Derivation of Mutation / selection equilibrium 

Consider a rare, recessive, deleterious allele a
    f(a) = q << 1   &   f(A) = p ~ 1
    f(Aa) =  µ  mutation rate (# new mutant alleles / gamete / generation)

The value of q is an equilibrium between loss of a due to selection
                                                                   & replacement of a by new mutation

change in f(a) due to selection: qs= -spq2 / (1 - sq2)
change in f(a) due to mutation: qµ= µp

Then q+  qµ=  µp -  spq2 / (1 - sq2)
                              µp  -  spq    [since  (1 - sq2)    1   if   q<<p ]
                            =  p (µ - sq2)

At equilibrium () q = 0 =  p (µ - s2)
                                                  µ - s2     [since 1 ]
                                      s2 =µ
  So             =  (µ / s)(1/2)


Text material © 2005 by Steven M. Carr