Deleterious alleles are maintained by recurrent mutation.
      A stable equilibrium(where q = 0) is reached
            when the rate of replacement (by mutation)
            balances the rate of removal (by selection).

       µ = frequency of new mutant alleles per locus per generation
           typical µ = 10^-6: 1 in 1,000,000 gametes has new mutant
                              _{_____}
            then =(µ / s)      [see derivation]

        Ex.: For a recessive lethal allele (s  = 1) with a mutation rate of µ = 10^-6
               then  =  û = (10^-6 / 1.0) = 0.001
 

mutational genetic load
    Lowering selection against alleles increases their frequency.
        Medical intervention has increased the frequency of heritable conditions
            in Homo (e.g., diabetes, myopia)
    Eugenics: 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
                          [See: S. J. Gould, "The Mismeasure of Man"]

       Is eugenics effective at reducing frequency of deleterious alleles?
            What proportion of 'deleterious alleles' are found in heterozygous carriers?

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

             if s = 1 as above, ratio is 1000 / 1 : most of variation is in heterozygotes,
                                                         not subject to selection

      Directional selection is balanced by influx of 'immigrant' alleles;
            a stable 'equilibrium' can be reached iff migration rate constant.

Consider an island adjacent to a mainland, with unidirectional migration to the island.
The fitness values of the AA, AB, and BB genotypes differ in the two environments,
      so that the allele frequencies differ between the mainland (q_m) and the island (q_i).
 

	AA	AB	BB
	W0	W1	W2	q
Island	1	1-t	1-2t	qi  0
Mainland	0	0	1	qm  1

 B has high fitness on mainland, and low fitness on island.
    [For this model only, allele A is semi-dominant to allele B,
        so we use t for the selection coefficient to avoid confusion]

      m = freq. of new migrants (with q_m) as fraction of residents (with q_i)
      if m << t    q_i = (m / t)(q_m)      [see derivation]

      Gene flow can hinder optimal adaptation of a population to local conditions.

     Ex: Water snakes (Natrix sipedon) live on islands in Lake Erie (Camin & Ehrlich 1958)
            Island Natrix mostly unbanded; on adjacent mainland, all banded.
            Banded snakes are non-cryptic on limestone islands, eaten by gulls
            Suppose   A = unbanded     B = banded   [AB are intermediate]
                   Let      q_m = 1.0     ["B" allele is fixed on mainland]
                               m   = 0.05   [5% of island snakes are new migrants]
                                 t    = 0.5     so  W₂ = 0  ["Banded" trait is lethal on island]
                   then    q_i = (0.05/0.5)(1) = 0.05
                 and     H_exp = 2pq = (2)(0.95)(0.05)  10%
                            i.e, about 10% of snakes show intermediate banding, despite strong selection

       => Recurrent migration can maintain a disadvantageous trait at high frequency.

      Inbreeding is the mating of (close) relatives
                         or, mating of individuals with at least one common ancestor

     F (Inbreeding Coefficient) = prob. of "identity by descent":
            Expectation that two alleles in an individual are
                exact genetic copies of an allele in the common ancestor
               or, proportion of population with two alleles identical by descent

                This is determined by the consanguinity (relatedness) of parents.

      Inbreeding reduces H_exp  by a proportion F
                (& increases the proportion of homozygotes). [see derivation]

            f(AB) = 2pq (1-F)
            f(BB) = q² + Fpq
            f(AA) = p² + Fpq

      Inbreeding affects genotype proportions,
            inbreeding does not affect allele frequencies.

      Inbreeding increases the frequency of individuals
            with deleterious recessive genetic diseases by F/q [see derivation]

          Ex.: if q = 10^-3 and F = 0.10 , F/q = 100
                  => 100-fold increase in f(BB) births

      Inbreeding coefficient of a population can be estimated from experimental data:

         F =  ( 2pq - H_obs ) / 2pq  [see derivation]

    &   q = 0.374 + (1/2)(0.400) = 0.574

Then F = (0.489 - 0.400) / (0.489) = 0.182
    which is intermediate between F_full-sib      = 0.250
                                                    &  F_1st-cousin = 0.125

    => Mice live in small family groups with close inbreeding
            [This is typical for small mammals]

         However, in combination with natural selection, inbreeding can be "advantageous":
           increases rate of evolution in the long-term (q 0 more quickly)
                    deleterious alleles are eliminated more quickly.
           increases phenotypic variance (homozygotes are more common).
                    advantageous alleles are also reinforced in homozygous form

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

      Sewall Wright (1889 - 1989): "Evolution and the Genetics of Populations"

      Stochastic q  > deterministic q in small populations:
            allele frequencies drift more rapidly in 'small' than 'large' populations.

