Genetic variation in populations can
be described by genotype and allele frequencies.
(not
"gene"
frequencies)
Consider a diploid autosomal
locus with two alleles and no dominance
(=>
semidominance: AA , Aa , aa
phenotypes are distinguishable)
# AA = x # Aa = y # aa = z x + y + z = N (sample size)
f(AA) = x / N f(Aa) = y / N f(aa) = z / N
f(A) = (2x + y) / 2N f(a) = (2z + y) / 2N
or f(A) = f(AA) + 1/2 f(Aa) f(a) = f(aa) + 1/2 f(Aa)
let p = f(A), q = f(a) p & q are allele frequencies
Properties of p & q
p + q = 1 p = 1  q q = 1  p
(p + q)^{2 }= p^{2} + 2pq + q^{2} = 1
(1  q)^{2} + 2(1  q)(q) + q^{2} = 1
p & q are interchangeable wrt [read, "with respect to"] A & a;
q is usually used for the
rarer,
recessive,
or deleterious (disadvantageous) allele;
BUT 'common'
& 'rare' are statistical properties
'dominant' & 'recessive' are
genotypic properties
'advantageous' & 'deleterious'
are phenotypic properties
*** combination of these properties is
possible ***
What happens to p & q in one generation of random mating?
Consider a population of monoecious organisms reproducing
by random union of
gametes
("tide pool" model)...
(1)
Determine the expectations
of
parental
alleles coming together in various genotype combinations.
[expectation: the
anticipated value of a variable probability]
The probability
, binomial
expansion , Punnet
Square methods
all
show
that expectation of f(AA) = p^{2}
expectation
of
f(Aa) = 2pq
expectation
of
f(aa) = q^{2}
(2) Redescribe allele frequencies among offspring (A' & a').
f(A') = f(AA) + 1/2 f(Aa)
= p^{2} + (1/2)(2pq) = p^{2} + pq
= p(p+q) = p' = p
f(a') = f(aa) + 1/2 f(Aa)
= q^{2} + (1/2)(2pq) = q^{2} + pq
= q(p+q) = q' = q
p^{2} : 2pq : q^{2} are HardyWeinberg proportions (cf. Mendelian ratios 1 : 2 : 1 )
The HardyWeinberg Theorem holds under "more realistic" conditions:
(1) multiple alleles / locus
p
+
q + r = 1
(p
+
q + r)^{2 }= p^{2} + 2pq + q^{2} + 2qr
+ r^{2} + 2pr = 1
The
proportion
of heterozygotes (H = 'heterozygosity')
is
a
measure of genetic variation at a locus.
H_{obs} = f(Aa) = observed heterozygosity
H_{exp} = 2pq = expected heterozygosity (for two
alleles)
H_{e} = 2pq + 2pr + 2qr = 1  (p^{2} + q^{2} + r^{2}) for three alleles
n
H_{e} = 1  (q_{i})^{2} for n alleles
i=1
where q_{i} = freq. of ith allele of n
alleles at a locus
Ex.: if q_{1} = 0.5, q_{2} = 0.3, & q_{3}
= 0.2
then H_{e} = 1  (0.5^{2} + 0.3^{2}
+ 0.2^{2}) = 0.62
(2) sexlinked loci
iff [read: "if and only
if"] allele frequencies in males and females are identical
If
frequencies
are initially unequal, they converge
over several generations.
(3) dioecious organisms
sexes
are
separate
HW
is
produced by random mating of individuals (random
union of genotypes).
expand (p^{2} 'AA' + 2pq 'AB' + q^{2}
'BB')^{2} :
nine possible 'matings' among genotypes
(See derivation)
No selfing
(selffertilization not possible)
Genotype proportions in natural
populations can be tested for HW conditions
H_{o}(null hypothesis): no outside
factors are acting.
Among North American whites:








f(M) = [(2)(1787) + 3039] / (2)(6129)= 0.539
f(N) = [(2)(1303) + 3039] / (2)(6129)= 0.461 = 1.0  0.539
Chisquare (^{2}) test:























































Chisquare test on combined data:






















*=> A mixture of populations, each of which shows
HardyWeinberg proportions,
will not show expected HardyWeinberg proportions
if the allele frequencies are different in the separate
populations.
Wahlund Effect: an artificial mixture of populations will have a deficiency of heterozygotes
The HardyWeinberg conditions are
the 'null hypothesis':
What
are the consequences of other genetic / evolutionary phenomena?
Five major factors:
1. Natural
selection
Change
of
allele frequencies (q) [read as 'delta q']
occurs
due
to differential effects of alleles on 'fitness'
Consequences
depend
on dominance of fitness
(see Lab #1)
Natural
Selection
is the principle concern of evolutionary theory
(& first half of this course)
2. Mutation
A and A' are interconverted at some rate µ .
If µ(AA') µ'(AA'), net change
will occur in one direction.
3. Gene flow
Net
movement
of alleles between populations occurs at some rate m .
(Im)migration
introduces new alleles, changes frequency of existing alleles.
4. Population structure
Inbreeding: preferential
mating of relatives at some rate F
(see Homework).
Nonrandom reproduction:
variable sex ratio, offspring number, population size
5. Statistical sampling error
Chance
fluctuations
occur in finite populations, especially those with small
size N.
Genetic drift: random
change of allele frequencies
over
time
& among populations (see Homework)