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A General Solution of the Wright–Fisher Model of Random Genetic Drift

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Abstract

We introduce a general solution concept for the Fokker–Planck (Kolmogorov) equation representing the diffusion limit of the Wright–Fisher model of random genetic drift for an arbitrary number of alleles at a single locus. This solution will continue beyond the transitions from the loss of alleles, that is, it will naturally extend to the boundary strata of the probability simplex on which the diffusion is defined. This also takes care of the degeneracy of the diffusion operator at the boundary. We shall then show the existence and uniqueness of a solution. From this solution, we can readily deduce information about the evolution of a Wright–Fisher population.

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Acknowledgments

We would like to thank the anonymous referee for helpful comments, in particular suggestions concerning SNPs and mitochondrial DNA, as mentioned in Sect. 4. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 267087.

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Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, 04103, Leipzig, Germany

    Tat Dat Tran, Julian Hofrichter & Jürgen Jost

  2. Department of Mathematics, Leipzig University, 04081, Leipzig, Germany

    Jürgen Jost

  3. Santa Fe Institute for the Sciences of Complexity, Santa Fe, NM, 87501, USA

    Jürgen Jost

Authors
  1. Tat Dat Tran
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  2. Julian Hofrichter
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  3. Jürgen Jost
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Corresponding author

Correspondence to Jürgen Jost.

Additional information

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 267087. T. D. Tran and J. Hofrichter have also been supported by the IMPRS “Mathematics in the Sciences”, and the material presented in this paper is largely based on the first author’s thesis.

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Tran, T.D., Hofrichter, J. & Jost, J. A General Solution of the Wright–Fisher Model of Random Genetic Drift. Differ Equ Dyn Syst 27, 467–492 (2019). https://doi.org/10.1007/s12591-016-0289-7

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