      Drift is most noticeable if s  0, and/or N small (< 10)  [N  1/s]
            genetic variance =  _q² = (q)(1 - q) / 2N
                if q = p = 0.5, then _q² = 1 / 8N

       q drifts among populations (among population variance increases);
                  eventually, half lose the allele, half fix it.

           Ex:  [Demonstration #3]
                      [Try: q = 0.5, W0 = W1 = W2 = 1.0, and N = 10, 50, 200, 1000;
                                repeat 10 trials each, note q at endpoint]

      **=> Variation is 'fixed' or 'lost'  &  populations will diverge by chance <=**

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

       What happens in the really long run?

      Effective Population Size (N_e)
            = size of an 'ideal' population with same genetic variation (measured as H)
                  as the observed 'real' population.
            = The 'real' population behaves evolutionarily like one of size N_e :
                  e.g., the population will drift like one of size N_e
            loosely,  the number of breeding individuals in the population

    Consider three special cases where N_e < or << N_obs  [the 'count' of individuals]:

      (1) Unequal sex ratio

     Ne = (4)(N_m)(N_f) / (N_m  + N_f)

                 where N_m & N_f are numbers of breeding males & females, respectively.

           "harem" structures in mammals (N_m << N_f)
             Ex.: if N_m  = 1 "alpha male" and  N_f = 200
                    then     N_e = (4)(1)(200)/(1 + 200)  4
              A single male elephant seal (Mirounga) does most of the breeding
                    Elephant seals have very low genetic variation

           eusocial (colonial) insects like ant & bees (N_f << N_m)
                     Ex.: if N_f = 1 "queen" and  N_m= 1,000 drones
                                   then     N_e = (4)(1)(1,000)/(1 + 1,000)  4
                    Hives are like single small families

      (2) Unequal reproductive success
             In stable population, N_{offspring/parent} = 1
             "Random" reproduction follows Poisson distribution (N = 1  1)
                  (some parents have 0, most have 1, some have 2, a few have 3 or more)
 

X		Ne =	Reproductive strategy
1	1	Nobs	Breeding success is random
1	0	2 x Nobs	A zoo-breeding strategy
1	>1	< Nobs	K-strategy, as in Homo
1	>>1	<< Nobs	r-strategy, as in Gadus

      (3) Population size variation over time

           N_e = harmonic mean of N  = inverse of arithmetic mean of inverses
                    [a harmonic mean is much closer to lowest value in series]
                            n
           N_e = n /  [ (1/N_i) ]  where N_i = pop size in i th generation
                           i=1

       Populations exist in changing environments:
                Populations are unlikely to be stable over very long periods of time
                  10^-2 forest fire / 10^-3 flood / 10^-4 ice age

           Ex.: if typical N = 1,000,000  & every 100th generation N = 10 :
                         then  N_e = (100) / [(99)(10^-6) + (1)(1/10)]  100 / 0.1 = 1,000

       Founder Effect & Bottlenecks:
                Populations are started by (very) small number of individuals  (N=2),
                    or undergo dramatic reduction in size (N < 10)).

         Ex.: Origin of Newfoundland moose (Alces):
                            2 bulls + 2 cows at Howley in 1904
                                [maximum 8 alleles / locus, likely less if single source population]
                            [1 bull + 1 cow at Gander in 1878 didn't succeed].

       Population cycles: Hudson Bay Co. trapping records (Elton 1925)
                Population densities of lynx, hare, muskrat cycle over several orders of magnitude
                Lynx cycle appears to "chase" hare cycle

The effect of drift on genetic variation in populations

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

           H = (4N_eµ) / (4N_eµ + 1) [the "neutralist equation"]

   Ex.: if µ= 10^-7   & N_e = 10⁶   then   N_eµ = 1   and H_exp = (0.4)/(0.4 + 1) = 0.29

            But typical H_obs 0.20  which suggests  N_e 10⁵
             Most natural populations have a much smaller effective size than their typically observed size.

             Ex.: Gadus morhua in W. Atlantic were confined to Flemish Cap during last Ice Age 8~10 KYBP
                    mtDNA sequence variation occurs as "star phylogeny":
                         most variants are rare and related to a common surviving genotype
                    Carr et al. 1995 estimated N_e = 3x10⁴  as compared with N_{ob s} 10⁹
                        Effective size is ca. 5 orders of magnitude smaller than observed

                 Marshall et al. (2004) showed that "genome-types" at Flemish Cap are distinctive

            Stochastic effects may be as or more important than deterministic processes in long-term evolution.