Diffusion approximations in population genetics and the rate of Muller’s ratchet

Abstract

The Wright–Fisher binomial model of allele frequency change is often approximated by a scaling limit in which selection, mutation and drift all decrease at the same 1/N rate. This construction restricts the applicability of the resulting ‘Wright–Fisher diffusion equation’ to the weak selection, weak mutation regime of evolution. We argue that diffusion approximations of the Wright–Fisher model can be used more generally, for instance in cases where genetic drift is much weaker than selection. One important example of this regime is Muller’s ratchet phenomenon, whereby deleterious mutations slowly but irreversibly accumulate through rare stochastic fluctuations. Using a modified diffusion equation we derive improved analytical estimates for the mean click time of the ratchet.