The Basis of Dominance

Dominance as a product of evolution

Fisher’s model. All discussions under this head must begin with Fisher’s theory of the evolution of dominance via modifier genes. He considered that it is rather a peculiarity of the wild type to be generally dominant than a peculiarity of the mutant to be recessive to the type from which it arose. He believed that in the absence of any mechanism to adjust the results, the intensity of any character would be proportional to the gene dose, so that for genotypes AA, Aa and aa the phenotypes would show 100%, 50% and 0% respectively of normal level of the character. He further considered that this result would be harmful not only to the homozygote aa but also to the heterozygote Aa. Over many generations, therefore, natural selection would have provided modifier genes to bring the heterozygote phenotype to equivalence with that of the normal homozygote. The modifier would act on the heterozygote but would not affect fitness by itself. Selection would act in favour of the modifier at a rate proportional to the product of two factors: the chance that the modifier was present in the heterozygous individual and the fitness advantage caused by its presence.

Difficulties with the Fisher model. Strictly speaking, if one wishes to explain the evolution of presentday wild type alleles starting from a condition in which they arose as recessives, modifier action should be invoked only in the (original) recessive homozygote — that is, if one wishes to say that the mutation was fully recessive at the beginning. But going by reasonable population sizes and what we know of mutation rates (about 1 in 10⁶ to 1 in 10⁵ per locus), the population is hardly likely to contain individuals that are homozygous for the mutant allele right from the start. Even if we say that the modifier acts on the heterozygote, which amounts to assuming that the mutation had a weak dominant effect at the beginning, the pool of individuals on whom selection can act remains miniscule because the mutation rate is extremely low. The number of relevant individuals in reasonably sized interbreeding groups could be so small that evolution for dominance, besides being slow, would run the risk of being overwhelmed by selection acting on the modifier gene itself.^7,8 Fisher discounted this possibility: he thought that populations would always be large enough and that one could envisage modifiers that had no direct effect on their own. It can be shown that if the modifier evolves because of a direct effect on fitness, selection for dominance need not be hampered by the fact that the mutation rate, or the pool of heterozygotes, is small.¹³

Haldane⁶ also criticized Fisher’s theory on grounds somewhat similar to those of Wright. He put forward an alternative explanation in which dominance had coevolved as a byproduct of selection in favour of buffering the genotype against environmental or genetic perturbations. Selection for buffering would gradually change the phenotypic value of the mutant heterozygote to the same level as in the wild-type homozygote. Much later, Haldane¹⁴ drew on his investigations into the evolution of industrial melanism (in which the melanic form is both advantageous and dominant) and suggested that dominance might have evolved from a low level, conceivably by a Fisher-type process, concomitantly with the spread of the allele. Note that in Haldane’s scheme the pool of heterozygotes is large and keeps increasing. However, in common with Fisher’s model both his proposals share the assumption that selection acts on the heterozygote.

The criticisms made by Wright and Haldane were on the grounds of plausibility. Besides that, there are two problems with Fisher’s model, both based on observational grounds. The first is that it does not explain why, in the heterozygous state, the presence of a strongly deleterious allele is not as apparent as the presence of a weak allele. In other words, the model cannot account for the observed negative correlation between the degree of dominance and the fitness of the mutant homozygote.¹⁵ The second problem is more serious. Fisher included in his explanation of dominance a feature that almost precluded any possibility of experimental verification, namely that the selection of modifier genes would require an enormous number of generations before the heterozygous phenotype became indistinguishable from that of the normal homozygote. This would mean that one could not study successive generations for a sufficiently long time to observe the gradual evolution of dominance.

