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The non-equilibrium allele frequency spectrum in a Poisson random field framework

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  • Kaj, Ingemar
  • Mugal, Carina F.

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.

Suggested Citation

  • Kaj, Ingemar & Mugal, Carina F., 2016. "The non-equilibrium allele frequency spectrum in a Poisson random field framework," Theoretical Population Biology, Elsevier, vol. 111(C), pages 51-64.
  • Handle: RePEc:eee:thpobi:v:111:y:2016:i:c:p:51-64
    DOI: 10.1016/j.tpb.2016.06.003
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    References listed on IDEAS

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    1. Chen, Hua, 2012. "The joint allele frequency spectrum of multiple populations: A coalescent theory approach," Theoretical Population Biology, Elsevier, vol. 81(2), pages 179-195.
    2. Živković, Daniel & Stephan, Wolfgang, 2011. "Analytical results on the neutral non-equilibrium allele frequency spectrum based on diffusion theory," Theoretical Population Biology, Elsevier, vol. 79(4), pages 184-191.
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