Selection for heterozygotes in small populations.

I N his mathematical analyses of evolutionary processes WRIGHT ( 193 1, 1937) has given particular attention to the interaction of population size with the factors of selection, mutation and migration. Here he introduced the powerful concept of the "gene frequency distribution" which may be thought of as referring either to similar loci in the same population or one locus in many populations. In the absence of mutation and migration, he showed that the gene frequency distribution assumes a constant form with all frequencies, except the two extremes, declining by a constant fraction each generation. In the absence of selection, the distribution becomes a rectangular one and intermediate frequencies decline by a fraction 1/2N each generation, where N is the effective population size. With both selection and mutation, he showed that an equilibrium distribution is reached in which the fixation of populations by chance is exactly balanced by those segregating anew because of mutation, and he derived a general formula for this distribution. With no selection the distribution takes the iorm +(q)+ ( q (1 q))"""-' where q is the gene frequency and U the mutation rate (if the latter is assumed equal for both alleles). The mean frequency of heterozygotes is then given by 4Nu/(8Nv f 1) and is therefore proportional to the population size, if Nv is small. The analysis of the effect of particular kinds of selection has been almost restricted to the case of constant decline without mutation, There WRIGHT and KERR (1954) and KIMURA (1957) have dealt with genes having an additive effect on selective advantage and KIMURA also dealt with recessive or dominant genes. But, in the study of genetic variation, by far the most interesting situation is that in which the heterozygote has a selective advantage over both homozygotes. REEVE (1955) has analyzed the effect of heterozygous advantage in very small populations, using the method of mating type matrices, but this method becomes cumbersome for all but the simplest mating systems. All these analyses have been concerned with the effects of differences in selective advantage within the inbred line or population. HAYMAN and MATHER (1953) dealt with the matter rather more generally by also taking into account competition between sublines of restricted size. The two situations differ fundamentally in that if we are dealing only with differences within a line, there is always a finite chance, with the exception of lethals balanced against one another,