Nonetheless, the model does lead to a specific prediction that is open to testing in an appropriate organism. It is obvious that concepts such as dominance and recessivity have no meaning in haploid organisms; in addition, less obviously, Fisher’s model implies that modifier genes for dominance, and hence the evolution of dominance, cannot occur in haploid organisms, and no dominance should be observed in a haploid organism that passes through occasional diploid or polyploid generations. Chlamydomonas reinhardtii is such an organism: it exists normally as a haploid organism, with one nucleus per cell, and with no more than a single copy of each chromosome, but it also has diploid or polyploid generations only occasionally. These occur too infrequently for the organism to experience the consequences of heterozygosis to drive the selection of suitable modifier genes. If Fisher’s hypothesis is correct, therefore, one should see no sign of dominance in diploid generations of C. reinhardtii.

What is observed is quite different. Orr¹⁶ examined numerous mutations in diploid cells and found that the great majority were recessive, exactly as would be expected from the theory of Kacser and Burns⁹ developed from Wright’s ideas, but inconsistent with that of Fisher. Obviously, if a property that can only evolve if it is selected in diploid (or polyploid) cells proves to be exactly the same in a species that is nearly always haploid, an explanation for that property that requires the longterm persistence of diploidy cannot be a universal explanation. Orr noted that his observations disposed not only of Fisher’s model but also of the two proposed by Haldane.⁶ Haldane’s models also required selection in heterozygotes, and therefore could not explain why mutant genes are as likely to be recessive in the very rare heterozygotes of a principally haploid species like C. reinhardtii as they are in species that are always diploid.

Wright’s theory: dominance as a physiological phenomenon

We have seen how Wright⁷ considered that the selection pressure would be too weak for Fisher’s explanation to work, and later8 he pointed out that the modifier genes were themselves subject to mutation, and thus in need of their own modifier genes to protect them from the effects of such mutations, and so on, with obvious possibilities of infinite regress. He argued that an explanation for dominance should be sought in mechanistic terms, meaning in terms of immediate physiology. Even with care to avoid reading more into Wright’s words than he intended, it is clear that his statement that the curve expressing the relation of the product to enzyme amount is a hyperbola, asymptotic at its upper limit. Doubling the quantity of enzyme will less than double the amount of product (in which the italicized words and phrases replace algebraic symbols in the original) contained two important ideas. First, there is nothing here that can be interpreted as a reference to genes, so we are dealing with an explanation at a physiological or biochemical level; second, he recognizes at the outset that biochemical responses to enzyme levels are nonlinear.

As we shall see, this makes his explanation of recessivity far closer to current ideas than anything that Fisher or Haldane wrote, even though Haldane was much better informed about biochemistry in general and enzymes in particular than either of his rivals: his book,¹⁷ written at about the time of the controversy, is a classic account of enzymes that is still worth reading for its essential insights more than 70 years afterwards.

Nonetheless, it was Fisher’s ideas that were generally accepted, not those of Haldane or Wright. As late as 1966, Wright’s view of recessivity as a physiological phenomenon was rejected as impossible;¹⁸ and in 1981 an important book¹⁹ by an author justifiably famous for the clarity and cogency of his writing could still contain some obscure pages presenting a vague exposition of Fisher’s ideas barely more intelligible or convincing than Fisher’s own. In the same year, however, Kacser and Burns⁹ provided a fully modern version of the physiological theory, in which they showed that dominance and recessivity follow automatically from the known kinetic behaviour of enzymes in isolation and when embedded in metabolic pathways.

Effect of enzyme activity on metabolic flux

For almost any enzyme in isolation, for example in a spectrophotometric assay, the rate of reaction at specified concentrations of substrates, products and any inhibitors or activators present in the mixture is proportional (at least approximately) to the concentration of the enzyme: if the activity of enzyme is doubled, the rate is doubled. Exceptions occur if the enzyme exists in several different states of association with different degrees of catalytic activity, and the experiment is done in the concentration range where the degree of association varies. Other exceptions arise if the enzyme is unstable at high dilution, a problem common enough in kinetic studies to oblige experimenters to take precautions to handle it properly. However, it is an artificial problem in the physiological context, because the high enzyme dilutions commonly used to make steady-state rates slow enough to study conveniently in the spectrophotometer are themselves unphysiological. Nonetheless, the usual proportional dependence of rate on enzyme concentration observed in steady-state kinetic experiments is misleading as a guide to behaviour in the cell, because in the cell the concentrations of substrates, products and so on are not constants fixed by an external agent, the experimenter, but variables that depend on the activities not only of the enzyme of interest but of all the enzymes in the system. The product of almost any enzyme is also the substrate of one or more other enzymes; the substrate of almost any enzyme is also the product of one or more other enzymes.

