Social selection in human populations: sufficient conditions for protection of deleterious alleles in a subdivided population

Population dynamics of wild type (A1) and the deleterious genes (A2) under social selection have been studied by considering a subdivided population where the i-th subpopulation consists of Ni individuals with relative size ci (= Ni/sigma i Ni, i = 1,2, ..., n). A social selection model is constructed by assuming that the fitness of an individual is determined by its own as well as the parental phenotypes and that the number of migrants (M) from the ith subpopulation is divided equally into other subpopulations including the ith subpopulation itself. It has been shown that the gene frequency change depends on the loss of fitness of an individual due to the trait (gamma), an affected parent in the ith subpopulation (beta i), the probability that the heterozygote develops the trait (h), and the migration rates mi (= M/Ni). For 0 less than h less than or equal to 1, a sufficient condition for protection of the deleterious allele from extinction also depends on all of these parameters. However, when mi much less than 1 for all i, the condition is beta i less than gamma/(1 - gamma) for some i, whereas when mi much greater than hgamma + beta i(1 - gamma)] for all i it is given by sigma i ci beta i less than -gamma/(1 - gamma). When h = 0, that condition is given by sigma ici beta i less than - gamma/(1 - gamma). Analyses also show that, when the deleterious alleles in a population are rare, the relative fitnesses of A1A1, A1A2, and A2A2 are given approximately by 1, 1-hS, and 1 - S, respectively, where S is the harmonic mean of Si = gamma + beta i(1 - gamma). Thus, under mutation-selection balance, the equilibrium frequency of deleterious alleles in the entire population is given by alpha/hS for 0 less than h less than or equal to 1 and square root alpha/S for h = 0, where alpha is the irreversible mutation rate from A1 to A2 in each generation. Population dynamics of rare deleterious genes under social selection can readily be studied by considering a finite population size.