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)
[Read "Suggestions for using the Website" for in this course]
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)