Fig. 1. Effect of an abrupt change in enzyme activity in a system of two enzymes in steady state. The system considered is shown in the inset at bottomright. In the initial steady state the two rates v₁ and v₂ are equal to one another at some arbitrary value J₀. If the activity of the second enzyme is abruptly decreased by 50%, this must produce a corresponding abrupt decrease in v₂ to 0.5J₀, but the system is then no longer in steady state because v₁ > v₂, and so the intermediate S is being released faster than it is being consumed. Its concentration [S] must therefore increase, with two effects: as S is the product of the first enzyme its increased concentration increases the product inhibition and so v₁ decreases; at the same time it is the substrate of the second enzyme, and so by the usual effects of substrates on enzymes the rate v₂ increases. Eventually a new steady state is reached in which the two rates are again equal, at a value smaller than J₀ but larger than 0.5J₀.

Consider now what will happen in response to a twofold decrease in the activity of the second of two enzymes in a pathway that constitutes the whole of a two-enzyme system (Fig. 1). The immediate effect will be the loss of any steady state that existed before the change, because the first enzyme will initially continue to work at the original rate, with the second working at only half that rate. The common intermediate, the product of the first enzyme and substrate of the second, is thus no longer in steady state, because it is being produced faster than it is being consumed: its concentration must therefore increase, but that has its own effects, increased inhibition of the first enzyme and increased saturation of the second. Eventually these will produce a new steady state in which the rates are again balanced, but notice that the increased inhibition of the first enzyme and increased saturation of the second mean that the new steady-state rate must be less than the original rate but greater than half the original rate. In other words the effect of decreasing the activity of the second enzyme is less than proportional. This result is general, applying to changes more or less than twofold, to changes in the activity of the first enzyme as well as to that of the second, and to systems with any number of enzymes greater than one. In all these cases the result of changing any enzyme activity is a less-than-proportional change in the steady-state flux through the system.

The original analysis was made by Kacser and Burns²⁰ before they analysed dominance and recessivity, and is now discussed in textbooks,^21,22 so here it is sufficient to give the main result. The effect on the metabolic flux of any enzyme activity can be expressed as a flux control coefficient, defined as the logarithmic derivative of the flux J with respect to a perturbation p divided by the logarithmic derivative of the enzyme activity v_i with respect to the same perturbation p:

(1) C^J_i = (∂lnJ/∂lnp)/(∂lnv_i/∂lnp) = (∂lnJ/∂ln v_i)

The anonymous perturbation p is introduced to avoid the mathematical looseness in the simpler form at the right derived from the fact that v_i is not a true independent variable of the system; the simpler form is acceptable as long as it is remembered that an external perturbation is implied even if not specified. Moreover, to the extent that enzyme activities are proportional to enzyme concentrations one can regard the enzyme concentration as the perturbation p: this was done in the original paper of Kacser and Burns²⁰ but in more recent work most authors have preferred to avoid an assumption that is not only unnecessary but also, more seriously, was responsible for a widespread misconception that metabolic control analysis deals only with changes in activity brought about by changes in enzyme concentration. By a more rigorous analysis along the lines of what was done above for a two-enzyme system, one can readily prove the fundamental property of flux control coefficients expressed by the following equation:

(2) Σⁿ_i=1 C^J_i = 1

in which n is the number of enzymes in the system. This is called the summation property for flux control coefficients, and expresses the fundamental idea that flux control is shared among all the enzymes in the system. As even the simplest real cell contains hundreds of different enzymes the average share must be very small. More realistically, perhaps, noting that most enzymes have a very small share of the control of flux through pathways other than those in which they occur, one can say that most of the control of the flux through any pathway is shared among the enzymes of that pathway. This still implies an average share of no more than 20% for most pathways, and often less.

