sum Test statistic used to compare the means. Computing gait signatures from RNN internal states To develop the gait signatures, we extract the activation and cell states from the LSTM (denoted “H” and “C” respectively) which evolve over time (the course of the gait cycle) as the kinematics of each trial are fed through the trained RNN. These H and C parameters represent how the model’s internal parameters change as it encodes the prediction of future kinematic trajectories. The selected 512-node LSTM layer had 512 H parameters and 512 C parameters. Time-varying gait signatures were computed by identifying dominant modes of variation in the internal states using principal components analysis (PCA). A single PCA operation was used to transform the internal states for all participants into a common basis. Consequently, inter-trial differences in the time-varying activations of the principal components (modes) reflect differences in the underlying dynamics of the individual(s). These activations constituted the time-varying (1500 sample) gait signatures, which had the same dimension as the RNN’s hidden layer (1024 units). However, the first six principal components accounted for ~72% of the variance in the internal states. To compare gait signatures within and between individuals, we phase averaged each trial’s signatures across strides. Rather than linearly interpolating the data between foot contact events before averaging, as is common in gait analysis [122,125–127] we computed a continuous phase using the first 3 gait signature modes for each trial using Von Mises interpolation [128]. Compared to averaging across linearly interpolated strides, phase averaging is expected to reduce the variance in the data at any point in the stride [66,90]. As the domain for interpolation, we estimated the time-varying phase for each trial separately using the Phaser algorithm, using the first 3 principal components as phase variables [66]. To align phase estimates across trials, we defined zero-phase as the maximum of the first principal component. Gait event estimation of phase averaged signatures The force plates embedded in the treadmill captured precise gait event timing information (left heel strike, right toe off, right heel strike, left toe off) across individuals’ trials which we represented as a vector of 1’s, 2’s, 3’s and 4’s, respectively (ground truth markings for the 4 gait events). We leveraged the Phaser algorithm again [66] to develop a phase estimator to transform these 4 gait events over time into gait events over phase. For each trial, we determined the mode phase that corresponded to each of the 4 gait events to gain a representation of where the 4 gait events occurred during phase averaged dynamics (0–2π) for each trial. Interpolation of unseen speed gait signatures to reconstruct kinematics To demonstrate the generalizability of gait dynamics, we show that linearly interpolating gait signatures to predict gait kinematics at new walking speeds is more accurate than linearly interpolating the kinematics themselves. We trained another RNN model with the same architecture and hyperparameters to the first model, however using only the 2 slowest and 2 fastest speeds from each participant (i.e., we held out the 2 middle speed trials from each participant). We then linearly interpolated the 2-middle speeds’ internal states and ran the data through the trained RNN to reconstruct or predict kinematics. We compared the original phase averaged kinematics to the predicted kinematics resulting from each of the two linear interpolations (dynamics and kinematics) using the coefficient of determination. Furthermore, even when we reduced the dimensionality of the model’s internal states from the full 1024 to the first 6 principal components (the selected dimension on the gait signatures), it still performed better than interpolating kinematics (also rank = 6). Biomechanical interpretation of principal components of the gait signature To reconstruct kinematics from the corresponding underlying dynamics (internal state representations), we restored our trained model’s weights to a new model using the ‘model.set’ and ‘model.get_weights’ Keras built in functions. The function ‘model.predict’ takes in the hidden state values (Hs) only (first 512 of the 1024 internal-state time trajectories) and predicts the corresponding kinematics for the provided internal states. Using this framework, we provided this new model with independent principal component representations of individuals’ hidden states and visualized the corresponding kinematics through stick figure movie representations of the resulting kinematics over the walking trials. Predicting time evolution of kinematics from an initial posture (self-driving) Our trained generative gait model can take in a single initial posture of size (6,1) corresponding to a single time point representation of each of the 6 joint angles to predict the next time step posture/kinematics using command ‘model.predict’. To make further predictions, the predicted value is used as the new initial condition (posture) and predictions are made on a one-time step basis in a similar fashion for a pre-specified prediction length (self-driving). However, it is important to note that even though our current framework only predicts one time-step in the future, the LSTM layer remains stateful through each gait cycle, allowing the model to learn much longer timescales as seen when self-driving the network (S7 Fig). Supporting information S1 Video. Stick figure movie demonstrating the RNN’s kinematic reconstruction of all the PCs of dynamics. https://doi.org/10.1371/journal.pcbi.1011556.s001 (MP4) S2 Video. Stick figure movie demonstrating that PC1 encodes dynamics driving hip flexion and extension. https://doi.org/10.1371/journal.pcbi.1011556.s002 (MP4) S3 Video. Stick figure movie demonstrating that PC2 encodes dynamics driving knee flexion and extension. https://doi.org/10.1371/journal.pcbi.1011556.s003 (MP4) S4 Video. Stick figure movie demonstrating that PC3 encodes dynamics driving postural coordination. https://doi.org/10.1371/journal.pcbi.1011556.s004 (MP4) S5 Video. Stick figure movie demonstrating the kinematic reconstruction when AB2’s PC1 projection is replaced with ST4’s impaired PC1 projection. https://doi.org/10.1371/journal.pcbi.1011556.s005 (MP4) S6 Video. Stick figure movie demonstrating the kinematic reconstruction when AB2’s PC3 projection is replaced with ST4’s impaired PC3 projection. https://doi.org/10.1371/journal.pcbi.1011556.s006 (MP4) S7 Video. Stick figure movie demonstrating the kinematic reconstruction when AB2’s PC4 projection is replaced with ST2’s impaired PC4 projection. https://doi.org/10.1371/journal.pcbi.1011556.s007 (MP4) S8 Video. Stick figure movie demonstrating the kinematic reconstruction when ST3’s impaired PC5 projection is replaced with AB2’s PC5 projection. https://doi.org/10.1371/journal.pcbi.1011556.s008 (MP4) S1 Fig. Comparison of model performance on training and validation loss (left), and long- and short-time prediction performance (right). In both plots, small dots represent the average values across trials for each random initialization of each model. Large dots and bars denote the average and standard deviation of model performance metrics across initializations. Left: Training and validation loss (RMSE) for all 12 hyperparameter pairs. Models in the lower-left consider are considered better. Right: Long- and short-time prediction performance (R2) for the 3 hyperparameter pairs with the lowest training and validation loss. Models in the upper-right corner are considered better. https://doi.org/10.1371/journal.pcbi.1011556.s009 (TIF) S2 Fig. RNN model training (green) and validation (blue) loss curves. https://doi.org/10.1371/journal.pcbi.1011556.s010 (TIF) S3 Fig. RNN dynamic learning generalizes across 12 leave-one-individual-out models. The minimum train loss (blue) and validation loss (green) was low (< 0.02 degrees2) for 5 models that were trained each with a single able-bodied individual held out of the training data (A, i) and 7 models each with a stroke individual held out of training data (A, ii). The magnitude and range of the test loss (evaluation of model on the held-out data) (orange) was higher than the respective minimum training and validation losses for both held-out able-bodied (A, i) and stroke (A, ii) models. The magnitude and range of test losses evaluated on held-out able-bodied individuals, however, were lower than models evaluated on held-out stroke data. The models generate external predictions (blue) of held-out test trials with higher R2 values than that of self-driven predictions (red) in models evaluated on both able-bodied (B, i) and stroke trials (B, ii). The models can generate external kinematic predictions (blue) of held-out able-bodied (B, i) trials better than that of stroke (B, ii). Self-driven predictions of stroke kinematics were generally very low (R2 values below 0.5). Models were incapable of generating self-driven predictions for 5 of 30 able-bodied trials and 16 of 42 stroke trials. (C) shows reference (black), externally driven (blue) and self-driven (red) phase averaged kinematic predictions for an exemplary able-bodied trial (C, i) and exemplary stroke trial (C, ii). Models can predict kinematics of held-out able-bodied trials better (higher R2) than that of stroke. https://doi.org/10.1371/journal.pcbi.1011556.s011 (TIF) S4 Fig. Cumulative proportion of variance explained by the first 100 principal components of gait dynamics. Six (6) principal components (PCs) explained 77% of the variance in the gait dynamics. The top 6 dominant PCs were used to develop the gait signature. https://doi.org/10.1371/journal.pcbi.1011556.s012 (TIF) S5 Fig. Support vector machine cross-validation classification accuracy of four different gait descriptors (discrete variables, gait signatures, kinematics, and a combination of kinematics & joint velocity) for discrimination between: a) gait group (able-bodied vs. stroke) and B) individuals. Using k = 25 folds, RNN gait signatures distinguished between impaired and unimpaired gait with 100% accuracy, along with the 26 discrete variables (100%, p = 1), whereas kinematic (92.67 ± 0.15%, p < 0.05) and kinematics & velocity (88.67 ± 0.17%, p <0.05) discrimination were significantly lower. Using k = 6 folds, SVM classification of individuals was most accurate using RNN gait signatures and discrete variables (100%), lower using kinematics (88.9 ± 0.13%, p = 0.061) and significantly lower using a kinematics and velocity (68.10 ± 0.16%, p < 0.05). https://doi.org/10.1371/journal.pcbi.1011556.s013 (TIF) S6 Fig. Relationship between participants’ similarity to able-bodied gait signatures and the RNN’s ability to predict gait kinematics over one gait cycle. There exists a negative correlation between self-driven R2 and Euclidean distance to the AB centroid that is statistically significant at the 0.05 level (S6 Fig, dots). Lower functioning stroke survivors are located further away from the able-bodied centroid, however remarkably all but one (outlier) of the R2 values are above 0.73 in both high and lower-functioning individuals. Even though the model can better predict able-bodied future kinematics better than stroke (R2 values above 0.8), the ability of our model to predict at least a single gait cycle of future stroke kinematics with R2 above 0.73 is promising. To provide a control for this analysis, we found the average R2 values between the self-driven prediction of the last full gait cycle of each trial and 5 randomly selected gait cycles from other randomly selected individuals in the same gait group (able-bodied vs. stroke) (S6 Fig, stars). This control would effectively reveal whether our model learned each individual’s specific gait pattern or was just producing arbitrary (averaged) gait patterns. We expect that if the model learned individuals specific gait patterns, then the distribution of the R2 values of the self-driven kinematic predictions would be significantly different to that of the averaged R2 values corresponding to arbitrary gait patterns. In fact, the averaged R2 of the generic gait cycle comparisons ranged from 0.56 to 0.95 (compared to 0.61–0.99 in self-driven predictions). Further, the distributions of the R2 outputs for both able-bodied and stroke individuals were significantly different (at the 0.05 level) to self-driven R2 outputs with p-values 5.5x10-6 and 5.4x10-3 respectively using the Wilcoxon Rank-Sum test. This reveals that even for stroke survivors the model has learned their dynamics in some capacity and the model isn’t simply predicting arbitrary gait patterns. https://doi.org/10.1371/journal.pcbi.1011556.s014 (TIF) S7 Fig. Graphical summary and pseudo code of our gait signatures framework and algorithm. A) Training dataset: Using the 6-dimensional gait trajectory, the inputs (green) were concurrent segments from the gait trajectory, each one 499-time steps long. The outputs (blue) were 1-shifted (in time) segments of inputs. B) Model architecture: Our model consisted of an input layer, a hidden layer composed of 512 LSTM units, and a 6-unit Dense output layer. C) Stateful training: The hidden state of an RNN at time t is a function of the input at time t and the hidden state at time t-1. The model starts with processing the first mini-batch, calculating a new hidden state at each t and predicting gait kinematics at time t+1 given kinematics data at t. At the end of the mini-batch processing, the model calculates MSE over the entire mini-batch to calculate error for backpropagation and to update model weights. The final hidden state h(t+L) is used as the initial hidden state for generating predictions for the next mini batch (L is the temporal length of each mini-batch). The hidden state is initialized as zero before processing the first mini-batch. https://doi.org/10.1371/journal.pcbi.1011556.s015 (TIF) S1 Table. Stroke participant’s demographics and clinical scores. https://doi.org/10.1371/journal.pcbi.1011556.s016 (TIF)

Neal, Maxwell;Thiruppathy, Deepan;Zengler, Karsten

doi: 10.1371/journal.pcbi.1011594pmid: 37903176

Introduction and background Bacteroides fragilis is a human commensal colon bacterium [1]. It is gram-negative, anaerobic but somewhat aerotolerant, and ferments dietary fibers and resistant starches [1,2]. The bacterium belongs to the phylum Bacteroidetes, which along with Firmicutes comprises 90% of the total gut microbiota [3], and is essentially universal in humans. B. fragilis can consume an assortment of carbohydrates, including host protein glycans [1,4]. The bacterium excretes volatile fatty acids (VFA) and other carboxylic fermentation products such as acetate, propionate, lactate, and fumarate. The regulation of liver and intestinal human metabolism is partially influenced by these products and their abundance ratios. These metabolites are therefore associated with the development of obesity, inflammatory bowel disease (Crohn’s disease and ulcerative colitis), and other diet associated disorders [5–8]. The metabolic associations are different for each of these fermentation products. Acetate can suppress appetite, reducing overall caloric intake. Propionate is known to inhibit lipogenesis, while acetate promotes it, in certain conditions. Both propionate and butyrate promote gluconeogenesis, affecting blood sugar homeostasis [6,8]. Thus, while carbohydrates promote overall VFA production in the gut, the effects on host metabolism vary significantly depending on the metabolic activity and fermentation profiles of specific bacteria. Bacteroides species primarily produce acetate and propionate, while members of the Firmicutes phylum produce more butyrate [1,5]. While the metabolism of the related B. thetaiotaomicron and its carbon utilization has been well studied and mathematically modeled [9], our knowledge of B. fragilis’ metabolism has been relatively limited. Therefore, the ability to predict the production flux ranges of these VFAs from the ubiquitous B. fragilis and its growth rates in various environments is critical for delineating B. fragilis’ and the other Bacteroides species contributions on host physiology. Current efforts have been faced with issues connecting individual observations about growth phenotypes or enzyme activities to form a complete understanding of B. fragilis’ metabolism and design strategies to modulate its behavior. Genome-scale metabolic models (GEMs) overcome this barrier by representing metabolism with a set of reactions derived from the organism’s genome [10]. Therefore, a GEM would create a comprehensive, quantitative model of metabolism, enabling the user to elucidate the interactions between B. fragilis’ metabolic components and optimize VFA production rates [11]. An annotated genome provides the GEM with enzymes and transporters that are available to the organism and associates them with biochemical reactions. During the manual curation of the GEM, enzyme functions and transporters may be added and corrected with available data in the literature. Manual curation is further supported by databases such as BiGG [12] and KEGG [13], which contain information on enzymes, reactions, and substrates for the purposes of metabolic modeling and analysis. Then lipidomics, proteomics, and other data are integrated into the GEM to describe the metabolites needed to produce the organism’s biomass, which yields the biomass reaction [14]. Furthermore, the metabolic rates in a GEM are constrained by the definition of the available resources in the media and by the reversibility of all reactions as inferred by thermodynamics. These constraints, assumptions, and reaction stoichiometries narrow down the possible ranges of metabolic reaction fluxes. Therefore, a GEM is able to identify which reactions and pathways must be active and at what rates under defined conditions. To date, GEMs have been used to predict the growth rate or ability to utilize a variety of nutrients, the maximal output rates of desired metabolites, and the minimal number of reactions needed to produce a desired output for thousands of species across all domains of life [12,15]. In this work, we reconstructed a GEM for Bacteroides fragilis strain 638R refined with experimental growth and metabolite utilization data. We subsequently deployed the model to analyze systems-level features of its metabolic network, its ability to produce different VFAs across conditions, and its efficiency in utilizing different carbon sources for growth. Together, our results outline B. fragilis’ potential metabolic role within the gut microbial community by identifying its possible inputs, outputs, and preferred substrates. We also compare our model to an existing semi-automatic reconstruction. The model, named iMN674 to follow convention, enables the integration of multi-omics data sets and the generation of hypotheses with respect to the metabolism and niches of B. fragilis. Results and discussion Model reconstruction The genome-scale metabolic reconstruction iMN674 of Bacteroides fragilis contains 1,109 metabolites distributed across three compartments, i.e. the extracellular space, the periplasm, and the cytosol. The model contains 1,362 metabolites in total associated with 1,634 reactions. The associated pathways of these are broken down in Fig 1. Of these reactions, 142 are exchange reactions, 253 are transporters, and the remaining 1,239 are metabolic. The reactions are associated with 674 genes excluding pseudogenes identifying non-enzymatic reactions, exchanges, and reactions with no identified gene. The complete model is available as a MATLAB file, COBRApy json file, and spreadsheet format (S1 and S2 Files). