A rational treatment of Mendelian genetics

John Porteous

doi:10.1186/1742-4682-1-6

A rational treatment of Mendelian genetics

Porteous, John 2004-08-31 00:00:00 Background: The key to a rational treatment of elementary Mendelian genetics, specifically to an understanding of the origin of dominant and recessive traits, lies in the facts that: (1) alleles of genes encode polypeptides; (2) most polypeptides are catalysts, i.e. enzymes or translocators; (3) the molecular components of all traits in all cells are the products of systems of enzymes, i.e. of fluxing metabolic pathways; (4) any flux to the molecular components of a trait responds non-linearly (non-additively) to graded mutations in the activity of any one of the enzymes at a catalytic locus in a metabolic system; (5) as the flux responds to graded changes in the activity of an enzyme, the concentrations of the molecular components of a trait also change. Conclusions: It is then possible to account rationally, and without misrepresenting Mendel, for: the origin of dominant and recessive traits; the occurrence of Mendel's 3(dominant):1(recessive) trait ratio; deviations from this ratio; the absence of dominant and recessive traits in some circumstances, the occurrence of a blending of traits in others; the frequent occurrence of pleiotropy and epistasis. 1. Background because Mendel defined the terms dominant and recessive The currently favoured explanation for the origin of Men- for traits or characters, it was illegitimate (and illogical) to del's dominant and recessive traits is untenable [1]. The call alleles dominant or recessive, and to represent them primary error in this current attempted explanation is the by the same letters used by Mendel to represent traits [1]. assumption that there is a direct, proportional, relation- ship in a diploid cell between a series of allegedly domi- (ii) A trait series written as (AA + 2Aa + aa) suggests, incor- nant and recessive alleles written as (AA + 2Aa + aa) and rectly, that dominant and recessive traits comprise two the dominant, hybrid and recessive traits written as (AA + aliquots, (A + A) or (a + a), of dominance or recessivity. 2Aa + aa). This assumption (Figure 2, in reference [1]) incorporates four fundamental faults: (iii) A failure to take account of the long established fact that the first non-nucleotide product of the expression of (i) A failure to distinguish between the parameters and the an allele is a polypeptide and that most polypeptides are variables of any system of interacting components, specif- enzymes or membrane-located translocators. ically between the determinants (alleles in modern termi- nology) and what is determined (the form of the trait or (iv) A failure to note that the components of all tangible characteristic expressed in a cell or organism). Thus, traits comprised the molecular products of metabolic Page 1 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 corresponding trait series will appear as (B + 2H + b). Mendel's notation (Aa) for a hybrid trait will be used in this article only when referring directly to Mendel's paper A, H [2]. 2. A rational explanation of Mendel's observations Our stated task was to explain logically how an allele series (UU + 2Uu + uu) is expressed as a series of qualita- tively distinguishable F2 traits (A + 2H + a) when F1 hybrids (H) are allowed to self-fertilise [1]. This is very simply achieved by correcting faults (iii) and (iv) in four 0 successive steps (sections 2.1–2.4) based on a paper pub- lished 23 years ago [3]. A fifth step (section 2.5) allows us 0 50 100 to go beyond that paper to explain how the trait ratio Relative enzyme activity 3(dominant):1(recessive) sometimes occurs and some- times does not. A sixth step (section 2.6), consistent with uu uU UU the earlier ones, explains why dominance and recessivity Allele constitution are not always observed. Section 2.7 validates an earlier section. Section 2.8 accounts for some aspects not dealt with in textbooks and reviews of genetics. A n Figure 2 accoun nt):1(r ting ece for Mendel's ssive) trait raob tio in his F2 p servation ofo a 3( pulations of plants domi- Accounting for Mendel's observation of a 3(domi- The treatment in this section 2 is extended in section 3 to nant):1(recessive) trait ratio in his F2 populations of plants. account for quantitatively different traits, in section 4 to Mendel's notations for a dominant trait, a hybrid and a reces- sive trait were (A), (Aa) and (a) respectively. For reasons illustrate some implications of the present treatment, and given in the preceding paper [1], a hybrid trait is represented in section 5 to account for pleiotropy and epistasis. Sec- in Figure 2 by (H). The molecular components of all traits are tion 6 defines the conditions that must be met if a rational synthesised by a metabolic pathway. When the activity of any account is to be given for the occurrence of dominant and one enzyme in a metabolic pathway is changed in discrete recessive traits. steps, the flux to a trait component responds in non-linear (non-additive) fashion [3]. If the flux response is quasi-hyper- 2.1. A generalised metabolic system bolic, as shown here, the hybrid trait (H) will be indistinguish- If: the first non-nucleotide product of expression of an able from the trait (A) expressed in the wild-type cell or allele is a polypeptide and most polypeptides are enzymes organism, even when the enzyme activity in the hybrid (H) [3,4], it follows that most mutations at any one gene locus has been reduced to 50% of the wild-type activity. Trait (a), will be distinguishable from both traits (A) and (H) only if the will result in the formation of a mutant enzyme at a cata- enzyme activity is further reduced to a sufficient extent. lytic locus in a metabolic pathway. This is true even if the Under these circumstances the trait series (A + 2H + a) functioning enzyme is composed of more than one becomes (3A + a); Mendel's 3(dominant):1(recessive) trait polypeptide, each specified by different genes. It then fol- ratio is accounted for without introducing arbitrary and lows that we need to ask how the concentration of a nor- inconsistent arguments [1]. mal molecular component of a trait will be affected by a mutation of any one enzyme within a metabolic system. In short, a systemic approach, outlined below, is obligatory. pathways, i.e., the products of sequences of enzyme-cata- This is the key to an understanding of the origin of domi- lysed reactions. nant and recessive traits, as first pointed out in the follow- ing two sentences: "When as geneticists, we consider Correction of the first two of these four faults has already substitutions of alleles at a locus, as biochemists, we con- been achieved (section 4 in reference [1]) by writing an sider alterations in catalytic parameters at an enzyme step. allele series as (UU + 2Uu + uu) and the corresponding - -. The effect on the phenotype of altering the genetic trait series as (A + 2H + a). In these statements (U) and (u) specification of a single enzyme - - - is unpredictable from are normal and mutant (not dominant and recessive) alle- a knowledge of events at that step alone and must involve les respectively. Mendel's notation (A) and (a) is used to the response of the system to alterations of single enzymes represent dominant and recessive traits but (H) replaces when they are embedded in the matrix of all other enzymes." Mendel's implausible notation (Aa) for a hybrid class of ([3]; p.641). trait [1]. Mutations at another gene locus, in the same or a different cell, will be written as (WW + Ww + ww); the Page 2 of 17 (page number not for citation purposes) Mendel's traits Flux to trait component Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 locus that concerns us here, irrespective of whether the E E E E E E 1 2 3 4 5 6 X S S S S S X (J) 0 1 2 3 4 5 6 cell is haploid, diploid or polyploid. v v v v v v 1 2 3 4 5 6 (6) The catalytic activity (v) at any one metabolic locus A Figure 1 segment of a model metabolic pathway can be left at its current value or changed to and main- A segment of a model metabolic pathway. This diagram tained at a new value by the experimentalist, e.g. by suita- shows those features, discussed in the text, that permit a sys- ble genetic manipulation of an allele. Each allele in these temic analysis of the response of any variable of a metabolic circumstances is therefore an internal parameter of the sys- system (e.g. a flux J or the concentration of any intracellular metabolite S) to changes in any one parameter of the system tem; it is accessible to modification by the direct and sole (e.g. an enzyme activity). Each S is an intracellular metabolite; intervention of the experimentalist [1]. each X is an extracellular metabolite. In a diploid cell, every E stands for a pair of enzymes (allozymes), each specified by (7) Because X and X are external to the system in Figure 0 6 one of the two alleles at a gene locus. Each E is then a locus 1, their concentrations can be fixed, and maintained at a of catalytic activity within a system of enzymes; each v stands chosen value, by the direct intervention of the experimen- for the individual reaction rates catalysed jointly by a pair of talist; they are external parameters of the metabolic system. allozymes in a diploid cell. Either or both allozymes at such a locus may be mutated. (8) In contrast to X and X , the concentrations of metab- 0 6 olites S to S within the system cannot be fixed and main- 1 5 tained at any desired value solely by the direct 2.2 Metabolic systems and steady states intervention of the experimentalist. The concentrations of Metabolic processes are facilitated by a succession of cata- S to S are internal variables of the system. (If a fixed 1 5 lysed steps; i.e. by enzyme-catalysed transformations of amount of any one of these metabolites were to be substrates to products or by carrier-catalysed translocation injected through the membrane into the system, contin- of metabolites across membranes. Because enzymes and ued metabolism would ensure that the new intracellular membrane-located carriers (or porters) are saturable cata- metabolite concentration could not be maintained). lysts that exhibit similar kinetics it is convenient in this article to refer only to enzymes and to represent both (9) By the same arguments, each reaction rate (v) and the kinds of catalysts by the letter E. Any segment of a flux (J) through the system are also variables of the sequence of enzyme-catalysed reactions can then be writ- system. ten as shown in Figure 1. (10) The magnitude of each variable of the system is There are ten important features of any such system. determined at all times by the magnitudes of all the parameters of the system and of its immediate environ- (1) Each enzyme, E to E , is embedded within a meta- ment. The variables comprise the concentrations (s , s , s , 1 6 1 2 3 bolic pathway, i.e. within a system of enzymes. s ,s ) of the intracellular metabolites shown in Figure 1 4 5 and any other intracellular metabolites; the individual (2) All components of this system except the external reaction rates v , v , v , v , v , v ; and the flux J through this 1 2 3 4 5 6 metabolites X and X are enclosed by a membrane. system of enzyme-catalysed steps. 0 6 (3) E and E may then represent membrane-located It follows that, provided we maintain the concentrations 1 6 enzymes or translocators. of X and X constant, the system depicted (Figure 1) will, 0 6 in time, come to a steady state such that: (4)X and X interact with only one enzyme, whereas each 0 6 internal metabolite (S , S , S , S , S ) interacts with two v = v = v = v = v = v = J (the flux through this system). 1 2 3 4 5 1 2 3 4 5 6 flanking enzymes. At the same time the concentration of each intracellular (5) In a haploid cell there will be one specimen of an metabolite S to S will settle to an individual steady value. 1 5 enzyme molecule (E) at each catalytic locus. In a diploid cell there will be two specimens of enzyme molecules 2.3. The response of the system variables to a change in (two allozymes) at each catalytic locus: one specified by any one system parameter the maternal allele, the other by the paternal allele, at the In a metabolic system, the product of any one enzyme-cat- corresponding gene locus or loci. The effective catalytic alysed reaction is the substrate for the immediately activity at each metabolic locus in a diploid will be, in the adjacent downstream enzyme (Figure 1). If, for any rea- simplest case, the sum of the two individual activities. It is son, the concentration of the common intermediate the single effective enzyme activity (v) at each catalytic metabolite of two adjacent enzymes is changed (for Page 3 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 example by mutation of one of the two adjacent tration of that molecular component of a trait respond, enzymes), the concentration of the other adjacent enzyme when any one enzyme activity was changed by mutation will not change but its activity will change in accordance in a series of finite steps? with the known response of an enzyme activity (at con- stant enzyme concentration) to a change in the concentra- It was shown, by experiment, that graded changes in the tion of its substrate or product. In other words, no matter activity of any one of four different enzymes in the how complicated that system may be, the activity of any arginine pathway resulted in a non-linear (quasi-hyper- one enzyme depends, at all times, on the activity of the bolic) response of the flux to arginine in constructed het- adjacent enzyme; and this is true for every pair of adjacent erokaryons of Neurospora crassa ([3], Figures 1a,1b,1c,1d). enzymes throughout the system (up to the point in the Similar non-linear (non-additive) flux responses were system where a terminal product is formed). observed when a series of mutations occurred in a single enzyme in four other metabolic pathways in four different [This last statement is obviously still true for the system in diploid or polyploid systems ([3], Figures 1e,1f,1g,1h). Figure 1 if we omit the words in parentheses but only Similar flux responses were observed during genetic is a terminal product. because the extracellular product X down-modulation of any one of five enzymes involved in X is not an intermediate metabolite, flanked by two adja- tryptophan synthesis in Saccharomyces cerevisiae [5]. The cent enzymes; it is not a substrate that is further metabo- same quasi-hyperbolic response of a defined flux to a lised by the system depicted. There are instances where an series of graded changes in one enzyme activity was intracellular terminal product is formed. We must observed in a haploid cell [6]. We can therefore dismiss therefore add the words in parentheses if the statement is the possibility that these non-linear responses (of a flux- to apply generally]. to-a-trait-component) were restricted to the systems inves- tigated by Kacser and Burns [3] or were in some way A finite change (by mutation) in any one allele at a locus related to the ploidy of the cells and organisms they will change the activity (v) of one enzyme at the corre- studied. sponding metabolic locus; but, for reasons just stated in the first paragraph of this section 2.3, the activity (v) of On the contrary, the various flux responses are a funda- each of the other enzymes will alter, the flux (J) will mental biochemical property of the fluxing metabolic sys- change, and the concentrations of all the metabolites (S - tem. It does not matter how the graded changes in activity S ) will also change, some more than others, until the sys- of any one enzyme are brought about. Mutation is one tem settles to a new steady state. way but not the only one. Graded replacement of a defec- tive gene that expressed the chloride translocator in the Thus, finite changes in the magnitude of any one of the cystic fibrosis mouse produced continuously non-additive internal or external parameters of the system will shift the responses of various functions associated with chloride original values of all the variables of the system to a new set transport, including the duration of the survival of the of steady-state values. But, providing the external parame- mouse [7]. Induced synthesis of graded concentrations of ters X and X are kept constant, we can be sure that a a single membrane-located enzyme resulted in continu- 0 6 change in any one selected internal parameter (an allele or ously non-linear changes in growth rate, glucose oxida- an enzyme) would be the sole cause of any changes in the tion, the uptake and phosphorylation of α-methyl glucose system variables. In short, we are obliged to adopt a by Escherichia coli cells [8]. whole-system (a systemic) approach if we want to under- stand how the flux to a trait component responds to a Stepwise decreases in cytochrome c oxidase activity (by change in any one internal or external parameter of the titrating rat muscle mitochondria with an enzyme-specific system, no matter how that change in a parameter value is inhibitor) had little effect on respiration until the enzyme brought about. We are here concerned with changes in activity was decreased to about 25% of normal; further any one internal parameter such as a mutation in one or decreases in this one enzyme activity caused a precipitous, both alleles of a diploid cell. continuously non-linear, decrease in mitochondrial respi- ration [9]. Other examples of non-linear (non-additive) Suppose the activity of any one of the enzymes E to E in responses of a defined flux to a change in activity of one 1 6 Figure 1 were to be changed stepwise (e.g. by a series of enzyme in a metabolising system have been recorded mutations of one or both alleles at a locus in a diploid) so [10], [[11], Figures 6.2,6.3,6.4,6.6.6.7,6.8]. The results of that the residual activity of the enzyme was decreased in these various "genetic" and "biochemical" experiments successive steps to, say, 75%, 60%, 45%, 25%, 0% of its illustrate the generality of the statement by Kacser and initial activity. How would the flux (flow) through the Burns [3] quoted in section 2.1 of this article. whole series of enzymes vary; i.e. how would the flux (to a trait component) respond, and how would the concen- Page 4 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 Biochemistry and genet Figure 6 ics merged thirty years ago Biochemistry and genetics merged thirty years ago. The symbol indicates the catalysed translocation of an extracellu- lar substrate or substrates (X ) and the subsequent intracellular catalysed transformations, including scavenging pathways, that form nucleoside triphosphate (NTP) precursors for the transcription process. Similarly, indicates the catalysed translocation of the extracellular substrates (X ) and the subsequent synthesis from (X ), and other intracellular substrates, of 2 2 the amino acid (AA) precursors for the translation process. The enzymes subsumed as E and E are involved in the final Ts Tl stages of the expression (transcription and translation) of genes g1, g2, g3, g4 - - etc as polypeptides (P , P , P , P - - etc). In 1 2 3 4 diploid cells a pair of proteins will be synthesised from each pair of alleles at a gene locus. Those pairs of polypeptides (pro- teins) that are catalytically active in a diploid cell are represented by the single symbols E , E , E , E - - - etc in this Figure 6. 1 2 3 4 Further details are given in Section 5.5. 