A GENERAL MODEL TO ACCOUNT FOR ENZYME VARIATION IN NATURAL POPULATIONS. III. MULTIPLE ALLELES

Recent biochemical explorations into electrophoretic classes of enzyme variants have revealed a wealth of previously suspected but undetected variation (Bernstein et al., 1973; Singh et al., 1975). The existence of this variation poses new problems for the hypothesis that natural selection is maintaining the variation by heterotic selection since the conditions for polymorphism under heterosis become incredibly restrictive as the number of alleles increases (Dr. Ken-ichi Kojima, pers. comm.). One way to quantify this idea would be to ask: Of the total set of all possible fitnesses for the n(n + 1)72 genotypes at a locus with n alleles, what fraction leads to a stable polymorphism with all alleles segregating? This fraction can be easily calculated by a Monte-Carlo simulation on the computer by assigning each of the n(n + 1)72 genotypes a fitness value uniformly and independently distributed on the unit interval and applying the usual criterion for the existence and stability of an internal equilibrium in a population with n alleles (see, e.g. Mandel, 1959). When this experiment is repeated a large number of times, an estimate of the proportion of the fitness space which yields stable internal equilibria results. Figure 1 gives the results of such a simulation and illustrates well the truth of Kojima's statement about the restrictive nature of the heterosis assumption. There is, however, a very unrealistic aspect of this argument. In biological sysstems, certain laws govern the relationships of fitnesses in related genotypes. This aspect is amply born out in the work on the quantitative genetics of fitness components (e.g., Mukai et al., 1972) which gives estimates of the correlation of the heterozygote fitness to the mean of the two parental homozygotes. Although the laws governing the relationships of fitnesses of related genotypes are not known, it is clear they can radically change the impression given by Figure 1. For example, if heterozygotes were always intermediate in fitness between the parental homozygotes, and if the environment were constant, no stable polymorphism would be possible and the subset of stable fitnesses would be empty. On the other hand, if some biological principle dictated that heterozygotes are invariably more fit than their parental homozygotes, the fraction of the space of fitnesses yielding stable points would be much larger than given in Figure 1. Finding constraints on the assignment of fitnesses which are biologically meaningful and which will allow the maintenance of large numbers of alleles should be a major area of investigation now that new genetic variation is being uncovered in natural populations. In this paper, the constraints suggested in the previous papers in this series (Gillespie and Langley, 1974; Gillespie, 1976), will be examined and shown to allow readily the stable existence of large numbers of alleles by balancing selection in a fluctuating environment.