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. An overview of the model’s reactions and basic properties. (A) In the major subsystem plot, subsystems associated with major macromolecule types were grouped. Transport associated reactions are shown in red. (B) The secondary subsystem bar plot, which expands the “other” category in the first plot. https://doi.org/10.1371/journal.pcbi.1011594.g001 Our model was generated following the procedure outlined by Thiele and Palsson [10], beginning with finding reactions associated with genes in the annotated genome on the KEGG database. Additional pathways not present in the database were gathered through a search of the literature (see the “Reference” column of the reactions sheet of S2 File for reactions added this way). The initial draft model was assembled using these gathered reactions, the necessary exchange reactions to provide metabolites, and an initial biomass composition estimate. This is followed by an iterative process of examining the model’s growth on known substrates, accuracy of ATP production, and other features. Deficits are fixed by delving back into the literature and finding additional transporters, reactions, and constraints to further refine the model. Nine reactions were included in the model without genome annotation, excluding non-enzymatic reactions. For comparison, iML1515, a highly comprehensive model of E. coli, contains 113 such reactions [16]. In iMN674, three of these were transporters for metabolites B. fragilis was experimentally shown to consume, but no suitable genes could be found to import these metabolites. Transport mechanisms in the Bacteroidetes phylum as a whole remain unclear for a number of carbohydrates and further work will be required to elucidate these processes [1]. In addition to missing annotation for genes involved in carbon utilization, there is also a lack of information on the genes required for replication. About half of the essential genes in B. fragilis 638R previously determined through transposon gene disruption have no identified function [17]. Our model has a sensitivty and specificity of 0.27 and 0.86 when predicting gene essentiality. Note that as so many essential genes lack annotation, they could not be included in the mode. S1 Table contains this comparison and a list of essential genes. One reaction in iMN674 with no gene association is thymidine monophosphate kinase (E.C. 2.7.4.9). This reaction is necessary for the production of dTDP from dTMP, and thereby the production of dTTP for DNA replication. No match for this enzyme in B. fragilis’ genome could be identified. The enzymes on either side of this reaction in the dTTP production pathway are present, so in consideration of the necessity of this enzyme for growth the reaction was included in the model without a gene association. A complete list of reactions without genome annotations is provided in S2 Table. These unknowns and gaps leave 25 percent of the reactions unable to carry flux. These reactions are generally associated with secondary metabolites and are entirely uninvolved in producing growth metabolites. For example, the genome contains several enzymes involved in the production of quorum sensing molecules, but the complete pathways have not been identified [18]. This renders the few known quorum sensing reactions disconnected from the metabolic network. There are also reactions associated with the degradation and export of various drugs. As these miscellaneous molecules are not otherwise part of the model, their associated reactions cannot carry flux. Many such reactions are secondary uses of more common enzymes which are added to automated reconstructions in a species independent manner. Additionally, this may also be a reflection of B. fragilis’ apparent proclivity to carry a diverse set of enzymes for a broader range of possibilities [1]. The biomass reaction provides the objective function in most simulations performed and its components define which pathways must be active under any growth conditions [10]. We defined a biomass reaction consisting of amino acids, lipids, carbohydrates, DNA and RNA monomers, cofactors, minerals, and the cycling of carrier molecules for these components like tRNAs. This information was compiled from data on B. fragilis and other members of the Bacteroides genus (see Methods - Biomass objective). The biomass reaction contains 98 metabolites, and is as detailed as possible to maximize the model’s accuracy. Its components are summarized in Fig 2. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. The biomass components consumed in the model. The central pie chart shows the percent of consumed metabolites that belong to each category. Next to each sector is a diagram representing how a typical component is represented in the biomass reaction. Water, hydrogen, and other minor components of these reactions are not shown. https://doi.org/10.1371/journal.pcbi.1011594.g002 Experimental nutrient utilization We validated B. fragilis 638R’s ability to utilize 44 nutrients in a dilute complex medium experimentally. The change in OD600 after two days was measured to determine which nutrients B. fragilis utilizes (see Methods - Nutrient utilization experiment). Then the model was simulated with and without each nutrient to confirm if the model also predicted an increase in growth rate. The model predicted that many metabolites would only be beneficial in conjunction with others, as elaborated upon in Results—Co-metabolism. Thus a dilute complex medium was chosen to reveal which nutrients could be utilized, without the need of testing thousands of combinations. The results of our growth experiment and other growth predictions are summarized in Fig 3. Our model achieves a sensitivity and specificity of 0.9 and 0.96 when predicting growth or no growth on these nutrients. The full results are in S3 Table. Full simulation details are in S4 Table. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. A comparison of in silico and experimental growth phenotypes. This bar chart compares predicted and experimental growth rates for five media from the literature. The legend below describes the carbon and nitrogen sources, with numbers corresponding to the bars in the main figure. Full details are present in S4 Table. Conditions 1 and 2 were found in Spence et al [19], Fig 2A and 2B. Conditions 3–5 are from Varel and Bryant [20], Fig 1 curves 1, 3, and 4. For most metabolites, the associated exchange reaction was allowed a maximal uptake of 20 millimoles per gram dry weight of organism per hour (mmol/gDW/h), except maltose which was constrained to 10 mmol/gDW/h as it contains two units of glucose. https://doi.org/10.1371/journal.pcbi.1011594.g003 Additionally, the model was curated to predict growth phenotypes in a variety of media routinely used to grow B. fragilis [19–23]. The ATP requirements of the model were optimized to constrain the growth rate within 20% of the values presented in Varel and Bryant and Spence et al [19,20], those used in Fig 3. This yielded a growth-associated ATP maintenance (GAM) coefficient of 24.9 and a non-growth (NGAM) coefficient of 38.9. Our GAM is significantly lower than the values in many other models (such as 75.5 in iML1515), due to us explicitly modeling ATP consumption in tRNA charging and amino sugar synthesis for biomass (see Methods - ATP maintenance coefficients). The mean percent error over the data from these sources was 14%. B. fragilis cannot use amino acids as the sole nitrogen source [20], and we found no growth improvement in the dilute complex medium when amino acids were added. With the incorporation of these growth rates from the literature and our metabolite utilization experiment, we have consolidated the available growth rate data and expanded the knowledge of which substrates B. fragilis may use. Subsequently, we analyzed carbon utilization pathways and determined condition-dependent fermentation profiles. VFA production The model captures the production of a variety of VFAs from glucose or other carbon sources. The fermentation products identified in Onderdonk and Gorbach and Frantz and McCallum (1979) [22,24] were examined to determine how much of each VFA could be produced at a fixed growth rate and carbon intake. Isovalerate was not included in the model, as the pathway that produces it has not been fully characterized. Table 1 displays these results. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Fermentation Product Output Rates (mmol/gDW/h) on a Minimal Glucose Medium at a Fixed Growth Rate. https://doi.org/10.1371/journal.pcbi.1011594.t001 To construct Table 1, the model was simulated with glucose, ammonia, sulfide, and minerals (see S4 Table). The growth rate was constrained to be at 95% of the optimal value, in the range [0.406, 0.428]. This allows for a broader range of metabolic choices to be represented rather than being limited to the one optimum. The model was then optimized to maximize the output of each metabolite along the rows. The corresponding column is the amount of each carbon containing product produced. Thus, row one represents the results of the simulation when ethanol production was the objective being maximized. The rows of this table thereby outline the range of possible fermentation product outputs. The main diagonal represents the maximum amount of each that may be produced. In each of the cases shown in Table 1, the model predicts that of the 120 mmol/gDW/h of carbon entering the system as 20 mmol/gDW/h of glucose, 93.8 mmol/gDW/h or 78% of the original carbon leaves the system as one of the above fermentation products or carbon dioxide; the rest becomes biomass. The VFA ratios in Frantz and McCallum (1979) [22] are attainable by the model without altering the optimal growth rate by more than 5%. These results outline the range of VFA production possibilities; any values between those shown in the table will also be possible. The actual in vivo ratios of output products must be determined by factors other than which pathways are available. These could include transcriptional control, translational control, or enzyme inhibition. The model predicts that acetate and propionate are partially decoupled. Acetate can be produced without propionate, but propionate production is acetate dependent. In general, the model primarily produced acetate and there were no conditions in which acetate flux was zero. The model could not produce butyrate at all, in line with the observed phenotype for this species. There are a few butyrate-utilizing enzymes present in the model, but the production pathway is incomplete. A similar result has been previously observed in the reconstruction of B. thetaiotaomicron’s metabolism [9]. Additionally, the model was analyzed to determine which metabolite would allow for the largest increase in the production of each fermentation product. When added to provide an additional flux of 10 mmol carbon per gram dry weight per hour (C-mmol/gDW/h), maltose provided the greatest increase in yield of all of the products except butyrate. The acetate to propionate ratio was most decreased by glucose and other carbohydrates. It was most increased by supplementation of amino acids and glucosamine. In general, nitrogenous compounds increased the acetate to propionate ratio while simple carbohydrates decreased the ratio. S5 Table contains the full results. This implies that propionate production increases most in nitrogen depleted, carbohydrate rich conditions, such as a low protein or high carbohydrate diet [25]. Growth efficiency on different carbon sources To predict which metabolites best promote growth of B. fragilis, the efficiency of B. fragilis’ growth on various carbon sources was determined (Fig 4). This efficiency, calculated as simulated biomass yield per millimole of carbon taken in, represents how well a given carbon source can be incorporated into biomass. Common carbohydrates like glucose, starch, and fructose are the most efficiently used by B. fragilis. Some of these can either be immediately hydrolyzed to glucose or are converted to glucose in the non-oxidative pentose phosphate pathway. Therefore, their efficiency is similar to that of glucose. On the lower end of the efficiency spectrum is fucose, which was predicted to have 26% the efficiency of glucose. Fucose is split into two molecules, dihydroxyacetone phosphate and L-lactaldehyde. The former is transformed into pyruvate via glycolysis, while the latter is converted into propanediol and exported, producing no further energy. Fucose-containing substrates such as 2’- and 3’-fucosyllactose similarly showed a reduced efficiency. These and other fucosylated oligosaccharides form 35–50% of human milk oligosaccharides [26]. The inefficiency of B. fragilis in metabolizing fucose-containing carbon sources may help explain why B. fragilis and other Bacteroidetes species are less common in the neonatal gut microbiome than in adults [27]. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Simulated growth yields per mole of carbon on different substrates. iMN674 was simulated on a minimal medium containing ammonia, sulfide, and minerals. The carbon source indicated in each bar was given a maximal intake flux of 120 C-mmol/gDW/h. The predicted growth rate was divided by the carbon flux to indicate the growth efficiency. Metabolites in green contain fucose. Metabolites in red have the same efficiency as glucose. https://doi.org/10.1371/journal.pcbi.1011594.g004 Two of the most efficient substrates identified by the simulation are maltose and glucose-6-phosphate. The maltose phosphorylase reaction splits the maltose units apart and yields glucose and glucose-1-phosphate. This can be converted to glucose 6-phosphate via phosphoglucomutase. Therefore, half of the carbohydrate in maltose enters glycolysis without ATP use, greatly improving the energy yield. Starch is instead hydrolyzed into glucose subunits and does not provide this benefit. Co-metabolism of fermentation products and amino acids While discussion of Bacteroides’ metabolism is typically focused on carbohydrates, amino acids and fermentation products from other microbes are generally available to these microorganisms in the colon. B. fragilis cannot utilize any single amino acid or mixture of amino acids as the sole nitrogen source [20] and the model reflects this. Our model further suggests that B. fragilis cannot produce ATP from acetate, propionate, butyrate, or ethanol. However, the presence of these fermentation end products in addition to other substrates still confers a growth advantage for B. fragilis. It has been previously observed that B. fragilis consumes acetate in conjunction with other organic molecules and is enriched by the consumption of alcohol, which is converted to acetate by the liver [28]. To investigate the potential for VFA co-metabolism, the model was simulated with a range of carbon sources in addition to a mixture of fermentation products, a mixture of amino acids, both mixtures, or neither, all provided at a flux of up to 20 mmol/gDW/h. The growth rates under these conditions were then compared. The addition of these molecules was considered to improve the growth rate if the rate increased by more than 0.001 h-1. Conditions where the model predicted growth with VFAs or amino acids are listed in Table 2. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Predicted Growth Rates for Metabolites Which Were Not a Sole Carbon Source, but Allowed Growth When Supplemented with VFAs or Amino Acids. https://doi.org/10.1371/journal.pcbi.1011594.t002 The VFA mixture, containing acetate, propionate, butyrate, ethanol, and pyruvate, did little to improve the growth rates of B. fragilis. When it did improve, it increased by only 6% on average, despite the massive increase in available carbon. The presence of fermentation products allowed for weak growth on lactate, malate, fumarate, succinate, and several amino acids, none of which could act as sole carbon sources. Generally, the VFAs were incorporated into acyl-CoA and used to assemble lipids. Since VFAs can not be used to produce the same range of biomass precursors as glucose, utilization of VFAs only marginally improved the growth rate of B. fragilis. The amino acid mixture had a greater impact on B. fragilis’ growth and resulted in a 15% increase in growth on average across conditions. This included a 30% increase in growth on glucose and glucose equivalents. In addition to improved growth in the presence of amino acids, we also observed a greater resilience of B. fragilis to gene knockouts. When glucose was the sole carbon source, 127 genes were essential for producing at least one biomass component. These genes were identified by deleting each gene and its associated reactions in the model one at a time to simulate single gene knockouts. If the model predicted no growth, the gene was labeled as essential. Each of these genes, on average, blocked the production of 9 biomass components when deleted. When amino acids were included, this effect dropped to 123 genes which blocked only 6 components on average. Even a relatively small amount of amino acids significantly improved the robustness of B. fragilis. This effect remained even when the maximum uptake of the amino acids was limited to 0.01 mmol/gDW/h. Table 3 identifies the genes that are no longer essential for growth with the addition of amino acids. There were six additional non-essential genes that decreased the growth rate by more than 10% when deleted in the glucose medium but not when supplemented with amino acids. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Differentially Essential Genes. https://doi.org/10.1371/journal.pcbi.1011594.t003 Each row in Table 3 corresponds to a gene in the model that is essential when grown on glucose as the sole carbon source but not when a small amount of amino acids is supplemented. The last column states which metabolites cannot be synthesized when that gene is deleted. One may expect that arginine, histidine, and proline production was restored by direct assimilation of those amino acids, but no transporter for these metabolites is currently annotated. Fig 5 shows how the availability of amino acids saves nucleotide metabolism from the effects of deleterious knockouts. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Purine biosynthesis is saved from gene knockouts by external amino acids. This figure shows an abbreviated view of purine synthesis in B. fragilis. The nitrogen in purine bases is taken from amino acids. These amino acids are produced in transamination reactions involving TCA intermediates. Thus, the gene knockouts corresponding to these transamination reactions (red Xs) are lethal, as the production of key amino acids and all purines is halted. However, if amino acids are made available in the environment (blue arrows), purine biosynthesis is able to proceed. Pyrimidine production is also aspartate dependent, so it is rescued similarly. https://doi.org/10.1371/journal.pcbi.1011594.g005 No genes changed in essentiality when a mixture of fermentation products was supplemented to the model. While VFAs can supplement glucose by forming acyl-CoA, the cell is still dependent on glucose for most biomass precursors. VFAs and amino acids together allowed for low levels of growth (0.11 h -1, 25% of that on glucose) without an additional carbon source. S5 Table contains the full results of these co-metabolism simulations. Overall, these results suggest that even though amino acids and fermentation end products are neither necessary nor sufficient for biomass production alone, they provide an advantage for B. fragilis. Even with only minute influxes, amino acids allowed flux through several inactive pathways and allowed growth when otherwise essential genes were knocked out. The gut microbiome is a complex environment, so it is highly beneficial for B. fragilis to be able to utilize a wide range of metabolites. B. fragilis’ possible niches and metabolism within the colon could therefore be much broader than originally thought and might not only be limited to catabolism of polysaccharides. Human oligosaccharide catabolism in B. fragilis The ability to digest HMOs is a major driving factor deciding which organisms are prevalent in the infant gut microbiome [26,29]. To further investigate the ability of B. fragilis to utilize components of human milk, reactions were added to represent the degradation of the 15 most abundant human milk oligosaccharides (HMOs) [30]. B. fragilis has been shown to digest some oligosaccharides in the periplasmic space [4], so these digestion reactions were assumed to be in the periplasm as well. B. fragilis possesses enzymes capable of digesting a broad range of oligosaccharides. Each bond in each of these compounds could be paired to an enzyme potentially capable of breaking it. S6 Table provides a list of compounds and the enzymes needed to digest them. For several bonds, there are multiple enzymes in the genome that could perform the necessary hydrolysis. This aligns with previous work that suggests B. fragilis readily consumes HMOs [31] and bovine milk oligosaccharides [32] and suggests that B. fragilis could digest them completely. Other bacteria such as Bifodobacterium and Enterococcus spp. are nonetheless able to outcompete B. fragilis when certain HMOs are increased in abundance in the infant gut microbiome [33], indicating that the interplay between these microbes is defined by something other than whether or not the oligosaccharides can be utilized. B. fragilis has been shown to be able to scavenge human N-linked glycans [4]. The ten most abundant N-linked glycans from human transferrin identified by Abu Bakar et al [34] were added as representatives. Again, all oligosaccharides were able to be fully divided into the component monomers in multiple ways. As human N-glycans share the same core and many repeated motifs, it is therefore likely that most such glycans can be used as a carbohydrate source for B. fragilis. The predicted ability to both liberate and consume the sialic acids in these oligosaccharides stands in contrast to the closely related B. thetaiotaomicron, which cannot consume the sialic acids it frees [35]. Sialic acids act as a carbon source for pathogenic Clostridioides difficile and Salmonella typhimurium [35], and have been shown in gnotobiotic mouse models to consume sialic acids liberated by B. thetaiotaomicron [36]. B. fragilis could therefore be competing with these pathogenic bacteria for sialic acid. Comparison of iMN674 to automated AGORA2 model Heinken et al recently published semi-automatic reconstructions of 7,302 human associated microorganisms [37], including a model for Bacteroides fragilis 638R. This model, while refined through some experimentally available growth data, did not undergo the same degree of curation as iMN674. Excluding exchange reactions, the AGORA2 model has 936 reactions without genome annotation, compared to 9 reactions in our model. Among these 936 reactions, 103 are indispensable for the model to grow, even when all exchanges are fully open. This number increases to 133 essential genes with no annotations when limited to a glucose, ammonia, and mineral medium. None of the 621 transporters present in the AGORA2 model have gene associations. Overall, more than 50 percent of the non-exchange reactions lack genome annotations. iMN674, which has genome support for all but 9 reactions, contains a smaller reaction network, but the degree of certainty in these reactions is much higher. Moreover, iMN674 has been refined and curated to accurately reflect experimental data. When comparing model-data agreement as in Fig 3, our model achieves a sensitivity and specificity of 0.90 and 0.96; the AGORA2 model’s sensitivity and specificity are 0.55 and 0.88. While semi-automated models provide a good tool for the community, well curated models such as iMN674 contain an experimentally refined and evidenced reconstruction that provides the highest accuracy to study the metabolism of the target organism. Model reconstruction The genome-scale metabolic reconstruction iMN674 of Bacteroides fragilis contains 1,109 metabolites distributed across three compartments, i.e. the extracellular space, the periplasm, and the cytosol. The model contains 1,362 metabolites in total associated with 1,634 reactions. The associated pathways of these are broken down in Fig 1. Of these reactions, 142 are exchange reactions, 253 are transporters, and the remaining 1,239 are metabolic. The reactions are associated with 674 genes excluding pseudogenes identifying non-enzymatic reactions, exchanges, and reactions with no identified gene. The complete model is available as a MATLAB file, COBRApy json file, and spreadsheet format (S1 and S2 Files). Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. An overview of the model’s reactions and basic properties. (A) In the major subsystem plot, subsystems associated with major macromolecule types were grouped. Transport associated reactions are shown in red. (B) The secondary subsystem bar plot, which expands the “other” category in the first plot. https://doi.org/10.1371/journal.pcbi.1011594.g001 Our model was generated following the procedure outlined by Thiele and Palsson [10], beginning with finding reactions associated with genes in the annotated genome on the KEGG database. Additional pathways not present in the database were gathered through a search of the literature (see the “Reference” column of the reactions sheet of S2 File for reactions added this way). The initial draft model was assembled using these gathered reactions, the necessary exchange reactions to provide metabolites, and an initial biomass composition estimate. This is followed by an iterative process of examining the model’s growth on known substrates, accuracy of ATP production, and other features. Deficits are fixed by delving back into the literature and finding additional transporters, reactions, and constraints to further refine the model. Nine reactions were included in the model without genome annotation, excluding non-enzymatic reactions. For comparison, iML1515, a highly comprehensive model of E. coli, contains 113 such reactions [16]. In iMN674, three of these were transporters for metabolites B. fragilis was experimentally shown to consume, but no suitable genes could be found to import these metabolites. Transport mechanisms in the Bacteroidetes phylum as a whole remain unclear for a number of carbohydrates and further work will be required to elucidate these processes [1]. In addition to missing annotation for genes involved in carbon utilization, there is also a lack of information on the genes required for replication. About half of the essential genes in B. fragilis 638R previously determined through transposon gene disruption have no identified function [17]. Our model has a sensitivty and specificity of 0.27 and 0.86 when predicting gene essentiality. Note that as so many essential genes lack annotation, they could not be included in the mode. S1 Table contains this comparison and a list of essential genes. One reaction in iMN674 with no gene association is thymidine monophosphate kinase (E.C. 2.7.4.9). This reaction is necessary for the production of dTDP from dTMP, and thereby the production of dTTP for DNA replication. No match for this enzyme in B. fragilis’ genome could be identified. The enzymes on either side of this reaction in the dTTP production pathway are present, so in consideration of the necessity of this enzyme for growth the reaction was included in the model without a gene association. A complete list of reactions without genome annotations is provided in S2 Table. These unknowns and gaps leave 25 percent of the reactions unable to carry flux. These reactions are generally associated with secondary metabolites and are entirely uninvolved in producing growth metabolites. For example, the genome contains several enzymes involved in the production of quorum sensing molecules, but the complete pathways have not been identified [18]. This renders the few known quorum sensing reactions disconnected from the metabolic network. There are also reactions associated with the degradation and export of various drugs. As these miscellaneous molecules are not otherwise part of the model, their associated reactions cannot carry flux. Many such reactions are secondary uses of more common enzymes which are added to automated reconstructions in a species independent manner. Additionally, this may also be a reflection of B. fragilis’ apparent proclivity to carry a diverse set of enzymes for a broader range of possibilities [1]. The biomass reaction provides the objective function in most simulations performed and its components define which pathways must be active under any growth conditions [10]. We defined a biomass reaction consisting of amino acids, lipids, carbohydrates, DNA and RNA monomers, cofactors, minerals, and the cycling of carrier molecules for these components like tRNAs. This information was compiled from data on B. fragilis and other members of the Bacteroides genus (see Methods - Biomass objective). The biomass reaction contains 98 metabolites, and is as detailed as possible to maximize the model’s accuracy. Its components are summarized in Fig 2. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. The biomass components consumed in the model. The central pie chart shows the percent of consumed metabolites that belong to each category. Next to each sector is a diagram representing how a typical component is represented in the biomass reaction. Water, hydrogen, and other minor components of these reactions are not shown. https://doi.org/10.1371/journal.pcbi.1011594.g002 Experimental nutrient utilization We validated B. fragilis 638R’s ability to utilize 44 nutrients in a dilute complex medium experimentally. The change in OD600 after two days was measured to determine which nutrients B. fragilis utilizes (see Methods - Nutrient utilization experiment). Then the model was simulated with and without each nutrient to confirm if the model also predicted an increase in growth rate. The model predicted that many metabolites would only be beneficial in conjunction with others, as elaborated upon in Results—Co-metabolism. Thus a dilute complex medium was chosen to reveal which nutrients could be utilized, without the need of testing thousands of combinations. The results of our growth experiment and other growth predictions are summarized in Fig 3. Our model achieves a sensitivity and specificity of 0.9 and 0.96 when predicting growth or no growth on these nutrients. The full results are in S3 Table. Full simulation details are in S4 Table. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. A comparison of in silico and experimental growth phenotypes. This bar chart compares predicted and experimental growth rates for five media from the literature. The legend below describes the carbon and nitrogen sources, with numbers corresponding to the bars in the main figure. Full details are present in S4 Table. Conditions 1 and 2 were found in Spence et al [19], Fig 2A and 2B. Conditions 3–5 are from Varel and Bryant [20], Fig 1 curves 1, 3, and 4. For most metabolites, the associated exchange reaction was allowed a maximal uptake of 20 millimoles per gram dry weight of organism per hour (mmol/gDW/h), except maltose which was constrained to 10 mmol/gDW/h as it contains two units of glucose. https://doi.org/10.1371/journal.pcbi.1011594.g003 Additionally, the model was curated to predict growth phenotypes in a variety of media routinely used to grow B. fragilis [19–23]. The ATP requirements of the model were optimized to constrain the growth rate within 20% of the values presented in Varel and Bryant and Spence et al [19,20], those used in Fig 3. This yielded a growth-associated ATP maintenance (GAM) coefficient of 24.9 and a non-growth (NGAM) coefficient of 38.9. Our GAM is significantly lower than the values in many other models (such as 75.5 in iML1515), due to us explicitly modeling ATP consumption in tRNA charging and amino sugar synthesis for biomass (see Methods - ATP maintenance coefficients). The mean percent error over the data from these sources was 14%. B. fragilis cannot use amino acids as the sole nitrogen source [20], and we found no growth improvement in the dilute complex medium when amino acids were added. With the incorporation of these growth rates from the literature and our metabolite utilization experiment, we have consolidated the available growth rate data and expanded the knowledge of which substrates B. fragilis may use. Subsequently, we analyzed carbon utilization pathways and determined condition-dependent fermentation profiles. VFA production The model captures the production of a variety of VFAs from glucose or other carbon sources. The fermentation products identified in Onderdonk and Gorbach and Frantz and McCallum (1979) [22,24] were examined to determine how much of each VFA could be produced at a fixed growth rate and carbon intake. Isovalerate was not included in the model, as the pathway that produces it has not been fully characterized. Table 1 displays these results. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Fermentation Product Output Rates (mmol/gDW/h) on a Minimal Glucose Medium at a Fixed Growth Rate. https://doi.org/10.1371/journal.pcbi.1011594.t001 To construct Table 1, the model was simulated with glucose, ammonia, sulfide, and minerals (see S4 Table). The growth rate was constrained to be at 95% of the optimal value, in the range [0.406, 0.428]. This allows for a broader range of metabolic choices to be represented rather than being limited to the one optimum. The model was then optimized to maximize the output of each metabolite along the rows. The corresponding column is the amount of each carbon containing product produced. Thus, row one represents the results of the simulation when ethanol production was the objective being maximized. The rows of this table thereby outline the range of possible fermentation product outputs. The main diagonal represents the maximum amount of each that may be produced. In each of the cases shown in Table 1, the model predicts that of the 120 mmol/gDW/h of carbon entering the system as 20 mmol/gDW/h of glucose, 93.8 mmol/gDW/h or 78% of the original carbon leaves the system as one of the above fermentation products or carbon dioxide; the rest becomes biomass. The VFA ratios in Frantz and McCallum (1979) [22] are attainable by the model without altering the optimal growth rate by more than 5%. These results outline the range of VFA production possibilities; any values between those shown in the table will also be possible. The actual in vivo ratios of output products must be determined by factors other than which pathways are available. These could include transcriptional control, translational control, or enzyme inhibition. The model predicts that acetate and propionate are partially decoupled. Acetate can be produced without propionate, but propionate production is acetate dependent. In general, the model primarily produced acetate and there were no conditions in which acetate flux was zero. The model could not produce butyrate at all, in line with the observed phenotype for this species. There are a few butyrate-utilizing enzymes present in the model, but the production pathway is incomplete. A similar result has been previously observed in the reconstruction of B. thetaiotaomicron’s metabolism [9]. Additionally, the model was analyzed to determine which metabolite would allow for the largest increase in the production of each fermentation product. When added to provide an additional flux of 10 mmol carbon per gram dry weight per hour (C-mmol/gDW/h), maltose provided the greatest increase in yield of all of the products except butyrate. The acetate to propionate ratio was most decreased by glucose and other carbohydrates. It was most increased by supplementation of amino acids and glucosamine. In general, nitrogenous compounds increased the acetate to propionate ratio while simple carbohydrates decreased the ratio. S5 Table contains the full results. This implies that propionate production increases most in nitrogen depleted, carbohydrate rich conditions, such as a low protein or high carbohydrate diet [25]. Growth efficiency on different carbon sources To predict which metabolites best promote growth of B. fragilis, the efficiency of B. fragilis’ growth on various carbon sources was determined (Fig 4). This efficiency, calculated as simulated biomass yield per millimole of carbon taken in, represents how well a given carbon source can be incorporated into biomass. Common carbohydrates like glucose, starch, and fructose are the most efficiently used by B. fragilis. Some of these can either be immediately hydrolyzed to glucose or are converted to glucose in the non-oxidative pentose phosphate pathway. Therefore, their efficiency is similar to that of glucose. On the lower end of the efficiency spectrum is fucose, which was predicted to have 26% the efficiency of glucose. Fucose is split into two molecules, dihydroxyacetone phosphate and L-lactaldehyde. The former is transformed into pyruvate via glycolysis, while the latter is converted into propanediol and exported, producing no further energy. Fucose-containing substrates such as 2’- and 3’-fucosyllactose similarly showed a reduced efficiency. These and other fucosylated oligosaccharides form 35–50% of human milk oligosaccharides [26]. The inefficiency of B. fragilis in metabolizing fucose-containing carbon sources may help explain why B. fragilis and other Bacteroidetes species are less common in the neonatal gut microbiome than in adults [27]. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Simulated growth yields per mole of carbon on different substrates. iMN674 was simulated on a minimal medium containing ammonia, sulfide, and minerals. The carbon source indicated in each bar was given a maximal intake flux of 120 C-mmol/gDW/h. The predicted growth rate was divided by the carbon flux to indicate the growth efficiency. Metabolites in green contain fucose. Metabolites in red have the same efficiency as glucose. https://doi.org/10.1371/journal.pcbi.1011594.g004 Two of the most efficient substrates identified by the simulation are maltose and glucose-6-phosphate. The maltose phosphorylase reaction splits the maltose units apart and yields glucose and glucose-1-phosphate. This can be converted to glucose 6-phosphate via phosphoglucomutase. Therefore, half of the carbohydrate in maltose enters glycolysis without ATP use, greatly improving the energy yield. Starch is instead hydrolyzed into glucose subunits and does not provide this benefit. Co-metabolism of fermentation products and amino acids While discussion of Bacteroides’ metabolism is typically focused on carbohydrates, amino acids and fermentation products from other microbes are generally available to these microorganisms in the colon. B. fragilis cannot utilize any single amino acid or mixture of amino acids as the sole nitrogen source [20] and the model reflects this. Our model further suggests that B. fragilis cannot produce ATP from acetate, propionate, butyrate, or ethanol. However, the presence of these fermentation end products in addition to other substrates still confers a growth advantage for B. fragilis. It has been previously observed that B. fragilis consumes acetate in conjunction with other organic molecules and is enriched by the consumption of alcohol, which is converted to acetate by the liver [28]. To investigate the potential for VFA co-metabolism, the model was simulated with a range of carbon sources in addition to a mixture of fermentation products, a mixture of amino acids, both mixtures, or neither, all provided at a flux of up to 20 mmol/gDW/h. The growth rates under these conditions were then compared. The addition of these molecules was considered to improve the growth rate if the rate increased by more than 0.001 h-1. Conditions where the model predicted growth with VFAs or amino acids are listed in Table 2. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Predicted Growth Rates for Metabolites Which Were Not a Sole Carbon Source, but Allowed Growth When Supplemented with VFAs or Amino Acids. https://doi.org/10.1371/journal.pcbi.1011594.t002 The VFA mixture, containing acetate, propionate, butyrate, ethanol, and pyruvate, did little to improve the growth rates of B. fragilis. When it did improve, it increased by only 6% on average, despite the massive increase in available carbon. The presence of fermentation products allowed for weak growth on lactate, malate, fumarate, succinate, and several amino acids, none of which could act as sole carbon sources. Generally, the VFAs were incorporated into acyl-CoA and used to assemble lipids. Since VFAs can not be used to produce the same range of biomass precursors as glucose, utilization of VFAs only marginally improved the growth rate of B. fragilis. The amino acid mixture had a greater impact on B. fragilis’ growth and resulted in a 15% increase in growth on average across conditions. This included a 30% increase in growth on glucose and glucose equivalents. In addition to improved growth in the presence of amino acids, we also observed a greater resilience of B. fragilis to gene knockouts. When glucose was the sole carbon source, 127 genes were essential for producing at least one biomass component. These genes were identified by deleting each gene and its associated reactions in the model one at a time to simulate single gene knockouts. If the model predicted no growth, the gene was labeled as essential. Each of these genes, on average, blocked the production of 9 biomass components when deleted. When amino acids were included, this effect dropped to 123 genes which blocked only 6 components on average. Even a relatively small amount of amino acids significantly improved the robustness of B. fragilis. This effect remained even when the maximum uptake of the amino acids was limited to 0.01 mmol/gDW/h. Table 3 identifies the genes that are no longer essential for growth with the addition of amino acids. There were six additional non-essential genes that decreased the growth rate by more than 10% when deleted in the glucose medium but not when supplemented with amino acids. