Mechanistic Explanations for Dominance The Case of Metabolism

During the early years of the twentieth century, one of the more prominent explanations for dominance, advocated by W. Bateson, R. C. Punnett, and others, was the 'presence-and-absence' hypothesis. Consider a situation in which a phenotype or trait comes about as a result of the action of a set of gene products. One can conceive of a situation in which each gene i plays a role X for the formation of a given phenotype P. The presence-absence hypothesis assumes that as long as there is one functional copy of gene i, then the role Xj is satisfied, and hence P remains at wild-type levels. With advances in physiological genetics, and subsequent advances in molecular biology, the presence-absence hypothesis has been abandoned. Nonetheless, we should note that the prime weakness in the presence-absence hypothesis was not necessarily its incomplete representation of the underlying mechanisms; all mechanistic representations of genotype-phenotype relations are at some level incomplete. By the standards of today, the presence-absence hypothesis is more a logical argument than a detailed exposition on mechanism. However, its major failure was that as a phenomenological description, it failed to fit much of the mounting experimental evidence; it was not general enough to encompass many exceptions.

Given present knowledge of the variety of molecular processes occurring in the cell, such as gene regulation, signal transduction, and metabolism, it would now seem unreasonable to use any specific mechanistic model as a general model for dominance modification. Depending on the underlying causes of a given phenotypic trait, the proximal causes for dominance modification and the constraints placed upon it can be very different from one case to the next. Hence, any given molecular explanation may serve as an instantiation of general concepts pertaining to dominance modification, but it is unlikely to serve as the grand model for modification. Nonetheless, if any given modification model is sufficiently well-developed, it can be used to test some general evolutionary questions.

The case study that has been most central to addressing dominance evolution revolves around phenotypes that are directly dependent on the functioning of metabolic enzymes. Shortly after Fisher proposed his hypothesis, both Wright and Haldane argued that the proximal causes for the manifestation of dominance would have to depend on the underlying physiology. At the time, enzymes and their biochemistry were at the forefront of mechanistic explanations for the inner workings of organisms. Accordingly, they also had a central role in Wright and Haldane's explanations. With the assumption that most genes code for enzymes, and the use of a simplified nonlinear model of enzyme pathways, Wright tried to address dominance from a physiological perspective. Based on his model, he hypothesized that the rate of substrate formation throughout a pathway will have a diminishing returns relation to enzyme concentrations (Figure 2 a). Under such circumstances, when enzyme concentrations are in the flat region of the

Enzyme activity (labeled by genotype)

(wild type)

BB Bb

Enzyme activity (labeled by genotype)

bb (wild type)

Figure 2 Physiological model for dominance based on enzyme activity and substrate accumulation. (a) Dominant wild type (Wright's physiological model), (b) recessive wild type (hypothetical model).

BB Bb

Enzyme activity (labeled by genotype)

bb (wild type)

Figure 2 Physiological model for dominance based on enzyme activity and substrate accumulation. (a) Dominant wild type (Wright's physiological model), (b) recessive wild type (hypothetical model).

curve, then reductions in enzyme concentration will have relatively small effects on substrate levels. Accordingly, let us assume that a wild-type homozygote AA has two copies of a gene coding for an enzyme E, while the mutant heterozygote Aa has one functional copy. Furthermore, let us assume that enzyme concentrations are proportional to copy numbers, and hence that the concentration of functional enzyme E, and consequently enzyme activity, is halved in Aa genotypes in comparison to AA. If the AA genotype lies deep in the plateau region of the curve in Figure 2a, then the reduction of enzyme concentrations in Aa genotypes to half of AA would not have an appreciable effect on phenotype, and the wildtype phenotype associated with AA would be considered dominant. In Wright's model, dominance of the wild type results from the convex shape of the relation between enzyme activity and phenotype. On the other hand, a concave curve such as the one in Figure 2b would lead to the wild type being recessive. In the context of Figure 2a, dominance of the wild type would evolve when wild-type enzyme activity is pushed to the flat end of the curve. Conversely, as exemplified by the a9a9 homozygote in Figure 2 a, a phenotype that is associated with the steeper part of the curve will tend toward codo-minance with respect to the null aa homozygote.

The physiological model discussed above ignores compensatory effects such as gene regulation, feedback, redundancy, and nonsequential pathway topologies. Nonetheless, as a model, it is a plausible starting point for addressing questions related to dominance and its evolutionary origins; in fact, it has been central to research on this topic. An important question that arises within this model, is the reason whereby wild-type enzyme activities should be at the high end of the curve. Haldane's suggestion was that organisms could evolve a 'factor of safety' against perturbations. Such perturbations could in principle be either environmental or genetic. Another explanation could be that high enzyme activities are simply required for individual fitness, and because of the convex shape of the curve, dominance evolves as a side effect of selection for high enzyme activity.

