A two-allele model with recurrent mutation

There's another way of looking at the interaction between drift and mutation. Suppose we have a set of populations with two alleles, $A_1$ and $A_2$. Suppose further that the rate of mutation from $A_1$ to $A_2$ is equal to the rate of mutation from $A_2$ to $A_1$.⁴ Call that rate $\mu$. In the absence of mutation a fraction $p_0$ of the populations would fix on $A_1$ and the rest would fix on $A_2$, where $p_0$ is the original frequency of $A_1$. With recurrent mutation, no population will ever be permanently fixed for one allele or the other. Instead we see the following:

When $4N\mu < 1$ the stationary distribution of allele frequencies is bowl-shaped, i.e, most populations have allele frequencies near 0 or 1. When $4N\mu > 1$, the stationary distribution of allele frequencies is hump-shaped, i.e., most populations have allele frequencies near 0.5. In other words if the population is ``small,'' drift dominates the distribution of allele frequencies and causes populations to become differentiated. If the population is ``large,'' mutation dominates and keeps the allele frequencies in the different populations similar to one another.

A population is large with respect to the drift-mutation process if $4N\mu > 1$, and it is small if $4N\mu < 1$. Notice that calling a population large or small is really just a convenient shorthand. There isn't much of a difference between the allele frequency distributions when $4N\mu = 0.9$ and when $4N\mu = 1.1$. Notice also that because mutation is typically rare, on the order of $10^{-5}$ or less per locus per generation for a protein-coding gene and on the order of $10^{-3}$ or less per locus for a microsatellite, a population must be pretty large ($> 25,000$ or $>250$) to be considered large with respect to the drift-migration process. Notice also that whether the population is ``large'' or ``small'' will depend on the loci that you're studying.