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Differentially Essential Genes. https://doi.org/10.1371/journal.pcbi.1011594.t003 Each row in Table 3 corresponds to a gene in the model that is essential when grown on glucose as the sole carbon source but not when a small amount of amino acids is supplemented. The last column states which metabolites cannot be synthesized when that gene is deleted. One may expect that arginine, histidine, and proline production was restored by direct assimilation of those amino acids, but no transporter for these metabolites is currently annotated. Fig 5 shows how the availability of amino acids saves nucleotide metabolism from the effects of deleterious knockouts. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Purine biosynthesis is saved from gene knockouts by external amino acids. This figure shows an abbreviated view of purine synthesis in B. fragilis. The nitrogen in purine bases is taken from amino acids. These amino acids are produced in transamination reactions involving TCA intermediates. Thus, the gene knockouts corresponding to these transamination reactions (red Xs) are lethal, as the production of key amino acids and all purines is halted. However, if amino acids are made available in the environment (blue arrows), purine biosynthesis is able to proceed. Pyrimidine production is also aspartate dependent, so it is rescued similarly. https://doi.org/10.1371/journal.pcbi.1011594.g005 No genes changed in essentiality when a mixture of fermentation products was supplemented to the model. While VFAs can supplement glucose by forming acyl-CoA, the cell is still dependent on glucose for most biomass precursors. VFAs and amino acids together allowed for low levels of growth (0.11 h -1, 25% of that on glucose) without an additional carbon source. S5 Table contains the full results of these co-metabolism simulations. Overall, these results suggest that even though amino acids and fermentation end products are neither necessary nor sufficient for biomass production alone, they provide an advantage for B. fragilis. Even with only minute influxes, amino acids allowed flux through several inactive pathways and allowed growth when otherwise essential genes were knocked out. The gut microbiome is a complex environment, so it is highly beneficial for B. fragilis to be able to utilize a wide range of metabolites. B. fragilis’ possible niches and metabolism within the colon could therefore be much broader than originally thought and might not only be limited to catabolism of polysaccharides. Human oligosaccharide catabolism in B. fragilis The ability to digest HMOs is a major driving factor deciding which organisms are prevalent in the infant gut microbiome [26,29]. To further investigate the ability of B. fragilis to utilize components of human milk, reactions were added to represent the degradation of the 15 most abundant human milk oligosaccharides (HMOs) [30]. B. fragilis has been shown to digest some oligosaccharides in the periplasmic space [4], so these digestion reactions were assumed to be in the periplasm as well. B. fragilis possesses enzymes capable of digesting a broad range of oligosaccharides. Each bond in each of these compounds could be paired to an enzyme potentially capable of breaking it. S6 Table provides a list of compounds and the enzymes needed to digest them. For several bonds, there are multiple enzymes in the genome that could perform the necessary hydrolysis. This aligns with previous work that suggests B. fragilis readily consumes HMOs [31] and bovine milk oligosaccharides [32] and suggests that B. fragilis could digest them completely. Other bacteria such as Bifodobacterium and Enterococcus spp. are nonetheless able to outcompete B. fragilis when certain HMOs are increased in abundance in the infant gut microbiome [33], indicating that the interplay between these microbes is defined by something other than whether or not the oligosaccharides can be utilized. B. fragilis has been shown to be able to scavenge human N-linked glycans [4]. The ten most abundant N-linked glycans from human transferrin identified by Abu Bakar et al [34] were added as representatives. Again, all oligosaccharides were able to be fully divided into the component monomers in multiple ways. As human N-glycans share the same core and many repeated motifs, it is therefore likely that most such glycans can be used as a carbohydrate source for B. fragilis. The predicted ability to both liberate and consume the sialic acids in these oligosaccharides stands in contrast to the closely related B. thetaiotaomicron, which cannot consume the sialic acids it frees [35]. Sialic acids act as a carbon source for pathogenic Clostridioides difficile and Salmonella typhimurium [35], and have been shown in gnotobiotic mouse models to consume sialic acids liberated by B. thetaiotaomicron [36]. B. fragilis could therefore be competing with these pathogenic bacteria for sialic acid. Comparison of iMN674 to automated AGORA2 model Heinken et al recently published semi-automatic reconstructions of 7,302 human associated microorganisms [37], including a model for Bacteroides fragilis 638R. This model, while refined through some experimentally available growth data, did not undergo the same degree of curation as iMN674. Excluding exchange reactions, the AGORA2 model has 936 reactions without genome annotation, compared to 9 reactions in our model. Among these 936 reactions, 103 are indispensable for the model to grow, even when all exchanges are fully open. This number increases to 133 essential genes with no annotations when limited to a glucose, ammonia, and mineral medium. None of the 621 transporters present in the AGORA2 model have gene associations. Overall, more than 50 percent of the non-exchange reactions lack genome annotations. iMN674, which has genome support for all but 9 reactions, contains a smaller reaction network, but the degree of certainty in these reactions is much higher. Moreover, iMN674 has been refined and curated to accurately reflect experimental data. When comparing model-data agreement as in Fig 3, our model achieves a sensitivity and specificity of 0.90 and 0.96; the AGORA2 model’s sensitivity and specificity are 0.55 and 0.88. While semi-automated models provide a good tool for the community, well curated models such as iMN674 contain an experimentally refined and evidenced reconstruction that provides the highest accuracy to study the metabolism of the target organism. Conclusion Here we present the genome-scale metabolic model iMN674 of the ubiquitous gut bacterium B. fragilis. The model was refined and validated by examining the growth phenotypes of the bacterium identified in the literature and discovered via nutrient utilization experiments. In addition, the ATP requirements for growth and stasis were calculated to provide realistic growth rates. Lastly, the model was shown to be able to produce fermentation products in experimentally observed ratios. The metabolic model allowed for the various fermentation products to be produced across a range of ratios at a fixed growth rate. This indicates that these ratios are controlled by some other feature of B. fragilis, such as transcriptional regulation or post-translational enzyme control. The fermentation yields of B. fragilis are influenced by the availability of carbohydrates and nitrogenous compounds in the media, highlighting the complex interplay between environmental conditions and metabolism for B. fragilis. The model demonstrates how even metabolites that cannot produce biomass by themselves can be co-metabolized by B. fragilis and could provide an ecological advantage for the bacteria to survive in diverse conditions. Overall, we demonstrated that iMN674 accurately captures the metabolic nature of B. fragilis and acts as a platform for further hypothesis generation regarding the microorganism, the effects of our diet on its growth, and its influence on the host. The gaps in iMN674, in particular lack of knowledge about carbohydrate transporters, highlight gaps in our understanding of B. fragilis’ metabolism. Additional work is needed to further validate the model and contextualize its results. As this knowledge is acquired, it will be continually incorporated into iMN674, allowing it to continually improve its accuracy in quantitative phenotypic prediction. Methods Initial model generation The initial draft models were generated using the RAVEN Toolbox version 2.5.0 (Fig 6, first panel. The rest of Fig 6 outlines the rest of the methods). [38]. A first draft was reconstructed via the KEGG species code method (getKEGGModelForOrganism(‘bfg’)), which generates a model based on the assigned protein homologies for the organism in question on KEGG (release 98) [39]. Another was made using a pre-trained hidden markov model to find homologous proteins in the organism’s genome (getKEGGModelForOrganism(’bfg’,bfg_’protein.faa’,’prok90_kegg94’). This was done using the same genome found for B. fragilis 638R (accession number GCA_000210835.1). The models made by either method were nearly identical and thus the gene-reaction associations from both were merged. These methods were chosen over using existing models as templates as few suitably phylogenetically related organisms with high-quality models could be found. Using KEGG allowed for a wider range of reactions to be considered, while template-based methods would only return the presumably limited set of reactions at the intersection of B. fragilis’ and the template’s metabolism. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. A flowchart outlining the general methodology from initial model generation to model analysis. https://doi.org/10.1371/journal.pcbi.1011594.g006 The RAVEN Toolbox’s output is in KEGG format. KEGG does not include transporters or detailed lipid reactions, so these were mapped to BiGG reactions (BiGG version 1.6). BiGG is a UC San Diego based database of GEMs in a mostly standardized format [12]. Additionally, uncommon reactions in B. fragilis not present in BiGG or KEGG had to be created and added manually, such as 2’-fucosyllactose degradation. Potential transporters and lipid reactions were identified by BLAST comparison to other gut bacteria models, including the E. coli model iML1515 [16] and the salmonella pan-reactome model iYS1720 [40]. Additional transporters were found in the Transporter Classification Database (2021 update) [41]. Custom reactions representing these transporters were created and added to the model. In the cases where B. fragilis is known to consume a metabolite but no suitable transporter could be found in the genome, the metabolite was assumed to enter the periplasm through porins then enter the cytosol. The reactions were then manually curated. To be included in the model, a protein coding sequence from the genome must be sufficiently similar to a prokaryotic gene applied to that reaction in another model or in the KEGG enzyme database. Similarity was assessed by BLAST, with cutoffs of identity > 30%, query cover > 95%, and e-value < 10−15. Enzymes with multiple possible roles were initially assumed to have all the possible functions assigned to it in KEGG unless refuted by evidence found in the literature. For example, the gene 5’/3’-nucleotidase SurE has the enzyme commission numbers 3.1.3.5, 3.13.6, and 3.6.1.11 [42]. Each of these are associated with the hydrolysis of phosphate from a variety of nucleotides, leading SurE to be associated with 30 reactions. Reactions with sufficient BLAST scores were rejected if the genome annotation or other evidence suggested the genes associated with them were more likely to be associated with another, incompatible enzyme. Genes with vague annotations or unclear enzymatic function were noted as low confidence. All metabolites in the model were paired with formulas and names. Complex cofactors that are not produced or consumed, but only cycled, were given the formula ‘R.’ These include tRNAs and acyl-carrier proteins. The metabolite formulas were kept consistent to prevent any reactions from producing or consuming any atoms except hydrogen. Molecule charge was not considered. Unbalanced or unclear reactions such as R08411 from KEGG were removed. All reactions were given descriptive names and subsystems. Reaction reversibility was assigned based on whether the majority of models in BiGG marked the reaction or similar reactions as reversible or not. All reversible reactions were given default bounds of +/- 1,000 mmol/gDW/h, while irreversible reactions had the lower bound set to 0 mmol/gDW/h. Exchange reactions were generally given bounds of [–20, 1,000] mmol/gDW/h. Biomass objective function Data on the molecular composition of B. fragilis is limited. Consequently, data from a variety of sources had to be considered to construct an estimate of the biomass composition. The proportions of the organism’s dry weight in each major category (lipids, carbohydrates, proteins, DNA, and RNA) were taken from Frantz and McCallum (1980) [43], which examined strain ATCC 23745. The data from the exponential growth phase was averaged and used. The data did not sum to 100%, so the remainder was assumed to be the cofactor, mineral, and metabolite contribution not measured in the study. Within the DNA category, the percent of each nucleotide was calculated by the percent of each found in the genome. Due to a lack of RNA-seq experiments, the relative amounts of each RNA monomer was assumed to be the same as in the genome. Within the carbohydrates category, the amounts were taken from Cherniak et al [44] and Kasper [45] then averaged. These data were calculated from the outer membrane of various B. fragilis strains and were taken to be representative of the overall composition. As carbohydrate monomers are incorporated into polymers via nucleotide carriers, they are included in the biomass reaction in the form carrier-carbohydrate -> released carrier or as the consumption of polymerized monomers. This forced flux through the nucleotide-sugar reactions, improving the realism of the model and its energy needs. The amino acid composition was found in Sok et al [46] for the related species B. ruminicola. This data did not measure the abundances of asparagine, glutamine, or tryptophan. The total did not add to 100%, so the unaccounted for abundance was divided equally between these. Similar to the carbohydrates, these metabolites were included in the form trna-amino acid -> free trna, with the amino acid being consumed. The lipids were drawn from Eiichi and Miyagawa [47], which measured lipid composition in B. fragilis. Its results detailed the percent composition of each chain length and desaturation level, but not the headgroups. Reactions were added in the model to have at least one representative from each chain type. The percent biomass assigned to each chain type was divided equally between all the lipids present in that category in the model. Lastly, cofactor, mineral, and vitamin consumption was taken from iML1515. The specific values were re-scaled to fit the metabolite and cofactor percentage used here. Some metabolites in this category were removed if they did not otherwise appear in the B. fragilis model and there was no experimental evidence to expect they were a necessary component. The final biomass objective function was calculated by normalizing the sum of weights of all these molecules to be 1 gram. For metabolites with carrier molecules, the mass used in the calculation was that of the free metabolite. Gap filling The draft model could not produce every biomass precursor. To fill these gaps, new reactions were added from BiGG and KEGG. The model was then run with all exchange reactions open and optimized for maximal production of each biomass metabolite. If a given reaction improved the number of biomass components that could be produced, it was appended to a list of new reactions to confirm via BLAST or literature search. This was repeated iteratively until the model could produce all precursors and therefore yield a non-zero flux through the biomass reaction. The process was repeated until the model could grow under all conditions found in the literature (see S7 Table). The draft model contained many reactions that could not carry flux. Many of these were secondary uses of enzymes. Such enzymes are a major source of dead-end metabolites and blocked reactions in GEMs [10,48]. Some reactions, including transporters of common metabolites and fragments of common pathways, could be connected to the rest of the metabolic network by gap filling. 399 reactions remain that cannot carry flux. 87 of them have dead end metabolites on both ends, leaving them entirely disconnected from the network. ATP maintenance coefficients The ATP maintenance coefficients related to growth and stasis (GAM and NGAM) were calculated by minimizing the error between growth rates found in the literature and predicted by the model. These included the growth rates shown in Varel and Bryant [20], Fig 1, and Spence et al [19] Fig 2A and 2B. Where peptides or casitone were mentioned, we activated all amino acid exchange reactions. S4 Table details the medium compositions for this process under Fig 3, but generally exchanges for main carbon and nitrogen sources were set to 20 mmol/gDW/h. This process is implemented in the script for Fig 3. At each iteration, the model was optimized in each condition and the growth rates were recorded. The overall error was defined as the maximum relative error between the model and experimental values, as to constrain the largest error. The error was then minimized using MATLAB’s fminsearch function with an initial estimation of [100,6]. This yielded a GAM and NGAM of 24.9 and 38.9. Repeating with a grid of initial guesses always returned these values. Nutrient utilization experiment Bacteroides fragilis 638R was grown from a glycerol stock in 50 mL of anaerobic BHI media (Teknova B9500). This media was supplemented with a 1% dilution of Remel R450951 vitamin K and hemin solution, 4 mM cysteine, 0.1% weight by volume yeast extract (RPI Y20020-250), and a 1% dilution of trace minerals (ATCC MD-TMS), and a pinch of resazurin. The bacteria were grown for two days before the experiment to reach exponential phase. Two liters of the same media but at 0.15x BHI concentration were mixed, degassed, and autoclaved as well in sealed anaerobic bottles. 0.15x BHI was selected to give low but non-zero growth, so that improvements from the added nutrient would be more visible. Autoclaved 16x125mm Hungate tubes with butyl stoppers were used to test the individual conditions. The tubes were individually opened, then a needle with flowing nitrogen placed into it to displace oxygen and provide positive pressure. Similar needles were placed into the 0.15x BHI media bottles after opening to ensure they remained anoxic. Using a serological pipette, 10ml of the media was placed in each tube before sealing. The tubes were allowed to rest until the color of resazurin dissipated. Then 100 microliters of a prepared 0.5 M solution of one of the nutrients was added, bringing the tube to 5 mM. Lastly, 100 microliters of the growing bacteria were added. This was repeated for each nutrient in triplicate. To the control tube 100 microliters of water was added in place of a nutrient. 200 microliters were drawn from each tube for an initial OD600 reading, and then the tubes were incubated at 37 C for 44 hours before taking a final OD600 reading. To decide if a set of tubes showed growth, the average change in OD600 for that condition was compared to the average OD600 change of the control tubes. If it was more than 0.05 higher, the condition was considered to have shown improved growth over the control. Otherwise it was considered to have shown no improvement. The gap filling was then repeated to allow for growth under any newly identified conditions. In the model, a growth increase of 0.01 or higher was considered to be a positive result. If the model predicted growth where the data suggested it should not, low confidence reactions and transporters were removed to eliminate the false positive. Computation All simulations were performed in MATLAB R2022b on an UBUNTU version 22.04.1 workstation. Optimizations were performed using the flux balance analysis feature of the Cobra Toolbox (version 3.4) [49] using the solver Gurobi 9.5.2. Chemical structures were generated using the Smi2Depict web interface of ChemDB [50]. FBC curation to make a standard reference for iMN674 was made using FROG analysis software (version 0.2.2) [51]. A MEMOTE report for the model was generated with MEMOTE 0.11.1 (S3 File) [52]. Initial model generation The initial draft models were generated using the RAVEN Toolbox version 2.5.0 (Fig 6, first panel. The rest of Fig 6 outlines the rest of the methods). [38]. A first draft was reconstructed via the KEGG species code method (getKEGGModelForOrganism(‘bfg’)), which generates a model based on the assigned protein homologies for the organism in question on KEGG (release 98) [39]. Another was made using a pre-trained hidden markov model to find homologous proteins in the organism’s genome (getKEGGModelForOrganism(’bfg’,bfg_’protein.faa’,’prok90_kegg94’). This was done using the same genome found for B. fragilis 638R (accession number GCA_000210835.1). The models made by either method were nearly identical and thus the gene-reaction associations from both were merged. These methods were chosen over using existing models as templates as few suitably phylogenetically related organisms with high-quality models could be found. Using KEGG allowed for a wider range of reactions to be considered, while template-based methods would only return the presumably limited set of reactions at the intersection of B. fragilis’ and the template’s metabolism. