Fixation of a deleterious allele under mutation pressure and finite selection intensity
Résumé
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. U.S.A. 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' produced irreversibly from A at small uniform rate . The relative fitnesses of the mutant homozygotes A' A', mutant heterozygotes A' A and wild-type homozygotes A A are 1-s , 1-h and 1, respectively, where it is assumed that ≪. 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 >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 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 Ns≪1, where N is the effective population size.
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