Mutation - Selection equilibrium

Derivation of Mutation / Selection equilibrium

Consider rare, recessive, deleterious allele a

f(a) = q << 1 & f(A) = p ~ 1
f(Aa) = µ mutation rate (# new mutant alleles / gamete / generation)

: equilibrium between loss of a due to selection
& replacement of a by new mutation

change in f(a) due to selection: q_s = -spq² / (1 - sq²) [complete dominance model]
change in f(a) due to mutation: q_µ = µp

Thenq_µ + q_s = µp - spq² / (1 - sq²)
µp - spq² [ (1 - sq²) 1 if q << p ]
= (p) (µ - sq²)

At equilibrium

, q = 0 = (p) (µ - s²)

  µ - s² [if p 1 ]
                              s²= µ
                                ²= µ / s
               = $($ µ / s )^0.5