Diffusion process calculations for mutant genes in nonstayionary populations

摘要

Diffusion process approximations were introduced into population genetics by Fisher and Wright and perfected by Kimura. Contrary to popular scientific opinion, these pioneers did not solve all of the interesting modeling problems. For instance, none of them has much to say about the stochastic dynamics of recessive disease genes. They are also more or less silent on the stochastic aspects of evolution in the presence of exponential population growth. The current paper uses Ito's formula to derive an infinite hierarchy of integral equations satisfied by the moments of a diffusion process. These integral equations can be converted into an infinite hierarchy of ordinary differential equations and solved either exactly or numerically. We illustrate some of the possibilities for dominant, neutral, and recessive models of inheritance by computing the moments of gene frequencies in the presence of exponential population growth.