2.4. A rational explanation for the origin of dominant and response of a flux to mutations in an allele, it is equally recessive traits certain that naming alleles as dominant or recessive will How did the observations of non-linear responses of indi- not provide the explanation [1]. We need to focus atten- vidual fluxes to graded changes in any one enzyme activity tion on the universally observed non-linear (often quasi- lead to a rational explanation for the origin of Mendel's hyperbolic) responses of the flux-to-a-trait-component dominant and recessive trait classes [2]? For reasons (and the concomitant change in concentration of that already given, we cannot arrive at the answers to this ques- component) when the activity of any one enzyme, within tion by relying on the illogical and illegitimate idea that a metabolic system of enzymes, is changed (decreased or alleles are themselves dominant or recessive. Such entities increased), in stages, by any means available (including have never existed and do not now exist. Alleles can only down-modulation by mutation and up-modulation by be normal or abnormal (i.e. normal or mutant). If the increasing the gene dose). ploidy of the cell cannot explain the non-additive Page 5 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 In this Section 2.4, and in Sections 2.5–2.7, consideration The paper by Kacser and Burns [3] thus explains, for the of the role of allele pairs (uu,uU,UU) in determining the first time in 115 years, how recessive traits arise from a suf- outcome of mutations or changes in gene dose is set aside; ficient decrease, by mutation, in one enzyme activity this role will be considered in Section 2.8. For the when that enzyme is embedded in a metabolic system. moment, attention is focussed on what can be learned The explanation depends on recognising that when from the non-linear response of a flux – to the molecular graded changes occur by mutation (in one, both or all of component(s) of a trait – when the activity of one enzyme the allozymes at any one metabolic locus in biochemical in a metabolic system is changed in graded steps by muta- pathways) there will be a non-linear response of the flux tion or by changes in gene dose. Figures 1a,1b,1c,1d in to the molecular component(s) of a defined trait; and Reference [3] showed that the flux to the normal trait concurrently a non-linear response of the concentrations component (arginine), and thus the concentration of of the normal molecular components of a trait (section arginine, was not significantly diminished before any one 2.3). of four enzyme activities was decreased by more than 50%. In Figures 1b,1d the enzyme activity was decreased Section 2.9 in reference [1] showed that it was difficult to to about 15% of normal activity in Neurospora crassa understand how Mendel's recessive traits (a) were dis- before any significant diminution in the flux to arginine played in 1/4 of his F2 population of plants (A + 2Aa + a) (and in the concentration of arginine) was detectable [3]; when these same recessive traits were not displayed in any further diminution of either enzyme activity caused a Mendel's hybrids (Aa). We have replaced Mendel's continuous but precipitous fall in the production of this implausible idea that his F1 hybrids (Aa) displayed only trait component. Similar characteristics were displayed by trait (A). We have substituted the plausible idea – based a diploid (Figure 1h in Reference [3]). Figure 2 represents on experimental evidence [3] – that, under certain condi- these observations. Flux response plots with these charac- tions, the F1 hybrid trait (H) is indistinguishable from trait teristics are quasi-hyperbolic and asymmetric in the sense (A). In the treatment advocated here, there is no problem that, over low ranges of enzyme activity, the flux (and the in understanding how 1/4 of the individual plants in the metabolite concentrations in that fluxing pathway) F2 population of genetically related plants (A + 2H + a) respond markedly to small increases or decreases in displayed the recessive trait (a). We can now also see why enzyme activity; on the other hand, over high ranges of Mendel emphasised the need to study crosses between enzyme activity, substantial changes in activity have a parental plants that displayed readily distinguishable trait small, if any, effect on the flux to a trait component and forms, e.g. red flowers (A) in one parent and white flowers on the concentrations of the molecular components of a (a) in the other [1]. Figure 2 shows that this distinction defined trait. A change in any "Flux-to-trait-component" would be possible only if the activity of one enzyme in the implies a change in the concentrations of those metabolic dominant trait plant was sufficiently diminished in the products that typify a defined trait. recessive trait plant. It was shown that a dominant trait (A) corresponded to Note too that trait dominance and trait recessivity are not the normal (100%) activity of the enzyme that was subse- independent phenomena (nor are they opposite, one to quently mutated to give lower activities [3]; i.e., the plot- the other). We cannot define a dominant trait except as an ting co-ordinate (wild-type enzyme activity versus trait A) alternative to a recessive trait; both traits must be observ- defined the terminus of the asymptote of the flux response able before we can identify either of them. The statements plot depicted in Figure 2. A hybrid (H) must then corre- in these last two sentences were obvious in Mendel's spond to any point on the asymptote of Figure 2 that original paper [2] but they have been inexplicably over- would not allow us (and would not have allowed Men- looked by many later authors. del) to distinguish a F1 hybrid (H) from its parent that dis- played a dominant trait (A). A recessive (a) must then 2.5. Mendel's 3(dominant):1(recessive) trait ratio occurs correspond to any point on the steeply falling part of the sometimes, not always flux-response plot (Figure 2) that would allow us (or Does this explanation for the origin of dominant and would have allowed Mendel) to distinguish the dominant recessive traits also account for the occurrence of Mendel's trait (A) and the hybrid (H) from the recessive trait (a), 3(dominant):1(recessive) trait ratio? The answer is yes. e.g. dominant trait red flowers and hybrid red flowers Does it also explain why this ratio is not always observed? from the recessive trait white flowers [1]. Note especially The answer is again, yes (although the original authors [3] that a recessive trait would not necessarily correspond to did not pose or answer these two questions). zero flux (a complete metabolic block and a complete absence of the normal, downstream, metabolic products) If the flux response plot is sufficiently asymmetric in Figure 2. (approaches a hyperbolic plot, as in Figure 2), the concen- tration of molecular components of a defined trait will Page 6 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 ratio in Mendel's, or any other F2 population of cells or organisms, depends entirely on an experimentally observed, sufficiently asymmetric, response of the flux (to the molecular components of defined trait) when changes 100 occur in enzyme activity at any one metabolic locus in a B fluxing biochemical pathway (Figure 1). It does not depend on the naïve and illegitimate assumption that H alleles are either dominant or recessive (Sections 3.2, 3.3, 4 in Reference [1]). Figure 2 illustrates one of a family of regularly non-linear (non-additive) response plots which exhibit various degrees of asymmetry [3]. Is the flux response always suf- ficiently asymmetric for the 3:1 trait ratio to be observed? It is not. A flux response was observed in one particular (diploid) metabolic system (Reference [3], Figure 1f) that was still clearly non-linear (non-additive) but not as 0 50 100 asymmetric as that shown in Figure 2. As in Figure 2, so in Relative enzyme activity Figure 3, a recessive trait (b) can be clearly distinguished from the dominant trait (B) because the concentrations of ww wW WW the molecular components of this trait were sufficiently Allele constitution different when one enzyme activity in the metabolic sys- tem is decreased to a sufficient extent. The trait displayed by the hybrid (H) is now distinguishable (rather than indis- Mendel's 3(domina occur Figure 3 nt):1(recessive) trait ratio does not always tinguishable) from the dominant trait (B) expressed in a Mendel's 3(dominant):1(recessive) trait ratio does not always genetically related normal cell or organism when, as in occur. Mendel's notation for a dominant trait, a hybrid and a recessive trait were (B), (Bb) and (b) respectively. For rea- Figure 2, the enzyme activity is decreased to an arbitrarily sons given in the preceding paper [1], the hybrid is repre- chosen 50% of the normal activity. The 3(domi- sented in Figure 3 by (H). When graded changes are made in nant):1(recessive) trait ratio will not then be observed any one enzyme in a metabolic pathway the response of the (Figure 2). A blend of traits (B) and (b) is possible in the flux through that pathway is always non-linear (non-additive) hybrid (H), for example when traits (B) and (b) are distin- but not always quasi-hyperbolic (Figure 2). Consequently guished by colour differences. when the enzyme activity at one metabolic locus is decreased in the heterozygote to (say) 50% of wild-type, the trait dis- 2.6. Dominant and recessive traits are not always observed played by the hybrid (H) is now distinguishable from the trait It is well known that dominance and recessivity are not (B) displayed by the wild type cell or organism and from the universally observed. Are they therefore of no signifi- trait (b) displayed by the homozygously mutant cell or organ- cance? Some authors have been tempted to think so. Their ism. Mendel's 3(dominant):1(recessive trait ratio will not be observed. The explanation is consistent with the explanation view is understandable because, before the work of Kacser for the observation of the 3:1 trait ratio in Figure 2 and and Burns [3], we lacked any credible explanation for the achieves what the currently favoured explanation of Mendel's occurrence of dominant and recessive traits. observations cannot achieve [1]. Can we now see why dominance and recessivity are not always observed? The answer is again, yes. Examination of Figure 2 and Figure 3 shows that it will be possible to observe dominant and recessive traits in genetically not be measurably different (when the activity of one related organisms only when the enzyme activity at a met- enzyme is decreased by, say, 50%) from the concentra- abolic locus is decreased from 100% to an activity tions of those same molecular components when the approaching, but not necessarily reaching, 0% activity. enzyme activity was 100%. When the response plot is of the kind shown in Figure 2, If the trait displayed by the hybrid (H) is indistinguishable it would be possible to decrease the expressed enzyme from the trait (A), as in Figure 2, the trait distribution in activity at a metabolic locus by at least 75%, perhaps by the F2 population (A + 2H + a) becomes 3(A) + (a); i.e. the 85%, without eliciting any detectable change in trait from trait ratio in this population will be 3(dominant):1(reces- that displayed by the wild-type or normal organism. In sive). This explanation for the occurrence of the 3:1 trait other words some mutations will not, apparently, display Page 7 of 17 (page number not for citation purposes) Mendel's traits Flux to trait component Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 Mendelian dominance and recessivity (dominant and ative enzyme activities other than 0, 50, 100 could be recessive traits). observed in a polyploid or heterokaryon (Figure 1a,1b,1c,1d,1e in Reference [3]). To account for the Only if the effective enzyme activity is decreased by at occurrence in a diploid of relative enzyme activities in least 95% in this instance (Figure 2), would clear domi- addition to those taking values of 0, 50, 100 (in Figures 2 nance and recessivity be noted. This is an extreme case; and 3, and in Figures 1f,1g,1h of Reference [3]), we need Figure 3 illustrates the other extreme. Between these to allow for allele pairs in addition to the three (UU, Uu, extremes, various degrees of asymmetry of flux response uu or WW, Ww, ww) in which the mutant alleles (u or w) plots may be observed (Figure 1 in Reference [3]). Never- express a catalytically inactive polypeptide. theless, unless: (i) the change in enzyme activity is meas- ured, (ii) it is realised that there is a non-additive The restriction to just three allele pairs in a diploid may be relationship between a change in any one enzyme activity traced to Sutton [1]. He wrote Mendel's F2 trait series (A + at a metabolic locus and a change in expressed trait, and 2Aa + a), incorrectly, as (AA + 2Aa + aa) and the number (iii) the shape of the flux response plot (Figure 2, Figure of distinguishable chromosome pairs as (AA + 2Aa + aa), 3) is revealed by plotting, it is simply not possible to state so establishing a false one-for-one relationship between that the system under investigation does or does not dis- pairs of chromosomes (AA or aa) and dominant or reces- play Mendelian dominance and recessivity. Terms such as sive traits (AA or aa). Sutton's notation for chromosome semi-dominance merely indicate that the flux response pairs was later transferred to allele pairs. In this article, plot is not quite asymmetric enough to be sure that a 50% dominant and recessive traits are represented, as Mendel reduction in enzyme activity produces a trait that is indis- did, by (A) and (a) respectively; alleles have been repre- tinguishable from the dominant trait. sented by different letters (e.g. UU, Uu, uu) in order to dis- tinguish alleles (parameters) from traits (variables). We 2.7. Is the Kacser & Burns treatment universally should allow for the situation where (U ) is a mutant of applicable? (U) that would express an allozyme activity lower than The change in the concentrations a normal metabolites has that expressed from (U) but not so low as that expressed been treated in the present article as the source of a change from (u); and where (u*) would be a mutant of (U) that in trait. This accords with the treatment in Figure 1 of ref- expresses an allozyme activity greater than that expressed erence [3]. Allowance should, however, be made for the by (u) = 0 in the traditional treatment but not so great as possibility that the change in concentration of a metabo- to merit the notation (U). The outcome of different lite is, in reality, a change in the concentration of a hypothetical crosses that involve different mutations of "signalling" metabolite (e.g. an allosteric activator or one both alleles at a given locus in genetically related dip- inhibitor of another enzyme in the pathway that gener- loid parents would then be as follows: ated the "signalling" metabolite, or in another pathway). Such mechanisms merely shift the cause of the change in (1) Repeated crosses (Uu × Uu) would give, on average, metabolite concentration to another part of the matrix of the allele series (UU + 2Uu + uu) thus permitting expres- intracellular metabolic pathways. In other words, the Kac- sion of no more than three distinctive enzyme activities at ser and Burns approach remains a valid explanation for the corresponding metabolic locus. the origin of dominant and recessive traits. (2) The cross (Uu* × Uu) would give the allele series (UU 2.8. Accounting for all the plotting points in Figures 2 and 3 + Uu + Uu* + uu*) in which two of the allele pairs differ In Figure 2, the relative enzyme activities (100, 50, 0) from those in the progeny of the first cross; and in which would be expressed from the series of allele pairs UU, Uu, three different heterozygotes are formed. uu in a diploid cell (Section 1) only if the mutant allele (u) † † was expressed as a catalytically inactive polypeptide. The (3) The cross (U u × Uu) would give the allele series (UU same considerations apply to the relative enzyme activi- + Uu + U u + uu) in which only one allele pair in the prog- ties expressed from the allele pairs WW, Ww, ww in Figure eny populations is identical with one of the allele pairs in 3. the progeny from the second cross. It is obvious that the continuously non-linear response (4) The cross (UU × Uu) would give, on average, the allele † † plots (Figures 2, 3; and References [3-10]]) could not be series (UU + UU + Uu + U u) which has only two allele constructed if these three allele pairs were the only ones pairs in common with the progeny of the third of these available to express a corresponding series of enzyme crosses of genetically related parents. activities. Figure 1 in Reference [3] showed that more than three distinct enzyme activities were observed in experi- (5) The cross (U u × Uu*) would give, on average, the † † mental practice in any one system. It is easy to see how rel- allele series (UU + U u* + Uu + uu*). Page 8 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 In the second and fourth crosses it was assumed that the then be zero at a metabolic locus and a "metabolic block" two heterozygous parents did possess exactly the same will occur at that locus. normal allele (U) at this particular locus so, among their progeny, the allele pair (UU) occurred. Analogously, Assembling the data from, for example, the second and among the progeny from the third cross, the allele pair third of the three hypothetical crosses between the genet- (uu) occurred. But, importantly, in each of crosses (2), (3) ically related parents described above gives an allele series † † and (4) three different heterozygotes occurred in each (UU, UU , U u, Uu, Uu*, uu*, uu). They would contribute † † progeny population (a heterozygote is defined in a dip- seven different allozyme pairs (EE, EE , E e, Ee, Ee*, ee*, loid by the occurrence of allele pairs other than those rep- ee) at one metabolic locus and seven different, single, resented here by UU or uu). The allele pairs in the enzyme activities (v), one from each pair of allozymes. heterozygotes in any one progeny population of these Given a range of enzyme activities in excess of the tradi- crosses (2), (3) and (4) are not all identical with those in tional three, a sufficient number of co-ordinates will be the progeny of another of these crosses. The parents in the available to establish a continuously non-additive plot of fifth cross did not share an identical allele; no two alleles the response of one defined flux (J) against changes in of a pair are then identical in the progeny. The allele pair enzyme activity (v) at one metabolic locus in genetically (Uu) occurs in all of the progeny of these five crosses but related cells or organisms (Figures 2, 3). There is no guar- only because one of two parents carried this allele pair or antee that all of these mutants will be generated in every because one parent carried allele (U) and the other carried case but since (U ) and (u*) each represent only one of allele (u). several possible mutations of allele (U), we may be rea- sonably confident of observing traits expressed from allele Cross (1) typifies events in self-fertilising organisms but is pairs in addition to, or instead of, those expressed from not typical of sexual reproduction in other organisms (cf the two traditional mutant pairs (Uu) and (uu). Assem- Figure 2 in reference [1]). Male and female parents that are bling sets of enzyme activity and flux (or metabolite con- identically heterozygous at any locus must be rare. Crosses centration) data from the progeny of different but (2)-(5) between two heterozygous parents will produce, genetically related parents then creates the non-linear flux under the circumstances noted above, truly homozygous response plots illustrated in Figures 2 and 3. All plotting allele pairs (such as UU and uu) but they will also points in the idealised Figures 2 and 3 should be regarded produce, on average, three different heterozygotes among as tokens for the experimental plots published earlier [3]. their progeny (four heterozygotes in the fifth cross). This simple explanation for the occurrence of more than The consequences are then as follows: From each locus in three co-ordinates for a plot of flux response against a diploid cell that expresses catalytic polypeptides, alloz- changes in enzyme activity (or gene dose) means that it is ymes (pairs of enzymes) will be expressed; one from the no longer acceptable to base arguments and conclusions gamete donated by the male parent the other from the on the assumed presence of only one heterozgote (Uu) in gamete donated by the female parent. For simplicity, it a diploid allele series at a locus, and on only one corre- will be assumed here that the combined allozyme activity sponding hybrid trait. Furthermore, statements that all at each catalytic locus in the metabolic pathways of the heterozygotes express 