title: Exact Markov chain and approximate diffusion solution for haploid genetic drift with one-way mutation.
abstract: The classical Kimura solution of the diffusion equation is investigated for a haploid random mating (Wright-Fisher) model, with one-way mutations and initial-value specified by the founder population. The validity of the transient diffusion solution is checked by exact Markov chain computations, using a Jordan decomposition of the transition matrix. The conclusion is that the one-way diffusion model mostly works well, although the rate of convergence depends on the initial allele frequency and the mutation rate. The diffusion approximation is poor for mutation rates so low that the non-fixation boundary is regular. When this happens we perturb the diffusion solution around the non-fixation boundary and obtain a more accurate approximation that takes quasi-fixation of the mutant allele into account. The main application is to quantify how fast a specific genetic variant of the infinite alleles model is lost. We also discuss extensions of the quasi-fixation approach to other models with small mutation rates.
authors.affiliation: Department of Mathematics, Div. of Mathematical Statistics, Stockholm University, Stockholm SE 106 91, Sweden. Electronic address: ola@math.su.se.
authors.forename: Ola
authors.initials: O
authors.lastname: Hössjer
authors.affiliation: Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, Ås 1432, Norway. Electronic address: peder.tyvand@nmbu.no.
authors.forename: Peder A
authors.initials: PA
authors.lastname: Tyvand
authors.affiliation: Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel. Electronic address: miloh@eng.tau.ac.il.
authors.forename: Touvia
authors.initials: T
authors.lastname: Miloh
mesh_terms: D040961:Genetic Drift; D006238:Haploidy; D008390:Markov Chains; D008957:Models, Genetic; D009154:Mutation
keywords: Diffusion; Jordan decomposition; Markov chain; Mutation; Quasi-fixation; Wright–Fisher model