title: The non-equilibrium allele frequency spectrum in a Poisson random field framework.
abstract: In population genetic studies, the allele frequency spectrum (AFS) efficiently summarizes genome-wide polymorphism data and shapes a variety of allele frequency-based summary statistics. While existing theory typically features equilibrium conditions, emerging methodology requires an analytical understanding of the build-up of the allele frequencies over time. In this work, we use the framework of Poisson random fields to derive new representations of the non-equilibrium AFS for the case of a Wright-Fisher population model with selection. In our approach, the AFS is a scaling-limit of the expectation of a Poisson stochastic integral and the representation of the non-equilibrium AFS arises in terms of a fixation time probability distribution. The known duality between the Wright-Fisher diffusion process and a birth and death process generalizing Kingman's coalescent yields an additional representation. The results carry over to the setting of a random sample drawn from the population and provide the non-equilibrium behavior of sample statistics. Our findings are consistent with and extend a previous approach where the non-equilibrium AFS solves a partial differential forward equation with a non-traditional boundary condition. Moreover, we provide a bridge to previous coalescent-based work, and hence tie several frameworks together. Since frequency-based summary statistics are widely used in population genetics, for example, to identify candidate loci of adaptive evolution, to infer the demographic history of a population, or to improve our understanding of the underlying mechanics of speciation events, the presented results are potentially useful for a broad range of topics.
authors.affiliation: Department of Mathematics, Uppsala University, Uppsala, Sweden. Electronic address: ikaj@math.uu.se.
authors.forename: Ingemar
authors.initials: I
authors.lastname: Kaj
authors.affiliation: Department of Ecology and Genetics, Uppsala University, Uppsala, Sweden. Electronic address: carina.mugal@ebc.uu.se.
authors.forename: Carina F
authors.initials: CF
authors.lastname: Mugal
mesh_terms: D005787:Gene Frequency; D005828:Genetics, Population; D008957:Models, Genetic; D011110:Polymorphism, Genetic
keywords: Coalescent theory; Duality relation; Natural selection; Non-equilibrium allele frequency spectrum; Poisson random field; Population genetics