50% (and only 50%) of the pheno- cell is the sum of the activities the two allozymes at each type expressed from the homozygous wild-type are based such locus. on the false idea that the mutant allele (u) always pro- duces a totally inactive enzyme. Figures The traditional allele series (UU + 2Uu + uu) in a diploid 1a,1b,1c,1d,1e,1f,1g,1h of Reference [3] depended upon will then generate the enzyme series (EE + 2Ee + ee) at one the availability of 5, 6 or 7 plotting points relating the flux metabolic locus in different, genetically related, individu- response to experimentally determined changes in als. This enzyme series provides two extreme combined enzyme activity (effectively to changes in allele constitu- allozyme activities, namely 100% (EE) and 0% (ee). There tion at a locus). In addition to the traditional heterozygote are no allele pairs at this locus that could provide <0% or (Uu), there must be a number of heterozygotes (e.g. UU , † † >100% enzyme activity. All other allele pairs, e.g. (UU ), U u, Uu*, uu*), and a corresponding a range of enzyme † † (U u), (U u*), (Uu*), (uu*), would provide combined activities (v), that account for the response of a flux (J) to allozyme activities that lie between the 100% and 0% val- a change in enzyme activity at one metabolic locus (Fig- ues just described. Only if (u) happens to be a null ures 1, 2, 3). In Figure 2, some of these additional hetero- mutant, will the heterozyote (Uu) express a single enzyme zygotes will establish the asymptote of the flux response activity (v) equal to 50% of the maximum available from plot. The trait expressed from any such heterozygote (UU). Only in this circumstance will the allele pair (uu) would be indistinguishable from the trait expressed from express two inactive polypeptides; the enzyme activity will the normal allele pair (UU); they could have accounted for the occurrence of Mendel's hybrids (Aa) which Page 9 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 appeared to display only the dominant trait (A). This is provide any justification for representing a trait by further evidence that the traditional treatment of elemen- twinned letters, e.g. (AA) or (aa). The single letters (A) and tary Mendelian genetics is inadequate and misleading [1]. (a) stood for qualitative differences in trait form in Men- del's work; they stand equally well for quantitative changes in a trait in modern work. The non-linear 3. Quantifiable differences between any two forms of a trait response plots of Kacser and Burns [3] apply to quantita- Differences in traits are generally and usefully described tive and to apparently qualitative changes in the pheno- by qualitative terms: type that arise from mutations of any one enzyme at a metabolic locus in a biochemical pathway. hirsute/bald; red flowers/white flowers; lithe/obese; mus- cular/"skinny"; slow/fleet; albino/black. Such descriptive 4. Implications of the systemic approach of terms do, however, disguise the obvious fact these appar- Kacser and Burns [3] ently qualitative differences in outward appearance are Figure 2 shows the response of the phenotype to changes based on quantitative differences in the concentrations of in enzyme activity at a metabolic locus or to changes in molecular products that contribute to the outward gene dose at the corresponding gene locus. It follows, if appearance or function of a cell or organism. the response plot takes this form, that increasing the dose of this particular gene in a wild-type haploid cell (or the These comments apply to the apparently qualitative dif- dose of the normal homozygous alleles in a wild-type dip- ferences examined by Mendel (Table 1 in reference [1]) loid or polyploid cell) is unlikely to produce a detectable and to those traits forms typified by a trait series (A + 2H change in the phenotype (e.g. an increase in the concen- + a) where (A) indicates the dominant trait form, (a) the tration of the trait component produced by a metabolic recessive trait form and (H) a hybrid trait that may be pathway; or a change in cell function associated with that indistinguishable (Figure 2) from the dominant traits (A) pathway). It was demonstrated that it was necessary, or distinguishable (Figure 3) from the dominant trait (B). under these circumstances, to increase concurrently the gene dose at each of no fewer than five loci if significant It should not therefore be supposed that the paper by Kac- increases in the flux (and in the concentration of meta- ser and Burns [3] provided an explanation only for the bolic product) was to be achieved [5]. The systemic occurrence of qualitative differences between any two approach to a rational explanation of the origins of dom- traits. On the contrary, a continuously variable response inant and recessive traits [3] has obvious implications for of each of several defined fluxes was brought about when biotechnologists. mutations of alleles at one locus changed the activity of one enzyme in a metabolic pathway (or when changes in Figure 2 (representing several plots in Reference [3]) also gene dose changed the concentration and thus the activity suggests that somatic recessive conditions (in contrast to of one enzyme in a metabolic pathway). so-called dominant conditions) could be ameliorated by partial gene replacement therapy. Experiments in the The flux responses were labelled "Flux to arginine", "Flux cystic fibrosis mouse model support this suggestion [7]; to biomass", "Flux to melanin", "Flux to products", "Flux they show that the systemic approach to the origins of to DNA repair" (Figure 1 in reference [3]). The molecular dominant and recessive traits has implications for medical compositions of "arginine", "biomass", "melanin", and genetics. "products" (of ethanol metabolism) were not changed. Their concentrations were changed as graded mutations at It was pointed out (section 2.6) that substantial decreases a gene locus caused graded changes in one enzyme activity in the dose of normal alleles at any one locus (or in the in those pathways that created arginine, biomass, mela- enzyme activity at the corresponding metabolic locus) nin, or the products (of ethanol metabolism). Similarly, a may not elicit detectable changes in the trait(s) of the cell. change in the "flux to DNA repair" was achieved by graded In other words, given a response plot approximating to increases in the dose of the gene specifying the synthesis that shown in Figure 2, traits – including associated cell of the "repair enzyme" that excises covalently-linked adja- functions – are inherently buffered against substantial cent thymines in DNA and allows incorporation of increases or decreases in the dose of any one gene, or thymidine in place of the excised pyrimidines. This against substantial changes in enzyme activity at the cor- "repair enzyme" activity is absent in Xeroderma pigmento- responding metabolic locus. This appears to be the prob- sum patients. able origin of the so-called "robustness" or buffering of chemotaxis against changes in enzyme kinetic constants Additional examples of quantitative changes in the con- [12-15]. centration of molecular components of a trait will be found in other publications [5-11]. None of these changes Page 10 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 genetic phenomena could perhaps also be explained. S S S S (J ) 1a 2a 3a 4a a Only two of the thirteen texts surveyed [1] gave a defini- tion, in their glossaries, of pleiotropy and epistasis. Both p q agreed that pleiotropy was a phenomenon where a change at one gene locus brought about a change in more than (J ) S S S S b 4b 3b 2b 1b one trait. Both attributed epistasis to an interaction A Figure 4 ccounting for the occurrence of pleiotropy between genes or their alleles. Neither of these descrip- Accounting for the occurrence of pleiotropy. One tions of pleiotropy and epistasis is particularly revealing. unbranched pathway is coupled to another by a conserved metabolite pair p and q. Such coupling is not uncommon in The following account, like those preceding it, does not cellular systems and is one source of pleiotropy. Mutation of depend on the fiction that all mutations generate inactive any one enzyme in one pathway will affect both fluxes (J and enzymes. Figure 1 is elaborated as shown in Figure 4. One J ) to a trait component and the concentrations of those trait pathway, like that shown in Figure 1, is now coupled to components. See also Figure 5. Figure 4, like Figure 1, illus- another analogous pathway by the conserved metabolite trates the need to adopt a systemic approach in attempts to pair (p, q). The sum of the concentrations of (p) and (q) is understand the responses of a metabolising system to changes in any enzyme activity brought about by mutation. constant (is conserved) but the ratio of the two concentra- tions (p/q) is a free variable. All the characteristics of the metabolic system in Figure 1 (Section 2), apply to each of the two fluxing pathways in Figure 4. Claims in the bio- chemical literature in the past that changes in the ratio (p/ This proposed explanation for metabolic buffering is q) controlled metabolic fluxes were and remain untena- quite general; it does not depend on the particular kinetic ble; one variable of a system cannot be said to control mechanisms that have been suggested to account for this another variable of the system. buffering [12]; it also suggests that there is no need to pos- tulate the presence of diagnostic "biological circuits" as Figure 1 may also be elaborated as shown in Figure 5. An the source of this buffering of the phenotype against input flux from X to S divides into two output fluxes 1 4 [16]. Of the input flux, a proportion (α) enters one of the mutations at a single locus. two output fluxes (J ) and a proportion (1-α) enters the Attempts to improve the concentration of metabolic prod- other output flux (J ). The magnitude of (α) is determined ucts by increasing the gene dose at one locus above that by the magnitudes of the activities of all the enzymes of available in the wild-type or normal cell could be success- the metabolic system; (α) is a systemic characteristic [17]. ful, at least to some self-limiting extent, if a response plot Again, all the characteristics of the model metabolic sys- like Figure 3 applies. Induced synthesis of one membrane- tem in Figure 1 (Section 2), apply to each of the two path- located enzyme activity to between 20% and 600% of ways that generate fluxes J and J shown in Figure 5. a b wild-type activity illustrates the possibility [8]. In this instance, plots like Figure 3 applied only to changes in the 5.2. The origin of pleiotropy explained uptake and phosphorylation of α-methyl glucoside; It will be obvious that a mutation of any one enzyme in changes in growth rates and glucose oxidation gave either of the two pathways of Figure 4 will cause changes response plots like Figure 2. The explanation for the differ- in the fluxes through both of the coupled pathways (and ence may lie in the suggestion [3] that shorter pathways the concentrations of metabolites in both pathways). Sim- will yield response plots like Figure 3, while the longer the ilarly, a mutation in any one enzyme of the input flux of pathway, the more likely is it that markedly asymmetric Figure 5 will affect the concentrations of metabolites in plots like Figure 2 will be observed. both output fluxes J and J . Pleiotropy (a change in more a b than one trait as a consequence of a single mutation), when it is detected, is thus seen to depend on mutating an 5. Expansions of the present treatment 5.1. Why mutating one enzyme in a metabolic pathway enzyme within a metabolic pathway, on the consequen- may alter more than one trait; and mutating more than tial changes in metabolite concentrations, and on the one enzyme may annul these changes in more than one structure and interdependence of biochemical pathways. trait Only if one of the enzymes in the input pathway shows If the explanation for the origin of dominant and recessive zero activity will both output fluxes (J and J ) cease (Fig- a b traits depends on realising that fluxing metabolic path- ure 5). ways generate the molecular components of all traits, and that mutating any one enzyme in these pathways alters 5.3. The origin of epistasis explained the flux and the concentrations of those normal metabolic Given a steady input flux from X to S (Figure 5), a muta- 1 4 products that are molecular components of a trait, other tion of one of the enzymes (E , E or any other enzyme 5a 6a Page 11 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 enzyme is mutated within a metabolic pathway. Whether pleiotropy or epistasis is detected, or not, will depend on 6a the severity of the mutation and on the nature of the flux S S (J ) 5a 6a a v E 5a 6a response plots (Figures 2, 3) as demonstrated in section 2. v v v E 2 3 4 5a X1 S2 S3 S4 E2 E3 E4 v5b E5b v6b 5.5. Biochemistry and genetics are not separable topics S S (J ) 5b 6b b Beadle and Tatum [18] isolated a series of mutants of Neu- 6b rospora crassa and tested their ability to grow on basal A Figure 5 ccounting for the occurrence of pleiotropy and epistasis medium or on basal medium supplemented with differ- Accounting for the occurrence of pleiotropy and epistasis. ent metabolites or cofactors. Wild-type Neurospora crassa Mutation of any one of enzymes E , E , E would affect both 2 3 4 grew on basal medium. Different isolated mutants would fluxes J and J to separate trait components. Mutation of any a b grow only if the basal medium was supplemented with one of enzymes E , E , etc would decrease flux J to a trait 5a 6a a the specific product of an enzyme rendered partially or component but increase J to another trait component; the fully inactive in one of the mutants. These brilliant concentrations of trait components in pathway J would observations led to the paradigm "one gene, one func- decrease, those in pathway J would increase. Epistasis would tion" [19,20], later to "one gene, one enzyme". These occur if a subsequent mutation occurred in any one of observations [18] made explicit what was implied by the enzymes E , E etc. A branched metabolic pathway is thus a 5b 6b potential source of pleiotropy and epistasis; see the text for observations of Garrod [21-24]] on inborn errors of further discussion. This diagram, like that in Figure 4, empha- metabolism namely: metabolism is catalysed by a sises the importance of adopting a systemic approach in sequence (or system) of different enzymes; and a suffi- understanding the potential effect, on a trait or traits, of a cient decrease (by mutation) in the activity of any one mutation in any one enzyme in enzyme-catalysed systems. enzyme may cause a change in the trait(s) or characteris- tic(s) of the system (e.g. the ability to grow, to accumulate cell mass [18]). Beadle [20] expressed surprise that Garrod's work had in this output limb) would decrease flux J and increase received so little attention. He wrote: "It is a fact both of flux J . The concentrations of metabolites in pathway J interest and historical importance that for many years b a would decrease and those in pathway J would increase, a Garrod's book had little influence on genetics. It was further example of a pleiotropic response to a single muta- widely known and cited by biochemists, and many genet- tion. But suppose that, following the mutation of E , a icists in the first two decades of the century knew of it and 5a mutation occurred in E or any other enzyme in this alter- the cases so beautifully described in it. Yet in the standard 6b native output limb. Clearly, the effect of the first mutation textbooks written in the twenties and thirties - - - - few on the cell characteristics would be at least partly nullified mention its cases or even give a reference to it. I have often by the second mutation – a phenomenon known as wondered why this was so. I suppose most geneticists epistasis and sometimes attributed in genetic texts to an were not yet inclined to think of hereditary traits in chem- interaction between genes but shown here to depend on ical terms. Certainly, biochemists with a few notable mutations of one or more enzymes, and on the structure exceptions such as the Onslows, Gortner and Haldane and interdependence of metabolic pathways. Only if the were not keenly aware of the intimate way in which genes activity of one of the enzymes in one of the two output direct the reactions of living systems that were the subject pathways is diminished to zero by mutation, will the of their science." products of that output limb downstream from the muta- tion be lost. This lack of attention to the implications of Garrod's work is all the more surprising when it is recalled that Bateson If the fluxes proceeded in the opposite direction to that [[25], p.133] pointed out that alkaptonuria (a change in shown in Figure 5 (so that two pathways merged into concentration of the normal metabolite, homogentisic one), mutation of an enzyme in one of the input fluxes acid, and one of Garrod's inborn errors of metabolism) followed by a mutation of an enzyme in the other input was an example of a Mendelian recessive trait or character; pathway could again elicit epistatic responses in the see also [[26], p.19]. In other words, some important system. aspects of genetics depended on recognising the role of changes in an enzyme activity, within a metabolic system, 5.4. Are pleiotropy and epistasis always detectable? in effecting a change in a trait. Particular but common metabolic structures (Figures 4, 5) provide the potential for pleiotropy and epistasis; i.e. The aphorism "one gene, one enzyme" was refined to changes in concentrations of normal metabolites when an "one allele, one polypeptide" after the elucidation of the Page 12 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 structure of DNA [27,28] and the rapid advances made in have other important functions (e.g. as hormones) and the next 10 or 15 years in elucidating the mechanisms of may be components of traits. expression of diploid alleles as pairs of polypeptides or proteins [29-32]] most of which are enzymes [3,4]. These X stands for all those initial extracellular substrates feed- more recent discoveries (Figure 6) emphasise what was ing the matrix of inter-dependent biochemical pathways implied by the work of Beadle and Tatum [18]: the that typify all functioning cells. It is these pathways that molecular components of dominant and recessive traits or generate the non-protein, non-polyribonucleotide, characteristics, in all biological forms, are generated by molecular products of all cell traits. fluxing metabolic pathways catalysed by sequences or sys- tems of enzymes. Dominant and recessive traits are not Each of these three major fluxing pathways (Figure 6) is the direct product of the expression of alleles as suggested catalysed by a succession of enzyme-catalysed reactions as by the currently favoured explanation of Mendel's obser- shown in Figure 1. The flux through any one of these path- vations (Figure 2 in Reference [1]); they are produced indi- ways will respond to a mutation of any one enzyme in the rectly by a system of enzymes (Figures 1, 4, 5, 6). pathway as shown in Figures 2, 3; any change in these fluxes could change the concentrations of the intermedi- Figure 6 depicts the direct relationship between any one ate metabolites or the final product (section 2.3); but, gene (g1, g2, g3, g4) and the synthesis of individual provided mutations do not alter the specificity of an polypeptides (P , P , P , P ) most of which, but not all, are enzyme, they will not change the existing molecular struc- 1 2 3 4 enzymes (E , E , E , E ). All polypeptides, catalytic and ture or composition of these metabolites. 