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. A flowchart outlining the general methodology from initial model generation to model analysis. https://doi.org/10.1371/journal.pcbi.1011594.g006 The RAVEN Toolbox’s output is in KEGG format. KEGG does not include transporters or detailed lipid reactions, so these were mapped to BiGG reactions (BiGG version 1.6). BiGG is a UC San Diego based database of GEMs in a mostly standardized format [12]. Additionally, uncommon reactions in B. fragilis not present in BiGG or KEGG had to be created and added manually, such as 2’-fucosyllactose degradation. Potential transporters and lipid reactions were identified by BLAST comparison to other gut bacteria models, including the E. coli model iML1515 [16] and the salmonella pan-reactome model iYS1720 [40]. Additional transporters were found in the Transporter Classification Database (2021 update) [41]. Custom reactions representing these transporters were created and added to the model. In the cases where B. fragilis is known to consume a metabolite but no suitable transporter could be found in the genome, the metabolite was assumed to enter the periplasm through porins then enter the cytosol. The reactions were then manually curated. To be included in the model, a protein coding sequence from the genome must be sufficiently similar to a prokaryotic gene applied to that reaction in another model or in the KEGG enzyme database. Similarity was assessed by BLAST, with cutoffs of identity > 30%, query cover > 95%, and e-value < 10−15. Enzymes with multiple possible roles were initially assumed to have all the possible functions assigned to it in KEGG unless refuted by evidence found in the literature. For example, the gene 5’/3’-nucleotidase SurE has the enzyme commission numbers 3.1.3.5, 3.13.6, and 3.6.1.11 [42]. Each of these are associated with the hydrolysis of phosphate from a variety of nucleotides, leading SurE to be associated with 30 reactions. Reactions with sufficient BLAST scores were rejected if the genome annotation or other evidence suggested the genes associated with them were more likely to be associated with another, incompatible enzyme. Genes with vague annotations or unclear enzymatic function were noted as low confidence. All metabolites in the model were paired with formulas and names. Complex cofactors that are not produced or consumed, but only cycled, were given the formula ‘R.’ These include tRNAs and acyl-carrier proteins. The metabolite formulas were kept consistent to prevent any reactions from producing or consuming any atoms except hydrogen. Molecule charge was not considered. Unbalanced or unclear reactions such as R08411 from KEGG were removed. All reactions were given descriptive names and subsystems. Reaction reversibility was assigned based on whether the majority of models in BiGG marked the reaction or similar reactions as reversible or not. All reversible reactions were given default bounds of +/- 1,000 mmol/gDW/h, while irreversible reactions had the lower bound set to 0 mmol/gDW/h. Exchange reactions were generally given bounds of [–20, 1,000] mmol/gDW/h. Biomass objective function Data on the molecular composition of B. fragilis is limited. Consequently, data from a variety of sources had to be considered to construct an estimate of the biomass composition. The proportions of the organism’s dry weight in each major category (lipids, carbohydrates, proteins, DNA, and RNA) were taken from Frantz and McCallum (1980) [43], which examined strain ATCC 23745. The data from the exponential growth phase was averaged and used. The data did not sum to 100%, so the remainder was assumed to be the cofactor, mineral, and metabolite contribution not measured in the study. Within the DNA category, the percent of each nucleotide was calculated by the percent of each found in the genome. Due to a lack of RNA-seq experiments, the relative amounts of each RNA monomer was assumed to be the same as in the genome. Within the carbohydrates category, the amounts were taken from Cherniak et al [44] and Kasper [45] then averaged. These data were calculated from the outer membrane of various B. fragilis strains and were taken to be representative of the overall composition. As carbohydrate monomers are incorporated into polymers via nucleotide carriers, they are included in the biomass reaction in the form carrier-carbohydrate -> released carrier or as the consumption of polymerized monomers. This forced flux through the nucleotide-sugar reactions, improving the realism of the model and its energy needs. The amino acid composition was found in Sok et al [46] for the related species B. ruminicola. This data did not measure the abundances of asparagine, glutamine, or tryptophan. The total did not add to 100%, so the unaccounted for abundance was divided equally between these. Similar to the carbohydrates, these metabolites were included in the form trna-amino acid -> free trna, with the amino acid being consumed. The lipids were drawn from Eiichi and Miyagawa [47], which measured lipid composition in B. fragilis. Its results detailed the percent composition of each chain length and desaturation level, but not the headgroups. Reactions were added in the model to have at least one representative from each chain type. The percent biomass assigned to each chain type was divided equally between all the lipids present in that category in the model. Lastly, cofactor, mineral, and vitamin consumption was taken from iML1515. The specific values were re-scaled to fit the metabolite and cofactor percentage used here. Some metabolites in this category were removed if they did not otherwise appear in the B. fragilis model and there was no experimental evidence to expect they were a necessary component. The final biomass objective function was calculated by normalizing the sum of weights of all these molecules to be 1 gram. For metabolites with carrier molecules, the mass used in the calculation was that of the free metabolite. Gap filling The draft model could not produce every biomass precursor. To fill these gaps, new reactions were added from BiGG and KEGG. The model was then run with all exchange reactions open and optimized for maximal production of each biomass metabolite. If a given reaction improved the number of biomass components that could be produced, it was appended to a list of new reactions to confirm via BLAST or literature search. This was repeated iteratively until the model could produce all precursors and therefore yield a non-zero flux through the biomass reaction. The process was repeated until the model could grow under all conditions found in the literature (see S7 Table). The draft model contained many reactions that could not carry flux. Many of these were secondary uses of enzymes. Such enzymes are a major source of dead-end metabolites and blocked reactions in GEMs [10,48]. Some reactions, including transporters of common metabolites and fragments of common pathways, could be connected to the rest of the metabolic network by gap filling. 399 reactions remain that cannot carry flux. 87 of them have dead end metabolites on both ends, leaving them entirely disconnected from the network. ATP maintenance coefficients The ATP maintenance coefficients related to growth and stasis (GAM and NGAM) were calculated by minimizing the error between growth rates found in the literature and predicted by the model. These included the growth rates shown in Varel and Bryant [20], Fig 1, and Spence et al [19] Fig 2A and 2B. Where peptides or casitone were mentioned, we activated all amino acid exchange reactions. S4 Table details the medium compositions for this process under Fig 3, but generally exchanges for main carbon and nitrogen sources were set to 20 mmol/gDW/h. This process is implemented in the script for Fig 3. At each iteration, the model was optimized in each condition and the growth rates were recorded. The overall error was defined as the maximum relative error between the model and experimental values, as to constrain the largest error. The error was then minimized using MATLAB’s fminsearch function with an initial estimation of [100,6]. This yielded a GAM and NGAM of 24.9 and 38.9. Repeating with a grid of initial guesses always returned these values. Nutrient utilization experiment Bacteroides fragilis 638R was grown from a glycerol stock in 50 mL of anaerobic BHI media (Teknova B9500). This media was supplemented with a 1% dilution of Remel R450951 vitamin K and hemin solution, 4 mM cysteine, 0.1% weight by volume yeast extract (RPI Y20020-250), and a 1% dilution of trace minerals (ATCC MD-TMS), and a pinch of resazurin. The bacteria were grown for two days before the experiment to reach exponential phase. Two liters of the same media but at 0.15x BHI concentration were mixed, degassed, and autoclaved as well in sealed anaerobic bottles. 0.15x BHI was selected to give low but non-zero growth, so that improvements from the added nutrient would be more visible. Autoclaved 16x125mm Hungate tubes with butyl stoppers were used to test the individual conditions. The tubes were individually opened, then a needle with flowing nitrogen placed into it to displace oxygen and provide positive pressure. Similar needles were placed into the 0.15x BHI media bottles after opening to ensure they remained anoxic. Using a serological pipette, 10ml of the media was placed in each tube before sealing. The tubes were allowed to rest until the color of resazurin dissipated. Then 100 microliters of a prepared 0.5 M solution of one of the nutrients was added, bringing the tube to 5 mM. Lastly, 100 microliters of the growing bacteria were added. This was repeated for each nutrient in triplicate. To the control tube 100 microliters of water was added in place of a nutrient. 200 microliters were drawn from each tube for an initial OD600 reading, and then the tubes were incubated at 37 C for 44 hours before taking a final OD600 reading. To decide if a set of tubes showed growth, the average change in OD600 for that condition was compared to the average OD600 change of the control tubes. If it was more than 0.05 higher, the condition was considered to have shown improved growth over the control. Otherwise it was considered to have shown no improvement. The gap filling was then repeated to allow for growth under any newly identified conditions. In the model, a growth increase of 0.01 or higher was considered to be a positive result. If the model predicted growth where the data suggested it should not, low confidence reactions and transporters were removed to eliminate the false positive. Computation All simulations were performed in MATLAB R2022b on an UBUNTU version 22.04.1 workstation. Optimizations were performed using the flux balance analysis feature of the Cobra Toolbox (version 3.4) [49] using the solver Gurobi 9.5.2. Chemical structures were generated using the Smi2Depict web interface of ChemDB [50]. FBC curation to make a standard reference for iMN674 was made using FROG analysis software (version 0.2.2) [51]. A MEMOTE report for the model was generated with MEMOTE 0.11.1 (S3 File) [52]. Supporting information S1 Table. Essential Genes. https://doi.org/10.1371/journal.pcbi.1011594.s001 (XLSX) S2 Table. Reactions Added Without Genome Annotation. https://doi.org/10.1371/journal.pcbi.1011594.s002 (XLSX) S3 Table. Nutrient Utilization Experiment and Model Agreement. https://doi.org/10.1371/journal.pcbi.1011594.s003 (XLSX) S4 Table. In Silico Media Conditions. https://doi.org/10.1371/journal.pcbi.1011594.s004 (XLSX) S5 Table. Supplementary Nutrients for Growth and VFA Production. Sheet 1 in this file shows how the growth on each exchange reaction in the model was altered by the addition of a fermentation product mixture, amino acid mixture, or both. Sheet 2 shows how different supplementary metabolites alter the potential for VFA production. https://doi.org/10.1371/journal.pcbi.1011594.s005 (XLSX) S6 Table. Enzymes Utilized in Human Oligosaccharide Degradation. https://doi.org/10.1371/journal.pcbi.1011594.s006 (XLSX) S7 Table. Media Conditions from Literature. This lists semi-or fully defined media conditions found in literature used in gap-filling the model. https://doi.org/10.1371/journal.pcbi.1011594.s007 (XLSX) S1 File. iMN674 in various Formats. This is the iMN674 model presented in.mat, COBRApy.json, and SBML.xml formats, set to model a glucose and mineral medium. https://doi.org/10.1371/journal.pcbi.1011594.s008 (ZIP) S2 File. iMN674 in Spreadsheet Format. https://doi.org/10.1371/journal.pcbi.1011594.s009 (XLSX) S3 File. MEMOTE Report. https://doi.org/10.1371/journal.pcbi.1011594.s010 (PDF) S4 File. FROG Report. This file contains the FBC curation of the model to present a reproducible model state for model behavior validation across systems. It also reports the effects of gene and reaction deletions in the provided medium. https://doi.org/10.1371/journal.pcbi.1011594.s011 (ZIP) S5 File. Figure Code. These files contain the code needed to reproduce the figures. https://doi.org/10.1371/journal.pcbi.1011594.s012 (ZIP) S6 File. Nutrient Utilization Raw Data. These show the initial and final OD600 values found in the bacterial growth experiment. https://doi.org/10.1371/journal.pcbi.1011594.s013 (XLSX)

He, Mengqi;Tang, Sanyi;Xiao, Yanni

doi: 10.1371/journal.pcbi.1011535pmid: 37851640

Introduction The COVID-19 pandemic has lasted for three years since the end of 2019. Due to the continuous variation of the virus strain and the dynamic adjustment of prevention and control measures, it is a great challenge to propose a dynamic model of infectious diseases to evaluate the effectiveness of non-pharmaceutical interventions (NPIs) [1]. In particular, before October 2022, due to China’s implementation of the dynamic zero-case policy, strong close contact tracing and isolation measures or even static management mode make almost all outbreaks be cleared in about 40 days. From the point view of mathematical modelling, increased quarantine/isolation rate and decreased contact rate have played an essential role in reducing new infections. However accurately quantifying the rate functions and examining their effects on infections remain unclear and fall within the scope of this study. Modelling the dynamics of infectious diseases is an essential tool to provide the quantitative basis for decision making during the COVID-19 pandemic. Traditionally, the intrinsic transmission mechanism of infectious diseases and the flow among individuals in various compartments are mainly described by ordinary/partial differential equations [2, 3], delay differential equations [4] and fractional differential equations [5]. In traditional mechanism-based models, researchers usually incorporated constant contact rate and quarantine/isolation rate for simplicity to analyze the transmission risk [6], model the impact of contact tracing and quarantine on the development of COVID-19 [7, 8] and evaluate the independent effectiveness of vaccines [9]. There are also a large number of literatures in which the specific functions were supposed to represent the dynamic changes in intensity of interventions for comparing the effectiveness of various control strategies [10], understanding the drivers of multiple waves of outbreaks [11] and exploring the transmission mechanism of COVID-19 with different intervention patterns [12]. Moreover, Wang et al. [13] considered a dynamic epidemiological model with a piecewise contact rate and quarantine rate to simulate the dynamics of the Omicron variant in Shanghai, and explored the feasibility of different control patterns in avoiding subsequent waves. Li et al. [14] developed a model with pulse population-wide nucleic acid screening, and simulated the changes of contact/quarantine rates over time by using exponential decline/increase functions, respectively, focusing on the impact of large-scale screening on the transmission dynamics of COVID-19 infection and the operation of medical resources. Note that the preset specific functions may not accurately capture the dynamic adjustment of intervention strategies. And the assumed rate functions may inevitably involve more parameters in the model, which brings significant challenges to data fitting and parameter estimation. This mechanism of preset rate functions inevitably causes that the outcomes are usually dependent on the particular types of rate functions due to various assumptions. Hence the data-driven inference of rate function is of great significance to quantify and assess the intensity of control interventions. Data-driven statistical models are widely used in biological, medicine, social science and other fields due to the flexibility and feasibility of the method [15, 16]. Especially in recent years, it has played an important role in simulating COVID-19 pandemic [17, 18]. For example, Sindhu et al. [19] proposed a three parametric model named as Exponentiated transformation of Gumbel Type-II (ETGT-II) for analyzing the number of deaths due to COVID-19 for Europe and China. In addition, there are several studies have developed different types of statistical models based on COVID-19 mortality data and evaluated the performance of the models [20, 21]. Rahman et al. [22] developed a seasonal Autoregressive Integrated Moving Average (ARIMA) model and eXtreme Gradient Boosting (XGBoost) model to simulate the overall trend of confirmed cases and deaths of COVID-19 infection in Bangladesh, and compared the accuracy of predictions of two methods. Külah et al. [23] considered Shifted Gaussian Mixture Model with Similarity-based Estimation (SGSE) to predict the development trend of COVID-19 pandemic for a specific country by examining similar behaviors in other countries. Note that these data-driven statistical methods do not incorporate prior transmission mechanisms, resulting in poor interpretability of simulation results, making it difficult to provide decision-making basis for optimizing control strategies. Data-driven deep learning is another powerful tool for analyzing the dynamics of COVID-19 pandemic. It is a nonlinear mathematical tool with powerful learning ability, and is widely used in natural language processing [24], fault detections [25], image recognitions [26] and reliability analysis [27–29]. During COVID-19 pandemic, neural networks are used to construct various simulation frameworks to predict the development trend of the epidemic [30, 31]. For example, Jin et al. [32] predicted COVID-19 infection based on multiple neural networks and reinforcement learning. Shafiq et al. [33] estimated the COVID-19 mortality rates in Italy by using maximum likelihood estimation and artificial neural network (ANN). Xu et al. [34] employed three different deep learning models, including the convolutional neural network (CNN), long short-term memory (LSTM) and convolutional neural network-long short-term memory (CNN-LSTM), to predict the number of new cases and forecast the spread of COVID-19 infection. Utku [35] developed a convolutional neural network-gated recurrent unit (CNN-GRU), based on hybrid deep learning model, to predict COVID-19 cross-country spread. Gautam [36] applied transfer learning to the LSTM network to learn the trends of new cases and new death of COVID-19 infection from case data in Italy and the United States and to make projections for other countries. However, the black box attribute of the algorithms makes it face uninterpretable risks, especially the end-to-end learning method cannot reveal the underlying transmission mechanism of epidemics or the impact of intervention measures on mitigating the disease spread. The main purpose of this study is to combine scattered observational data with deep learning