The free energy method and the Wright–Fisher model with 2 alleles

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.

[1]  W. Ewens.  Mathematical Population Genetics , 1980 .

[2]  D. Kinderlehrer,et al.  Free energy and the Fokker-Planck equation , 1997 .

[3]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[4]  Tat Dat Tran,et al.  A General Solution of the Wright–Fisher Model of Random Genetic Drift , 2012, 1207.6623.

[5]  F. Hérau,et al.  Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential , 2004 .

[6]  Jürgen Jost,et al.  An introduction to the mathematical structure of the Wright–Fisher model of population genetics , 2012, Theory in Biosciences.

[7]  B. Houchmandzadeh,et al.  Exact results for a noise-induced bistable system. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  G. A. Watterson,et al.  Reversibility and the age of an allele. II. Two-allele models, with selection and mutation. , 1977, Theoretical population biology.

[9]  Benny Levikson.  The age distribution of Markov processes , 1977 .

[10]  Bahram Houchmandzadeh,et al.  Alternative to the diffusion equation in population genetics. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  M. Nagasawa,et al.  An application of time reversal of Markov processes to a problem of population genetics , 1979, Advances in Applied Probability.

[12]  Simone Calogero,et al.  Exponential Convergence to Equilibrium for Kinetic Fokker-Planck Equations , 2010, 1009.5086.

[13]  Y. Iwasa,et al.  Free fitness that always increases in evolution. , 1988, Journal of theoretical biology.

[14]  M. Norman,et al.  Error estimate for the diffusion approximation of the Wright--Fisher model. , 1977, Proceedings of the National Academy of Sciences of the United States of America.

[15]  Tat Dat Tran,et al.  Information Geometry and Population Genetics , 2017 .

[16]  F. A. Chalub,et al.  The frequency-dependent Wright–Fisher model: diffusive and non-diffusive approximations , 2011, Journal of mathematical biology.

[17]  C. Villani,et al.  On the trend to global equilibrium in spatially inhomogeneous entropy‐dissipating systems: The linear Fokker‐Planck equation , 2001 .

[18]  Giuseppe Toscani,et al.  Exponential convergence toward equilibrium for homogeneous Fokker–Planck‐type equations , 1998 .

[19]  A J McKane,et al.  Singular solutions of the diffusion equation of population genetics. , 2007, Journal of theoretical biology.

[20]  Giuseppe Toscani,et al.  ON CONVEX SOBOLEV INEQUALITIES AND THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR FOKKER-PLANCK TYPE EQUATIONS , 2001 .

[21]  G. Kallianpur.  Stochastic differential equations and diffusion processes , 1981 .

[22]  Cédric Villani,et al.  On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation , 2005 .