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1) Two alleles model
 Let us assume a gene containing two mutually exclusive
alleles:A1,A2.
 There are three possible combinations:A1/A1,A1/A2 and A2/A2,
with initial frequencies: u,v and w.
 u+v+w=1
 Let us compute their density after one division
2) First generation
Mating type
Offsprings
frequency
A1/A1xA1/A2
0.5A1/A1+ 0.5A1/A2
2uv
A1/A1xA2/A2
A1/A2
2uw
A1/A1xA1/A1
A1/A1
U2
A1/A2xA1/A2
0.25A1/A1+0.5A1/A20.25+ A2/A2 V2
A1/A2xA2/A2
0.5A1/A2+0.5A2/A2
2vw
A2/A2xA2/A2
A2/A2
w2
3) In the next generation
 Hardy-Weinberg Equilibrium
A1/ A1  u 2  uw  0.25v 2  (u  0.5v)2  p12
A2 / A1  uv  2uw  0.5v 2  vw  2(u  0.5v)( w  0.5v)  2 p1 p2
A2 / A2  0.25v 2  vw  w2  ( w  0.5v) 2  p22
4) X linked loci
 If an allele is present in the X chromosome, the situation is more
complicated.
 If qn is the allele frequency in women in the generation n, and rn
the allele frequency in mens.
rn  qn-1
qn  0.5qn-1  0.5rn-1
2 / 3qn  1/ 3rn  2 / 3(0.5qn-1  0.5rn-1 )  1/ 3qn-1
 2 / 3qn-1  1/ 3rn-1  p
5)
Dynamics to equilibrium
3
1
3
1
p  p  0.5qn-1  0.5rn-1 - p  p
2
2
2
2
3
1
 0.5qn-1  0.5rn-1 - (2 / 3qn-1  1/ 3rn-1 )  p
2
2
1
1
1
  qn-1  p    qn-1 - p 
2
2
2
qn - p  qn -
6) When the Hardy-Weinberg Law Fails to
Apply
 Mutation
 The frequency of gene B and its allele b will not remain in Hardy-Weinberg

equilibrium if the rate of mutation of B -> b (or vice versa) changes.
By itself, mutation probably plays only a minor role in evolution; the rates are
simply too low. But evolution absolutely depends on mutations because this is
the only way that new alleles are created. After being shuffled in various
combinations with the rest of the gene pool, these provide the raw material on
which natural selection can act.
 Gene Migration
 Many species are made up of local populations whose members tend to breed

within the group. Each local population can develop a gene pool distinct from
that of other local populations.
However, members of one population may breed with occasional immigrants
from an adjacent population of the same species. This can introduce new genes
or alter existing gene frequencies in the residents. In many plants and some
animals, gene migration can occur not only between subpopulations of the same
species but also between different (but still related) species. This is called
hybridization. If the hybrids later breed with one of the parental types, new genes
are passed into the gene pool of that parent population. This process, is called
introgression.
 Genetic Drift
 As we have seen, interbreeding often is limited to the members of local
populations. If the population is small, hardy-Weinberg may be violated. Chance
alone may eliminate certain members out of proportion to their numbers in the
population. In such cases, the frequency of an allele may begin to drift toward
higher or lower values. Ultimately, the allele may represent 100% of the gene
pool or, just as likely, disappear from it. Drift produces evolutionary change, but
there is no guarantee that the new population will be more fit than the original
one. Evolution by drift is aimless, not adaptive.
Controversial when first proposed (Kimura 1968) , Incontrovertible in
2001:
DNA sequence polymorphisms are abundant
In Eukarya, most of the genome is noncoding
most sequence polymorphism lies in noncoding regions
Most sequence polymorphisms appear selectively neutral
 Nonrandom Mating
 One of the cornerstones of the Hardy-Weinberg equilibrium is that mating in the

population must be random. If individuals (usually females) are choosy in their
selection of mates the gene frequencies may become altered. Darwin called this
sexual selection.
Assortative mating -Humans seldom mate at random preferring phenotypes like
themselves (e.g., size, age, ethnicity). This is called assortative mating.
 Natural Selection
 If individuals having certain genes are better able to produce mature offspring
than those without them, the frequency of those genes will increase. This is
simple expressing Darwin's natural selection in terms of alterations in the gene
pool. (Darwin knew nothing of genes.) Natural selection results from
differential mortality and/or
differential fecundity.


 Mortality Selection
 Certain genotypes are less successful than others in surviving through to the end
of their reproductive period.
 The evolutionary impact of mortality selection can be felt anytime from the
formation of a new zygote to the end (if there is one) of the organism's period of
fertility. Mortality selection is simply another way of describing Darwin's criteria
of fitness: survival.
 Fecundity Selection
 Certain phenotypes (thus genotypes) may make a disproportionate contribution
to the gene pool of the next generation by producing a disproportionate number
of young. Such fecundity selection is another way of describing another criterion
of fitness described by Darwin: family size. In each of these examples of natural
selection certain phenotypes are better able than others to contribute their genes
to the next generation. Thus, by Darwin's standards, they are more fit. The
outcome is a gradual change in the gene frequencies in that population.
7) Effect of Natural Selection on Gene
Frequencies.
 Let us define the frequency of each genotype in the population
as: w, and the initial allele distribution as p and q for A1 and A2.
wA1/ A1  1- r
wA1/ A 2  1
wA 2 / A 2  1- s
8) Fitness
The average fitness is:
W  (1- r ) p 2  2 pq  (1- s)q 2  1- rp 2 - sq 2
P  ([(1- r ) p 2  pq]/ W ) - p  pq[ s - (r  s) p]/ W
0,1


[ p, q ]  
1, 0
 s /(r  s), r /(r  s)

9) Hetrozygote Advantage
s (1-r) p 2n +p n q n
s
p n+1 =
r+s
W
r+s
s
=[(1-r)p 2n +p n q n ]W]/W=
r+s
s
=[(1-r)p 2n +p n q n ](1-rp n2 -sq n2 )]/W
r+s
1-rp n -sqn
s
=
[p n ]
W
r+s
The difference decreases to zero only for positive r and s. Thus the
scenario in which both alleles can survive is Hetrozygote
Advantage