Hardy-Weinberg Equilibrium
•
Rules of probability
•
A simple population model
•
Mechanisms of evolutionary
change
Evolution & Genetics
???
The Modern Synthesis
R.A. Fisher
T. Dobzhansky
Sewell Wright
J.B.S.Haldane
“Population Genetics”
Population Genetics
Study of how genes behave in populations
Involves description and also prediction
Description: Fr(red) = 7/20 = 0.35
Alleles & Genotypes
S
F
Heterozygotes
(FS)
Homozygotes
(FF or SS)
Fr(FS genotype) = 7/14 = 0.500
Fr(F allele) = 11/28 = 0.393 Fr(S) = 0.607
Alleles & Genotypes
S
F
Population Genetics is the study of how
genes behave in populations
Evolution = change in allele frequencies
Theory: Can we predict changes in allele
and genotype frequencies?
Mathematical Models
(don’t be frightened)
p’ =
[2(p2) + 1(2pq)]
2[p2 + q2 + 2pq]
Rules of Probability
The probability of randomly
encountering an item of a certain
type is equal to the frequency of that
type in the population.
The probability of rolling a 6 with a
single die is 1/6.
Rules of Probability
Addition Rule: The probability that
either of two mutually exclusive
events will occur is equal to the sum
of their independent probabilities of
occurrence
The probability of rolling a 6 or a 1
with a single die is 1/6 + 1/6 = 1/3.
The sum of all possible outcomes = 1
Rules of Probability
Multiplication Rule: The probability
that two independent events will both
occur is equal to the product of their
independent probabilities of
occurrence.
The probability of rolling a 6 and
another 6 with two dice is (1/6)*(1/6)
= 1/36.
Rules of Probability
What is the probability of
rolling an 11 with two dice?
Possibility 1: Roll a 5 and a 6
Probability = (1/6)*(1/6) = 1/36
Possibility 2: Roll a 6 and a 5
Probability = (1/6)*(1/6) = 1/36
Total Prob. = (1/36) + (1/36) = 1/18
Hardy-Weinberg Equilibrium
Evolution = change in allele frequencies
What happens to allele and
genotype frequencies over time?
(simple null model)
Hardy-Weinberg Equilibrium
Assumptions about the population

Large Population

No immigration or emigration

No mutation

Random mating

Random reproductive success
(i.e., no selection)
Hardy-Weinberg Equilibrium
Figure 5.1: Basic
population cycle
Hardy-Weinberg Equilibrium
Single locus with two alleles (A and a)
Fr(A) = p
gene pool
Fr(a) = q
p + q = 1
a
a
a
A
A
a
A
A
a
A
A
a
A a
a
A
a
A A A a a
aA a
a
A
a
A
a a a
A a A
a A A a a A
A A
Hardy-Weinberg Equilibrium
Single locus with two alleles (A and a)
Fr(A) = p
Fr(a) = q
p + q = 1
Create a new generation by randomly
combining gametes (random mating)
Hardy-Weinberg Equilibrium
Single locus with two alleles (A and a)
Fr(A) = p
Fr(a) = q
p + q = 1
Probability of creating an AA individual?
Probability that 1st allele is an A = p
Probability that 2nd allele is an A = p
Probability that both alleles are A =
Fr(AA) =
2
p
Fr(aa) =
p2
2
q
Hardy-Weinberg Equilibrium
Single locus with two alleles (A and a)
Fr(A) = p
Fr(a) = q
p + q = 1
What is probability of an Aa heterozygote?
Prob. of A from dad and a from mom = pq
Prob. of a from dad and A from mom = pq
Fr(Aa) = 2pq
Hardy-Weinberg Equilibrium
Single locus with two alleles (A and a)
Fr(A) = p
Fr(a) = q
p + q = 1
If assumptions of Hardy-Weinberg hold, then
we can predict the genotype freq’s in next gen:
Fr(AA) = p2
Fr(aa) = q2
Fr(Aa) = 2pq
p2 + q2 + 2pq = 1
Hardy-Weinberg Equilibrium
Single locus with two alleles (A and a)
Fr(A) = p
Fr(a) = q
p + q = 1
What are allele freq.’s in next generation?
Fr(A)’ = p’ =
[2(Fr(AA)) + 1(Fr(Aa))]
Total
Hardy-Weinberg Equilibrium
Single locus with two alleles (A and a)
Fr(A) = p
Fr(a) = q
p + q = 1
What are allele freq.’s in next generation?
Fr(A)’ = p’ =
[2(p2) + 1(2pq)]
Total
AA,aa and Aa individuals,
each with 2 alleles
Hardy-Weinberg Equilibrium
Single locus with two alleles (A and a)
Fr(A) = p
Fr(a) = q
p + q = 1
What are allele freq.’s in next generation?
Fr(A)’ = p’ =
=
[2(p2) + 1(2pq)]
[2p(p + q)]
2
=
2p
2[p2 + q2 + 2pq]
2
= p
Hardy-Weinberg Equilibrium
Conclusions of the null model


Genotypes occur in predictable
frequencies
Allele frequencies do not change over
time (i.e., evolution does not occur)
Hardy-Weinberg Equilibrium
Violations of the Assumptions

Small Population --> Genetic Drift

Immigration/emigration --> Gene Flow

Mutation --> Mutation Pressure

Non-random mating --> Pop. Structure

Differential RS --> Natural Selection
Hardy-Weinberg Equilibrium
Fig. 5.10:
Mechanisms of
evolutionary
change
Testing for HWE
Apple maggot fly (Rhagoletis pomonella)
McPheron et al. 1988. Nature 336:64-66
Testing for HWE
Locus: b-hydroxyacid dehydrogenase (Had)
Genotype
No.
100/100
260
125/125
180
100/125
360
800
Pop’n in HWE?
Box 5.5
Step 1: Calculate observed allele frequencies
Fr(100) = (2*260 + 1*360)/(2*800) = 0.55 (= p)
Fr(125) = (2*180 + 1*360)/(2*800) = 0.45 (= q)
Testing for HWE
Locus: b-hydroxyacid dehydrogenase (Had)
Genotype
No.
Exp.
100/100
260
242
p2*N
125/125
180
162
q2*N
100/125
360
396
2*p*q*N
Step 2: Calculate expected genotype numbers
Testing for HWE
Locus: b-hydroxyacid dehydrogenase (Had)
Genotype
No.
Exp.
Stat.
100/100
260
242
1.34
125/125
180
162
3.27
100/125
360
396
2.00
Step 3: Compare observe to expected
χ =
2
Σ
(O-E)2
=
6.61
E
df = 1
crit = 3.84
Testing for HWE
Locus: b-hydroxyacid dehydrogenase (Had)
Genotype
No.
Exp.
100/100
260
242
125/125
180
162
100/125
360
396
Conclude: Population deviates from HWE
Fewer heterozygotes than expected
Why?
Testing for HWE
Rhagoletis pomonella
Fruit type
preference:
Apples vs.
hawthorn
Non-random
mating
Deficiency of
heterozygotes