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).
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
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
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'
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 (qm) and the island (qi).
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 qm)
as fraction of residents (with qi)
if m << t qi = (m / t)(qm) [see derivation]
Gene flow can hinder optimal adaptation of a population to local conditions.
Water snakes (Natrix
sipedon) live on islands in
Lake Erie (Camin & Ehrlich
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 qm = 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 qi = (0.05/0.5)(1) = 0.05
and Hexp = 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
or, mating of individuals with at least one common ancestor
F (Inbreeding Coefficient) = prob. of "identity by descent":
Probability that two alleles are exact genetic copies of an allele in the common ancestor
This is determined by the consanguinity (relatedness) of parents.
Inbreeding reduces Hexp by a
(& increases the proportion of homozygotes). [see derivation]
f(AB) = 2pq (1-F)
f(BB) = q2 + Fpq
f(AA) = p2 + Fpq
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 - Hobs ) / 2pq [see derivation]
since p = 0.226 +
(1/2)(0.400) = 0.426
& q = 0.374 + (1/2)(0.400) = 0.574
Then F = (0.489 - 0.400) /
(0.489) = 0.182
which is intermediate between Ffull-sib = 0.250
& F1st-cousin = 0.125
live in small family groups with close inbreeding
[This is typical for small mammals]
combination with natural selection, inbreeding can be
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,
(cf. deterministic q [predictable, due to selection, mutation, migration)
Sewall Wright (1889 - 1989): "Evolution and the Genetics of Populations"
Stochastic q is greater than
deterministic q in small populations:
allele frequencies drift more in 'small' than 'large' populations.
Drift is most noticeable if s 0, and/or N small (< 10) [N 1/s]
q drifts between
generations (variation decreases within
populations over time) ;
eventually, allele is lost (q = 0) or fixed (q = 1) (50:50 odds)
Ex: NatSel Laboratory Exercise
[Try: q = 0.5, W0 = W1 = W2 = 1.0, and N = 10, 50, 200, 1000]
q drifts among
populations (variation increases among
populations over time);
eventually, half lose the allele, half fix it.
**=> Variation is 'fixed' or 'lost' & populations will diverge by chance <=**
"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 (Ne)
= 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 Ne :
the population will drift like one of size Ne
loosely, the number of breeding individuals in the population
Consider three special cases where Ne < or << Nobs [the 'count' of individuals]:
(1) Unequal sex ratio
Ne = (4)(Nm)(Nf) /
(Nm + Nf)
where Nm & Nf are numbers of breeding males & females, respectively.
"harem" structures in mammals (Nm <<
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
[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 like single small families
In stable population, Noffspring/parent = 1
"Random" reproduction follows Poisson distribution (N = 1 1)
(some parents have 0, most have 1, some have 2 or 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
Ne = harmonic mean of N = inverse of
arithmetic mean of inverses
[a harmonic mean is much closer to lowest value in series]
Ne = n / [ (1/Ni) ] where Ni = pop size in i th generation
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 Ne = (100) / [(99)(10-6) + (1)(1/10)] 100 / 0.1 = 1,000
Founder Effect & Bottlenecks:
Populations are started by (very) small number of individuals,
or undergo dramatic reduction in size.
Ex.: Origin of Newfoundland moose (Alces):
2 bulls + 2 cows at Howley in 1904
[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
are more variable (higher H) than smaller
if s = 0: H reflects balance between loss of alleles by drift
and replacement by mutation
H = (4Neµ) / (4Neµ + 1)
Ex.: if µ= 10-7 & Ne = 106 then Neµ = 1 and Hexp = (0.4)/(0.4 + 1) = 0.29
But typical Hobs
0.20 which suggests Ne
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 Ice Age 8 ~ 10,000 YBP
mtDNA sequence variation occurs as "star phylogeny":
most variants are rare and related to a common surviving genotype
Carr et al. 1995 estimated Ne = 3x104 as compared with Nobs 109
Effective size is ca. 5 orders of magnitude smaller than observed
may often be more important than deterministic processes in