Drift and mutation

Remember that in the absence of mutation

$$f_{t+1} = \left(\frac{1}{2N}\right) + \left(1 - \frac{1}{2N}\right)f_t \quad ,$$ (1)

One way of modeling mutation is to assume that every time a mutation occurs it introduces a new allele into the population. This model is referred to as the infinite alleles model, because it implicitly assumes that there is potentially an infinite number of alleles. Under this model we need to make only one simple modification to equation (1). We have to multiply the expression on the right by the probability that neither allele mutated:

$$f_{t+1} = \left(\left(\frac{1}{2N}\right) + \left(1 - \frac{1}{2N}\right)f_t\right)(1-\mu)^2 \quad ,$$ (2)

where $\mu$ is the mutation rate, i.e., the probability that an allele in an offspring is different from the allele it was derived from in a parent.

So where do we go from here? Well, if you think about it, mutation is always introducing new alleles that, by definition, are different from any of the alleles currently in the population. It stands to reason, therefore, that we'll never be in a situation where all of the alleles in a population are identical by descent as they would be in the absence of mutation. In other words we expect there to be an equilibrium between loss of diversity through genetic drift and the introduction of diversity through mutation.² From the definition of an equilibrium,

$$\begin{eqnarray*} \hat f &=& \left(\left(\frac{1}{2N}\right) + \left(1 - \fra... ...1 - 2\mu}{1 + 4N\mu - 2\mu} \\ &\approx& \frac{1}{4N\mu + 1} \end{eqnarray*}$$

Since $f$ is the probability that two alleles chosen at random are identical by descent within our population, $1-f$ is the probability that two alleles chosen at random are not identical by descent in our population. So $1-f = 4N\mu/(4N\mu + 1)$ is a reasonable measure of the genetic diversity within the population. Notice that as $N$ increases, the genetic diversity maintained in the population also increases. This shouldn't be too surprising. The rate at which diversity is lost declines as population size increases so larger populations should retain more diversity than small ones.³