1 2 3 4 non-catalytic, are synthesised in this way. Most attention is concentrated on the pathway initiated X , X and X in Figure 6 are immediately identified as by X for the simple reason that this pathway stands for all 1 2 3 1 extracellular parameters of a cell system. X stands for the matrix of interdependent biochemical fluxes that gen- those substrates that lead, through a series of enzyme-cat- erate such a wide range of the non-protein (and non- alysed reactions, to the synthesis of nucleoside polyribonucleotide) molecular components of cell traits triphosphates (NTPs) and their subsequent incorporation (e.g. skin pigments, membrane lipids, chlorophyll, xan- into mRNA. Note that mRNA is a terminal product of this thocyanins, non-peptide hormones, neural transmitters, pathway. It is a coding entity, a proxy for DNA. Each chitin, serum cholesterol, peptidoglycans, etc, etc). mRNA specifies the order of incorporation of individual amino acids into a polypeptide, but no individual mRNA If any one of the three major pathways shown in Figure 6 molecule participates as a substrate in the subsequent is coupled to another pathway (Figure 4) or contains a steps of the catalysed formation of a polypeptide. The branch (Figure 5) there will be, potentially detectable, control of the overall expression of a gene as a polypeptide pleiotropic and epistatic responses to mutations of any of is therefore necessarily treated in Metabolic Control Anal- the pathway enzymes (section 5.3). Such pathway ysis as a cascade of two fluxing metabolic pathways, one coupling and branching is a common feature of the path- that starts at X , the other that starts at X [33]. ways that start with one of the extracellular substrates typ- 3 2 ified by X . X stands for those extracellular substrates that lead, through a series of enzyme-catalysed reactions, to the syn- If the implications of the work of Beadle and Tatum [18] thesis de novo of amino acids (AAs) and their subsequent were not fully realised at the time, Figure 6 might have incorporation, along with any existing amino acids, into a suggested that a fresh approach to an understanding of the polypeptide (P). In a haploid cell, one polypeptide is syn- origins of dominant and recessive traits was needed. The thesised from each gene locus. In a diploid, one polypep- currently favoured explanation for Mendel's findings ([1], tide is synthesised from each of two alleles at a gene locus. Figure 2) does not take account of the biochemical path- If these pairs of polypeptides are catalytically active, each ways of the synthesis of enzymes (Figure 6) established enzyme in a diploid cell (E , E , E , etc) consists of a pair 30–40 years ago, does not acknowledge that the molecu- 1 2 3 of allozymes, one of each pair specified by the allele lar components of all traits are synthesised by systems of derived from the male parent, the other specified by allele enzymes, does not take account of the change in concen- derived from the female parent. Each pair of allozymes, tration of molecular components of traits when any one whether normal or mutated, exhibits only one measura- enzyme is mutated, and fails to distinguish the system ble activity (v) at a catalytic locus in a metabolic pathway. parameters (alleles) from the system variables (traits). If the pairs of polypeptides (P) synthesised by a diploid cell are not catalytically active they will not, of course, play Note that changes in the concentrations of external a direct role in catalysing a metabolic pathway. They may metabolites (whether they are substrates like X , X , X in 1 2 3 Figure 6, or extracellular inhibitors or activators of intrac- Page 13 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 ellular enzymes) may effect changes in intracellular (v) The argument that changes in concentration of a trait metabolism and consequently modify the effects of a component may nevertheless be revealed as a qualitative mutation. This topic is not immediately relevant in the change in that trait. present article but is a notable feature of Metabolic Con- trol Analysis. Descriptions of the role of the Combined (vi) A demonstration that both alleles (normal or mutant) Response Coefficient (R) in permitting extracellular at a locus in a diploid are generally expressed. If the nor- effectors to modulate intracellular metabolism (and thus mal allele expresses a catalytically active polypeptide, the effects of a mutation) will be found elsewhere [11,34- many mutants of this allele will express an enzyme with 36]. lower activity; a mutated enzyme with zero activity is an extreme case. If pleiotropic and epistatic responses to a mutation are as common as is suggested (sections 5.1–5.4), the question (vii) The demonstration that an explanation of Mendel's then arises: how do we account for Mendelian segregation observations cannot be based on an allele series contain- of traits during sexual reproduction? The answer lies in the ing only three terms (e.g. uu, 2uU, UU) one of which is a fact that a mutation at a biochemical locus, within the unique heterozygote (uU). matrix of interdependent pathways, has its most obvious effect on the most closely associated pathways. Distant (viii) A demonstration that dominant and recessive traits pathways (on the scale of cellular dimensions) will be less cannot be generated by those polypeptides that are not obviously affected. Kacser and Burns (Reference [3], enzymes embedded in a system of enzymes. p.649) pointed out that "This apparent independence of most characters makes simple Mendelian genetics possi- (ix) Rejection of the unjustified traditional claim that a ble, but conceals the fact that there is universal pleiotropy. hybrid (H) expresses a dominant trait (A) because the All characters should be viewed as 'quantitative' since, in (allegedly) recessive allele (u) in a heterozygote (Uu) is principle, variation anywhere in the genome affects every always completely ineffective or because the allegedly character." Section 3 in the present article emphasised the dominant allele (U) suppresses the allegedly recessive importance of quantitative changes in cell traits. The con- allele (u) in the heterozygote [1]. siderations in this paragraph are germane to the apparent absence of a detectable change of phenotype in some so- (x) Rejection of the traditional, unsubstantiated and called 'knock-out' experiments. implausible claim that one so-called dominant allele in a heterozygote is as effective as two such alleles in the wild- 6. Conditions that must be met to explain type cell [1]. dominance and recessivity The explanation advocated in this article for the origins of It was also shown that pleiotropy and epistasis can be dominant and recessive traits from normal and mutant explained by taking a similar system approach to that used alleles in a diploid is based on: in explaining the origin of dominant and recessive traits. (i) An obligatory distinction, by notation and nomencla- It is then apparent that, to account rationally for Mendel's ture, between the variables (traits) and the parameters observations of dominant and recessive traits, a minimum (alleles and enzymes) of genetic/biochemical systems. of four conditions must be met. (ii) The contention that the molecular components of all (i) Alleles must be distinguished by notation, nomencla- traits are the products of fluxing metabolic systems (Fig- ture and concept from traits; functions of components of ures 1, 4, 5, 6). the genotype must be distinguished from properties of components of the phenotype. Traits alone may be dom- (iii) Experimental evidence for an inevitable non-linear inant or recessive. response of a flux (through a metabolic system of enzymes) to graded changes in the activity of any one of (ii) Alleles cannot be called "dominant" or "recessive". those enzymes [3], evidence that is supported by a (When alleles are so called, the flaws present in the current number of independent observations [5-11]. attempts to explain Mendel's observations will inevitably re-appear [1]). (iv) A demonstration that dominant and recessive traits arise from changes in the concentration of the normal (iii) It must be shown how dominant traits become distin- molecular components of a defined trait. guishable from recessive traits in the same cell or organ- ism (Figure 2, 3). Page 14 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 (iv) It must be shown how a hybrid trait sometimes hybrid in Figure 2 is replaced mentally and temporarily by becomes indistinguishable from the dominant trait and (Aa), it will be clear why Mendel postulated that his sometimes does not (Figures 2, 3). The first circumstance hybrids (Aa) displayed trait (A) and not trait (a). If the will account for Mendel's 3(dominant):1(recessive) trait same exercise is repeated in Figure 3 by replacing (H) tem- ratio; the second for exceptions to this ratio. porarily by (Bb), it will be clear why Mendel observed an anomalous blending of flower colours in the hybrids If all four conditions are be met; the first two conditions when he crossed parental bean plants bearing different must first be met. The treatment given in sections 2–5 flower colours. meets each of these requirements. The treatment of elementary Mendelian genetics advo- 7. Conclusions cated here is based on the work of Kacser and Burns [3]. Kacser and Burns [3] provided the basis for a rational So far as the present author is aware, that paper has not explanation for the origin of dominant and recessive traits been described by any student textbook of "classical" or that arose from mutations of alleles at any one gene locus "molecular" genetics published in the intervening 23 in a diploid or polyploid cell (sections 2.3, 2.4). Inherent years. Orr [37] did not see the full significance of the Kac- in this explanation, as set out above (sections 2.5, 2.6), are ser and Burns paper [3]. Darden [[38], p. 72] declared that further explanations for the occurrence of the 3(domi- "(trying) to unravel the complex relations between nant):1(recessive) trait ratio in some situations in a dip- mutant alleles and enzymes (Kacser and Burns, 1981) - - - loid (Figure 2), for the absence of this trait ratio from is not a major research topic in genetics." other situations (Figure 3), for the absence of dominant and recessive traits in yet others and for the appearance of Several possible reasons for this failure to see the merits of a blend of parental traits in some heterozygotes. These five the Kacser and Burns paper [3] may be worth considera- demonstrations are internally consistent. In contrast to tion. They include: the currently favoured attempt to explain Mendel's results [1], no arbitrary assumptions are introduced (section 2.8) (1) Persistent misrepresentations of Mendel's paper, and † † to explain how heterozygous allele pairs (e.g. UU , U u, incorporation of these distortions into currently favoured Uu*, uu*) may produce a trait that is indistinguishable explanations of Mendel's observations [1]. from the trait expressed from the "homozygous" allele pairs (UU). (2) A failure to recognise the consequences of not distin- guishing between the function of the alleles and the prop- In other words, provided: erties of traits in attempting to explain Mendel's results. Normal and mutant alleles specify the kind (and order of (a) all current misrepresentations of Mendel's paper [1] incorporation) of amino acids into polypeptides (most are first discarded, but not all are enzymes). Dominance and recessivity are a reflection of changes in the concentration(s) of the molec- (b) alleles are distinguished by notation and nomencla- ular component(s) of a trait when an enzyme is mutated ture from the traits they specify, within a fluxing metabolic pathway. (c) alleles are regarded as normal or mutant (but not (3) Tardy recognition of the need to adopt the systemic dominant or recessive), it is possible to provide a rational approach of Metabolic Control Analysis in explaining the and internally consistent explanation for the origin of response of the variables of a biological system to pertur- Mendel's dominant and recessive traits, for the occurrence bations of the magnitude any one system parameter. of his 3:1 trait ratio, and for exceptions to these observa- tions noted by later investigators. The same systemic (4) A reluctance to accept a change in concepts even when approach is applicable to current problems in biotechnol- currently accepted representations of Mendel's results are ogy and medical genetics (section 4). It also explains the demonstrably untenable. origins of pleiotropy and epistasis (section 5); and chal- lenges the assumption that a mutation in a non-catalytic (5) Elucidation of the double helical structure of DNA protein provides an example of Mendel's dominant and (Figure 6) and all that followed in the next 10–15 years recessive traits [1]. imposed profound changes on genetics but was not per- haps always taken into account. Mendel found, by experiment, that the proportions of plant forms in each of his F2 populations was represented (6) A determination in some quarters to regard genetics as by (A + 2Aa + a). In the present paper these proportions an autonomous subject. It has been obvious at least since have been written as (A + 2H + a). If the symbol (H) for a the work of Beadle and Tatum [18] that such claims can- Page 15 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 11. Fell DA: Understanding the Control of Metabolism London and Miami: not be sustained. Genetics is intimately related to, and in Portland Press; 1997. some respects dependent upon, biochemistry. The 12. Barkai N, Leibler S: Robustness in simple biochemical converse is equally true. Genetics and biochemistry are networks. Nature 1997, 387:913-917. 13. Hartwell L: A robust view of biochemical pathways. Nature not separable topics in biology. 1997, 387:855-857. 14. Cohen P: Weathering the storm. New Scientist 2000 Online Confer- ence Reports 6:. It is significant that Kacser & Burns were also one of two 15. Ferber D: The spice of life. NewScientist Online Conference Report sets of authors who initiated the systemic approach to the control of metabolite concentrations and fluxes [39,40]. 16. Kacser H: The control of enzyme systems in vivo: elasticity analysis of the steady state. Biochem Soc Trans 1983, 11:35-40. This approach was elaborated by the original authors and 17. Fell DA, Sauro JM: Metabolic control analysis. Additional rela- many others. For some accounts and reviews, see tionships between elasticities and control coefficients. Eur J [11,36,41-44]. Biochem 1985, 148:555-561. 18. Beadle GW, Tatum EL: Genetic control of biochemical reac- tions in Neurospora. Proc Natl Acad Sci USA 1941, 27:499-506. 8. A correction 19. Beadle GW: Genetics and metabolism in Neurospora. Physiol Rev 1945, 25:643-663. In an earlier paper [45] it was stated that Mendel had 20. Beadle GW: Chemical genetics. In: Genetics in the 20th Century inferred the presence of segregating particles. These partic- Edited by: Dunn LC. The Macmillan Company, New York; 1951. ulate determinants were then represented by (A) and (a). 21. Garrod AE: About Alkaptonuria. Lancet 1901, ii:1484-1486. 22. Garrod AE: The incidence of Alkaptonuria. Lancet 1902, These statements are here formally withdrawn. They were ii:1616-1620. consistent with textbook treatments of Mendelian genet- 23. Garrod AE: Croonian Lectures to the Royal Society of Physi- cians: Inborn Errors of Metabolism. Lancet 1908, ii:1-7. 1432– ics [1] but a subsequent reading of Mendel's original 145; 173–179; 214–220 paper revealed that these statements, and others that 24. Garrod AE: Inborn Errors of Metabolism (a revision of the occur frequently in the recent reviews of Mendel's paper Croonian Lectures, 1909). . reprinted in reference [26] 25. Bateson W: Report of the Evolution Committee of the Royal Society 1902, and in current textbooks, were incorrect and misleading. 1(III):125-160. A history of the misunderstandings and misrepresenta- 26. Harris H: Garrod's Inborn Errors of Metabolism Oxford: University tions that have sustained the currently favoured depiction Press; 1963. 27. Watson JD, Crick FHC: Molecular structure of the nucleic of Mendelian genetics [1] will be presented elsewhere. A acids. A structure for deoxyribonucleic acid. Nature 1953, paper setting out the concepts of parameters and variables 171:737-738. 28. Watson JD, Crick FHC: Genetical implications of the structure will also be submitted. of deoxyribonucleic acid. Nature 1953, 171:964-967. 29. Crick FHC: On protein synthesis. Symp Soc Exp Biol 1958, Acknowledgements 12:138-163. 30. Crick FHC: The Genetic Code III. Sci Amer 1966, 215(4):55-62. I thank Dr Colin Pearson for his support during the preparation of this and 31. Taylor JH: Molecular Genetics II New York and London: Academic the preceding paper, Dr Denys Wheatley for temporary accommodation Press; 1967. during a logistic exercise and Dr Paul Agutter for valuable suggested mod- 32. Yanofsky C: Gene Structure and Protein Structure. Sci Amer ifications to the drafts of these two papers. 1967, 216(5):80-94. 33. Westerhoff HV, Koster JG, van Workum M, Rudd KE: On the con- trol of gene expression. In: Control of Metabolic Processes Edited by: References Cornish-Bowden A, Cárdenas ML. New York. Plenum Press; 1. Porteous JW: We still fail to account for Mendel's 1990:399-412. observations. Theor Biol Med Modelling 2004 in press. 34. Kacser H, Porteous JW: Control of Metabolism: What do we 2. Mendel G: Versuche über Pflanzen-Hybriden. Verhandlungen des have to measure? Trends Biochem Sci 1987, 12:5-14. Naturforschenden Vereines in Brunn 1866, 4(Abhandlungen):3-47. 35. Porteous JW: A theory that works. In: Control of Metabolic Processes 3. Kacser H, Burns JA: The Molecular Basis of Dominance. Genetics Edited by: Cornish-Bowden A, Cárdenas ML. New York: Plenum 1981, 97:639-666. Press; 1990:51-67. 4. Nirenberg MW: The Genetic Code II. Sci Amer 1963, 36. Cornish-Bowden A: Fundamentals of enzyme kinetics London: Portland 208(3):80-94. Press; 1995. 5. Niederberger P, Prasad R, Miozzari G, Kacser H: A strategy for 37. Orr HA: A test of Fisher's theory of dominance. Proc Natl Acad increasing an in vivo flux by genetic manipulations. The tryp- Sci USA 1991, 88:11413-11415. tophan system of yeast. Biochem J 1992, 287:473-479. 38. Darden L: Theory change in science Oxford and New York: Oxford 6. Dean AM, Dykhuisen DE, Hartl DL: Fitness as a function of β- University Press; 1991. galactosidase activity in Escherichia coli. Genet Res 1986, 48:1-8. 39. Heinrich R, Rapoport TA: Linear theory of enzymatic chains; its 7. Dorin JR, Farley R, Webb S, Smith SN, Farini E, Delaney SJ, Wain- application for the analysis of the cross-over theorem and wright BJ, Alton EWFW, Porteous DJ: A demonstration using the glycolysis of human erythrocytes. Acta Biol Med Germanica mouse models that successful gene therapy for cystic fibrosis 1973, 341:479-494. requires only partial gene correction. Gene Therapy 1996, 40. Kacser H, Burns JA: The control of flux. Symp Soc Exp Biol 1973, 3:797-801. 27:65-104. 8. Ruyter GJG, Postma PW, van Dam K: Control of glucose metab- 41. Fell DA: Metabolic Control Analysis: a survey of its theoreti- Glc olism by EnzymeII of the phosphoenolpyruvate-depend- cal and experimental development. Biochem J 1992, ent phosphotransferase system in Escherichia coli. J Bact 1991, 286:313-330. 173(19):6184-6191. 42. Kell DB, Westerhoff HV: Metabolic control theory; its role in 9. Letellier T, Heinrich R, Malgat M, Mazat J-P: The kinetic basis of microbiology and biotechnology. FEMS Microbiol Rev 1986, threshhold effects observed in mitochondrial diseases: a sys- 39:305-320. temic approach. Biochem J 1994, 302:171-174. 43. Teusink B, Baganz F, Westerhoff HV, Oliver SG: Metabolic Control 10. Small JR, Kacser H: Responses of metabolic systems to large Analysis as a tool in the elucidation of novel genes. Methods changes in enzyme activities and effectors. 1. The linear Microbiol 1998, 26:298-336. treatment of unbranched chains. Eur J Biochem 1993, 213:613-624. Page 16 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 44. Wildermuth MC: Metabolic control analysis: biological applica- tions and insights. Genome Biology 2000, 1(6):1031.1-1031.5. 45. Porteous JW: Dominance – One hundred and fifteen years after Mendel's paper. J Theor Biol 1996, 182:223-232. Publish with Bio Med Central and every scientist can read your work free of charge "BioMed Central will be the most significant development for disseminating the results of biomedical researc h in our lifetime." 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Publisher: Springer Journals
Copyright: Copyright © 2004 by Porteous; licensee BioMed Central Ltd.