and epidemic models, in order to avoid assuming the specific rate functions in advance and make neural networks follow the rules of epidemic systems in the process of learning. This mechanism of physics-informed neural network (PINN) may provide a flexible computational framework for scientific problems [37, 38]. By applying a data-driven module to extend an epidemiological model with control interventions derived from first principles, we implement the time-dependent parameters that quantify the intensity of prevention and control measures as different neural networks, and then embed the epidemiological model into the neural network through adding the residuals of the equations to the loss function, and develop an extended transmission-dynamics-informed neural network algorithm framework. We simulate the COVID-19 epidemic evolution trend in Xi’an, Guangzhou, Yangzhou, Hainan and Xinjiang with TDINN, and discover the temporal evolution pattern of time-dependent parameters reflecting the dynamic adjustment of the control strategies based on the epidemic curves in these regions. Furthermore, we reconstruct the dynamic evolution trend of time-dependent parameters through specific functions and provide interpretability analysis for the output of deep learning. Finally, We also test the fitting performance of the TDINN algorithm on the COVID-19 infection with multiple waves in Liaoning province. In this study, we will develop a TDINN algorithm that integrates epidemic data, deep learning, and epidemiological models to identify the intensity of interventions during COVID-19 pandemic. It is worth noting that the TDINN algorithm guides neural networks to adhere to epidemic system rules during the learning process and meanwhile avoids the pre-assumption of modeling contact/quarantine rates with specific functions. The proposed algorithm can not only fit the multi-source epidemic data well, but also reconstruct the epidemic development trend with incomplete reported data. Further, we successfully trace the temporal evolution patterns of the contact rate and quarantine rate, and perform the interpretability analysis of the time-dependent rate functions inferred by TDINN algorithm. Methods Data We obtained the daily reported number of confirmed cases for Xi’an outbreak from December 9th, 2021 to January 20th, 2022, Guangzhou outbreak from May 21st, 2021 to June 18th, 2021, and for Yangzhou outbreak from July 28th, 2021 to August 26th, 2021 from Health Commissions of Shaanxi [39], Guangdong [40] and Jiangsu provinces [41], respectively. In addition, we collected the daily reported number of confirmed cases from August 1st, 2022 to September 23rd, 2022 in Hainan province [42] and form August 4th, 2022 to September 26th, 2022 in Xinjiang Uygur Autonomous Region [43], respectively. Data information includes the number of daily reported cases in the community() and in the quarantined zone(). It is important to note that the numbers of daily reported case in the community or quarantined zone are incomplete for Hainan and Xinjiang, but we have complete daily reported case numbers () in these two regions. Moreover, we also obtained the daily reported number of confirmed cases for Liaoning outbreak from 6th March 2022 to 21st May 2022 from Health Commissions of Liaoning provinces [44], where the data information only contains a column of daily reported case numbers () and shows multi-wave outbreaks. Detailed data are shown in Fig 1a–1d. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Multi-source epidemic data and the framework of transmission dynamic model. (a) Epidemic data of COVID-19 infection in Liaoning province from 6th March 2022 to 21st May 2022; (b) Epidemic data of COVID-19 infection in Xi’an from 9th December 2021 to 20th January 2022, in Guangzhou from 21st May to 18th June 2021, and in Yangzhou from 28th July to 26th August 2021; (c) Epidemic data of COVID-19 infection in Hainan from August 1st to September 23rd, 2022; (d) Epidemic data of COVID-19 infection in Xinjiang from August 4th to September 26th, 2022; (e) Flow diagram among epidemiological classes. https://doi.org/10.1371/journal.pcbi.1011535.g001 For Xi’an, Guangzhou, and Yangzhou, we can calculate the cumulative reported cases in the community()(or quarantined zone ()) based on the daily reported cases in the community (or quarantined zone), while for Hainan, Xinjiang and Liaoning, we only obtain the cumulative reported cases(). Therefore, in this study, we have access to three categories of reported data sets, which are as the follows: Set 1: , , , , for Xi’an, Guangzhou, Yangzhou; Set 2: , , , , for Hainan, Xinjiang; Set 3: , , for Liaoning. The model During COVID-19 pandemic, China’s government has mainly adopted the dynamic zero-case policy, i.e., strict close contact tracking and isolation, high-frequency and large-scale nucleic acid screening, closed management and etc, to quickly respond to the outbreak. These powerful NPIs effectively make most infected people not go through the complete process from infection to incubation period, and then to asymptomatic or symptomatic, that is, patients may be detected at every stage after infection. Therefore, this study simulates the transmission mode and evolution dynamics of COVID-19 infection based on the classic deterministic Susceptible-Infected-Removed (SIR) type epidemiological model [2]. Then we extend the simplest SIR-type dynamic model by including contact tracing and isolation, and the flow diagram is shown in Fig 1e. Given an outbreak taking off in a city, the city can usually be divided into two regions according to different intensity of control measures: free region (or community) and quarantined region. Consequently, we stratify the population in the free (quarantined) region into the susceptible class S (Sq) and the infected class Ic (Iq), and the removed class (denoted by R). Note that here we do not distinguish the individuals in the removed class in free or quarantined region since they can not be re-infected within a relatively short duration, and then consider a single compartment. Here we use the subscripts ‘q’ to represent quarantined population, i.e. Sq and Iq represent quarantined susceptible class and quarantined infected class, respectively. To model the continuously adjusted intervention measures, we assume the time-dependent contact rate and quarantine rate, denoted by c(t) and q(t), respectively. The transmission probability of per contact is supposed to be β. Then, the quarantined individuals, if infected (or uninfected), move to the compartment Iq (or Sq) at a rate of βc(t)q(t) (or (1 − β)c(t)q(t)). Those who are not quarantined, if infected, will move to the compartment Ic at a rate of βc(t)(1 − q(t)). According to the fact that the quarantined individuals do not return to the susceptible population before the end of outbreak, then we ignore the rate of transition from Sq to S class. Hence we have the following ordinary differential equations: (1) where N represents the total population of the region, the recovery rate of infected individuals in community (quarantined region) denoted by γ(δq), and the definitions and values of all parameters used in the model are given in Table 1. Here we consider three additional auxiliary compartments to record cumulative reported cases (), the cumulative reported cases in the community () (or the quarantined region ()). The dynamics of these three compartments are driven by the following equations: (2) Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Parameter definitions and estimation for model (1). https://doi.org/10.1371/journal.pcbi.1011535.t001 Parameter estimation It is known that fully connected deep neural networks with arbitrary nonlinear activation functions are universal approximators [45], we then use three independent neural networks with time t as input to represent the time-dependent contact rate c(t), quarantined rate q(t) and each state variable in model (1) respectively. So we have where U is a vector of all epidemiological categories considered in model (1), i.e., U = (S, Ic, Sq, Iq, R). Here cNN, qNN, UNN represent neural network operators and (Θc, Θq, ΘU) is a parameter set composed of network weights and biases. Based on the method of physics-informed neural networks proposed in [37], we integrate three different neural networks to obtain an extended transmission-dynamics-informed neural network(TDINN), shown in Fig 2. The neural networks in the purple shaded area are used to infer the time-dependent contact rate c(t) and quarantine rate q(t). The neural network in the green shaded area is used to fit the available data and approximately solve model (1). The approximated network solution of model (1) can be defined as The next critical step is to embed the information of transmission dynamics into the neural network to constrain the output(solutions) to satisfy the observational data and the ODE system, which is achieved by constructing a loss function corresponding to reported data and epidemiological models. Specifically, the output of the neural network at the temporal nodes should be as close as possible to the observed data. In addition, we enforce the neural network to satisfy the ODE system at the temporal nodes . This can be achieved by using automatic differentiation to calculate the residual error of the ODE system at , so is also called “residual points”. Here, let represent the number of observed data and represent the number of residual points. It is worth noting that residual points can be arbitrarily sampled in the entire computational domain. To measure the mismatch between the outputs from neural network/ODE systems and the observed data, we define the loss function as follows [37]: (3) where MSE stands for mean square error, MSEdata is used to measure the degree of matching between the output of the neural network and the observed data, and MSEode, as a penalty term, describes whether the solution learned by the neural network satisfies the ODE system. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Schematic diagram of transmission-dynamics-informed neural network. Different neural networks are used to represent the state variables (green shaded area) and time-dependent parameters (purple shaded area) of model (1). The symbols “σ” and “” represent the activation function and the automatic differentiation operator, respectively. https://doi.org/10.1371/journal.pcbi.1011535.g002 The first term MSEdata in the loss function (3) has different expressions based on the three categories of available datasets. For the data in Set 1, For the data in Set 2, the MSEdata becomes For the data in Set 3, the MSEdata becomes Where , , , , and represent the approximate solution of the neural network. Combining ordinary differential Eqs (1) and (2), we give the residual form of each component as follows: therefore, we have Finally, we simultaneously learn the network parameters and infer the unknown parameters of the model (1) by training the neural network to minimize the loss function (3). We use TDINN algorithm for fitting and parameter inferring based on the data available in different regions. The algorithm is implemented in Python using Tensorflow [46], an open source library for deep learning computations. We found empirically that the neural network structures used to solve model (1) and inferred time-dependent parameters c(t) and q(t) may be different due to the different sample sizes of observed data in various regions. The corresponding depth and width of neural networks are given in Table 2. We use the hyperbolic tangent function tanh(x) as the activation function σ shown in Fig 2. For the optimization of the loss function (3), we use a gradient-based optimizer such as the Adam optimizer [47], whose learning rate is set to be 0.001 by default, and the number of training iterations for each region is listed in Table 2. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Hyperparameters for the problems in this study. https://doi.org/10.1371/journal.pcbi.1011535.t002 Data We obtained the daily reported number of confirmed cases for Xi’an outbreak from December 9th, 2021 to January 20th, 2022, Guangzhou outbreak from May 21st, 2021 to June 18th, 2021, and for Yangzhou outbreak from July 28th, 2021 to August 26th, 2021 from Health Commissions of Shaanxi [39], Guangdong [40] and Jiangsu provinces [41], respectively. In addition, we collected the daily reported number of confirmed cases from August 1st, 2022 to September 23rd, 2022 in Hainan province [42] and form August 4th, 2022 to September 26th, 2022 in Xinjiang Uygur Autonomous Region [43], respectively. Data information includes the number of daily reported cases in the community() and in the quarantined zone(). It is important to note that the numbers of daily reported case in the community or quarantined zone are incomplete for Hainan and Xinjiang, but we have complete daily reported case numbers () in these two regions. Moreover, we also obtained the daily reported number of confirmed cases for Liaoning outbreak from 6th March 2022 to 21st May 2022 from Health Commissions of Liaoning provinces [44], where the data information only contains a column of daily reported case numbers () and shows multi-wave outbreaks. Detailed data are shown in Fig 1a–1d. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Multi-source epidemic data and the framework of transmission dynamic model. (a) Epidemic data of COVID-19 infection in Liaoning province from 6th March 2022 to 21st May 2022; (b) Epidemic data of COVID-19 infection in Xi’an from 9th December 2021 to 20th January 2022, in Guangzhou from 21st May to 18th June 2021, and in Yangzhou from 28th July to 26th August 2021; (c) Epidemic data of COVID-19 infection in Hainan from August 1st to September 23rd, 2022; (d) Epidemic data of COVID-19 infection in Xinjiang from August 4th to September 26th, 2022; (e) Flow diagram among epidemiological classes. https://doi.org/10.1371/journal.pcbi.1011535.g001 For Xi’an, Guangzhou, and Yangzhou, we can calculate the cumulative reported cases in the community()(or quarantined zone ()) based on the daily reported cases in the community (or quarantined zone), while for Hainan, Xinjiang and Liaoning, we only obtain the cumulative reported cases(). Therefore, in this study, we have access to three categories of reported data sets, which are as the follows: Set 1: , , , , for Xi’an, Guangzhou, Yangzhou; Set 2: , , , , for Hainan, Xinjiang; Set 3: , , for Liaoning. The model During COVID-19 pandemic, China’s government has mainly adopted the dynamic zero-case policy, i.e., strict close contact tracking and isolation, high-frequency and large-scale nucleic acid screening, closed management and etc, to quickly respond to the outbreak. These powerful NPIs effectively make most infected people not go through the complete process from infection to incubation period, and then to asymptomatic or symptomatic, that is, patients may be detected at every stage after infection. Therefore, this study simulates the transmission mode and evolution dynamics of COVID-19 infection based on the classic deterministic Susceptible-Infected-Removed (SIR) type epidemiological model [2]. Then we extend the simplest SIR-type dynamic model by including contact tracing and isolation, and the flow diagram is shown in Fig 1e. Given an outbreak taking off in a city, the city can usually be divided into two regions according to different intensity of control measures: free region (or community) and quarantined region. Consequently, we stratify the population in the free (quarantined) region into the susceptible class S (Sq) and the infected class Ic (Iq), and the removed class (denoted by R). Note that here we do not distinguish the individuals in the removed class in free or quarantined region since they can not be re-infected within a relatively short duration, and then consider a single compartment. Here we use the subscripts ‘q’ to represent quarantined population, i.e. Sq and Iq represent quarantined susceptible class and quarantined infected class, respectively. To model the continuously adjusted intervention measures, we assume the time-dependent contact rate and quarantine rate, denoted by c(t) and q(t), respectively. The transmission probability of per contact is supposed to be β. Then, the quarantined individuals, if infected (or uninfected), move to the compartment Iq (or Sq) at a rate of βc(t)q(t) (or (1 − β)c(t)q(t)). Those who are not quarantined, if infected, will move to the compartment Ic at a rate of βc(t)(1 − q(t)). According to the fact that the quarantined individuals do not return to the susceptible population before the end of outbreak, then we ignore the rate of transition from Sq to S class. Hence we have the following ordinary differential equations: (1) where N represents the total population of the region, the recovery rate of infected individuals in community (quarantined region) denoted by γ(δq), and the definitions and values of all parameters used in the model are given in Table 1. Here we consider three additional auxiliary compartments to record cumulative reported cases (), the cumulative reported cases in the community () (or the quarantined region ()). The dynamics of these three compartments are driven by the following equations: (2) Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Parameter definitions and estimation for model (1). https://doi.org/10.1371/journal.pcbi.1011535.t001 Parameter estimation It is known that fully connected deep neural networks with arbitrary nonlinear activation functions are universal approximators [45], we then use three independent neural networks with time t as input to represent the time-dependent contact rate c(t), quarantined rate q(t) and each state variable in model (1) respectively. So we have where U is a vector of all epidemiological categories considered in model (1), i.e., U = (S, Ic, Sq, Iq, R). Here cNN, qNN, UNN represent neural network operators and (Θc, Θq, ΘU) is a parameter set composed of network weights and biases. Based on the method of physics-informed neural networks proposed in [37], we integrate three different neural networks to obtain an extended transmission-dynamics-informed neural network(TDINN), shown in Fig 2. The neural networks in the purple shaded area are used to infer the time-dependent contact rate c(t) and quarantine rate q(t). The neural network in the green shaded area is used to fit the available data and approximately solve model (1). The approximated network solution of model (1) can be defined as The next critical step is to embed the information of transmission dynamics into the neural network to constrain the output(solutions) to satisfy the observational data and the ODE system, which is achieved by constructing a loss function corresponding to reported data and epidemiological models. Specifically, the output of the neural network at the temporal nodes should be as close as possible to the observed data. In addition, we enforce the neural network to satisfy the ODE system at the temporal nodes . This can be achieved by using automatic differentiation to calculate the residual error of the ODE system at , so is also called “residual points”. Here, let represent the number of observed data and represent the number of residual points. It is worth noting that residual points can be arbitrarily sampled in the entire computational domain. To measure the mismatch between the outputs from neural network/ODE systems and the observed data, we define the loss function as follows [37]: (3) where MSE stands for mean square error, MSEdata is used to measure the degree of matching between the output of the neural network and the observed data, and MSEode, as a penalty term, describes whether the solution learned by the neural network satisfies the ODE system. