Abstract
We systematically investigate the Wright–Fisher model of population genetics with the free energy functional formalism of statistical mechanics and in the light of recent mathematical work on the connection between Fokker–Planck equations and free energy functionals. In statistical physics, entropy increases, or equivalently, free energy decreases, and the asymptotic state is given by a Gibbs-type distribution. This also works for the Wright–Fisher model when rewritten in divergence to identify the correct free energy functional. We not only recover the known results about the stationary distribution, that is, the asymptotic equilibrium state of the model, in the presence of positive mutation rates and possibly also selection, but can also provide detailed formulae for the rate of convergence towards that stationary distribution. In the present paper, the method is illustrated for the simplest case only, that of two alleles.
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Notes
We should point out an essential incompatibility between the mathematical and the biological terminology here. Mathematically, in a Fokker–Planck-type equation, the leading part which contains second derivatives w.r.t. the spatial variables is called the diffusion part, and an additional first term, which may or may not be present, is called a drift term. In the biological model, random genetic drift, which is the most important component of the Wright–Fisher model, causes the diffusion, and not the drift term in the Fokker–Planck equation.
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Acknowledgments
We are grateful to Yoh Iwasa for drawing our attention to Iwasa (1988). We also thank two anonymous referees for their constructive and useful suggestions to improve the first version of this paper. The research leading to these results has received funding from the European Research Council under the European Union Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 267087.
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Tran, T.D., Hofrichter, J. & Jost, J. The free energy method and the Wright–Fisher model with 2 alleles. Theory Biosci. 134, 83–92 (2015). https://doi.org/10.1007/s12064-015-0218-2
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DOI: https://doi.org/10.1007/s12064-015-0218-2