Subject: Biomedicine; Biomedicine general; Systems Biology; Physiological, Cellular and Medical Topics; Statistics for Life Sciences, Medicine, Health Sciences
eISSN: 1742-4682
DOI: 10.1186/1742-4682-1-6
pmid: 15339331
Publisher site: See Article on Publisher Site

Abstract

Background: The key to a rational treatment of elementary Mendelian genetics, specifically to an understanding of the origin of dominant and recessive traits, lies in the facts that: (1) alleles of genes encode polypeptides; (2) most polypeptides are catalysts, i.e. enzymes or translocators; (3) the molecular components of all traits in all cells are the products of systems of enzymes, i.e. of fluxing metabolic pathways; (4) any flux to the molecular components of a trait responds non-linearly (non-additively) to graded mutations in the activity of any one of the enzymes at a catalytic locus in a metabolic system; (5) as the flux responds to graded changes in the activity of an enzyme, the concentrations of the molecular components of a trait also change. Conclusions: It is then possible to account rationally, and without misrepresenting Mendel, for: the origin of dominant and recessive traits; the occurrence of Mendel's 3(dominant):1(recessive) trait ratio; deviations from this ratio; the absence of dominant and recessive traits in some circumstances, the occurrence of a blending of traits in others; the frequent occurrence of pleiotropy and epistasis. 1. Background because Mendel defined the terms dominant and recessive The currently favoured explanation for the origin of Men- for traits or characters, it was illegitimate (and illogical) to del's dominant and recessive traits is untenable [1]. The call alleles dominant or recessive, and to represent them primary error in this current attempted explanation is the by the same letters used by Mendel to represent traits [1]. assumption that there is a direct, proportional, relation- ship in a diploid cell between a series of allegedly domi- (ii) A trait series written as (AA + 2Aa + aa) suggests, incor- nant and recessive alleles written as (AA + 2Aa + aa) and rectly, that dominant and recessive traits comprise two the dominant, hybrid and recessive traits written as (AA + aliquots, (A + A) or (a + a), of dominance or recessivity. 2Aa + aa). This assumption (Figure 2, in reference [1]) incorporates four fundamental faults: (iii) A failure to take account of the long established fact that the first non-nucleotide product of the expression of (i) A failure to distinguish between the parameters and the an allele is a polypeptide and that most polypeptides are variables of any system of interacting components, specif- enzymes or membrane-located translocators. ically between the determinants (alleles in modern termi- nology) and what is determined (the form of the trait or (iv) A failure to note that the components of all tangible characteristic expressed in a cell or organism). Thus, traits comprised the molecular products of metabolic Page 1 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 corresponding trait series will appear as (B + 2H + b). Mendel's notation (Aa) for a hybrid trait will be used in this article only when referring directly to Mendel's paper A, H [2]. 2. A rational explanation of Mendel's observations Our stated task was to explain logically how an allele series (UU + 2Uu + uu) is expressed as a series of qualita- tively distinguishable F2 traits (A + 2H + a) when F1 hybrids (H) are allowed to self-fertilise [1]. This is very simply achieved by correcting faults (iii) and (iv) in four 0 successive steps (sections 2.1–2.4) based on a paper pub- lished 23 years ago [3]. A fifth step (section 2.5) allows us 0 50 100 to go beyond that paper to explain how the trait ratio Relative enzyme activity 3(dominant):1(recessive) sometimes occurs and some- times does not. A sixth step (section 2.6), consistent with uu uU UU the earlier ones, explains why dominance and recessivity Allele constitution are not always observed. Section 2.7 validates an earlier section. Section 2.8 accounts for some aspects not dealt with in textbooks and reviews of genetics. A n Figure 2 accoun nt):1(r ting ece for Mendel's ssive) trait raob tio in his F2 p servation ofo a 3( pulations of plants domi- Accounting for Mendel's observation of a 3(domi- The treatment in this section 2 is extended in section 3 to nant):1(recessive) trait ratio in his F2 populations of plants. account for quantitatively different traits, in section 4 to Mendel's notations for a dominant trait, a hybrid and a reces- sive trait were (A), (Aa) and (a) respectively. For reasons illustrate some implications of the present treatment, and given in the preceding paper [1], a hybrid trait is represented in section 5 to account for pleiotropy and epistasis. Sec- in Figure 2 by (H). The molecular components of all traits are tion 6 defines the conditions that must be met if a rational synthesised by a metabolic pathway. When the activity of any account is to be given for the occurrence of dominant and one enzyme in a metabolic pathway is changed in discrete recessive traits. steps, the flux to a trait component responds in non-linear (non-additive) fashion [3]. If the flux response is quasi-hyper- 2.1. A generalised metabolic system bolic, as shown here, the hybrid trait (H) will be indistinguish- If: the first non-nucleotide product of expression of an able from the trait (A) expressed in the wild-type cell or allele is a polypeptide and most polypeptides are enzymes organism, even when the enzyme activity in the hybrid (H) [3,4], it follows that most mutations at any one gene locus has been reduced to 50% of the wild-type activity. Trait (a), will be distinguishable from both traits (A) and (H) only if the will result in the formation of a mutant enzyme at a cata- enzyme activity is further reduced to a sufficient extent. lytic locus in a metabolic pathway. This is true even if the Under these circumstances the trait series (A + 2H + a) functioning enzyme is composed of more than one becomes (3A + a); Mendel's 3(dominant):1(recessive) trait polypeptide, each specified by different genes. It then fol- ratio is accounted for without introducing arbitrary and lows that we need to ask how the concentration of a nor- inconsistent arguments [1]. mal molecular component of a trait will be affected by a mutation of any one enzyme within a metabolic system. In short, a systemic approach, outlined below, is obligatory. pathways, i.e., the products of sequences of enzyme-cata- This is the key to an understanding of the origin of domi- lysed reactions. nant and recessive traits, as first pointed out in the follow- ing two sentences: "When as geneticists, we consider Correction of the first two of these four faults has already substitutions of alleles at a locus, as biochemists, we con- been achieved (section 4 in reference [1]) by writing an sider alterations in catalytic parameters at an enzyme step. allele series as (UU + 2Uu + uu) and the corresponding - -. The effect on the phenotype of altering the genetic trait series as (A + 2H + a). In these statements (U) and (u) specification of a single enzyme - - - is unpredictable from are normal and mutant (not dominant and recessive) alle- a knowledge of events at that step alone and must involve les respectively. Mendel's notation (A) and (a) is used to the response of the system to alterations of single enzymes represent dominant and recessive traits but (H) replaces when they are embedded in the matrix of all other enzymes." Mendel's implausible notation (Aa) for a hybrid class of ([3]; p.641). trait [1]. Mutations at another gene locus, in the same or a different cell, will be written as (WW + Ww + ww); the Page 2 of 17 (page number not for citation purposes) Mendel's traits Flux to trait component Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 locus that concerns us here, irrespective of whether the E E E E E E 1 2 3 4 5 6 X S S S S S X (J) 0 1 2 3 4 5 6 cell is haploid, diploid or polyploid. v v v v v v 1 2 3 4 5 6 (6) The catalytic activity (v) at any one metabolic locus A Figure 1 segment of a model metabolic pathway can be left at its current value or changed to and main- A segment of a model metabolic pathway. This diagram tained at a new value by the experimentalist, e.g. by suita- shows those features, discussed in the text, that permit a sys- ble genetic manipulation of an allele. Each allele in these temic analysis of the response of any variable of a metabolic circumstances is therefore an internal parameter of the sys- system (e.g. a flux J or the concentration of any intracellular metabolite S) to changes in any one parameter of the system tem; it is accessible to modification by the direct and sole (e.g. an enzyme activity). Each S is an intracellular metabolite; intervention of the experimentalist [1]. each X is an extracellular metabolite. In a diploid cell, every E stands for a pair of enzymes (allozymes), each specified by (7) Because X and X are external to the system in Figure 0 6 one of the two alleles at a gene locus. Each E is then a locus 1, their concentrations can be fixed, and maintained at a of catalytic activity within a system of enzymes; each v stands chosen value, by the direct intervention of the experimen- for the individual reaction rates catalysed jointly by a pair of talist; they are external parameters of the metabolic system. allozymes in a diploid cell. Either or both allozymes at such a locus may be mutated. (8) In contrast to X and X , the concentrations of metab- 0 6 olites S to S within the system cannot be fixed and main- 1 5 tained at any desired value solely by the direct 2.2 Metabolic systems and steady states intervention of the experimentalist. The concentrations of Metabolic processes are facilitated by a succession of cata- S to S are internal variables of the system. (If a fixed 1 5 lysed steps; i.e. by enzyme-catalysed transformations of amount of any one of these metabolites were to be substrates to products or by carrier-catalysed translocation injected through the membrane into the system, contin- of metabolites across membranes. Because enzymes and ued metabolism would ensure that the new intracellular membrane-located carriers (or porters) are saturable cata- metabolite concentration could not be maintained). lysts that exhibit similar kinetics it is convenient in this article to refer only to enzymes and to represent both (9) By the same arguments, each reaction rate (v) and the kinds of catalysts by the letter E. Any segment of a flux (J) through the system are also variables of the sequence of enzyme-catalysed reactions can then be writ- system. ten as shown in Figure 1. (10) The magnitude of each variable of the system is There are ten important features of any such system. determined at all times by the magnitudes of all the parameters of the system and of its immediate environ- (1) Each enzyme, E to E , is embedded within a meta- ment. The variables comprise the concentrations (s , s , s , 1 6 1 2 3 bolic pathway, i.e. within a system of enzymes. s ,s ) of the intracellular metabolites shown in Figure 1 4 5 and any other intracellular metabolites; the individual (2) All components of this system except the external reaction rates v , v , v , v , v , v ; and the flux J through this 1 2 3 4 5 6 metabolites X and X are enclosed by a membrane. system of enzyme-catalysed steps. 0 6 (3) E and E may then represent membrane-located It follows that, provided we maintain the concentrations 1 6 enzymes or translocators. of X and X constant, the system depicted (Figure 1) will, 0 6 in time, come to a steady state such that: (4)X and X interact with only one enzyme, whereas each 0 6 internal metabolite (S , S , S , S , S ) interacts with two v = v = v = v = v = v = J (the flux through this system). 1 2 3 4 5 1 2 3 4 5 6 flanking enzymes. At the same time the concentration of each intracellular (5) In a haploid cell there will be one specimen of an metabolite S to S will settle to an individual steady value. 1 5 enzyme molecule (E) at each catalytic locus. In a diploid cell there will be two specimens of enzyme molecules 2.3. The response of the system variables to a change in (two allozymes) at each catalytic locus: one specified by any one system parameter the maternal allele, the other by the paternal allele, at the In a metabolic system, the product of any one enzyme-cat- corresponding gene locus or loci. The effective catalytic alysed reaction is the substrate for the immediately activity at each metabolic locus in a diploid will be, in the adjacent downstream enzyme (Figure 1). If, for any rea- simplest case, the sum of the two individual activities. It is son, the concentration of the common intermediate the single effective enzyme activity (v) at each catalytic metabolite of two adjacent enzymes is changed (for Page 3 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 example by mutation of one of the two adjacent tration of that molecular component of a trait respond, enzymes), the concentration of the other adjacent enzyme when any one enzyme activity was changed by mutation will not change but its activity will change in accordance in a series of finite steps? with the known response of an enzyme activity (at con- stant enzyme concentration) to a change in the concentra- It was shown, by experiment, that graded changes in the tion of its substrate or product. In other words, no matter activity of any one of four different enzymes in the how complicated that system may be, the activity of any arginine pathway resulted in a non-linear (quasi-hyper- one enzyme depends, at all times, on the activity of the bolic) response of the flux to arginine in constructed het- adjacent enzyme; and this is true for every pair of adjacent erokaryons of Neurospora crassa ([3], Figures 1a,1b,1c,1d). enzymes throughout the system (up to the point in the Similar non-linear (non-additive) flux responses were system where a terminal product is formed). observed when a series of mutations occurred in a single enzyme in four other metabolic pathways in four different [This last statement is obviously still true for the system in diploid or polyploid systems ([3], Figures 1e,1f,1g,1h). Figure 1 if we omit the words in parentheses but only Similar flux responses were observed during genetic is a terminal product. because the extracellular product X down-modulation of any one of five enzymes involved in X is not an intermediate metabolite, flanked by two adja- tryptophan synthesis in Saccharomyces cerevisiae [5]. The cent enzymes; it is not a substrate that is further metabo- same quasi-hyperbolic response of a defined flux to a lised by the system depicted. There are instances where an series of graded changes in one enzyme activity was intracellular terminal product is formed. We must observed in a haploid cell [6]. We can therefore dismiss therefore add the words in parentheses if the statement is the possibility that these non-linear responses (of a flux- to apply generally]. to-a-trait-component) were restricted to the systems inves- tigated by Kacser and Burns [3] or were in some way A finite change (by mutation) in any one allele at a locus related to the ploidy of the cells and organisms they will change the activity (v) of one enzyme at the corre- studied. sponding metabolic locus; but, for reasons just stated in the first paragraph of this section 2.3, the activity (v) of On the contrary, the various flux responses are a funda- each of the other enzymes will alter, the flux (J) will mental biochemical property of the fluxing metabolic sys- change, and the concentrations of all the metabolites (S - tem. It does not matter how the graded changes in activity S ) will also change, some more than others, until the sys- of any one enzyme are brought about. Mutation is one tem settles to a new steady state. way but not the only one. Graded replacement of a defec- tive gene that expressed the chloride translocator in the Thus, finite changes in the magnitude of any one of the cystic fibrosis mouse produced continuously non-additive internal or external parameters of the system will shift the responses of various functions associated with chloride original values of all the variables of the system to a new set transport, including the duration of the survival of the of steady-state values. But, providing the external parame- mouse [7]. Induced synthesis of graded concentrations of ters X and X are kept constant, we can be sure that a a single membrane-located enzyme resulted in continu- 0 6 change in any one selected internal parameter (an allele or ously non-linear changes in growth rate, glucose oxida- an enzyme) would be the sole cause of any changes in the tion, the uptake and phosphorylation of α-methyl glucose system variables. In short, we are obliged to adopt a by Escherichia coli cells [8]. whole-system (a systemic) approach if we want to under- stand how the flux to a trait component responds to a Stepwise decreases in cytochrome c oxidase activity (by change in any one internal or external parameter of the titrating rat muscle mitochondria with an enzyme-specific system, no matter how that change in a parameter value is inhibitor) had little effect on respiration until the enzyme brought about. We are here concerned with changes in activity was decreased to about 25% of normal; further any one internal parameter such as a mutation in one or decreases in this one enzyme activity caused a precipitous, both alleles of a diploid cell. continuously non-linear, decrease in mitochondrial respi- ration [9]. Other examples of non-linear (non-additive) Suppose the activity of any one of the enzymes E to E in responses of a defined flux to a change in activity of one 1 6 Figure 1 were to be changed stepwise (e.g. by a series of enzyme in a metabolising system have been recorded mutations of one or both alleles at a locus in a diploid) so [10], [[11], Figures 6.2,6.3,6.4,6.6.6.7,6.8]. The results of that the residual activity of the enzyme was decreased in these various "genetic" and "biochemical" experiments successive steps to, say, 75%, 60%, 45%, 25%, 0% of its illustrate the generality of the statement by Kacser and initial activity. How would the flux (flow) through the Burns [3] quoted in section 2.1 of this article. whole series of enzymes vary; i.e. how would the flux (to a trait component) respond, and how would the concen- Page 4 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 Biochemistry and genet Figure 6 ics merged thirty years ago Biochemistry and genetics merged thirty years ago. The symbol indicates the catalysed translocation of an extracellu- lar substrate or substrates (X ) and the subsequent intracellular catalysed transformations, including scavenging pathways, that form nucleoside triphosphate (NTP) precursors for the transcription process. Similarly, indicates the catalysed translocation of the extracellular substrates (X ) and the subsequent synthesis from (X ), and other intracellular substrates, of 2 2 the amino acid (AA) precursors for the translation process. The enzymes subsumed as E and E are involved in the final Ts Tl stages of the expression (transcription and translation) of genes g1, g2, g3, g4 - - etc as polypeptides (P , P , P , P - - etc). In 1 2 3 4 diploid cells a pair of proteins will be synthesised from each pair of alleles at a gene locus. Those pairs of polypeptides (pro- teins) that are catalytically active in a diploid cell are represented by the single symbols E , E , E , E - - - etc in this Figure 6. 1 2 3 4 Further details are given in Section 5.5. 