Homozygosity of selected alleles in subdivided populations

The mean homozygosity is a useful measure of genetic variation in a natural population and it is of interest to determine the effects of migration and selection on this quantity. It has been shown that some features of the mean homozygosity are independent of migration patterns when alleles are selectively neutral. Here, we examine the mean homozygosity in a structured population to determine if these features are still independent in the presence of selection. We consider the equilibrium mean homozygosity within a single finite subpopulation for a gene with K different allelic types. Each possible genotype is assigned a distinct selection coefficient. Usually the mean homozygosity cannot be explicitly determined in a structured population when selection is operating, even for fairly simple migration patterns, but an approximation can be made. We have found four properties of the mean homozygosity that are independent of migration patterns even when selection is acting. These are the values of the mean homozygosity and of its derivative with respect to the total migration rate into the subpopulation, when migration rates are either very small or very large. These four properties can be combined into a single continuous approximation that is applicable to any migration scheme. The approximation is found to be very accurate when compared to the exact solutions for several population structures and selection schemes.