2010/11/30 by Michael Assaf, Mauro Mobilia
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · Physics and Astronomy · Social Sciences · #Allele #Allele frequency #Biology #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #Fixation (population genetics) #Gene #Genetics #Heavy traffic approximation #Locus (genetics) #Markov chain #Mathematical and Theoretical Epidemiology and Ecology Models #Mathematics #Mutant #Population #Statistics #cond-mat.stat-mech #nlin.AO #q-bio.PE #q-bio.QM
paper · pdf · doi:10.1016/j.jtbi.2011.01.025
published as J. Theor. Biol. 275, 93-103 (2011) · 26 pages, 5 figures. Accepted by the Journal of Theoretical Biology
arxiv created 2011/01/19 · openalex publication_date 2011/01/26 · arxiv updated 2011/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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. Vol.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 v. The relative fitnesses of the mutant homozygotes A'A', mutant heterozygotes A'A and wild-type homozygotes AA are 1-s, 1-h and 1, respectively, where it is assumed that v<< s. Here, we adopt an approach based on the direct treatment of the underlying Markov chain (birth-death process) obeyed by the allele frequency (whose dynamics is prescribed by the Moran model), which allows to accurately account for the 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 s is finite, we demonstrate that our results are superior to those of the diffusion theory that is shown to be an accurate approximation only when Ne s2 << 1, where Ne is the effective population size.