2.4. A rational explanation for the origin of dominant and response of a flux to mutations in an allele, it is equally recessive traits certain that naming alleles as dominant or recessive will How did the observations of non-linear responses of indi- not provide the explanation [1]. We need to focus atten- vidual fluxes to graded changes in any one enzyme activity tion on the universally observed non-linear (often quasi- lead to a rational explanation for the origin of Mendel's hyperbolic) responses of the flux-to-a-trait-component dominant and recessive trait classes [2]? For reasons (and the concomitant change in concentration of that already given, we cannot arrive at the answers to this ques- component) when the activity of any one enzyme, within tion by relying on the illogical and illegitimate idea that a metabolic system of enzymes, is changed (decreased or alleles are themselves dominant or recessive. Such entities increased), in stages, by any means available (including have never existed and do not now exist. Alleles can only down-modulation by mutation and up-modulation by be normal or abnormal (i.e. normal or mutant). If the increasing the gene dose). ploidy of the cell cannot explain the non-additive Page 5 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 In this Section 2.4, and in Sections 2.5–2.7, consideration The paper by Kacser and Burns [3] thus explains, for the of the role of allele pairs (uu,uU,UU) in determining the first time in 115 years, how recessive traits arise from a suf- outcome of mutations or changes in gene dose is set aside; ficient decrease, by mutation, in one enzyme activity this role will be considered in Section 2.8. For the when that enzyme is embedded in a metabolic system. moment, attention is focussed on what can be learned The explanation depends on recognising that when from the non-linear response of a flux – to the molecular graded changes occur by mutation (in one, both or all of component(s) of a trait – when the activity of one enzyme the allozymes at any one metabolic locus in biochemical in a metabolic system is changed in graded steps by muta- pathways) there will be a non-linear response of the flux tion or by changes in gene dose. Figures 1a,1b,1c,1d in to the molecular component(s) of a defined trait; and Reference [3] showed that the flux to the normal trait concurrently a non-linear response of the concentrations component (arginine), and thus the concentration of of the normal molecular components of a trait (section arginine, was not significantly diminished before any one 2.3). of four enzyme activities was decreased by more than 50%. In Figures 1b,1d the enzyme activity was decreased Section 2.9 in reference [1] showed that it was difficult to to about 15% of normal activity in Neurospora crassa understand how Mendel's recessive traits (a) were dis- before any significant diminution in the flux to arginine played in 1/4 of his F2 population of plants (A + 2Aa + a) (and in the concentration of arginine) was detectable [3]; when these same recessive traits were not displayed in any further diminution of either enzyme activity caused a Mendel's hybrids (Aa). We have replaced Mendel's continuous but precipitous fall in the production of this implausible idea that his F1 hybrids (Aa) displayed only trait component. Similar characteristics were displayed by trait (A). We have substituted the plausible idea – based a diploid (Figure 1h in Reference [3]). Figure 2 represents on experimental evidence [3] – that, under certain condi- these observations. Flux response plots with these charac- tions, the F1 hybrid trait (H) is indistinguishable from trait teristics are quasi-hyperbolic and asymmetric in the sense (A). In the treatment advocated here, there is no problem that, over low ranges of enzyme activity, the flux (and the in understanding how 1/4 of the individual plants in the metabolite concentrations in that fluxing pathway) F2 population of genetically related plants (A + 2H + a) respond markedly to small increases or decreases in displayed the recessive trait (a). We can now also see why enzyme activity; on the other hand, over high ranges of Mendel emphasised the need to study crosses between enzyme activity, substantial changes in activity have a parental plants that displayed readily distinguishable trait small, if any, effect on the flux to a trait component and forms, e.g. red flowers (A) in one parent and white flowers on the concentrations of the molecular components of a (a) in the other [1]. Figure 2 shows that this distinction defined trait. A change in any "Flux-to-trait-component" would be possible only if the activity of one enzyme in the implies a change in the concentrations of those metabolic dominant trait plant was sufficiently diminished in the products that typify a defined trait. recessive trait plant. It was shown that a dominant trait (A) corresponded to Note too that trait dominance and trait recessivity are not the normal (100%) activity of the enzyme that was subse- independent phenomena (nor are they opposite, one to quently mutated to give lower activities [3]; i.e., the plot- the other). We cannot define a dominant trait except as an ting co-ordinate (wild-type enzyme activity versus trait A) alternative to a recessive trait; both traits must be observ- defined the terminus of the asymptote of the flux response able before we can identify either of them. The statements plot depicted in Figure 2. A hybrid (H) must then corre- in these last two sentences were obvious in Mendel's spond to any point on the asymptote of Figure 2 that original paper [2] but they have been inexplicably over- would not allow us (and would not have allowed Men- looked by many later authors. del) to distinguish a F1 hybrid (H) from its parent that dis- played a dominant trait (A). A recessive (a) must then 2.5. Mendel's 3(dominant):1(recessive) trait ratio occurs correspond to any point on the steeply falling part of the sometimes, not always flux-response plot (Figure 2) that would allow us (or Does this explanation for the origin of dominant and would have allowed Mendel) to distinguish the dominant recessive traits also account for the occurrence of Mendel's trait (A) and the hybrid (H) from the recessive trait (a), 3(dominant):1(recessive) trait ratio? The answer is yes. e.g. dominant trait red flowers and hybrid red flowers Does it also explain why this ratio is not always observed? from the recessive trait white flowers [1]. Note especially The answer is again, yes (although the original authors [3] that a recessive trait would not necessarily correspond to did not pose or answer these two questions). zero flux (a complete metabolic block and a complete absence of the normal, downstream, metabolic products) If the flux response plot is sufficiently asymmetric in Figure 2. (approaches a hyperbolic plot, as in Figure 2), the concen- tration of molecular components of a defined trait will Page 6 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 ratio in Mendel's, or any other F2 population of cells or organisms, depends entirely on an experimentally observed, sufficiently asymmetric, response of the flux (to the molecular components of defined trait) when changes 100 occur in enzyme activity at any one metabolic locus in a B fluxing biochemical pathway (Figure 1). It does not depend on the naïve and illegitimate assumption that H alleles are either dominant or recessive (Sections 3.2, 3.3, 4 in Reference [1]). Figure 2 illustrates one of a family of regularly non-linear (non-additive) response plots which exhibit various degrees of asymmetry [3]. Is the flux response always suf- ficiently asymmetric for the 3:1 trait ratio to be observed? It is not. A flux response was observed in one particular (diploid) metabolic system (Reference [3], Figure 1f) that was still clearly non-linear (non-additive) but not as 0 50 100 asymmetric as that shown in Figure 2. As in Figure 2, so in Relative enzyme activity Figure 3, a recessive trait (b) can be clearly distinguished from the dominant trait (B) because the concentrations of ww wW WW the molecular components of this trait were sufficiently Allele constitution different when one enzyme activity in the metabolic sys- tem is decreased to a sufficient extent. The trait displayed by the hybrid (H) is now distinguishable (rather than indis- Mendel's 3(domina occur Figure 3 nt):1(recessive) trait ratio does not always tinguishable) from the dominant trait (B) expressed in a Mendel's 3(dominant):1(recessive) trait ratio does not always genetically related normal cell or organism when, as in occur. Mendel's notation for a dominant trait, a hybrid and a recessive trait were (B), (Bb) and (b) respectively. For rea- Figure 2, the enzyme activity is decreased to an arbitrarily sons given in the preceding paper [1], the hybrid is repre- chosen 50% of the normal activity. The 3(domi- sented in Figure 3 by (H). When graded changes are made in nant):1(recessive) trait ratio will not then be observed any one enzyme in a metabolic pathway the response of the (Figure 2). A blend of traits (B) and (b) is possible in the flux through that pathway is always non-linear (non-additive) hybrid (H), for example when traits (B) and (b) are distin- but not always quasi-hyperbolic (Figure 2). Consequently guished by colour differences. when the enzyme activity at one metabolic locus is decreased in the heterozygote to (say) 50% of wild-type, the trait dis- 2.6. Dominant and recessive traits are not always observed played by the hybrid (H) is now distinguishable from the trait It is well known that dominance and recessivity are not (B) displayed by the wild type cell or organism and from the universally observed. Are they therefore of no signifi- trait (b) displayed by the homozygously mutant cell or organ- cance? Some authors have been tempted to think so. Their ism. Mendel's 3(dominant):1(recessive trait ratio will not be observed. The explanation is consistent with the explanation view is understandable because, before the work of Kacser for the observation of the 3:1 trait ratio in Figure 2 and and Burns [3], we lacked any credible explanation for the achieves what the currently favoured explanation of Mendel's occurrence of dominant and recessive traits. observations cannot achieve [1]. Can we now see why dominance and recessivity are not always observed? The answer is again, yes. Examination of Figure 2 and Figure 3 shows that it will be possible to observe dominant and recessive traits in genetically not be measurably different (when the activity of one related organisms only when the enzyme activity at a met- enzyme is decreased by, say, 50%) from the concentra- abolic locus is decreased from 100% to an activity tions of those same molecular components when the approaching, but not necessarily reaching, 0% activity. enzyme activity was 100%. When the response plot is of the kind shown in Figure 2, If the trait displayed by the hybrid (H) is indistinguishable it would be possible to decrease the expressed enzyme from the trait (A), as in Figure 2, the trait distribution in activity at a metabolic locus by at least 75%, perhaps by the F2 population (A + 2H + a) becomes 3(A) + (a); i.e. the 85%, without eliciting any detectable change in trait from trait ratio in this population will be 3(dominant):1(reces- that displayed by the wild-type or normal organism. In sive). This explanation for the occurrence of the 3:1 trait other words some mutations will not, apparently, display Page 7 of 17 (page number not for citation purposes) Mendel's traits Flux to trait component Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 Mendelian dominance and recessivity (dominant and ative enzyme activities other than 0, 50, 100 could be recessive traits). observed in a polyploid or heterokaryon (Figure 1a,1b,1c,1d,1e in Reference [3]). To account for the Only if the effective enzyme activity is decreased by at occurrence in a diploid of relative enzyme activities in least 95% in this instance (Figure 2), would clear domi- addition to those taking values of 0, 50, 100 (in Figures 2 nance and recessivity be noted. This is an extreme case; and 3, and in Figures 1f,1g,1h of Reference [3]), we need Figure 3 illustrates the other extreme. Between these to allow for allele pairs in addition to the three (UU, Uu, extremes, various degrees of asymmetry of flux response uu or WW, Ww, ww) in which the mutant alleles (u or w) plots may be observed (Figure 1 in Reference [3]). Never- express a catalytically inactive polypeptide. theless, unless: (i) the change in enzyme activity is meas- ured, (ii) it is realised that there is a non-additive The restriction to just three allele pairs in a diploid may be relationship between a change in any one enzyme activity traced to Sutton [1]. He wrote Mendel's F2 trait series (A + at a metabolic locus and a change in expressed trait, and 2Aa + a), incorrectly, as (AA + 2Aa + aa) and the number (iii) the shape of the flux response plot (Figure 2, Figure of distinguishable chromosome pairs as (AA + 2Aa + aa), 3) is revealed by plotting, it is simply not possible to state so establishing a false one-for-one relationship between that the system under investigation does or does not dis- pairs of chromosomes (AA or aa) and dominant or reces- play Mendelian dominance and recessivity. Terms such as sive traits (AA or aa). Sutton's notation for chromosome semi-dominance merely indicate that the flux response pairs was later transferred to allele pairs. In this article, plot is not quite asymmetric enough to be sure that a 50% dominant and recessive traits are represented, as Mendel reduction in enzyme activity produces a trait that is indis- did, by (A) and (a) respectively; alleles have been repre- tinguishable from the dominant trait. sented by different letters (e.g. UU, Uu, uu) in order to dis- tinguish alleles (parameters) from traits (variables). We 2.7. Is the Kacser & Burns treatment universally should allow for the situation where (U ) is a mutant of applicable? (U) that would express an allozyme activity lower than The change in the concentrations a normal metabolites has that expressed from (U) but not so low as that expressed been treated in the present article as the source of a change from (u); and where (u*) would be a mutant of (U) that in trait. This accords with the treatment in Figure 1 of ref- expresses an allozyme activity greater than that expressed erence [3]. Allowance should, however, be made for the by (u) = 0 in the traditional treatment but not so great as possibility that the change in concentration of a metabo- to merit the notation (U). The outcome of different lite is, in reality, a change in the concentration of a hypothetical crosses that involve different mutations of "signalling" metabolite (e.g. an allosteric activator or one both alleles at a given locus in genetically related dip- inhibitor of another enzyme in the pathway that gener- loid parents would then be as follows: ated the "signalling" metabolite, or in another pathway). Such mechanisms merely shift the cause of the change in (1) Repeated crosses (Uu × Uu) would give, on average, metabolite concentration to another part of the matrix of the allele series (UU + 2Uu + uu) thus permitting expres- intracellular metabolic pathways. In other words, the Kac- sion of no more than three distinctive enzyme activities at ser and Burns approach remains a valid explanation for the corresponding metabolic locus. the origin of dominant and recessive traits. (2) The cross (Uu* × Uu) would give the allele series (UU 2.8. Accounting for all the plotting points in Figures 2 and 3 + Uu + Uu* + uu*) in which two of the allele pairs differ In Figure 2, the relative enzyme activities (100, 50, 0) from those in the progeny of the first cross; and in which would be expressed from the series of allele pairs UU, Uu, three different heterozygotes are formed. uu in a diploid cell (Section 1) only if the mutant allele (u) † † was expressed as a catalytically inactive polypeptide. The (3) The cross (U u × Uu) would give the allele series (UU same considerations apply to the relative enzyme activi- + Uu + U u + uu) in which only one allele pair in the prog- ties expressed from the allele pairs WW, Ww, ww in Figure eny populations is identical with one of the allele pairs in 3. the progeny from the second cross. It is obvious that the continuously non-linear response (4) The cross (UU × Uu) would give, on average, the allele † † plots (Figures 2, 3; and References [3-10]]) could not be series (UU + UU + Uu + U u) which has only two allele constructed if these three allele pairs were the only ones pairs in common with the progeny of the third of these available to express a corresponding series of enzyme crosses of genetically related parents. activities. Figure 1 in Reference [3] showed that more than three distinct enzyme activities were observed in experi- (5) The cross (U u × Uu*) would give, on average, the † † mental practice in any one system. It is easy to see how rel- allele series (UU + U u* + Uu + uu*). Page 8 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 In the second and fourth crosses it was assumed that the then be zero at a metabolic locus and a "metabolic block" two heterozygous parents did possess exactly the same will occur at that locus. normal allele (U) at this particular locus so, among their progeny, the allele pair (UU) occurred. Analogously, Assembling the data from, for example, the second and among the progeny from the third cross, the allele pair third of the three hypothetical crosses between the genet- (uu) occurred. But, importantly, in each of crosses (2), (3) ically related parents described above gives an allele series † † and (4) three different heterozygotes occurred in each (UU, UU , U u, Uu, Uu*, uu*, uu). They would contribute † † progeny population (a heterozygote is defined in a dip- seven different allozyme pairs (EE, EE , E e, Ee, Ee*, ee*, loid by the occurrence of allele pairs other than those rep- ee) at one metabolic locus and seven different, single, resented here by UU or uu). The allele pairs in the enzyme activities (v), one from each pair of allozymes. heterozygotes in any one progeny population of these Given a range of enzyme activities in excess of the tradi- crosses (2), (3) and (4) are not all identical with those in tional three, a sufficient number of co-ordinates will be the progeny of another of these crosses. The parents in the available to establish a continuously non-additive plot of fifth cross did not share an identical allele; no two alleles the response of one defined flux (J) against changes in of a pair are then identical in the progeny. The allele pair enzyme activity (v) at one metabolic locus in genetically (Uu) occurs in all of the progeny of these five crosses but related cells or organisms (Figures 2, 3). There is no guar- only because one of two parents carried this allele pair or antee that all of these mutants will be generated in every because one parent carried allele (U) and the other carried case but since (U ) and (u*) each represent only one of allele (u). several possible mutations of allele (U), we may be rea- sonably confident of observing traits expressed from allele Cross (1) typifies events in self-fertilising organisms but is pairs in addition to, or instead of, those expressed from not typical of sexual reproduction in other organisms (cf the two traditional mutant pairs (Uu) and (uu). Assem- Figure 2 in reference [1]). Male and female parents that are bling sets of enzyme activity and flux (or metabolite con- identically heterozygous at any locus must be rare. Crosses centration) data from the progeny of different but (2)-(5) between two heterozygous parents will produce, genetically related parents then creates the non-linear flux under the circumstances noted above, truly homozygous response plots illustrated in Figures 2 and 3. All plotting allele pairs (such as UU and uu) but they will also points in the idealised Figures 2 and 3 should be regarded produce, on average, three different heterozygotes among as tokens for the experimental plots published earlier [3]. their progeny (four heterozygotes in the fifth cross). This simple explanation for the occurrence of more than The consequences are then as follows: From each locus in three co-ordinates for a plot of flux response against a diploid cell that expresses catalytic polypeptides, alloz- changes in enzyme activity (or gene dose) means that it is ymes (pairs of enzymes) will be expressed; one from the no longer acceptable to base arguments and conclusions gamete donated by the male parent the other from the on the assumed presence of only one heterozgote (Uu) in gamete donated by the female parent. For simplicity, it a diploid allele series at a locus, and on only one corre- will be assumed here that the combined allozyme activity sponding hybrid trait. Furthermore, statements that all at each catalytic locus in the metabolic pathways of the heterozygotes express 50% (and only 50%) of the pheno- cell