# Mutation-drift equilibrium and among loci variance in heterozygosity

At mutation-drift balance, the increased heterozygosity brought by new mutations is exactly equal to the loss of heterozygosity due to genetic drift. At equilibrium, the expected heterozygosity for a given locus is:

$$\hat h = \frac{4Nu}{1+4Nu}$$

The expected heterozygosity is only the first moment of the whole distribution of locus specific heterozygosity. What is the (among loci) variance in heterozygosity?

## 1 Answer

In The Distribution of Gene Frequencies in Populations, Sewall Wrights derivates a continuous approximation to the probability distribution for the frequency of an allele in a panmictic population under the pressure of drift and mutation on a bi-allelic locus. In his paper, he reports several version of the same equation. Below is one of them, where $\phi(q)$ is the probability density of the one specified allele to be at frequency $q$.

$$\phi(q) = C q^{4N\nu -1}(1-q)^{4N\mu -1}$$

, where $C=\frac{\Gamma(4N\mu+4N\nu)}{\Gamma (4N\mu) \cdot \Gamma (4N\nu)}$ is a constant, $\mu$ and $\nu$ and the forward and backward mutation rate The heterozygosity is then a simple one-to-one function of the allele frequency.