Effects of mutation on selection limits in finite populations with multiple alleles

The ultimate response to directional selection (i.e., the selection limit) under recurrent mutation is analyzed by a diffusion approximation for a population in which there are k possible alleles at a locus. The limit mainly depends on two scaled parameters S (= 4Ns sigma a) and theta (= 4Nu) and k, the number of alleles, where N is the effective population size, u is the mutation rate, s is the selection coefficient, and sigma 2a is the variance of allelic effects. When the selection pressure is weak (S less than or equal to 0.5), the limit is given approximately by 2S sigma a[1 - (1 + c2)/k]/(theta + 1) for additive effects of alleles, where c is the coefficient of variation of the mutation rates among alleles. For strong selection, other approximations are devised to analyze the limit in different parameter regions. The effect of mutation on selection limits largely relies on the potential of mutation to introduce new and better alleles into the population. This effect is, however, bounded under the present model. Unequal mutation rates among alleles tend to reduce the selection limit, and can have a substantial effect only for small numbers of alleles and weak selection. The selection limit decreases as the mutation rate increases.