is the sum of the activities the two allozymes at each type expressed from the homozygous wild-type are based such locus. on the false idea that the mutant allele (u) always pro- duces a totally inactive enzyme. Figures The traditional allele series (UU + 2Uu + uu) in a diploid 1a,1b,1c,1d,1e,1f,1g,1h of Reference [3] depended upon will then generate the enzyme series (EE + 2Ee + ee) at one the availability of 5, 6 or 7 plotting points relating the flux metabolic locus in different, genetically related, individu- response to experimentally determined changes in als. This enzyme series provides two extreme combined enzyme activity (effectively to changes in allele constitu- allozyme activities, namely 100% (EE) and 0% (ee). There tion at a locus). In addition to the traditional heterozygote are no allele pairs at this locus that could provide <0% or (Uu), there must be a number of heterozygotes (e.g. UU , † † >100% enzyme activity. All other allele pairs, e.g. (UU ), U u, Uu*, uu*), and a corresponding a range of enzyme † † (U u), (U u*), (Uu*), (uu*), would provide combined activities (v), that account for the response of a flux (J) to allozyme activities that lie between the 100% and 0% val- a change in enzyme activity at one metabolic locus (Fig- ues just described. Only if (u) happens to be a null ures 1, 2, 3). In Figure 2, some of these additional hetero- mutant, will the heterozyote (Uu) express a single enzyme zygotes will establish the asymptote of the flux response activity (v) equal to 50% of the maximum available from plot. The trait expressed from any such heterozygote (UU). Only in this circumstance will the allele pair (uu) would be indistinguishable from the trait expressed from express two inactive polypeptides; the enzyme activity will the normal allele pair (UU); they could have accounted for the occurrence of Mendel's hybrids (Aa) which Page 9 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 appeared to display only the dominant trait (A). This is provide any justification for representing a trait by further evidence that the traditional treatment of elemen- twinned letters, e.g. (AA) or (aa). The single letters (A) and tary Mendelian genetics is inadequate and misleading [1]. (a) stood for qualitative differences in trait form in Men- del's work; they stand equally well for quantitative changes in a trait in modern work. The non-linear 3. Quantifiable differences between any two forms of a trait response plots of Kacser and Burns [3] apply to quantita- Differences in traits are generally and usefully described tive and to apparently qualitative changes in the pheno- by qualitative terms: type that arise from mutations of any one enzyme at a metabolic locus in a biochemical pathway. hirsute/bald; red flowers/white flowers; lithe/obese; mus- cular/"skinny"; slow/fleet; albino/black. Such descriptive 4. Implications of the systemic approach of terms do, however, disguise the obvious fact these appar- Kacser and Burns [3] ently qualitative differences in outward appearance are Figure 2 shows the response of the phenotype to changes based on quantitative differences in the concentrations of in enzyme activity at a metabolic locus or to changes in molecular products that contribute to the outward gene dose at the corresponding gene locus. It follows, if appearance or function of a cell or organism. the response plot takes this form, that increasing the dose of this particular gene in a wild-type haploid cell (or the These comments apply to the apparently qualitative dif- dose of the normal homozygous alleles in a wild-type dip- ferences examined by Mendel (Table 1 in reference [1]) loid or polyploid cell) is unlikely to produce a detectable and to those traits forms typified by a trait series (A + 2H change in the phenotype (e.g. an increase in the concen- + a) where (A) indicates the dominant trait form, (a) the tration of the trait component produced by a metabolic recessive trait form and (H) a hybrid trait that may be pathway; or a change in cell function associated with that indistinguishable (Figure 2) from the dominant traits (A) pathway). It was demonstrated that it was necessary, or distinguishable (Figure 3) from the dominant trait (B). under these circumstances, to increase concurrently the gene dose at each of no fewer than five loci if significant It should not therefore be supposed that the paper by Kac- increases in the flux (and in the concentration of meta- ser and Burns [3] provided an explanation only for the bolic product) was to be achieved [5]. The systemic occurrence of qualitative differences between any two approach to a rational explanation of the origins of dom- traits. On the contrary, a continuously variable response inant and recessive traits [3] has obvious implications for of each of several defined fluxes was brought about when biotechnologists. mutations of alleles at one locus changed the activity of one enzyme in a metabolic pathway (or when changes in Figure 2 (representing several plots in Reference [3]) also gene dose changed the concentration and thus the activity suggests that somatic recessive conditions (in contrast to of one enzyme in a metabolic pathway). so-called dominant conditions) could be ameliorated by partial gene replacement therapy. Experiments in the The flux responses were labelled "Flux to arginine", "Flux cystic fibrosis mouse model support this suggestion [7]; to biomass", "Flux to melanin", "Flux to products", "Flux they show that the systemic approach to the origins of to DNA repair" (Figure 1 in reference [3]). The molecular dominant and recessive traits has implications for medical compositions of "arginine", "biomass", "melanin", and genetics. "products" (of ethanol metabolism) were not changed. Their concentrations were changed as graded mutations at It was pointed out (section 2.6) that substantial decreases a gene locus caused graded changes in one enzyme activity in the dose of normal alleles at any one locus (or in the in those pathways that created arginine, biomass, mela- enzyme activity at the corresponding metabolic locus) nin, or the products (of ethanol metabolism). Similarly, a may not elicit detectable changes in the trait(s) of the cell. change in the "flux to DNA repair" was achieved by graded In other words, given a response plot approximating to increases in the dose of the gene specifying the synthesis that shown in Figure 2, traits – including associated cell of the "repair enzyme" that excises covalently-linked adja- functions – are inherently buffered against substantial cent thymines in DNA and allows incorporation of increases or decreases in the dose of any one gene, or thymidine in place of the excised pyrimidines. This against substantial changes in enzyme activity at the cor- "repair enzyme" activity is absent in Xeroderma pigmento- responding metabolic locus. This appears to be the prob- sum patients. able origin of the so-called "robustness" or buffering of chemotaxis against changes in enzyme kinetic constants Additional examples of quantitative changes in the con- [12-15]. centration of molecular components of a trait will be found in other publications [5-11]. None of these changes Page 10 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 genetic phenomena could perhaps also be explained. S S S S (J ) 1a 2a 3a 4a a Only two of the thirteen texts surveyed [1] gave a defini- tion, in their glossaries, of pleiotropy and epistasis. Both p q agreed that pleiotropy was a phenomenon where a change at one gene locus brought about a change in more than (J ) S S S S b 4b 3b 2b 1b one trait. Both attributed epistasis to an interaction A Figure 4 ccounting for the occurrence of pleiotropy between genes or their alleles. Neither of these descrip- Accounting for the occurrence of pleiotropy. One tions of pleiotropy and epistasis is particularly revealing. unbranched pathway is coupled to another by a conserved metabolite pair p and q. Such coupling is not uncommon in The following account, like those preceding it, does not cellular systems and is one source of pleiotropy. Mutation of depend on the fiction that all mutations generate inactive any one enzyme in one pathway will affect both fluxes (J and enzymes. Figure 1 is elaborated as shown in Figure 4. One J ) to a trait component and the concentrations of those trait pathway, like that shown in Figure 1, is now coupled to components. See also Figure 5. Figure 4, like Figure 1, illus- another analogous pathway by the conserved metabolite trates the need to adopt a systemic approach in attempts to pair (p, q). The sum of the concentrations of (p) and (q) is understand the responses of a metabolising system to changes in any enzyme activity brought about by mutation. constant (is conserved) but the ratio of the two concentra- tions (p/q) is a free variable. All the characteristics of the metabolic system in Figure 1 (Section 2), apply to each of the two fluxing pathways in Figure 4. Claims in the bio- chemical literature in the past that changes in the ratio (p/ This proposed explanation for metabolic buffering is q) controlled metabolic fluxes were and remain untena- quite general; it does not depend on the particular kinetic ble; one variable of a system cannot be said to control mechanisms that have been suggested to account for this another variable of the system. buffering [12]; it also suggests that there is no need to pos- tulate the presence of diagnostic "biological circuits" as Figure 1 may also be elaborated as shown in Figure 5. An the source of this buffering of the phenotype against input flux from X to S divides into two output fluxes 1 4 [16]. Of the input flux, a proportion (α) enters one of the mutations at a single locus. two output fluxes (J ) and a proportion (1-α) enters the Attempts to improve the concentration of metabolic prod- other output flux (J ). The magnitude of (α) is determined ucts by increasing the gene dose at one locus above that by the magnitudes of the activities of all the enzymes of available in the wild-type or normal cell could be success- the metabolic system; (α) is a systemic characteristic [17]. ful, at least to some self-limiting extent, if a response plot Again, all the characteristics of the model metabolic sys- like Figure 3 applies. Induced synthesis of one membrane- tem in Figure 1 (Section 2), apply to each of the two path- located enzyme activity to between 20% and 600% of ways that generate fluxes J and J shown in Figure 5. a b wild-type activity illustrates the possibility [8]. In this instance, plots like Figure 3 applied only to changes in the 5.2. The origin of pleiotropy explained uptake and phosphorylation of α-methyl glucoside; It will be obvious that a mutation of any one enzyme in changes in growth rates and glucose oxidation gave either of the two pathways of Figure 4 will cause changes response plots like Figure 2. The explanation for the differ- in the fluxes through both of the coupled pathways (and ence may lie in the suggestion [3] that shorter pathways the concentrations of metabolites in both pathways). Sim- will yield response plots like Figure 3, while the longer the ilarly, a mutation in any one enzyme of the input flux of pathway, the more likely is it that markedly asymmetric Figure 5 will affect the concentrations of metabolites in plots like Figure 2 will be observed. both output fluxes J and J . Pleiotropy (a change in more a b than one trait as a consequence of a single mutation), when it is detected, is thus seen to depend on mutating an 5. Expansions of the present treatment 5.1. Why mutating one enzyme in a metabolic pathway enzyme within a metabolic pathway, on the consequen- may alter more than one trait; and mutating more than tial changes in metabolite concentrations, and on the one enzyme may annul these changes in more than one structure and interdependence of biochemical pathways. trait Only if one of the enzymes in the input pathway shows If the explanation for the origin of dominant and recessive zero activity will both output fluxes (J and J ) cease (Fig- a b traits depends on realising that fluxing metabolic path- ure 5). ways generate the molecular components of all traits, and that mutating any one enzyme in these pathways alters 5.3. The origin of epistasis explained the flux and the concentrations of those normal metabolic Given a steady input flux from X to S (Figure 5), a muta- 1 4 products that are molecular components of a trait, other tion of one of the enzymes (E , E or any other enzyme 5a 6a Page 11 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 enzyme is mutated within a metabolic pathway. Whether pleiotropy or epistasis is detected, or not, will depend on 6a the severity of the mutation and on the nature of the flux S S (J ) 5a 6a a v E 5a 6a response plots (Figures 2, 3) as demonstrated in section 2. v v v E 2 3 4 5a X1 S2 S3 S4 E2 E3 E4 v5b E5b v6b 5.5. Biochemistry and genetics are not separable topics S S (J ) 5b 6b b Beadle and Tatum [18] isolated a series of mutants of Neu- 6b rospora crassa and tested their ability to grow on basal A Figure 5 ccounting for the occurrence of pleiotropy and epistasis medium or on basal medium supplemented with differ- Accounting for the occurrence of pleiotropy and epistasis. ent metabolites or cofactors. Wild-type Neurospora crassa Mutation of any one of enzymes E , E , E would affect both 2 3 4 grew on basal medium. Different isolated mutants would fluxes J and J to separate trait components. Mutation of any a b grow only if the basal medium was supplemented with one of enzymes E , E , etc would decrease flux J to a trait 5a 6a a the specific product of an enzyme rendered partially or component but increase J to another trait component; the fully inactive in one of the mutants. These brilliant concentrations of trait components in pathway J would observations led to the paradigm "one gene, one func- decrease, those in pathway J would increase. Epistasis would tion" [19,20], later to "one gene, one enzyme". These occur if a subsequent mutation occurred in any one of observations [18] made explicit what was implied by the enzymes E , E etc. A branched metabolic pathway is thus a 5b 6b potential source of pleiotropy and epistasis; see the text for observations of Garrod [21-24]] on inborn errors of further discussion. This diagram, like that in Figure 4, empha- metabolism namely: metabolism is catalysed by a sises the importance of adopting a systemic approach in sequence (or system) of different enzymes; and a suffi- understanding the potential effect, on a trait or traits, of a cient decrease (by mutation) in the activity of any one mutation in any one enzyme in enzyme-catalysed systems. enzyme may cause a change in the trait(s) or characteris- tic(s) of the system (e.g. the ability to grow, to accumulate cell mass [18]). Beadle [20] expressed surprise that Garrod's work had in this output limb) would decrease flux J and increase received so little attention. He wrote: "It is a fact both of flux J . The concentrations of metabolites in pathway J interest and historical importance that for many years b a would decrease and those in pathway J would increase, a Garrod's book had little influence on genetics. It was further example of a pleiotropic response to a single muta- widely known and cited by biochemists, and many genet- tion. But suppose that, following the mutation of E , a icists in the first two decades of the century knew of it and 5a mutation occurred in E or any other enzyme in this alter- the cases so beautifully described in it. Yet in the standard 6b native output limb. Clearly, the effect of the first mutation textbooks written in the twenties and thirties - - - - few on the cell characteristics would be at least partly nullified mention its cases or even give a reference to it. I have often by the second mutation – a phenomenon known as wondered why this was so. I suppose most geneticists epistasis and sometimes attributed in genetic texts to an were not yet inclined to think of hereditary traits in chem- interaction between genes but shown here to depend on ical terms. Certainly, biochemists with a few notable mutations of one or more enzymes, and on the structure exceptions such as the Onslows, Gortner and Haldane and interdependence of metabolic pathways. Only if the were not keenly aware of the intimate way in which genes activity of one of the enzymes in one of the two output direct the reactions of living systems that were the subject pathways is diminished to zero by mutation, will the of their science." products of that output limb downstream from the muta- tion be lost. This lack of attention to the implications of Garrod's work is all the more surprising when it is recalled that Bateson If the fluxes proceeded in the opposite direction to that [[25], p.133] pointed out that alkaptonuria (a change in shown in Figure 5 (so that two pathways merged into concentration of the normal metabolite, homogentisic one), mutation of an enzyme in one of the input fluxes acid, and one of Garrod's inborn errors of metabolism) followed by a mutation of an enzyme in the other input was an example of a Mendelian recessive trait or character; pathway could again elicit epistatic responses in the see also [[26], p.19]. In other words, some important system. aspects of genetics depended on recognising the role of changes in an enzyme activity, within a metabolic system, 5.4. Are pleiotropy and epistasis always detectable? in effecting a change in a trait. Particular but common metabolic structures (Figures 4, 5) provide the potential for pleiotropy and epistasis; i.e. The aphorism "one gene, one enzyme" was refined to changes in concentrations of normal metabolites when an "one allele, one polypeptide" after the elucidation of the Page 12 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 structure of DNA [27,28] and the rapid advances made in have other important functions (e.g. as hormones) and the next 10 or 15 years in elucidating the mechanisms of may be components of traits. expression of diploid alleles as pairs of polypeptides or proteins [29-32]] most of which are enzymes [3,4]. These X stands for all those initial extracellular substrates feed- more recent discoveries (Figure 6) emphasise what was ing the matrix of inter-dependent biochemical pathways implied by the work of Beadle and Tatum [18]: the that typify all functioning cells. It is these pathways that molecular components of dominant and recessive traits or generate the non-protein, non-polyribonucleotide, characteristics, in all biological forms, are generated by molecular products of all cell traits. fluxing metabolic pathways catalysed by sequences or sys- tems of enzymes. Dominant and recessive traits are not Each of these three major fluxing pathways (Figure 6) is the direct product of the expression of alleles as suggested catalysed by a succession of enzyme-catalysed reactions as by the currently favoured explanation of Mendel's obser- shown in Figure 1. The flux through any one of these path- vations (Figure 2 in Reference [1]); they are produced indi- ways will respond to a mutation of any one enzyme in the rectly by a system of enzymes (Figures 1, 4, 5, 6). pathway as shown in Figures 2, 3; any change in these fluxes could change the concentrations of the intermedi- Figure 6 depicts the direct relationship between any one ate metabolites or the final product (section 2.3); but, gene (g1, g2, g3, g4) and the synthesis of individual provided mutations do not alter the specificity of an polypeptides (P , P , P , P ) most of which, but not all, are enzyme, they will not change the existing molecular struc- 1 2 3 4 enzymes (E , E , E , E ). All polypeptides, catalytic and ture or composition of these metabolites. 