Biallelic Mutation-Drift Diffusion in the Limit of Small Scaled Mutation Rates

The evolution of the allelic proportion $x$ of a biallelic locus subject to the forces of mutation and drift is investigated in a diffusion model, assuming small scaled mutation rates. The overall scaled mutation rate is parametrized with $\theta=(\mu_1+\mu_0)N$ and the ratio of mutation rates with $\alpha=\mu_1/(\mu_1+\mu_0)=1-\beta$. The equilibrium density of this process is beta with parameters $\alpha\theta$ and $\beta\theta$. Away from equilibrium, the transition density can be expanded into a series of modified Jacobi polynomials. If the scaled mutation rates are small, i.e., $\theta \ll 1$, it may be assumed that polymorphism derives from mutations at the boundaries. A model, where the interior dynamics conform to the pure drift diffusion model and the mutations are entering from the boundaries is derived. In equilibrium, the density of the proportion of polymorphic alleles, \ie\ $x$ within the polymorphic region $[1/N,1-1/N]$, is $\alpha\beta\theta(\tfrac1x+\tfrac1{1-x})=\tfrac{\alpha\beta\theta}{x(1-x)}$, while the mutation bias $\alpha$ influences the proportion of monomorphic alleles at 0 and 1. Analogous to the expansion with modified Jacobi polynomials, a series expansion of the transition density is derived, which is connected to Kimura's well known solution of the pure drift model using Gegenbauer polynomials. Two temporal and two spatial regions are separated. The eigenvectors representing the spatial component within the polymorphic region depend neither on the on the scaled mutation rate $\theta$ nor on the mutation bias $\alpha$. Therefore parameter changes, e.g., growing or shrinking populations or changes in the mutation bias, can be modeled relatively easily, without the change of the eigenfunctions necessary for the series expansion with Jacobi polynomials.