As an alternative explanation for dominance, Kacser and Burns suggested that in metabolic systems, one could explain dominance without making recourse to evolutionary explanations. Their argument is based on models that are at first sight similar to Wright's model. However, there is a fundamental difference. In Wright's model, levels of dominance can be modified by changing the enzyme activity associated with the 'wild-type' homozygote. The Kacser and Burns model (from here on abbreviated as the KB model) places a crucial constraint on this modification. In the latter case, dominance can be modified at any given locus in the same sense as in Wright's model. However, the KB model also suggests that modifications that result in increased phenotypic sensitivity to any enzyme are compensated by decreased sensitivity to other enzymes. Furthermore, the model also implies that the more enzymes are involved in a pathway, the smaller the average effect ofeach enzyme on the phenotype. Based on these premises, Kacser and Burns concluded that dominance - which depends on insensitivity of the phenotype to enzyme concentration changes - is a de facto property of metabolic pathways and that evolutionary explanations based on dominance modification are not necessary.

Many of the mechanistic details of Wright's model do not reflect what is known of enzyme kinetics today. However, his model does preserve some of the nonlinear aspects of chemical transformations. On the other hand, the KB model is more akin to modern conceptions of enzyme kinetics. However, the model contains some approximations that eliminate important nonlinearities in enzyme kinetics - namely, enzyme saturation. The consequence is that the scope of the KB model is much more limited than originally intended and, in hindsight, not a good candidate for investigating the evolutionary origin of dominance. For example, Bagheri and Wagner have suggested that if one considers simple models of enzyme kinetics, with the possibility of enzyme saturation, one finds that system robustness or dominance is not a de facto property of enzyme systems. In particular, the compensatory constraints on dominance modification that are postulated in the KB model are eliminated when the system allows for saturation. In fact, the modulation of enzyme saturation levels allows for dominance modification, and hence, dominance evolution. Figure 3 shows an example of a ten-enzyme pathway, which contrary to the KB model is modeled with the possibility of enzyme saturation. The pathway consists of a constant input and an output substrate S10, whose rate of production (flux, dS10/dt) is the phenotype of interest (Figure 3a). As indicated by Figure 3b, each enzyme-catalyzed reaction can be conceptualized as a process by which a substrate and an enzyme join to form an enzymesubstrate complex, which subsequently dissociates into a product and the unaltered enzyme. Saturation levels can be altered by mutations that either change concentration levels or the dissociation constant kcat. Flux for such a model can be obtained by numerical integration of the resulting system of differential equations. Figure 3 c shows the flux function for two realizations of a ten-enzyme pathway. The green lines trace a pathway whose enzymes possess relatively high kcat values. The black lines trace a pathway with low kcat values. Within each case, flux responses to changes in enzymes 2-10 are very similar, and the corresponding curves are almost indistinguishable. Responses to enzyme 1 are slightly more sensitive. What is important to note in this figure is that we are observing two different endpoints of possible evolutionary trajectories. In the low kcat case, the wildtype phenotype is codominant with respect to mutations that halve the concentration of 'any' enzyme. In contrast, in the high kcat case, the wild-type phenotype is dominant with respect to mutations that halve the concentration of any enzyme. Hence, one can conceptualize dominance evolution as an evolutionary process whereby kcat levels have been gradually modified from low to high values. High kcat values result in low enzyme saturation levels, and consequently, higher dominance levels for the wild type.

A further consequence of the model in Figure 3 c is the important observation that there is a slight difference between the wild-type AA phenotypes of the high and low kcat pathways. This means that if one is selecting for the high-flux phenotype, one can select modifiers due to their direct fitness effects rather than their dominance modification effects. Direct effects can be relatively small, but they are not dependent on the frequency of a alleles. Meanwhile, modifying effects can be larger, but nonetheless they occur less frequently due to their dependence on the frequency of a alleles. Hence, selection for modifiers with multiple effects would circumvent some of the frequency sensitivity problems associated with modifier evolution. A schematic drawing of such a scenario is shown in Figure 4, which corresponds to the enzyme kinetic model in Figure 3 . Dominance evolution in such a scenario is more likely to be successful than in cases

(Wild type is dominant) (Wild type is codominant)

Figure 3 Model of dominance modification for the flux phenotype of a ten-enzyme pathway. (a) Ten-enzyme sequential pathway, (b) details of each enzyme-catalyzed reaction, (c) flux phenotype of pathway as a function of enzyme concentrations.

Fitness

Phenotype aa Aa AA

aa Aa AA

Initial condition aa Aa AA

Evolved condition

Figure 4 Example of dominance evolution due to a modifier with multiple effects.

where modifiers only have a dominance modification effect (e.g., Figure 1a). Interestingly, despite the lack of an explicit enzyme kinetic representation, the mathematical properties of Wright's model are similar to the enzyme kinetic model presented here; namely, Wright's model allows for the possibility of dominance evolution through multiple effects, while the KB model does not.

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