Mendelian Genetics in Populations – Hardy Weinberg equilibrium
The Hardy-Weinberg equilibrium principle yields two fundamental conclusions:
Conclusion 1: The allele frequencies in a population will not
change, generation after generation.
Conclusion 2: If the allele frequencies in a population are given
by p and q, the genotype frequencies will be
given by p2, 2pq, q2.
Mendelian Genetics in Populations – Hardy Weinberg equilibrium
The Hardy-Weinberg equilibrium has 5 assumptions
1: There is no selection
2: There is no mutation
3: There is no migration (gene flow)
4: There are no chance events (Chance events are unlikely to
occur in a very large population. In a large
population, eggs and sperm ‘collide’ at their
actual frequencies of p and q.)
5: Individuals choose their mates at random.
If all of these assumptions are met, then nothing happens!
The population does not change.
Mendelian Genetics in Populations - Selection
Patterns of Selection
• Selection for Recessive and Dominant alleles
• Selection for Heterozygotes and Homozygotes
• Frequency Dependent Selection
Selection on Recessive and Dominant alleles
Gene called - l locus
Genotypes: +/+ - phenotypically normal
+/l - phenotypically normal
l/l - lethal
Peter Dawson (1970) starts 2 experimental
populations with heterozygotes (+/l) as founders.
Frequency of + and l are:
The frequency of an allele in the next generation
can be calculated by:
From Box 6.5
From Box 6.3
Trilobium castaneum (flour beetles)
Fitness
What is fitness, w?
Your text book: Fitness is the extent to which an individual contributes genes to
future generations.
-an individuals score on a measure of performance expected to
correlated with genetic contribution to future generations.
e.g. lifetime reproductive success
chance of survival to adulthood (reproduction)
Absolute fitness - is the total number of surviving offspring an individual produces
during its lifetime (lifetime reproductive success).
Survivorship is a component of fitness.
Absolute fitness = the chance of survival * the # of offspring.
Relative Fitness
Absolute fitness is standardize to obtain Relative fitness.
The genotype with the highest absolute fitness has a relative fitness of 1.0.
For every other genotype, their relative fitness is:
𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑓𝑖𝑡𝑛𝑒𝑠𝑠
𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑓𝑖𝑡𝑛𝑒𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑖𝑡𝑡𝑒𝑠𝑡 𝑔𝑒𝑛𝑜𝑡𝑦𝑝𝑒
Relative Fitness
Example
A beetle is polymorphic for color. It comes in black, brown and yellow morphs.
Birds and lizards prey on them, so that due to differences in survivorship, the
fitness of the color morphs differ.
CB CB black - 67% chance of survival to adulthood
CB CY brown - 93% chance of survival to adulthood
CY CY yellow - 11% chance of survival to adulthood
Assuming they have identical numbers of offspring, what is the relative fitness of
each genotype?
w CB CB - .67/.93 = 0.72
w CB CY = 1.0
w CY CY -.11/.93 = 0.12
Absolute Fitness
Problem:
A species of snake has two forms: brown and green.
They are determined by a single locus with two alleles.
GG = brown, Gg = brown, gg = green
• Brown phenotypes are cryptic and have a 80% chance of surviving
to reproduce.
• Green phenotypes have a 60% chance of surviving to reproduce.
• The brown phenotype produces an average of 13 offspring.
• The green phenotype produces an average of 20 offspring.
What are the absolute and relative fitnesses of each genotype?
Absolute Fitness
Problem:
A species of snake has two forms: brown and green.
They are determined by a single locus with two alleles.
GG = brown, Gg = brown, gg = green
Brown phenotypes have a 80% chance of surviving to reproduce.
Green phenotypes have a 60% chance of surviving to reproduce.
The brown phenotype produces an average of 13 offspring.
The green phenotype produces an average of 20 offspring.
What are the absolute and relative finesses of each genotype?
Absolute fitness of Brown phenotypes GG and Gg
.80 chance of survival x 13 offspring = 10.4 expected offspring.
