Mutation-Drift Equilibrium

Genetic variation is input by mutation and removed by genetic drift. A natural way to quantify genetic variation is by population heterozygosity, H. This is the frequency that two randomly selection gene copies from the population differ from each other.

The rate of mutation and drift over the same unit of time, most convenient is per generation, can be used to generate an equilibrium prediction. Two gene copies are sampled from gene copies in the previous generation. There is a 1/2N chance that they were both sampled from the same gene copy in the generation before. In this case they would be identical to each other and not contribute to heterozygosity. If the population size is kept constant, then if one gene copy gives rise to two copies in the next generation then another gene copy is lost from the population. The gene copy that was lost was potentially heterozygous with the gene copy that was amplified; genetic drift results in a reduction of heterozygosity at a rate of 1/2N per generation.

If a gene copy mutates then, assuming each mutation results in a new SNP in a DNA sequence, it results in a difference between gene sequences and increases heterozygosity. There are two chances, one for each of two alleles sampled from the previous generation, for mutation to occur with a total per generation rate of 2μ. Population heterozygosity is increased at a rate of 2μ per generation.

At equilibrium the rate of increase of heterozygosity by mutation and the rate of decrease by drift are equal.

[math]2 \mu = H \frac{1}{2N}[/math]

If there is no variation in the population then genetic drift has no effect. Genetic drift can only act to reduce the heterozygosity that exists in the popualtion, so 1/(2N) is multiplied by H.

By rearranging this predicts

[math]H = 4 N \mu[/math].