1 2 3 4 non-catalytic, are synthesised in this way. Most attention is concentrated on the pathway initiated X , X and X in Figure 6 are immediately identified as by X for the simple reason that this pathway stands for all 1 2 3 1 extracellular parameters of a cell system. X stands for the matrix of interdependent biochemical fluxes that gen- those substrates that lead, through a series of enzyme-cat- erate such a wide range of the non-protein (and non- alysed reactions, to the synthesis of nucleoside polyribonucleotide) molecular components of cell traits triphosphates (NTPs) and their subsequent incorporation (e.g. skin pigments, membrane lipids, chlorophyll, xan- into mRNA. Note that mRNA is a terminal product of this thocyanins, non-peptide hormones, neural transmitters, pathway. It is a coding entity, a proxy for DNA. Each chitin, serum cholesterol, peptidoglycans, etc, etc). mRNA specifies the order of incorporation of individual amino acids into a polypeptide, but no individual mRNA If any one of the three major pathways shown in Figure 6 molecule participates as a substrate in the subsequent is coupled to another pathway (Figure 4) or contains a steps of the catalysed formation of a polypeptide. The branch (Figure 5) there will be, potentially detectable, control of the overall expression of a gene as a polypeptide pleiotropic and epistatic responses to mutations of any of is therefore necessarily treated in Metabolic Control Anal- the pathway enzymes (section 5.3). Such pathway ysis as a cascade of two fluxing metabolic pathways, one coupling and branching is a common feature of the path- that starts at X , the other that starts at X [33]. ways that start with one of the extracellular substrates typ- 3 2 ified by X . X stands for those extracellular substrates that lead, through a series of enzyme-catalysed reactions, to the syn- If the implications of the work of Beadle and Tatum [18] thesis de novo of amino acids (AAs) and their subsequent were not fully realised at the time, Figure 6 might have incorporation, along with any existing amino acids, into a suggested that a fresh approach to an understanding of the polypeptide (P). In a haploid cell, one polypeptide is syn- origins of dominant and recessive traits was needed. The thesised from each gene locus. In a diploid, one polypep- currently favoured explanation for Mendel's findings ([1], tide is synthesised from each of two alleles at a gene locus. Figure 2) does not take account of the biochemical path- If these pairs of polypeptides are catalytically active, each ways of the synthesis of enzymes (Figure 6) established enzyme in a diploid cell (E , E , E , etc) consists of a pair 30–40 years ago, does not acknowledge that the molecu- 1 2 3 of allozymes, one of each pair specified by the allele lar components of all traits are synthesised by systems of derived from the male parent, the other specified by allele enzymes, does not take account of the change in concen- derived from the female parent. Each pair of allozymes, tration of molecular components of traits when any one whether normal or mutated, exhibits only one measura- enzyme is mutated, and fails to distinguish the system ble activity (v) at a catalytic locus in a metabolic pathway. parameters (alleles) from the system variables (traits). If the pairs of polypeptides (P) synthesised by a diploid cell are not catalytically active they will not, of course, play Note that changes in the concentrations of external a direct role in catalysing a metabolic pathway. They may metabolites (whether they are substrates like X , X , X in 1 2 3 Figure 6, or extracellular inhibitors or activators of intrac- Page 13 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 ellular enzymes) may effect changes in intracellular (v) The argument that changes in concentration of a trait metabolism and consequently modify the effects of a component may nevertheless be revealed as a qualitative mutation. This topic is not immediately relevant in the change in that trait. present article but is a notable feature of Metabolic Con- trol Analysis. Descriptions of the role of the Combined (vi) A demonstration that both alleles (normal or mutant) Response Coefficient (R) in permitting extracellular at a locus in a diploid are generally expressed. If the nor- effectors to modulate intracellular metabolism (and thus mal allele expresses a catalytically active polypeptide, the effects of a mutation) will be found elsewhere [11,34- many mutants of this allele will express an enzyme with 36]. lower activity; a mutated enzyme with zero activity is an extreme case. If pleiotropic and epistatic responses to a mutation are as common as is suggested (sections 5.1–5.4), the question (vii) The demonstration that an explanation of Mendel's then arises: how do we account for Mendelian segregation observations cannot be based on an allele series contain- of traits during sexual reproduction? The answer lies in the ing only three terms (e.g. uu, 2uU, UU) one of which is a fact that a mutation at a biochemical locus, within the unique heterozygote (uU). matrix of interdependent pathways, has its most obvious effect on the most closely associated pathways. Distant (viii) A demonstration that dominant and recessive traits pathways (on the scale of cellular dimensions) will be less cannot be generated by those polypeptides that are not obviously affected. Kacser and Burns (Reference [3], enzymes embedded in a system of enzymes. p.649) pointed out that "This apparent independence of most characters makes simple Mendelian genetics possi- (ix) Rejection of the unjustified traditional claim that a ble, but conceals the fact that there is universal pleiotropy. hybrid (H) expresses a dominant trait (A) because the All characters should be viewed as 'quantitative' since, in (allegedly) recessive allele (u) in a heterozygote (Uu) is principle, variation anywhere in the genome affects every always completely ineffective or because the allegedly character." Section 3 in the present article emphasised the dominant allele (U) suppresses the allegedly recessive importance of quantitative changes in cell traits. The con- allele (u) in the heterozygote [1]. siderations in this paragraph are germane to the apparent absence of a detectable change of phenotype in some so- (x) Rejection of the traditional, unsubstantiated and called 'knock-out' experiments. implausible claim that one so-called dominant allele in a heterozygote is as effective as two such alleles in the wild- 6. Conditions that must be met to explain type cell [1]. dominance and recessivity The explanation advocated in this article for the origins of It was also shown that pleiotropy and epistasis can be dominant and recessive traits from normal and mutant explained by taking a similar system approach to that used alleles in a diploid is based on: in explaining the origin of dominant and recessive traits. (i) An obligatory distinction, by notation and nomencla- It is then apparent that, to account rationally for Mendel's ture, between the variables (traits) and the parameters observations of dominant and recessive traits, a minimum (alleles and enzymes) of genetic/biochemical systems. of four conditions must be met. (ii) The contention that the molecular components of all (i) Alleles must be distinguished by notation, nomencla- traits are the products of fluxing metabolic systems (Fig- ture and concept from traits; functions of components of ures 1, 4, 5, 6). the genotype must be distinguished from properties of components of the phenotype. Traits alone may be dom- (iii) Experimental evidence for an inevitable non-linear inant or recessive. response of a flux (through a metabolic system of enzymes) to graded changes in the activity of any one of (ii) Alleles cannot be called "dominant" or "recessive". those enzymes [3], evidence that is supported by a (When alleles are so called, the flaws present in the current number of independent observations [5-11]. attempts to explain Mendel's observations will inevitably re-appear [1]). (iv) A demonstration that dominant and recessive traits arise from changes in the concentration of the normal (iii) It must be shown how dominant traits become distin- molecular components of a defined trait. guishable from recessive traits in the same cell or organ- ism (Figure 2, 3). Page 14 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 (iv) It must be shown how a hybrid trait sometimes hybrid in Figure 2 is replaced mentally and temporarily by becomes indistinguishable from the dominant trait and (Aa), it will be clear why Mendel postulated that his sometimes does not (Figures 2, 3). The first circumstance hybrids (Aa) displayed trait (A) and not trait (a). If the will account for Mendel's 3(dominant):1(recessive) trait same exercise is repeated in Figure 3 by replacing (H) tem- ratio; the second for exceptions to this ratio. porarily by (Bb), it will be clear why Mendel observed an anomalous blending of flower colours in the hybrids If all four conditions are be met; the first two conditions when he crossed parental bean plants bearing different must first be met. The treatment given in sections 2–5 flower colours. meets each of these requirements. The treatment of elementary Mendelian genetics advo- 7. Conclusions cated here is based on the work of Kacser and Burns [3]. Kacser and Burns [3] provided the basis for a rational So far as the present author is aware, that paper has not explanation for the origin of dominant and recessive traits been described by any student textbook of "classical" or that arose from mutations of alleles at any one gene locus "molecular" genetics published in the intervening 23 in a diploid or polyploid cell (sections 2.3, 2.4). Inherent years. Orr [37] did not see the full significance of the Kac- in this explanation, as set out above (sections 2.5, 2.6), are ser and Burns paper [3]. Darden [[38], p. 72] declared that further explanations for the occurrence of the 3(domi- "(trying) to unravel the complex relations between nant):1(recessive) trait ratio in some situations in a dip- mutant alleles and enzymes (Kacser and Burns, 1981) - - - loid (Figure 2), for the absence of this trait ratio from is not a major research topic in genetics." other situations (Figure 3), for the absence of dominant and recessive traits in yet others and for the appearance of Several possible reasons for this failure to see the merits of a blend of parental traits in some heterozygotes. These five the Kacser and Burns paper [3] may be worth considera- demonstrations are internally consistent. In contrast to tion. They include: the currently favoured attempt to explain Mendel's results [1], no arbitrary assumptions are introduced (section 2.8) (1) Persistent misrepresentations of Mendel's paper, and † † to explain how heterozygous allele pairs (e.g. UU , U u, incorporation of these distortions into currently favoured Uu*, uu*) may produce a trait that is indistinguishable explanations of Mendel's observations [1]. from the trait expressed from the "homozygous" allele pairs (UU). (2) A failure to recognise the consequences of not distin- guishing between the function of the alleles and the prop- In other words, provided: erties of traits in attempting to explain Mendel's results. Normal and mutant alleles specify the kind (and order of (a) all current misrepresentations of Mendel's paper [1] incorporation) of amino acids into polypeptides (most are first discarded, but not all are enzymes). Dominance and recessivity are a reflection of changes in the concentration(s) of the molec- (b) alleles are distinguished by notation and nomencla- ular component(s) of a trait when an enzyme is mutated ture from the traits they specify, within a fluxing metabolic pathway. (c) alleles are regarded as normal or mutant (but not (3) Tardy recognition of the need to adopt the systemic dominant or recessive), it is possible to provide a rational approach of Metabolic Control Analysis in explaining the and internally consistent explanation for the origin of response of the variables of a biological system to pertur- Mendel's dominant and recessive traits, for the occurrence bations of the magnitude any one system parameter. of his 3:1 trait ratio, and for exceptions to these observa- tions noted by later investigators. The same systemic (4) A reluctance to accept a change in concepts even when approach is applicable to current problems in biotechnol- currently accepted representations of Mendel's results are ogy and medical genetics (section 4). It also explains the demonstrably untenable. origins of pleiotropy and epistasis (section 5); and chal- lenges the assumption that a mutation in a non-catalytic (5) Elucidation of the double helical structure of DNA protein provides an example of Mendel's dominant and (Figure 6) and all that followed in the next 10–15 years recessive traits [1]. imposed profound changes on genetics but was not per- haps always taken into account. Mendel found, by experiment, that the proportions of plant forms in each of his F2 populations was represented (6) A determination in some quarters to regard genetics as by (A + 2Aa + a). In the present paper these proportions an autonomous subject. It has been obvious at least since have been written as (A + 2H + a). If the symbol (H) for a the work of Beadle and Tatum [18] that such claims can- Page 15 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 11. Fell DA: Understanding the Control of Metabolism London and Miami: not be sustained. Genetics is intimately related to, and in Portland Press; 1997. some respects dependent upon, biochemistry. The 12. Barkai N, Leibler S: Robustness in simple biochemical converse is equally true. Genetics and biochemistry are networks. Nature 1997, 387:913-917. 13. Hartwell L: A robust view of biochemical pathways. Nature not separable topics in biology. 1997, 387:855-857. 14. Cohen P: Weathering the storm. New Scientist 2000 Online Confer- ence Reports 6:. It is significant that Kacser & Burns were also one of two 15. Ferber D: The spice of life. NewScientist Online Conference Report sets of authors who initiated the systemic approach to the control of metabolite concentrations and fluxes [39,40]. 16. Kacser H: The control of enzyme systems in vivo: elasticity analysis of the steady state. Biochem Soc Trans 1983, 11:35-40. This approach was elaborated by the original authors and 17. Fell DA, Sauro JM: Metabolic control analysis. Additional rela- many others. For some accounts and reviews, see tionships between elasticities and control coefficients. Eur J [11,36,41-44]. Biochem 1985, 148:555-561. 18. Beadle GW, Tatum EL: Genetic control of biochemical reac- tions in Neurospora. Proc Natl Acad Sci USA 1941, 27:499-506. 8. A correction 19. Beadle GW: Genetics and metabolism in Neurospora. Physiol Rev 1945, 25:643-663. In an earlier paper [45] it was stated that Mendel had 20. Beadle GW: Chemical genetics. In: Genetics in the 20th Century inferred the presence of segregating particles. These partic- Edited by: Dunn LC. The Macmillan Company, New York; 1951. ulate determinants were then represented by (A) and (a). 21. Garrod AE: About Alkaptonuria. Lancet 1901, ii:1484-1486. 22. Garrod AE: The incidence of Alkaptonuria. Lancet 1902, These statements are here formally withdrawn. They were ii:1616-1620. consistent with textbook treatments of Mendelian genet- 23. Garrod AE: Croonian Lectures to the Royal Society of Physi- cians: Inborn Errors of Metabolism. Lancet 1908, ii:1-7. 1432– ics [1] but a subsequent reading of Mendel's original 145; 173–179; 214–220 paper revealed that these statements, and others that 24. Garrod AE: Inborn Errors of Metabolism (a revision of the occur frequently in the recent reviews of Mendel's paper Croonian Lectures, 1909). . reprinted in reference [26] 25. Bateson W: Report of the Evolution Committee of the Royal Society 1902, and in current textbooks, were incorrect and misleading. 1(III):125-160. A history of the misunderstandings and misrepresenta- 26. Harris H: Garrod's Inborn Errors of Metabolism Oxford: University tions that have sustained the currently favoured depiction Press; 1963. 27. Watson JD, Crick FHC: Molecular structure of the nucleic of Mendelian genetics [1] will be presented elsewhere. A acids. A structure for deoxyribonucleic acid. Nature 1953, paper setting out the concepts of parameters and variables 171:737-738. 28. Watson JD, Crick FHC: Genetical implications of the structure will also be submitted. of deoxyribonucleic acid. Nature 1953, 171:964-967. 29. Crick FHC: On protein synthesis. Symp Soc Exp Biol 1958, Acknowledgements 12:138-163. 30. Crick FHC: The Genetic Code III. Sci Amer 1966, 215(4):55-62. I thank Dr Colin Pearson for his support during the preparation of this and 31. Taylor JH: Molecular Genetics II New York and London: Academic the preceding paper, Dr Denys Wheatley for temporary accommodation Press; 1967. during a logistic exercise and Dr Paul Agutter for valuable suggested mod- 32. Yanofsky C: Gene Structure and Protein Structure. Sci Amer ifications to the drafts of these two papers. 1967, 216(5):80-94. 33. Westerhoff HV, Koster JG, van Workum M, Rudd KE: On the con- trol of gene expression. In: Control of Metabolic Processes Edited by: References Cornish-Bowden A, Cárdenas ML. New York. Plenum Press; 1. Porteous JW: We still fail to account for Mendel's 1990:399-412. observations. Theor Biol Med Modelling 2004 in press. 34. Kacser H, Porteous JW: Control of Metabolism: What do we 2. Mendel G: Versuche über Pflanzen-Hybriden. Verhandlungen des have to measure? Trends Biochem Sci 1987, 12:5-14. Naturforschenden Vereines in Brunn 1866, 4(Abhandlungen):3-47. 35. Porteous JW: A theory that works. In: Control of Metabolic Processes 3. Kacser H, Burns JA: The Molecular Basis of Dominance. Genetics Edited by: Cornish-Bowden A, Cárdenas ML. New York: Plenum 1981, 97:639-666. Press; 1990:51-67. 4. Nirenberg MW: The Genetic Code II. Sci Amer 1963, 36. Cornish-Bowden A: Fundamentals of enzyme kinetics London: Portland 208(3):80-94. Press; 1995. 5. Niederberger P, Prasad R, Miozzari G, Kacser H: A strategy for 37. Orr HA: A test of Fisher's theory of dominance. Proc Natl Acad increasing an in vivo flux by genetic manipulations. The tryp- Sci USA 1991, 88:11413-11415. tophan system of yeast. Biochem J 1992, 287:473-479. 38. Darden L: Theory change in science Oxford and New York: Oxford 6. Dean AM, Dykhuisen DE, Hartl DL: Fitness as a function of β- University Press; 1991. galactosidase activity in Escherichia coli. Genet Res 1986, 48:1-8. 39. Heinrich R, Rapoport TA: Linear theory of enzymatic chains; its 7. Dorin JR, Farley R, Webb S, Smith SN, Farini E, Delaney SJ, Wain- application for the analysis of the cross-over theorem and wright BJ, Alton EWFW, Porteous DJ: A demonstration using the glycolysis of human erythrocytes. Acta Biol Med Germanica mouse models that successful gene therapy for cystic fibrosis 1973, 341:479-494. requires only partial gene correction. Gene Therapy 1996, 40. Kacser H, Burns JA: The control of flux. Symp Soc Exp Biol 1973, 3:797-801. 27:65-104. 8. Ruyter GJG, Postma PW, van Dam K: Control of glucose metab- 41. Fell DA: Metabolic Control Analysis: a survey of its theoreti- Glc olism by EnzymeII of the phosphoenolpyruvate-depend- cal and experimental development. Biochem J 1992, ent phosphotransferase system in Escherichia coli. J Bact 1991, 286:313-330. 173(19):6184-6191. 42. Kell DB, Westerhoff HV: Metabolic control theory; its role in 9. Letellier T, Heinrich R, Malgat M, Mazat J-P: The kinetic basis of microbiology and biotechnology. FEMS Microbiol Rev 1986, threshhold effects observed in mitochondrial diseases: a sys- 39:305-320. temic approach. Biochem J 1994, 302:171-174. 43. Teusink B, Baganz F, Westerhoff HV, Oliver SG: Metabolic Control 10. Small JR, Kacser H: Responses of metabolic systems to large Analysis as a tool in the elucidation of novel genes. Methods changes in enzyme activities and effectors. 1. The linear Microbiol 1998, 26:298-336. treatment of unbranched chains. Eur J Biochem 1993, 213:613-624. Page 16 of 17 (page number not for citation purposes) Theoretical Biology and Medical Modelling 2004, 1:6 http://www.tbiomed.com/content/1/1/6 44. Wildermuth MC: Metabolic control analysis: biological applica- tions and insights. Genome Biology 2000, 1(6):1031.1-1031.5. 45. Porteous JW: Dominance – One hundred and fifteen years after Mendel's paper. J Theor Biol 1996, 182:223-232. Publish with Bio Med Central and every scientist can read your work free of charge "BioMed Central will be the most significant development for disseminating the results of biomedical researc h in our lifetime." 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A rational treatment of Mendelian genetics

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A rational treatment of Mendelian genetics

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