Genetic Drift 

      Genetic Drift is stochastic q [unpredictable, random]
                      (cf. deterministic q [predictable, due to selection, mutation, and (or) migration]

      Sewall Wright (1889 - 1988): "Evolution and the Genetics of Populations" (1931)

      Stochastic q greater than deterministic q in small populations:
            allele frequencies drift more in 'small' than 'large' populations

      Drift most noticeable if s 0, and (or) N small (< 10) and (or)  N 1/s
            Wright - Fisher Model for haploid organisms, finite size N (or 2N ~ diploid), discrete generations [NS 02-01smc]

       q drifts between generations (variation decreases within populations over time);
                  eventually, allele is lost (q = 0) or fixed (q = 1)

           Ex: Matlab model of Cepaea nemoralis

       q drifts among populations (variance increases among populations over time) [NS 02-02smc]
                  eventually, half lose the allele, half fix it

      **=> Variation 'fixed' or 'lost'  &  populations diverge due to chance <=**

      Evolutionary significance
            "Gambler's Dilemma" : if you play long enough, you win or lose everything
            All populations finite: many very small, somewhere or sometime
            Evolution occurs on vast time scales: "one in a million chance" a certainty (or a rate)
            Reproductive success of individuals in variable: "The race is not to the swift ..."

       What happens in the really long run?

      Effective Population Size (Ne)
            size of 'ideal' population Nexp with same [evolutionary / population genetic] behavior as 'real' population with Nobs
                    same genetic variation (measured as Hobs)
                    same inbreeding coefficient F [see derivation]
                    same 'genetic drift' (measured as genetic variance)
                            in any single generation
σ2 = (pq)/2N

over time t

              ~ approximately the number of breeding individuals in the local population 

    Consider four special cases where Ne << Nobs  [the observed 'count' of individuals]:

      (1) Unequal sex ratio

     Ne = (4)(Nm)(Nf) / (Nm + Nf)    [see derivation]
                 where Nm & Nf = counts of breeding males & females, respectively 

           "harem" structures in mammals (Nm << Nf)
            Ex.: if Nm  = 1 "Alpha Male" and  Nf = 200
                    then     Ne = (4)(1)(200)/(1 + 200) 4
              A single male elephant seal (Mirounga) does most of the breeding [NS 02-04]
                    [Elephant seals have very low genetic variation]

           eusocial (colonial) insects like ant & bees (Nf << Nm)
                     Ex.: if Nf = 1 "queen" and  Nm= 1,000 drones
                                   then     Ne = (4)(1)(1,000)/(1 + 1,000) 4
                    Hives are very small families

      (2) Unequal reproductive success
             In stable population, Noffspring / parent = 1
           "Random" reproduction follows Poisson distribution (N = 1  1):
                  Characteristics: =
σ2 ; if = 1,  σ2 = σ
                  ~ 1/3 of (female) parents have 0, ~1/3 have 1, ~1/3 have 2 or more

σ Ne = Reproductive strategy
Random breeding success
2x Nobs
A conservation strategy
> 1
< Nobs K-strategy, as in Homo
>> 1
<< Nobs r-strategy, as in Gadus

        Conservation strategy: All animals bred equally, gene pool conserved
: keep N constant near carrying capacity (K) by investing parental care
         r-strategy: maximize chance of success with high reproductive rate (r),
                            little parental investment 

      (3) Population size variation over time

           Ne = harmonic mean of N  = inverse of arithmetic mean of inverses
                                                          [much closer to smallest value in series]
           Ne = n /  [ Σ(1/Ni) ]  where Ni = pop size in i th generation 

       Populations exist in fluctuating environments:
                Populations unlikely to be stable over very long periods of time:
                  101~2 forest fire , 102~3 flood , 10-4 ice age

           Ex.: if typical N = 1,000,000  & every 100th generation 'bottlenecks' to N = 10 :
                         then  Ne = (100) / [(99)(10-6) + (1)(1/10)]  100 / 0.1 = 1,000 [HOMEWORK]

       Founder Effect & Bottlenecks:
                Populations started by (very) small number of individuals,
                    or undergo dramatic reduction in size [NS 02-03]

                Ex.: Founder Origin of Newfoundland moose (Alces):
                            2 bulls + 2 cows at Howley in 1904
                            [1 bull + 1 cow at Gander in 1878 didn't succeed].

                Ex.: Bottlenecks in population cycles: Hudson Bay Co. trapping records (Elton 1925)
                        Population densities of lynx, hare, muskrat cycle over several orders of magnitude
                        Explained as Lynx cycle "chasing" hare cycle

    (4) Non-uniform population dispersion

            For a population with a landscape architecture defined as
= density = # individuals / unit area
σ = mean dispersion between adult & offspring birthplace
                        (if normally distributed, then variance =
σ2 )            

            Ne = (4π)ẟσ2

                Ex.: In old-field Deer Mice (Peromyscus maniculatus),
    ẟ = 6 / 104 m-2, σ = 114 m, so σ2 = 1.3x104 m2
                       Ne = 98 / "neighborhood"
                            [cf: Canadian Rules football field = 6x103 m2:
= 3.4 mice / field
                 BUT if mice live in barns at 4X the density & 1/10 the dispersion:
Ne = 4

The effect of drift on genetic variation in populations

          Larger populations more variable (higher H) than smaller
                  if s = 0Heterozygosity (H) reflects balance between
                                loss of alleles by drift and replacement by mutation

           The Neutral Equation: H = (4Neµ) / (4Neµ + 1)
                    Ex.: if µ= 10-6   & Ne = 106   then   Neµ = 1   and Hexp = (4)/(4 + 1) = 0.80
                    But for protein electrophoretic data, typical Hobs 0.20  which suggests  Ne 105

                    Natural populations have much smaller Ne than observed Nc count

            Stochastic effects may often be more important than deterministic processes in evolution.

Text material © 2021 by Steven M. Carr