Absolute fitness of Green phenotype gg
.60 chance of survival x 20 offspring = 12 expected offspring.
w(gg) = 1
w(Gg) = w(GG) = 10.4/12 = 0.87
Selection on Recessive and Dominant alleles
1) When a recessive allele is
common (and a dominant allele
is rare), evolution is rapid.
Why?
If the dominant and recessive allele
differ in fitness, then there will be a
rapid reduction in the recessive allele
if it has lower fitness.
2) When a dominant allele is common
(and a lethal recessive allele is rare),
evolution is slow.
Even lethal recessive alleles will
persist because they “hide” in
heterozygotes. They are virtually
impossible to eliminate.
If the dominant allele is lethal, what happens? That allele is eliminated in one generation.
Selection on Recessive and Dominant alleles
2) When a dominant allele is common
(and a recessive allele is rare),
evolution is slow.
Even lethal recessive alleles will
persist because they “hide” in
heterozygotes. They are virtually
impossible to eliminate.
Selection on Recessive and Dominant alleles
When selection favors a recessive
allele, evolution is slow at first. Why?
Most recessive alleles are in heterozygotes
and cannot be selected for.
Once recessive alleles become present as
homozygotes, the rate of evolution
increases dramatically.
The recessive allele goes to fixation (= 1.0)
Selection for Heterozygotes and Homozygotes
Heterozygote superiority = heterosis = overdominance
Fitness of the heterozygote is greater than that of either
homozygote.
Selection for Heterozygotes
Mukai and Burdick (1959)
Single locus, two alleles
VV, VL – viable
LL - lethal
2 experimental populations
V = 0.5
L = 0.5
But viable allele
does not go close
to fixation, as
Dawson’s data did.
Results begin to look like Dawson’s
data with flour beetles.
Selection on Homozygotes
Mukai and Burdick (1959)
Using model from Box 6.5 and
the following fitnesses:
VV
VL
LL
0.735
1.0
0
Starting with V = 0.975, both
experimental data support the model and
previous experiments.
Selection for Heterozygotes
Generalizations about heterozygote superiority:
1. Alleles will reach equilibrium at frequencies other than
those predicted by H-W
2. Because selection is favoring heterozygotes, both alleles
will be maintained over time – instead of one allele being
fixed and one eliminated (selection is maintaining
genetic variability).
3. Because both alleles are actively maintained over time,
all three genotypes will be actively maintained over time.
4. The general term for population with multiple
phenotypes = balanced polymorphism
heterosis is one mechanism that can produce this pattern
(there are others)
Selection for Homozygotes
Heterozygote inferiority = underdominance
Heterozygotes have lower fitness than do either
homozygote.
 Because heterozygotes have low fitness, most matings will
take place among homozygotes.
 Whichever allele (dominant or recessive doesn’t matter) is
most common in a population will be selectively favored and
will be fixed.
Selection for Homozygotes
Do thought experiment: imagine population with two alleles A and a;
Selection acts against heterozygotes (assume no heterozygotes survive)
a. If A is more common than a, population will have high frequency of AA and
lower frequency of aa.
b. Assume random mating: who mates with whom?
i. Because they’re more common, AA individuals will most likely mate with AA.
ii. Because they’re rare, aa individuals are more likely to mate with AA than with aa.
iii. Because most aa mate with AA, they produce heterozygous offspring.
c. What happens to a alleles? a alleles will be found in heterozygotes and
eliminated.
d. What happens to the A allele? A becomes fixed.
e. Same will happen if a more common than A.
Note that fixation only depends on initial frequency, not on dominance or
recessiveness.
Selection on Homozygotes
Foster et al. (1972)
Don’t get caught up with how researchers created the “alleles”.
Frequencies of 0.5 for
each allele is unstable.
As soon as frequency shifts above
or below 0.5, the allele rapidly
goes to fixation or 0. (Consider
outcome of thought experiment).
