The equilibrium gene frequencies that result from mutation-selection balance can be calculated as follows:

Consider a single locus, at which there is initially one allele, a . The gene has a tendency to mutate to a dominant allele, A . We must specify the mutation rate and the selection coefficient (fitness) of the genotypes. Define m as the mutation rate from a to A per generation; we ignore back mutation.

The frequency of a = q , of A = p . Finally, we define the fitnesses as follows:

Evolution in this case will proceed to an equilibrium frequency of the gene A (we can write the stable equilibrium frequency as p*). If the frequency of A is higher than the equilibrium, natural selection removes more A genes than mutation creates and the frequency decreases; vice versa if the frequency is lower than the equilibrium. At the equilibrium, the rate of loss of A genes by selection equals their rate of gain by mutation.

We can use that statement to calculate the equilibrium gene frequency p*.

• What is the rate per generation of creation of A genes by mutation?

Each new A gene originates by mutation from an a gene and the chance that any one a gene mutates to an A gene is the mutation rate m . A proportion (1-p ) of the genes in the population are a genes. Therefore

rate of A genes arising by mutation = m (1-p )

• What is the rate at which A genes are eliminated?

Each A gene has a (1-s ) chance of surviving, or an s chance of dying. A proportion p of the genes in the population are A . Therefore

total rate of loss of A genes by selection = ps .

At the equilibrium gene frequency (p*)

rate of gain of A gene = rate of loss of A gene

m (1-p*) = p*s

Which can be multiplied out

m - mp* = p*s

p* = m /(s + m)

Of the two terms in the denominator, the mutation rate will usually be much less than the selection coefficient. With these values s +m = s and the expression is therefore usually given in the approximate forms

p* = m/s

The simple result is that the equilibrium gene frequency of the mutation is equal to the ratio of its mutation rate to its selective disadvantage. The result is intuitive: the equilibrium is the balance between the rates of creation and elimination of the gene. To obtain the result, we used an argument about an equilibrium. We noticed that at the equilibrium the rate of loss of the gene equals the rate of gain and used that to work out the exact result.