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A Wright–Fisher graph model and the impact of directional selection on genetic variation

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  • Kaj, Ingemar
  • Mugal, Carina F.
  • Müller-Widmann, Rebekka

Abstract

We introduce a multi-allele Wright–Fisher model with mutation and selection such that allele frequencies at a single locus are traced by the path of a hybrid jump–diffusion process. The state space of the process is given by the vertices and edges of a topological graph, i.e. edges are unit intervals. Vertices represent monomorphic population states and positions on the edges mark the biallelic proportions of ancestral and derived alleles during polymorphic segments. In this setting, mutations can only occur at monomorphic loci. We derive the stationary distribution in mutation–selection–drift equilibrium and obtain the expected allele frequency spectrum under large population size scaling. For the extended model with multiple independent loci we derive rigorous upper bounds for a wide class of associated measures of genetic variation. Within this framework we present mathematically precise arguments to conclude that the presence of directional selection reduces the magnitude of genetic variation, as constrained by the bounds for neutral evolution.

Suggested Citation

  • Kaj, Ingemar & Mugal, Carina F. & Müller-Widmann, Rebekka, 2024. "A Wright–Fisher graph model and the impact of directional selection on genetic variation," Theoretical Population Biology, Elsevier, vol. 159(C), pages 13-24.
    • Handle: RePEc:eee:thpobi:v:159:y:2024:i:c:p:13-24
    DOI: 10.1016/j.tpb.2024.07.004
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    References listed on IDEAS

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    1. Burden, Conrad J. & Tang, Yurong, 2016. "An approximate stationary solution for multi-allele neutral diffusion with low mutation rates," Theoretical Population Biology, Elsevier, vol. 112(C), pages 22-32.
    2. Ferguson, Jake M. & Buzbas, Erkan Ozge, 2018. "Inference from the stationary distribution of allele frequencies in a family of Wright–Fisher models with two levels of genetic variability," Theoretical Population Biology, Elsevier, vol. 122(C), pages 78-87.
    3. 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.
    4. Pontz, Martin & Feldman, Marcus W., 2020. "Loss of genetic variation in the two-locus multiallelic haploid model," Theoretical Population Biology, Elsevier, vol. 136(C), pages 12-21.
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