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Schematic diagram of transmission-dynamics-informed neural network. Different neural networks are used to represent the state variables (green shaded area) and time-dependent parameters (purple shaded area) of model (1). The symbols “σ” and “” represent the activation function and the automatic differentiation operator, respectively. https://doi.org/10.1371/journal.pcbi.1011535.g002 The first term MSEdata in the loss function (3) has different expressions based on the three categories of available datasets. For the data in Set 1, For the data in Set 2, the MSEdata becomes For the data in Set 3, the MSEdata becomes Where , , , , and represent the approximate solution of the neural network. Combining ordinary differential Eqs (1) and (2), we give the residual form of each component as follows: therefore, we have Finally, we simultaneously learn the network parameters and infer the unknown parameters of the model (1) by training the neural network to minimize the loss function (3). We use TDINN algorithm for fitting and parameter inferring based on the data available in different regions. The algorithm is implemented in Python using Tensorflow [46], an open source library for deep learning computations. We found empirically that the neural network structures used to solve model (1) and inferred time-dependent parameters c(t) and q(t) may be different due to the different sample sizes of observed data in various regions. The corresponding depth and width of neural networks are given in Table 2. We use the hyperbolic tangent function tanh(x) as the activation function σ shown in Fig 2. For the optimization of the loss function (3), we use a gradient-based optimizer such as the Adam optimizer [47], whose learning rate is set to be 0.001 by default, and the number of training iterations for each region is listed in Table 2. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Hyperparameters for the problems in this study. https://doi.org/10.1371/journal.pcbi.1011535.t002 Results Model calibration For Xi’an, Guangzhou and Yangzhou, we fitted the daily reported cases from communities and from quarantined zone through the TDINN algorithm, while for Hainan and Xinjiang, we further fitted daily reported cases. We present the best fitting results in Figs 3a, 3b, 3e, 3f, 3i, 3j, 4a–4c and 4f–4h (green solid lines), and the inferred time-dependent parameters c(t) and q(t) for each region in Figs 3c, 3d, 3g, 3h, 3k, 3m, 4d, 4e, 4i and 4j (magenta pentagrams), respectively. In addition, the estimated parameter values are listed in Table 1. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Data fitting and inference of the time-dependent parameters by TDINN algorithm for the local outbreaks in Xi’an, Guangzhou, and Yangzhou. (a)-(b), (e)-(f) and (i)-(j) show the fitting results in Xi’an, Guangzhou and Yangzhou, respectively, where the cyan and purple solid dots represent the daily reported data from communities and quarantined population respectively, green solid curves represent the best fitting results by TDINN, the dashed curves represent the corresponding solution curves after substituting various combinations of the family of functions (4) and (5) into model (1). (c)-(d), (g)-(h) and (k)-(m) show the inference and fitting results of the time-dependent contact rate c(t) and quarantined rate q(t) in Xi’an, Guangzhou and Yangzhou, respectively, where the magenta pentagrams represent the inference results of c(t) and q(t) by TDINN and the solid curves represent the fitting results of c(t) and q(t) based on different functions in (4) and (5). https://doi.org/10.1371/journal.pcbi.1011535.g003 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Data fitting and inference of the time-dependent parameters by TDINN algorithm for the local outbreaks in Hainan and Xinjiang. (a)-(c) and (f)-(h) show the fitting results in Hainan and Xinjiang, respectively, where the cyan solid dots represent the daily reported data from communities, the purple solid dots represent the daily reported data from quarantined population and the red solid dots represent the daily reported data, green solid curves represent the best fitting results by TDINN, the dashed curves represent the corresponding solution curves after substituting various combinations of the family of functions (4) and (5) into model (1). (d)-(e) and (i)-(j) show the inference and fitting results of the time-dependent contact rate c(t) and quarantined rate q(t) in Hainan and Xinjiang, respectively, where the magenta pentagrams represent the inference results of c(t) and q(t) by TDINN and the solid curves represent the fitting results of c(t) and q(t) based on different functions in (4) and (5). https://doi.org/10.1371/journal.pcbi.1011535.g004 As we can see from Figs 3 and 4, the TDINN algorithm can fit the daily reported cases from communities and from quarantined zones very well, and can also automatically capture the temporal variations of contact rate and quarantine rate under different epidemic patterns for different regions. It is worth noting that although part of data on the daily confirmed cases from communities and from quarantined zone in Hainan and Xinjiang are available, our algorithm can still accurately simulate the complete epidemic evolution trend for two regions. These numerical simulation results indicate that the proposed TDINN algorithm can not only adapt well to multi-source epidemic data in different regions, but also extract relevant information that can quantify the intensity of control interventions. Moreover, the TDINN algorithm can infer the unobserved dynamics of epidemic based on sparse and noisy observation data, thereby reconstructing the complete epidemic development process. It is worth noting here that although we do not have any prior information on the contact rate c(t) and quarantine rate q(t), that is, we do not assume the specific function expressions for c(t) and q(t) in advance, the variations in the contact rate and quarantine rate over time in different regions can completely be extracted from the multi-source epidemic data. From Figs 3c, 3d, 3g, 3h, 3k, 3m, 4d, 4e, 4i and 4j (magenta pentagrams), we can find that c(t) and q(t) inferred by TDINN algorithm show regional dependent, that is, the temporal evolution curves of c(t) and q(t) corresponding to different regions show significantly different behaviors in terms of shape, indicating differences in strength of implementation and execution of control intervention strategies to alleviate the COVID-19 infection in each region. This difference makes the epidemic curves in various regions exhibit diversity in terms of peak values and peak times, which further demonstrating the importance of capturing the underlying efficacy of intervention to quickly realize dynamic zero-case policy at that time. In addition, based on the inferred c(t) and q(t), we find that contact rate c(t) shows a downward trend (shown in Figs 3c, 3g, 3k, 4d and 4i, while quarantine rate q(t) shows an upward trend (shown in Figs 3d, 3h, 3m, 4e and 4j. This is associated with the fact that once an outbreak taking off, China’s dynamic zero-case policy leads to an increase in the quarantine rate and the contact rate decline due to local lockdown and the enhanced close contact tracing and quarantine measure. Then, an interesting question raised from this observation is whether we can describe the temporal evolution patterns of c(t) and q(t) with specific functions to better quantify the evolution of the interventions, and consequently enhance the interpretability of deep learning. Interpretability analysis of time-dependent parameters Note that the contact rate and quarantine rate resulting from TDINN inference are two abstract time series without particular expressions. Therefore, it is worth formulating the appropriate functions for contact rate and quarantine rate, which can describe deep learning’s inference results and reveal the temporal evolution process of interventions. These functions not only aid in better understanding the behavior of deep learning during the inference process, but also improve the prediction accuracy and interpretability of model, providing guidance for designing more effective prevention and control strategies. Note that the increasing/decreasing pattern of time series may be associated with various formulas of rate functions. Here, we consider the contact rate c(t) and quarantine rate q(t) as a family of functions, with each family comprising three distinct forms denoted as c1(t), c2(t), c3(t) and q1(t), q2(t), q3(t), respectively. The explicit expressions for these functions are assumed as follows: (4) and (5) Here, the functions c1(t) and q1(t) are derived from existing literatures [48–50]. Parameter c0i is the initial contact rate, parameter cbi represents the minimum contact rate, and parameter r1i denotes the exponential decreasing rate of the contact rate, i = 1, 2, 3. Parameter q0i is the initial quarantine rate, parameter qmi denotes the maximum quarantine rate with the intervention being implemented, and parameter r2i represents the exponential increasing rate of quarantine rate, i = 1, 2, 3. In contrast to the exponential decay/increasing functions of c1(t) and q1(t), the sustained strengthening control strategies is described by the Gaussian decay functions [51] of c2(t) and q2(t). Additionally, the construction of c3(t) and q3(t) is based on the analytical solution of the Rosenzweig model [52], where m and n are interference constants. Then, an interesting question is which function in the family of functions (4) and (5) can accurately describe the temporal evolution trends of c(t) and q(t) inferred by TDINN. To address this question, we initially begin by considering the time series corresponding to c(t) and q(t) learned from the TDINN algorithm as observed data, denoting them as and , respectively, where . Next, we fit the functions in (4) and (5) to the observed data and , estimate the corresponding unknown parameters, and select the appropriate function formula based on the statistical criterion. This is equivalent to solving the optimization problem: (6) with Parameters θ and ϑ represent the unknown parameter vectors in (4) and (5), respectively. Then, we utilize the least squares(LS) method to solve the optimization problem (6), and consequently obtain the estimated values of the unknown parameters in the family of functions (4) and (5), as listed in Table 3. The fitting results for each region are shown in Figs 3c, 3d, 3g, 3h, 3k, 3m, 4d, 4e, 4i and 4j (solid lines), respectively. Finally, we determine the optimal function form to accurately capture the temporal evolution of contact rate c(t) and quarantine rate q(t) inferred by the TDINN algorithm based on the criterion of minimizing the root mean squared error (RMSE). We computed the root mean square errors ( and ) which are generated by different functions in (4) and (5) when fitting the observed data and , where Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Parameter definitions and estimation for functions ci(t) and qi(t) (i = 1,2,3). https://doi.org/10.1371/journal.pcbi.1011535.t003 Note that, the functions with the smallest RMSEci and RMSEqi were selected as the best candidates. According to Figs 5a, 5b, 5d, 5e, 5g, 5h, 6a, 6b, 6d and 6e, we can draw the following conclusions: for Xi’an: , ; for Guangzhou: , ; for Yangzhou: , ; for Hainan: , ; for Xinjiang: , . Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. The optimal contact/quarantine rates from the family of functions (4) and (5) for Xi’an, Guangzhou and Yangzhou. (a, d, g) Root mean square error(), corresponding to fitting the time-dependent contact rate learned by TDINN algorithm using c1(t), c2(t) and c3(t) in Xi’an, Guangzhou and Yangzhou. (b, e, h) Root mean square error(), corresponding to fitting the time-dependent quarantine rate learned by TDINN algorithm using q1(t), q2(t) and q3(t) in Xi’an, Guangzhou and Yangzhou. (c, f, i) Average root mean square error (), corresponding to fitting epidemic data using model (1) based on various combinations of the family of functions (4) and (5) in Xi’an, Guangzhou and Yangzhou. https://doi.org/10.1371/journal.pcbi.1011535.g005 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. The optimal contact/quarantine rates from the family of functions (4) and (5) for Hainan and Xinjiang. (a, d) Root mean square error(), corresponding to fitting the time-dependent contact rate learned by TDINN algorithm using c1(t), c2(t) and c3(t) in Hainan and Xinjiang. (b, e) Root mean square error(), corresponding to fitting the time-dependent quarantine rate learned by TDINN algorithm using q1(t), q2(t) and q3(t) in Hainan and Xinjiang. (c, f) Average root mean square error (), corresponding to fitting epidemic data using model (1) based on various combinations of the family of functions (4) and (5) in Hainan and Xinjiang. https://doi.org/10.1371/journal.pcbi.1011535.g006 Based on the above results, we can select the optimal functions to quantify the evolution of the interventions in each region, that is, for Xi’an, Guangzhou, Yangzhou, Hainan and Xinjiang, the optimal rate functions are c2(t) and q2(t), c3(t) and q2(t), c3(t) and q2(t), c3(t) and q3(t), c2(t) and q1(t) respectively. To further validate our conclusions, we substituted various functions into the model (1) and re-fitted the multi-source data for each region by using the estimated parameters in Tables 1 and 3. The fitting results are presented in Figs 3a, 3b, 3e, 3f, 3i, 3j, 4a–4c and 4f–4h (dotted lines). Note that here we consider the average root mean square error (, i, j = 1, 2, 3) as a metric to evaluate the fitting performance of model (1) with different function combinations in the family of functions (4) and (5) for multi-source data in each region. For Xi’an, Guangzhou and Yangzhou, for Hainan and Xinjiang, where , are the predicted values by solving model (1). Considering all possible combinations of rate functions, the smaller value indicates the better fitting effect of model (1) on multi-source data. In the following, we summarized how the varied with respect to different choices of contact rate and quarantine rate for each region in Figs 5c, 5f, 5i, 6c and 6f. According to Figs 5c, 5f, 5i, 6c and 6f, the corresponding values for each region exhibit the following relationship: for Xi’an: ; for Guangzhou: ; for Yangzhou: ; for Hainan: ; for Xinjiang: . The above results show that selecting c2(t) and q2(t) (or c3(t) and q2(t), c3(t) and q2(t), c3(t) and q3(t), c2(t) and q1(t)) as the contact rate and quarantine rate leads to the smallest ARMSE value for Xi’an (or Guangzhou, Yangzhou, Hainan, Xinjiang). This indicates that model (1) can accurately replicate the development process of the COVID-19 epidemic in Xi’an(or Guangzhou, Yangzhou, Hainan, Xinjiang) under this combination, which further validates our previous conclusion that c2(t) and q2(t) (or c3(t) and q2(t), c3(t) and q2(t), c3(t) and q3(t), c2(t) and q1(t)) are the optimal functions for quantifying the evolution of control interventions in Xi’an(or Guangzhou, Yangzhou, Hainan, Xinjiang). Based on the optimal rate functions for each region (see Figs 5 and 6 for detail), we can find that it is difficult to construct a universal function combination to quantify the control intervention strategies implemented in different regions. That is to say, in order to response the outbreak, the pattern of epidemic prevention and control in one region cannot be directly applied to another region. Ideally, we should flexibly adjust and develop appropriate prevention and control measures according to the actual situation of different regions. According to the above analysis, we utilized the time series inferred by the TDINN algorithm to get the optimal contact rate and quarantine rate from the family of functions (4) and (5), which enabled us to accurately quantify the strength of control measures in each region. Further, it is worth noting all parameters in the rate functions have realistic meanings, then the selected rate functions help to enhance the interpretability of the time series inferred by deep learning. In addition, we can achieve the best fitting effect after substituting the optimal contact rate and quarantine rate into the model (1), which further validates that this method of quantifying the dynamic evolution of interventions is feasible. This method can also aid in improving our understanding of how control strategies are dynamically adjusted when fighting against epidemics. Simulation of the multiple epidemic waves To further illustrate the effectiveness of our proposed method, we also apply the proposed TDINN algorithm to the simulation of multiple waves of COVID-19 infection. To do this, we simulated the dynamics of the epidemic based on daily reported cases in Liaoning province and visualize the simulation results in Fig 7. The simulation results show that the TDINN algorithm can not only fit the epidemic data containing multiple waves well (see Fig 7a), but also capture the information on strengthening and relaxation of intervention measures, that is, the inferred contact rate and quarantine rate exhibit fluctuations as shown in Fig 7b and 7c. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. Data fitting and inference of the time-dependent parameters by TDINN algorithm for multiple waves of COVID-19 infection in Liaoning province. (a) shows the fitting results for the available data in Liaoning, where the purple solid dots represent the daily reported data, green solid curves represent the best fitting results by TDINN. (b) and (c) show the inferred time-dependent contact rate c(t) and quarantine rate q(t) by TDINN, respectively. https://doi.org/10.1371/journal.pcbi.1011535.g007 In fact, as the epidemic initially took off, we observed an increase in quarantine rate and a decrease in contact rate due to enhanced intervention measures to mitigate epidemic. While the outbreak was subsiding, the gradual relaxation of control interventions led to the quarantine rate decline and the contact rate increase, and thereby possibly inducing a resurgence of epidemic. As a consequence, comparing the inferred contact rate and quarantine rate with the time series of daily reported cases containing multiple epidemic waves (Fig 7a–7c), we can observe a feedback loop: epidemic taking off → quarantine rate increasing and contact rate decreasing → epidemic subsiding → quarantine rate decreasing and contact rate increasing → epidemic resurging, which drives multiple COVID-19 epidemic waves as observed in Liaoning province. It is worth noting that the inferred time series on contact rate and quarantine rate in Fig 7b and 7c exhibit complicated behaviors, which are difficult to simulate accurately through the family of functions (4) and (5). Model calibration For Xi’an, Guangzhou and Yangzhou, we fitted the daily reported cases from communities and from quarantined zone through the TDINN algorithm, while for Hainan and Xinjiang, we further fitted daily reported cases. We present the best fitting results in Figs 3a, 3b, 3e, 3f, 3i, 3j, 4a–4c and 4f–4h (green solid lines), and the inferred time-dependent parameters c(t) and q(t) for each region in Figs 3c, 3d, 3g, 3h, 3k, 3m, 4d, 4e, 4i and 4j (magenta pentagrams), respectively. In addition, the estimated parameter values are listed in Table 1. