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 > deterministic q in small populations:
            allele frequencies "drift" more in 'smaller' than 'larger' populations

      Drift most noticeable if s 0, and (or) N small (< 10) such that  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, any allele lost (q = 0) or fixed (q = 1) [all alleles lost, except one]

           Ex: Matlab model of Cepaea nemoralis snails

       q drifts among populations (variance increases among populations over time) [NS 02-02smc]
                  eventually, all fixed for one or another allele

      **=> 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 rate (or certainty)
            Reproductive success of individuals in variable: 0, 1, 2 ... / parent

       What happens in really long run?

      Effective Population Size (Ne)
            size of 'ideal' population Nexp with same evolutionary 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 number of breeding individuals in local population 

    Consider four special cases where Ne << Nobs  [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
 
            Single "Alpha" male elephant seal (Mirounga) does almost all breeding [NS 02-04]
                    [Elephant seals have very low genetic variation]

           eusocial (colonial) insects like ants & bees (Nf << Nm)
                     Ex.: if Nf = 1 "Queen" and  Nm= 1,000 drones
                                   then     Ne = (4)(1)(1,000)/(1 + 1,000) 4
                    Hives = 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
 

N
σ Ne = Reproductive strategy
1
1
Nobs
Random breeding success
1
0
2x Nobs
A conservation strategy
1
> 1
< Nobs K-strategy, as in Homo
1
>> 1
<< Nobs r-strategy, as in Gadus

        Conservation strategy: All animals bred equally, gene pool conserved
        K
-strategy
: keep N constant near carrying capacity (K) by investing parental care
         r-strategy: maximize 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]
                           n
           Ne = n /  [ Σ(1/Ni) ]  where Ni = pop size in i th generation 
                           i=1

       Populations exist in fluctuating environments:
                Population numbers unstable 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 / [10-4 + 10-1] 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 introduced 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 arctic lynx, hare, muskrat cycle over several orders of magnitude
                        Does Lynx cycle "chase" hare cycle ?

    (4) Non-uniform population dispersion

            For population with 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 = 1.3 x 104 m2
                       Ne = 98 / wild "neighborhood"
                            [cf: Canadian Rules football field = 6 x 103 m2:
= 3.4 mice / field
              
                 cf. Mus mice in barns at 4X density & 1/10 dispersion:
Ne = 4

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 © 2022 by Steven M. Carr