Average time until fixation of a mutant allele in a finite population under continued mutation pressure: Studies by analytical, numerical, and pseudo-sampling methods

Abstract

We consider a single locus, and denote by A the wild-type allele and by A' the mutant allele that is produced irreversibly in each generation from A at the rate v. Let 1 + s, 1 + h, and 1 be, respectively, the relative fitnesses of mutant homozygote A'A', mutant heterozygote A'A, and wild-type homozygote AA. Then, it is shown, on the basis of the diffusion equation method, that the average time until fixation of the mutant allele (A') in a randomly mating population of effective size N(e), given that the initial frequency is p, is [Formula: see text] in which B(x) = (S/2)x(2) + Hx(1 - x), S = 4N(e)s, H = 4N(e)h, and V = 4N(e)v. Of particular interest are the cases in which the mutant allele is deleterious (s = -s', s' > 0). Three cases are considered; the mutant is: (i) completely dominant s = h = -s', (ii) completely recessive s = -s', h = 0, and (iii) semidominant s = -s', h = -s'/2, in which s' is the selection coefficient against the mutant homozygote. It is shown that the average time until fixation is shorter when the deleterious mutant allele is dominant than when it is recessive if 4N(e)v is larger than 1. On the other hand, the situation is reversed if 4N(e)v is smaller than 1. It is also shown that for a mutant allele for which N(e)s' > 10, it takes such a long time until fixation that we can practically ignore the occurrence of random fixation of a deleterious allele under continued mutation pressure. To supplement the analytical treatment, extensive simulation experiments were performed by using a device called the pseudo-sampling variable, which can enormously accelerate the process of simulation by a computer. This method simulates the diffusion process itself rather than the binominal sampling process (in population genetics the diffusion model is usually regarded as an approximation of the discrete binomial sampling process).