Selection on Homozygotes
Foster et al. (1972)
If fitness of homozygotes
are unequal, then
unstable equilibrium
point shifts in favor of
the allele with greater
fitness.
Heterozygote inferiority leads to
a loss of genetic diversity within a
population.
Heterozygote inferiority may help maintain genetic diversity
among populations. How?
A: By driving different alleles to fixation in different populations.
Frequency Dependent Selection
What happens if the direction of selection changes over time?
1. When selection constantly favors one allele over another,
that can result in fixation of that allele.
2. A stable equilibrium would be achieved if the
heterozygotes were favored.
3. Negative frequency-dependent selection creates a
condition of stable equilibrium by favoring one allele over
another, but the favoritism alternates and is mediated by
behavior!
The punch line: In negative frequency dependent selection,
rare alleles are selectively favored.
Frequency Dependent Selection
Based on observations of Smithson and Macnair (1997)
 Naïve bumblebees visit stands of orchids to
sample flowers.
 If a bee visits a purple flower and receives
no reward, it looks next in a yellow flower.
 If it finds nothing in the yellow flower, it
looks next in a purple flower.
 Finding nothing in the purple flower sends
it back to the yellow flower.
 Bees visit equal numbers of purple and
yellow flowers.
 As a consequence, orchids with the less
common color are visited more often.
 More visits translates into greater
reproductive success.
 The allele responsible for the rare color
increases in frequency.
Frequency Dependent Selection
Experiment by Gigord et al. (2001)
 Artificial arrays of potted orchids.
 Frequency of yellow flowers was varied.
 When frequency of yellow flowers was
low, reproductive success was large.
Frequency Dependent Selection
Experiment by Gigord et al. (2001)
 Artificial arrays of potted orchids.
 Frequency of yellow flowers was varied.
 When frequency of yellow flowers was low,
reproductive success was large.
 When frequency of yellow flowers was
high, reproductive success was low.
Frequency Dependent Selection
Experiment by Gigord et al. (2001)
 Artificial arrays of potted orchids.
 Frequency of yellow flowers was varied.
 When frequency of yellow flowers was low,
reproductive success was large.
 When frequency of yellow flowers was
high, reproductive success was low.
 Dashed vertical lines indicate predicted
equilibrium frequencies for the yellow
allele as a function of:
1) allele frequency and 2) fitness.
 Observed frequency of yellow flowers in
20 natural populations was 69± 3%
Negative frequency dependent selection maintains genetic diversity in populations.
Mutation
The second assumption of Hardy-Weinberg equilibrium is –
No mutation
 Mutation is ultimately the origin of the genetic variation that
selection acts on – it provides the raw material for evolution.
 By itself, though (i.e., in the absence of selection for or against
the mutant allele), does mutation cause evolutionary change?
Figure 6-26
AA
Aa
0.8098 0.18014
Allele frequencies
change ever so slightly.
aa
0.010018
Genotype frequencies
change ever so slightly.
A high rate of
mutation, but
within the
known range.
Is this evolution?
Figure 6-27
Is this evolution?
Eventually
But it is not adaptive evolution.
Mutation
Mutation is an important source of variation for selection.
 Used inbred, homozygous stock.
 Two conditions
• Two cages w/o NaCl in the diet.
• Four cages with access to media containing
0, 1, 2, 3, 4, 5, 6% NaCl
5% and 6% NaCl lethal
 After 30 generations flies divided among cages with
0, 1, 2, 3, 4, 5, 6% NaCl in diet.
Mutation
Flies with access to all 6
NaCl conc. survive and
produce progeny on
each of the 6 NaCl conc.
= adaptive evolution
through mutation and
selection.
5% & 6% NaCl is
lethal to the
original stock
Mutation
Even unstressed flies developed
mutations in the absence of
selection that permitted them
to survive and produce progeny
up to 5% NaCl
5% & 6% NaCl is
lethal to the
original stock
Mutation-Selection balance
 Most mutations are at least mildly deleterious, so selection acts to
eliminate them.
