Mutation / selection equilibrium

      Deleterious alleles maintained by recurrent mutation.
          stable equilibrium(read as "q hat," where q  = 0) reached
                when rate of replacement (by mutation)
                balances rate of removal (by selection).

       µ = frequency of new mutant alleles per locus per generation
           µ = 10-6 : 1 in 1,000,000 gametes has new mutant 

            then  =  (µ / s) \sqrt{⋯}    [see derivation below]

        Ex.: For recessive lethal allele (s  = 1) with mutation rate of µ = 10-6
               then  =   10-6/ 1.0 \sqrt{⋯}   =    10-6 \sqrt{⋯} =  0.001
 

Mutational Genetic Load
    Lowered selection against deleterious allele increases frequency
        Medical intervention increases frequency of heritable conditions
            in Homo (e.g., diabetes, myopia)
    Eugenics: 1920s ~ 1960s social policy
            Modification of human condition by selective breeding
            'positive eugenics': encouraging people with "good genes" to breed
            'negative eugenics': discouraging people with "bad genes'' from breeding
                   e.g., immigration control, compulsory sterilization, and worse
                   [See: SJ Gould, "The Mismeasure of Man"]

       Is eugenics effective at reducing frequency of deleterious alleles and (or) genotypes?
            What proportion of 'deleterious alleles' occur in heterozygous carriers?

       (2pq) / 2q2 = p/q  1/q    (if  q << 1)

             for s = 1, ratio 1000 / 1 : most variant alleles in heterozygotes,
                                                    not subject to selection

Conclusion: if most mutations are rare (u < 0.001) & selectively disadvantageous ( s > 0.01)
                      mutation will not maintain population variation at high levels observed:

                       For u = 10-6, note that  is reciprocally proportional to s = 0.100 & 0.001

                                                                                       for paired values of s & µ 

  Mutation - selection equilibrium

HOMEWORK: Suppose presbyopia due to recessive allele p at a single locus, with mutation rate µ = 10-5 from P p.
                               Suppose presbyopia is historically associated with selection coefficient s = 0.1,
                                   & vision correction has reduced selection by 90%.
                          1) What are the former vs new equilibrium frequencies of the p allele?
                         2) What are the former vs new equilibrium frequencies of persons with presbyopia ?
                         3) EXTRA CREDIT: How many generations would it take to get there?

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

Then   qµ  +  qs  =  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

Text material © 2024 by Steven M. Carr