Some invariant properties of a geographically structured finite population: distribution of heterozygotes under irreversible mutation

Abstract

SUMMARY: It is shown that the distribution of the sum of heterozygotes, due to mutant gene(s), that appear in a finite population is invariant under geographical structure, provided that the mutant gene has additive effect on fitness and migration does not change the genetic constitution as a whole population. The expected number of heterozygotes is 2N when Ns = 0 and gradually rises to 4N as Ns increases provided s remains small, where N = the total population size and s = selective advantage of a mutant gene. The distribution of the heterozygosity summed over those generations in which the gene frequency in the entire population is specified, is also shown to be invariant. In the case of a neutral mutant, the density is equal to 4(1 − Y) where Y is the frequency of the mutant in the whole population, and in the selectively advantageous case, it is approximately equal to a constant function 4, provided that the population size times selection coefficient is sufficiently large. These quantities conditional on the fixation of the mutant are shown to be invariant and some special cases are obtained explicitly.