Mutation–selection balance

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The mutation–selection balance is a classic result in population genetics first derived in the 1920s by John Burdon Sanderson Haldane and R.A. Fisher.

A genetic variant that is deleterious will not necessarily disappear immediately from a population. Its frequency, when it first appears in a population of N individuals, will be 1/N (or 1/2N in a diploid population), and this frequency might drift up and down a bit before returning to zero. If the population is large enough, or if the mutation rate \mu is high enough, i.e., if \mu*N is high enough, then one has to consider additional mutations. In a hypothetical infinite population, the frequency will never return to zero. Instead, it will reach an equilibrium value that reflects the balance between mutation (pushing the frequency upward) and selection (pushing it downward), thus the name mutation–selection balance.

If 's' is the deleterious selection coefficient (the decrease in relative fitness), then the equilibrium frequency 'f' of an allele in mutation–selection balance is approximately f \approx \mu/s in haploids, or for the case of a dominant allele in diploids. For a recessive allele in a diploid population, f \approx \sqrt{\mu \over{s}}. A useful approximation for alleles of intermediate dominance is that f \approx \mu/(sh), where h is the coefficient of dominance. These formulae are all approximate because they ignore back-mutation, typically a trivial effect. These equations are also inexact because they use the assumption that the frequency of the mutation is small to simplify the derivation.

The mutation–selection balance has the practical use of allowing estimates of mutation rates from data on deleterious alleles (see examples on pp. 85–89 of Crow, 1986). For population geneticists, it provides a simple model for thinking about how variation persists in natural populations.

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