There is a complication in this argument that needs to be made explicit. In referring to shares we are tacitly assuming that all the control coefficients in the sum shown in eqn. (1) are positive, but that is not strictly true. The flux control coefficient of an enzyme for a flux through a part of the metabolic network not in series with the reaction that it catalyses can certainly be negative, and often is. However, in nearly all real cases such negative flux control coefficients are small enough to have little effect on the general argument. Instead of saying that control of flux through any pathway is exactly shared among all the enzymes of the system we must say that most of it is shared among the enzymes of the pathway. This statement is not very different from what we said in the previous paragraph.

Before leaving the question of negative control coefficients, we should note that these can be defined for other system variables as well as fluxes; for example, the effect of enzyme activities on any metabolite concentration [S] can be expressed by concentration control coefficients with definitions very similar to that in eqn. (1). The resulting quantities are often not merely negative but large in magnitude as well, and the summation property corresponding to eqn. (2) has a righthand side of zero, not unity. In such a case the idea of sharing loses all meaning, so we should emphasize that the idea of sharing flux control comes not merely from the existence of a summation property but from two additional points, one theoretical and general, the other from observation: first the sum is unity; second, the individual elements in the sum are mostly positive and, when negative, are normally small. On a preliminary view the summation property in eqn. (2) already provides the explanation of dominance and recessivity, as it suggests that changing any enzyme activity will produce a much less than proportional change in metabolic flux, and hence a much less than proportional change in metabolic output, such as the amount of a pigment produced by a pea plant. However, that is too simple, as it applies only to small — strictly infinitesimal — changes in activity, whereas we need to consider the effects of changes in activity of the order of twofold. For increases in activity the naive view remains valid, but for decreases, which are more relevant to heterozygotes in diploid organisms, it does not, because any flux control coefficient typically increases when the enzyme activity is decreased, eventually reaching a value of 1 when the activity is close to zero.

Both simple models and experimental observations in numerous cases indicate that for finite changes in activity the curve relating metabolic flux to the activity of any enzyme in the pathway resembles a rectangular hyperbola. In the simplest case where all enzymes operate far enough below saturation for the kinetics with respect to to their substrates to be of first-order the curve is exactly a rectangular hyperbola; it remains quite similar to one in the more realistic case where some or all of the enzymes operate in a range with detectable saturation. In Michaelis–Menten terms this implies substrate concentrations of the order of the relevant Michaelis constants. What eqn. (2) now means is that most enzymes are located near or on the flat part of the hyperbola in the state corresponding to the normal homozygote. If any particular enzyme happens to have a flux control coefficient close to unity — unusual, but not impossible — then all the others in the pathway need to move further to the right along the curve.

Consider now the expected metabolic fluxes J_AA, J_Aa and J_aa for the normal homozygote, heterozygote and abnormal homozygote respectively of the gene for an enzyme normally located in the typical position on the hyperbola, as shown in Fig. 2. It is immediately evident not only that J_AA ≅ J_Aa >> J_aa = 0, but also, more important, that this relation will survive even quite large variations in the assumption about where the normal enzyme lies on the curve: it can be displaced any distance to the right without invalidating the relationship, and significantly to the left also.

This, in essence, is the explanation of dominance and recessivity proposed by Kacser and Burns (1981). It may be objected that it predicts only that the heterozygote phenotype will be more similar to the normal homozygote phenotype than to the abnormal homozygote phenotype, not that it will be identical to it. This is true, but in practice the difference would not be likely to have been noticed in Mendel’s observations, or indeed in most observations of gross phenotypes since then. In summary, therefore, the theory of Kacser and Burns (1981), which was an elaboration and extension of that suggested by Wright (1934), is the only one currently available that provides a satisfactory explanation of dominance and recessivity.