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Data fitting and inference of the time-dependent parameters by TDINN algorithm for the local outbreaks in Xi’an, Guangzhou, and Yangzhou. (a)-(b), (e)-(f) and (i)-(j) show the fitting results in Xi’an, Guangzhou and Yangzhou, respectively, where the cyan and purple solid dots represent the daily reported data from communities and quarantined population respectively, green solid curves represent the best fitting results by TDINN, the dashed curves represent the corresponding solution curves after substituting various combinations of the family of functions (4) and (5) into model (1). (c)-(d), (g)-(h) and (k)-(m) show the inference and fitting results of the time-dependent contact rate c(t) and quarantined rate q(t) in Xi’an, Guangzhou and Yangzhou, respectively, where the magenta pentagrams represent the inference results of c(t) and q(t) by TDINN and the solid curves represent the fitting results of c(t) and q(t) based on different functions in (4) and (5). https://doi.org/10.1371/journal.pcbi.1011535.g003 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Data fitting and inference of the time-dependent parameters by TDINN algorithm for the local outbreaks in Hainan and Xinjiang. (a)-(c) and (f)-(h) show the fitting results in Hainan and Xinjiang, respectively, where the cyan solid dots represent the daily reported data from communities, the purple solid dots represent the daily reported data from quarantined population and the red solid dots represent the daily reported data, green solid curves represent the best fitting results by TDINN, the dashed curves represent the corresponding solution curves after substituting various combinations of the family of functions (4) and (5) into model (1). (d)-(e) and (i)-(j) show the inference and fitting results of the time-dependent contact rate c(t) and quarantined rate q(t) in Hainan and Xinjiang, respectively, where the magenta pentagrams represent the inference results of c(t) and q(t) by TDINN and the solid curves represent the fitting results of c(t) and q(t) based on different functions in (4) and (5). https://doi.org/10.1371/journal.pcbi.1011535.g004 As we can see from Figs 3 and 4, the TDINN algorithm can fit the daily reported cases from communities and from quarantined zones very well, and can also automatically capture the temporal variations of contact rate and quarantine rate under different epidemic patterns for different regions. It is worth noting that although part of data on the daily confirmed cases from communities and from quarantined zone in Hainan and Xinjiang are available, our algorithm can still accurately simulate the complete epidemic evolution trend for two regions. These numerical simulation results indicate that the proposed TDINN algorithm can not only adapt well to multi-source epidemic data in different regions, but also extract relevant information that can quantify the intensity of control interventions. Moreover, the TDINN algorithm can infer the unobserved dynamics of epidemic based on sparse and noisy observation data, thereby reconstructing the complete epidemic development process. It is worth noting here that although we do not have any prior information on the contact rate c(t) and quarantine rate q(t), that is, we do not assume the specific function expressions for c(t) and q(t) in advance, the variations in the contact rate and quarantine rate over time in different regions can completely be extracted from the multi-source epidemic data. From Figs 3c, 3d, 3g, 3h, 3k, 3m, 4d, 4e, 4i and 4j (magenta pentagrams), we can find that c(t) and q(t) inferred by TDINN algorithm show regional dependent, that is, the temporal evolution curves of c(t) and q(t) corresponding to different regions show significantly different behaviors in terms of shape, indicating differences in strength of implementation and execution of control intervention strategies to alleviate the COVID-19 infection in each region. This difference makes the epidemic curves in various regions exhibit diversity in terms of peak values and peak times, which further demonstrating the importance of capturing the underlying efficacy of intervention to quickly realize dynamic zero-case policy at that time. In addition, based on the inferred c(t) and q(t), we find that contact rate c(t) shows a downward trend (shown in Figs 3c, 3g, 3k, 4d and 4i, while quarantine rate q(t) shows an upward trend (shown in Figs 3d, 3h, 3m, 4e and 4j. This is associated with the fact that once an outbreak taking off, China’s dynamic zero-case policy leads to an increase in the quarantine rate and the contact rate decline due to local lockdown and the enhanced close contact tracing and quarantine measure. Then, an interesting question raised from this observation is whether we can describe the temporal evolution patterns of c(t) and q(t) with specific functions to better quantify the evolution of the interventions, and consequently enhance the interpretability of deep learning. Interpretability analysis of time-dependent parameters Note that the contact rate and quarantine rate resulting from TDINN inference are two abstract time series without particular expressions. Therefore, it is worth formulating the appropriate functions for contact rate and quarantine rate, which can describe deep learning’s inference results and reveal the temporal evolution process of interventions. These functions not only aid in better understanding the behavior of deep learning during the inference process, but also improve the prediction accuracy and interpretability of model, providing guidance for designing more effective prevention and control strategies. Note that the increasing/decreasing pattern of time series may be associated with various formulas of rate functions. Here, we consider the contact rate c(t) and quarantine rate q(t) as a family of functions, with each family comprising three distinct forms denoted as c1(t), c2(t), c3(t) and q1(t), q2(t), q3(t), respectively. The explicit expressions for these functions are assumed as follows: (4) and (5) Here, the functions c1(t) and q1(t) are derived from existing literatures [48–50]. Parameter c0i is the initial contact rate, parameter cbi represents the minimum contact rate, and parameter r1i denotes the exponential decreasing rate of the contact rate, i = 1, 2, 3. Parameter q0i is the initial quarantine rate, parameter qmi denotes the maximum quarantine rate with the intervention being implemented, and parameter r2i represents the exponential increasing rate of quarantine rate, i = 1, 2, 3. In contrast to the exponential decay/increasing functions of c1(t) and q1(t), the sustained strengthening control strategies is described by the Gaussian decay functions [51] of c2(t) and q2(t). Additionally, the construction of c3(t) and q3(t) is based on the analytical solution of the Rosenzweig model [52], where m and n are interference constants. Then, an interesting question is which function in the family of functions (4) and (5) can accurately describe the temporal evolution trends of c(t) and q(t) inferred by TDINN. To address this question, we initially begin by considering the time series corresponding to c(t) and q(t) learned from the TDINN algorithm as observed data, denoting them as and , respectively, where . Next, we fit the functions in (4) and (5) to the observed data and , estimate the corresponding unknown parameters, and select the appropriate function formula based on the statistical criterion. This is equivalent to solving the optimization problem: (6) with Parameters θ and ϑ represent the unknown parameter vectors in (4) and (5), respectively. Then, we utilize the least squares(LS) method to solve the optimization problem (6), and consequently obtain the estimated values of the unknown parameters in the family of functions (4) and (5), as listed in Table 3. The fitting results for each region are shown in Figs 3c, 3d, 3g, 3h, 3k, 3m, 4d, 4e, 4i and 4j (solid lines), respectively. Finally, we determine the optimal function form to accurately capture the temporal evolution of contact rate c(t) and quarantine rate q(t) inferred by the TDINN algorithm based on the criterion of minimizing the root mean squared error (RMSE). We computed the root mean square errors ( and ) which are generated by different functions in (4) and (5) when fitting the observed data and , where Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Parameter definitions and estimation for functions ci(t) and qi(t) (i = 1,2,3). https://doi.org/10.1371/journal.pcbi.1011535.t003 Note that, the functions with the smallest RMSEci and RMSEqi were selected as the best candidates. According to Figs 5a, 5b, 5d, 5e, 5g, 5h, 6a, 6b, 6d and 6e, we can draw the following conclusions: for Xi’an: , ; for Guangzhou: , ; for Yangzhou: , ; for Hainan: , ; for Xinjiang: , . Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. The optimal contact/quarantine rates from the family of functions (4) and (5) for Xi’an, Guangzhou and Yangzhou. (a, d, g) Root mean square error(), corresponding to fitting the time-dependent contact rate learned by TDINN algorithm using c1(t), c2(t) and c3(t) in Xi’an, Guangzhou and Yangzhou. (b, e, h) Root mean square error(), corresponding to fitting the time-dependent quarantine rate learned by TDINN algorithm using q1(t), q2(t) and q3(t) in Xi’an, Guangzhou and Yangzhou. (c, f, i) Average root mean square error (), corresponding to fitting epidemic data using model (1) based on various combinations of the family of functions (4) and (5) in Xi’an, Guangzhou and Yangzhou. https://doi.org/10.1371/journal.pcbi.1011535.g005 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. The optimal contact/quarantine rates from the family of functions (4) and (5) for Hainan and Xinjiang. (a, d) Root mean square error(), corresponding to fitting the time-dependent contact rate learned by TDINN algorithm using c1(t), c2(t) and c3(t) in Hainan and Xinjiang. (b, e) Root mean square error(), corresponding to fitting the time-dependent quarantine rate learned by TDINN algorithm using q1(t), q2(t) and q3(t) in Hainan and Xinjiang. (c, f) Average root mean square error (), corresponding to fitting epidemic data using model (1) based on various combinations of the family of functions (4) and (5) in Hainan and Xinjiang. https://doi.org/10.1371/journal.pcbi.1011535.g006 Based on the above results, we can select the optimal functions to quantify the evolution of the interventions in each region, that is, for Xi’an, Guangzhou, Yangzhou, Hainan and Xinjiang, the optimal rate functions are c2(t) and q2(t), c3(t) and q2(t), c3(t) and q2(t), c3(t) and q3(t), c2(t) and q1(t) respectively. To further validate our conclusions, we substituted various functions into the model (1) and re-fitted the multi-source data for each region by using the estimated parameters in Tables 1 and 3. The fitting results are presented in Figs 3a, 3b, 3e, 3f, 3i, 3j, 4a–4c and 4f–4h (dotted lines). Note that here we consider the average root mean square error (, i, j = 1, 2, 3) as a metric to evaluate the fitting performance of model (1) with different function combinations in the family of functions (4) and (5) for multi-source data in each region. For Xi’an, Guangzhou and Yangzhou, for Hainan and Xinjiang, where , are the predicted values by solving model (1). Considering all possible combinations of rate functions, the smaller value indicates the better fitting effect of model (1) on multi-source data. In the following, we summarized how the varied with respect to different choices of contact rate and quarantine rate for each region in Figs 5c, 5f, 5i, 6c and 6f. According to Figs 5c, 5f, 5i, 6c and 6f, the corresponding values for each region exhibit the following relationship: for Xi’an: ; for Guangzhou: ; for Yangzhou: ; for Hainan: ; for Xinjiang: . The above results show that selecting c2(t) and q2(t) (or c3(t) and q2(t), c3(t) and q2(t), c3(t) and q3(t), c2(t) and q1(t)) as the contact rate and quarantine rate leads to the smallest ARMSE value for Xi’an (or Guangzhou, Yangzhou, Hainan, Xinjiang). This indicates that model (1) can accurately replicate the development process of the COVID-19 epidemic in Xi’an(or Guangzhou, Yangzhou, Hainan, Xinjiang) under this combination, which further validates our previous conclusion that c2(t) and q2(t) (or c3(t) and q2(t), c3(t) and q2(t), c3(t) and q3(t), c2(t) and q1(t)) are the optimal functions for quantifying the evolution of control interventions in Xi’an(or Guangzhou, Yangzhou, Hainan, Xinjiang). Based on the optimal rate functions for each region (see Figs 5 and 6 for detail), we can find that it is difficult to construct a universal function combination to quantify the control intervention strategies implemented in different regions. That is to say, in order to response the outbreak, the pattern of epidemic prevention and control in one region cannot be directly applied to another region. Ideally, we should flexibly adjust and develop appropriate prevention and control measures according to the actual situation of different regions. According to the above analysis, we utilized the time series inferred by the TDINN algorithm to get the optimal contact rate and quarantine rate from the family of functions (4) and (5), which enabled us to accurately quantify the strength of control measures in each region. Further, it is worth noting all parameters in the rate functions have realistic meanings, then the selected rate functions help to enhance the interpretability of the time series inferred by deep learning. In addition, we can achieve the best fitting effect after substituting the optimal contact rate and quarantine rate into the model (1), which further validates that this method of quantifying the dynamic evolution of interventions is feasible. This method can also aid in improving our understanding of how control strategies are dynamically adjusted when fighting against epidemics. Simulation of the multiple epidemic waves To further illustrate the effectiveness of our proposed method, we also apply the proposed TDINN algorithm to the simulation of multiple waves of COVID-19 infection. To do this, we simulated the dynamics of the epidemic based on daily reported cases in Liaoning province and visualize the simulation results in Fig 7. The simulation results show that the TDINN algorithm can not only fit the epidemic data containing multiple waves well (see Fig 7a), but also capture the information on strengthening and relaxation of intervention measures, that is, the inferred contact rate and quarantine rate exhibit fluctuations as shown in Fig 7b and 7c. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. Data fitting and inference of the time-dependent parameters by TDINN algorithm for multiple waves of COVID-19 infection in Liaoning province. (a) shows the fitting results for the available data in Liaoning, where the purple solid dots represent the daily reported data, green solid curves represent the best fitting results by TDINN. (b) and (c) show the inferred time-dependent contact rate c(t) and quarantine rate q(t) by TDINN, respectively. https://doi.org/10.1371/journal.pcbi.1011535.g007 In fact, as the epidemic initially took off, we observed an increase in quarantine rate and a decrease in contact rate due to enhanced intervention measures to mitigate epidemic. While the outbreak was subsiding, the gradual relaxation of control interventions led to the quarantine rate decline and the contact rate increase, and thereby possibly inducing a resurgence of epidemic. As a consequence, comparing the inferred contact rate and quarantine rate with the time series of daily reported cases containing multiple epidemic waves (Fig 7a–7c), we can observe a feedback loop: epidemic taking off → quarantine rate increasing and contact rate decreasing → epidemic subsiding → quarantine rate decreasing and contact rate increasing → epidemic resurging, which drives multiple COVID-19 epidemic waves as observed in Liaoning province. It is worth noting that the inferred time series on contact rate and quarantine rate in Fig 7b and 7c exhibit complicated behaviors, which are difficult to simulate accurately through the family of functions (4) and (5). Discussion and conclusion During the COVID-19 pandemic, control measures played an important role in mitigating the disease spread. In particular, massive contact tracing following prompt quarantine and isolation showed decisive effect in dynamic clearing of the COVID-19 epidemic in China. Hence quantifying the dynamic contact rate and quarantine rate and estimate their impacts remain challenging. In this study, we integrated data-driven deep learning and dynamics-driven first principle modeling, and proposed an extended transmission-dynamics-informed neural network (TDINN) algorithm by encoding SIR-type compartment model into the neural networks, in order to obtain the time-dependent rate functions of mechanistic models. With the developed TDINN algorithm, we simulated the dynamics of COVID-19 infection in Xi’an, Guangzhou, Yangzhou, Hainan, Xinjiang and Liaoning province, by simultaneously inferring the unknown time-independent and time-dependent parameters. The TDINN algorithm enables us to successfully encode the contact rate and quarantine rate derived from deep neural networks into the compartment model, as well as integrating the transmission dynamic model into the deep neural networks. It is important to note that the TDINN algorithm overcomes some disadvantages of traditional transmission dynamic models for simulating the development process of the COVID-19 epidemic. For example, in the classic compartment model, the contact rate and quarantine rate are usually assumed to be constant or particular time-dependent functions, respectively, to describe the intensities of control interventions [10, 12]. That is, to simulate outbreaks in different regions, we need to pre-set various particular parameter values and/or time-dependent functions to quantify the continuously adjusted control measures in different regions, which significantly limits the performance of the transmission dynamic models. In contrast, our proposed TDINN algorithm can effectively overcome this disadvantage as it associates the transmission dynamic model with deep neural networks through the universal approximation property of neural networks [45] and can capture information on contact rate and quarantine rate from the epidemic data without assuming the particular formula for the rate functions in advance. Despite the structure of the considered transmission dynamic model (1) in the TDINN algorithm is quite simple, the model (1) incorporates time-dependent contact rate and quarantine rate inferred by neural networks, allowing us to well fit multi-source data for different regions that included daily reported cases in the community and in the quarantined zone (see Fig 3a, 3b, 3e, 3f, 3i and 3j), as well as daily reported cases with multiple epidemic waves (Fig 7). In addition, by using TDINN algorithm we can also reconstruct the epidemic process even if the data are insufficient (Fig 4a–4c and 4f–4h) and obtain the temporal evolution patterns of contact rate c(t) and quarantine rate q(t). The estimations of contact/quarantine rates show the regional-dependent (see Figs 3c, 3d, 3g, 3h, 3k, 3m, 4d, 4e, 4i, 4j, 7b and 7c), which indicates that there are differences in efficacy of control intervention strategies adopted in various regions. It further reveals why it is difficult to accurately quantify the strength of control measures through a specific function, that is, pre-setting the particular type of functions may not describe the actual contact rate and quarantine rate. It is interesting to observe the high consistency in the evolutionary trend of the contact rate and quarantine rate extracted by the TDINN algorithm from a single wave of epidemic (such as Xi’an, Guangzhou, Yangzhou, Hainan and Xinjiang), where the contact rate gradually decreases and the quarantine rate gradually increases (shown in Figs 3c, 3d, 3g, 3h, 3k, 3m, 4d, 4e, 4i and 4j). This suggests that reducing the contact rate and/or increasing the quarantine rate can significantly be associated with decrease in the daily reported cases, which agrees well with previous studies [53, 54]. In addition, Liaoning outbreak experienced two epidemic waves, which is related with the continuous strengthening and relaxation of control interventions, corresponding to the oscillations of the contact and quarantine rates (shown in Fig 7). In return, the shifting of the contact and quarantine rates can also affect the transmission dynamics of COVID-19 pandemic. This generates a feedback loop between the changes in the intensity of control measures and the epidemic shifting, which is the key to drive the fluctuations of the epidemics and is in line with observations of the existing study [11]. A key highlight of this study is that we can select the best combination from a family of functions (4) and (5) to accurately simulate the time series for contact and quarantine rate (c(t) and q(t)) learned by TDINN algorithm (Figs 5 and 6). The selection enables us to comprehensively explore the evolution trend of COVID-19 epidemic outbreak in different regions, and study the impact of various intervention strategies on the spread of infectious diseases. In addition, the selected rate functions based on the time series inferred by deep learning have reasonable meanings. In this study, we proposed the TDINN algorithm, which not only extends the traditional transmission dynamic model by embedding the time-dependent functions learned from the deep neural network, but also extends the neural network by embedding the information of the transmission dynamic model. The novel approach enables us to well integrate the advantages of the transmission mechanism model and the deep neural network. Compared with traditional dynamic models, the TDINN algorithm has better data learning ability and inference ability of unknown rate functions. Compared with end-to-end deep learning, our main results are more interpretable due to the incorporation of known propagation mechanisms. Furthermore, this method can be easily extended to more complex compartment models to study other aspects of emerging infectious diseases. Our study has some limitations. The transmission dynamic model (1) we considered is fairly simple and may overlook the impact of important factors such as the capacity of healthcare infrastructure, behavioral responses to epidemics, and vaccination on the development of the COVID-19 infection, but we hope the approaches, integrating transmission dynamics with deep learning, are able to be applied more generally. In addition, for the contact rate and quarantine rate inferred from the multiple epidemic waves, it is difficult to accurately simulate their temporal evolution patterns through smooth functions. We leave this for future work.