Fixation of a deleterious allele under mutation pressure and finite selection intensity

The mean fixation time of a deleterious mutant allele is studied beyond the diffusion approximation. As in Kimura's classical work [M. Kimura, Proc. Natl. Acad. Sci. USA. 77, 522 (1980)], that was motivated by the problem of fixation in the presence of amorphic or hypermorphic mutations, we consider a diallelic model at a single locus comprising a wild-type A and a mutant allele A′A′ produced irreversibly from A at small uniform rate v  . The relative fitnesses of the mutant homozygotes A′A′A′A′, mutant heterozygotes A′AA′A and wild-type homozygotes AAAA are 1−s, 1−h   and 1, respectively, where it is assumed that v⪡sv⪡s. Here, we employ a WKB theory and directly treat the underlying Markov chain (formulated as a birth–death process) obeyed by the allele frequency (whose dynamics is prescribed by the Moran model). Importantly, this approach allows to accurately account for effects of large fluctuations. After a general description of the theory, we focus on the case of a deleterious mutant allele (i.e. s>0s>0) and discuss three situations: when the mutant is (i) completely dominant (s=h); (ii) completely recessive (h=0), and (iii) semi-dominant (h=s/2). Our theoretical predictions for the mean fixation time and the quasi-stationary distribution of the mutant population in the coexistence state, are shown to be in excellent agreement with numerical simulations. Furthermore, when s   is finite, we demonstrate that our results are superior to those of the diffusion theory, while the latter is shown to be an accurate approximation only when Nes2⪡1Nes2⪡1, where Ne is the effective population size.