Julian Hofrichter

An introduction to the mathematical structure of the Wright–Fisher model of population genetics

In this paper, we develop the mathematical structure of the Wright–Fisher model for evolution of the relative frequencies of two alleles at a diploid locus under random genetic drift in a population of fixed size in its simplest form, that is, without mutation or selection. We establish a new concept of a global solution for the diffusion approximation (Fokker–Planck equation), prove its existence and uniqueness and then show how one can easily derive all the essential properties of this random genetic drift process from our solution. Thus, our solution turns out to be superior to the local solution constructed by Kimura.

A general solution of the Wright-Fisher model of random genetic drift

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.

The evolution of moment generating functions for the Wright-Fisher model of population genetics

We derive and apply a partial differential equation for the moment generating function of the Wright-Fisher model of population genetics.

The uniqueness of hierarchically extended backward solutions of the Wright–Fisher model

ABSTRACT The diffusion approximation of the Wright-Fisher model of population genetics leads to partial differentiable equations, the Kolmogorov forward and backward equations, with a leading term that degenerates at the boundary. This degeneracy has the consequence that standard PDE tools do not apply, and solutions lack regularity properties. In this paper, we develop a regularizing blow-up scheme for the iteratively extended global solutions of the backward Kolmogorov equation presented in a previous paper, which are constructed from a known class of solutions, and establish their uniqueness for the stationary case. As the model describes the random genetic drift of several alleles at the same locus from a backward perspective, the occurring singularities result from the loss of an allele. While in an analytical approach, this provides substantial difficulties, from a biological or geometric perspective, this is a natural process that can be analyzed in detail. The presented scheme regularizes the solution via a carefully constructed iterative transformation of the domain.

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.

Large Deviation Theory

This chapter applies Wentzell’s theory of large deviation s to the Wright–Fisher model, using the approach of Papangelou (Athens conference on applied probability and time series analysis. Lecture notes in statistics, vol 114. Springer, New York, pp 245–252, 1996; Papangelou, Ann Appl Probab, 8(1):182–192, 1998; Papangelou, Stochastic processes and related topics. Trends in mathematics. Birkhauser, Boston, pp 315–330, 1998; Papangelou, Ann Appl Probab 10(4):1259–1273, 2000). For a different approach to the large deviation principle for exit times in population genetics, we refer the reader to Morrow and Sawyer (Ann Probab 17(3):1124–1146, 1989) and Morrow (Ann Appl Probab 2(4):857–905, 1992). As customary, we shall abbreviate Large Deviation Principle as LDP .

The Backward Equation

The backward solution u(p, t) expresses the probability of having started in some p ∈ Δ n at the negative time t conditional upon being in a certain state u(p, 0) = f(p) at time t = 0, i.e. having reached the corresponding (generalised) target set . It becomes a parabolic equation upon time reversal, that is, replacing t by − t. We can then treat u(p, 0) = f(p) as the initial condition at time t = 0. In view of the biological model behind the Kolmogorov backward equation , however, we shall work with negative time and call u(p, 0) = f(p) a final condition.

The Wright–Fisher Model

The Wright–Fisher model considers the effects of sampling for the distribution of alleles across discrete generations. Although the model is usually formulated for diploid populations, and some of the interesting effects occurring in generalizations depend on that diploidy, the formal scheme emerges already for haploid populations. In the basic version, with which we start here, there is a single genetic locus that can be occupied by different alleles, that is, alternative variants of a gene. In the haploid case, it is occupied by a single allele, whereas in the diploid case, there are two alleles at the locus. Biologically, diploidy expresses the fact that one allele is inherited from the mother and the other from the father. However, the distinction between female and male individuals is irrelevant for the basic model.

Moment Generating and Free Energy Functionals

In this section, we will construct the moment generating function for the Wright–Fisher model and derive a partial differential equation that it satisfies. This differential equation encodes all the moment evolution equation s from the Sect. 4.3

The Forward Equation

In this chapter, we treat the Kolmogorov forward equation for the diffusion approximation of the (n + 1)-allelic 1-locus Wright–Fisher model, without mutation and selection. We recall the basic definitions.