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An approximate stationary solution for multi-allele neutral diffusion with low mutation rates

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  • Burden, Conrad J.
  • Tang, Yurong

Abstract

We address the problem of determining the stationary distribution of the multi-allelic, neutral-evolution Wright–Fisher model in the diffusion limit. A full solution to this problem for an arbitrary K×K mutation rate matrix involves solving for the stationary solution of a forward Kolmogorov equation over a (K−1)-dimensional simplex, and remains intractable. In most practical situations mutations rates are slow on the scale of the diffusion limit and the solution is heavily concentrated on the corners and edges of the simplex. In this paper we present a practical approximate solution for slow mutation rates in the form of a set of line densities along the edges of the simplex. The method of solution relies on parameterising the general non-reversible rate matrix as the sum of a reversible part and a set of (K−1)(K−2)/2 independent terms corresponding to fluxes of probability along closed paths around faces of the simplex. The solution is potentially a first step in estimating non-reversible evolutionary rate matrices from observed allele frequency spectra.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:thpobi:v:112:y:2016:i:c:p:22-32
    DOI: 10.1016/j.tpb.2016.07.005
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    References listed on IDEAS

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    1. Vogl, Claus, 2014. "Estimating the scaled mutation rate and mutation bias with site frequency data," Theoretical Population Biology, Elsevier, vol. 98(C), pages 19-27.
    2. Vogl, Claus & Bergman, Juraj, 2015. "Inference of directional selection and mutation parameters assuming equilibrium," Theoretical Population Biology, Elsevier, vol. 106(C), pages 71-82.
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    Cited by:

    1. Wenkai Huang & Feng Zhan, 2023. "A Novel Probabilistic Diffusion Model Based on the Weak Selection Mimicry Theory for the Generation of Hypnotic Songs," Mathematics, MDPI, vol. 11(15), pages 1-26, July.
    2. Schrempf, Dominik & Hobolth, Asger, 2017. "An alternative derivation of the stationary distribution of the multivariate neutral Wright–Fisher model for low mutation rates with a view to mutation rate estimation from site frequency data," Theoretical Population Biology, Elsevier, vol. 114(C), pages 88-94.
    3. Burden, Conrad J. & Griffiths, Robert C., 2018. "Stationary distribution of a 2-island 2-allele Wright–Fisher diffusion model with slow mutation and migration rates," Theoretical Population Biology, Elsevier, vol. 124(C), pages 70-80.
    4. Burden, Conrad J. & Tang, Yurong, 2017. "Rate matrix estimation from site frequency data," Theoretical Population Biology, Elsevier, vol. 113(C), pages 23-33.
    5. Vogl, Claus & Mikula, Lynette C. & Burden, Conrad J., 2020. "Maximum likelihood estimators for scaled mutation rates in an equilibrium mutation–drift model," Theoretical Population Biology, Elsevier, vol. 134(C), pages 106-118.

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