 Yet, deleterious alleles remain in populations at frequencies
higher than predicted by H-W – why?
A: Mutation is an ongoing process - new deleterious alleles are
always being created.
 If alleles are being created by mutation at the same rate
they’re eliminated by selection, the allele will achieve an
equilibrium frequency.
Mutation-Selection balance
 We can calculate the equilibrium frequency with a simple
equation.
𝑢
𝑞=
For recessive alleles
𝑠
𝑞 = the equilibrium frequency of the mutant allele
µ = the mutation rate (the rate at which the new allele is
created)
s = the selection coefficient = a number between 0 and 1 that
measures the strength of selection acting against the
allele. (s = 1-w)
If the favored phenotype produces 100 progeny, and only 90 are
produced by the less favorable phenotype, then the fitness (w) of the
less favorable phenotype is 0.9, and s = 1-w = 0.1.
When the mutant allele is dominant, then
If s = 1 then 𝑞 = 𝑢
𝑢
𝑞=
𝑠
Mutation-Selection balance
 We can calculate the equilibrium frequency with a simple
equation.
 When mutation rates are low and
selection against the allele is high, the
equilibrium frequency will be low.
low 𝑞 =
𝑢↓
𝑠↑
 When mutation rates are high and
selection is relatively weak, the
equilibrium frequency will be high.
high 𝑞 =
𝑢↑
𝑠↓
Mutation-Selection balance
 Cystic Fibrosis – is it maintained by selection-balance?
1. CF is caused by recessive loss-of-function
Lung damage by cystic fibrosis (right).
mutations in the CFTR (cystic fibrosis
transmembrane conductance regulator) gene.
a. CFTR is a cell surface protein expressed in
mucus membrane linings
b. Acts as, among other things, a chloride
channel (inability to regulate chloride
concentrations causes many of the
problems associated with CF).
c. Gene also plays a role in allowing cells of
lung lining to ingest and destroy
Pseudomonas aeruginosa bacteria.
i. Inability to destroy bacteria means individuals
homozygous for mutant CFTRs have chronic P.
aeruginosa lung infections.
ii. Ultimately leads to severe lung damage and
early death.
A lung epithelial cell ingesting
P. aeruginosa. Cells in red are
inside cell and surrounded by
CFTR.
Mutation-Selection balance
Selection against loss-of-function mutations in CFTR is strong.
a. Until recently, very few individuals with CF lived to
reproductive age.
b. Many of those who do survive into 30s and 40s are sterile.
 Mutant CFTR allele remains present at frequencies higher
than predicted by H-W .
 CF present 1 in 2,500 newborns of European ancestry.
What is the allele frequency of the mutant allele?
Mutation-Selection balance
𝑞=
𝑢
𝑠
a. Assume selection coefficient of 1 (homozygotes never
reproduce)
b. Determine the mutation rate that would be required to
produce equilibrium frequency of about 0.02.
Result = mutation rate of 4 x 10-4
 Actual mutation rate seems to be about 6.7 x 10-7.
 This is substantially lower than would be required for
mutation-selection balance.
Is something else is going on?
Mutation-Selection balance
 Piers et al. (1998) hypothesized that CF heterozygotes are
resistant to typhoid fever and therefore exhibit heterozygote
superiority.
 The hypothesis is that typhoid bacteria (Salmonella typhi) use
CFTR protein as a point of entry into cells:
• Homozygous wild-type individuals will be susceptible to
typhoid.
• Homozygous mutant individuals will have CF.
• Heterozygotes will be resistant to typhoid and not have CF.
Mutation-Selection balance
 Piers et al. (1998) tested
hypothesis by constructing mouse
cells with 3 different CFTR
genotypes using wild type and
most common mutant CFTR allele,
then exposing cells to S. typhi.
ΔF508 is a common, single-codon deletion in
the CFTR allele.
Results strongly suggests that CF, like sickle-cell anemia,
is maintained by heterozygote superiority.