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One-locus diploid model
One-locus diploid model
Most animals and plants are diploid.
Selection acts in a large
population only in the
diploid phase (not in the
haploid phase), with
random mating, no
migration, and no mutation
To complete our understanding of how natural
selection effects evolutionary change, we turn
to diploid models of selection.
One-locus diploid model
Let p[t] equal the
frequency of the A allele
among adult diploids
p[t]
q[t]
p[t]
q[t]
One-locus diploid model
p[t]
q[t]
p[t] will also equal the
frequency of the A allele
among the haploid gametes
of these adults.
p[t]
q[t]
AA
Aa
aa
p[t]2
2p[t]q[t]
q[t]2
Assuming random mating
(or random union of
gametes), diploid offspring
will be in Hardy-Weinberg
proportions.
One-locus diploid model
One-locus diploid model
Genotype frequencies after selection:
p[t]
q[t]
Each diploid genotype has
its own relative fitness
p[t]
q[t]
AA
p[t]2
Aa
aa
2p[t]q[t]
q[t]2
WAA
WAa
Waa
One-locus diploid model
freq(AA) =
p2WAA
}
2
2
p WAA + 2pqWAa + q Waa }
freq(Aa) =
2pqWAa
p2WAA + 2pqWAa + q2Waa
freq(aa) =
q2Waa
p2WAA + 2pqWAa + q2Waa
relative freq after selection
mean fitness,W
!
One-locus diploid model
Genotype frequencies
Allele frequencies
100%
p2WAA
freq(A)
freq(Aa)
2
2
p WAA + 2pqWAa + q Waa
p2WAA + pqWAa
p2WAA + 2pqWAa + q2Waa
2pqWAa
2
p WAA + 2pqWAa + q2Waa
q2Waa
p2WAA + 2pqWAa + q2Waa
2
p[ t + 1] =
50%
freq(Aa)
50%
freq(Aa)
100%
freq(aa)
p[ t ] W AA + p[ t ] q[ t ] W Aa
2
2
p[ t ] W AA + 2 p[ t ] q[ t ] W Aa + q[ t ] W aa
freq(a)
pqWAa + q2Waa
p2WAA + 2pqWAa + q2Waa
!
This equation describes the change in frequency of
allele A from one diploid generation to the next.
One-locus diploid model
One-locus diploid model
Forms of selection
Directional selection
Directional Selection:
• Favoring allele A: WAA > WAa > Waa
• Favoring allele a: WAA < WAa < Waa
Heterozygote advantage (or overdominance):
• WAA < WAa > Waa
Heterozygote disadvantage (or underdominance):
• WAA > WAa < Waa
(Which forms of selection were present in the haploid
selection model?)
As fitnesses become more similar, spread of A
becomes slower
One-locus diploid model
One-locus diploid model
Directional selection
Directional selection
s = 0.4
h = 1/2
s = 0.2
h = 1/2
Fitnesses are often defined
relative to one homozygote:
• WAA = 1 + s
s = 0.1
h = 1/2
• WAa = 1 + h s
• Waa = 1
• s is the selection coefficient
favoring allele A
Roughly, as the fitness differences decrease by a factor of x, the time
it takes for A to spread increases by a factor of x
• h is the dominance
coefficient of allele A
One-locus diploid model
One-locus diploid model
A digression on dominance
A digression on dominance
Imagine dominance as a tug-of-war between two alleles.
Dominance is NOT a characteristic of an allele -- it reflects an
interaction between two alleles.
Allele A might be dominant with respect to allele a, but recessive
with respect to allele a’.
a
A
A is additive
(h = 1/2)
Dominance also depends on the phenotype being measured (e.g.,
the sickle cell allele is nearly dominant with respect to avoiding
malaria, but it is nearly recessive with respect to anemia).
Waa
(1)
WAa
(1+s/2)
WAA
(1+s)
One-locus diploid model
One-locus diploid model
A digression on dominance
A digression on dominance
Imagine dominance as a tug-of-war between two alleles.
a
A
Imagine dominance as a tug-of-war between two alleles.
a
A
A is fully dominant
(h = 1)
Waa
(1)
WAa =WAA
(1+s)
A is partially dominant
(1/2 < h < 1)
Waa
(1)
WAa WAA
(1+hs) (1+s)
One-locus diploid model
One-locus diploid model
A digression on dominance
A digression on dominance
Imagine dominance as a tug-of-war between two alleles.
a
A
Imagine dominance as a tug-of-war between two alleles.
a
A
A is recessive
(h = 0)
WAa =Waa
(1)
A is partially recessive
(0 < h < 1/2)
Waa WAa
(1) (1+hs)
WAA
(1+s)
WAA
(1+s)
One-locus diploid model
One-locus diploid model
A digression on dominance
A digression on dominance
Imagine dominance as a tug-of-war between two alleles.
a
A
When A is recessive,
a is dominant.
WAa =Waa
(1)
WAA
(1+s)
In evolutionary biology, we typically focus on a reference allele,
which often is the most common allele within a population
(wildtype allele). Whether the reference allele is dominant,
recessive, or otherwise depends on which other alleles are
present in the population.
a
A
One-locus diploid model
One-locus diploid model
A digression on dominance
Directional selection
Just as we can measure the dominance coefficient of allele A, we
can also measure the dominance coefficient of allele a.
WAA = 1 + s
WAa = 1 + h s
Allele A “dominates” if h = 1
WAA = 1
WAa = 1 – h s
Waa = 1
Allele a “dominates” if h = 1 Waa = 1 – s
s is the selection coefficient
s is the selection coefficient
favouring allele A
acting against allele a
h is the dominance
coefficient of allele A
h is the dominance
coefficient of allele a
If WAA < WAa < Waa, selection favors the a allele
One-locus diploid model
One-locus diploid model
Heterozygote advantage
Heterozygote disadvantage
If the heterozygote has the
highest fitness, A
approaches an intermediate
frequency, and stays there.
This value is called a
polymorphic equilibrium.
The exact value of the polymorphic equilibrium depends on the
fitnesses.
If the heterozygote has the
lowest fitness, the A allele
rises in frequency if it is
common or decreases in
frequency if it is rare.
The cutoff between where A
rises and falls is called an
unstable polymorphic
equilibrium.
The exact value of the cutoff depends on the fitnesses.
(Note: The population never splits in two.)
One-locus diploid model
One-locus diploid model
Equilibria
Equilibria
An equilibrium is defined as a point at which allele
frequencies don’t change from one generation to the
next.
That is, where p[t] = p[t+1]
p[t]2WAA + pqWAa
p[t] = p[t + 1] =
p[t]2WAA + 2pqWAa + q[t]2Waa
p[t] = p[t + 1] =
p[t]2WAA + pqWAa
p[t]2WAA + 2pqWAa + q[t]2Waa
Solving, we get three equilibria:
p=0
p=1
WAa −Waa
p=
2WAa −WAA −Waa
One-locus diploid model
One-locus diploid model
Equilibria
p=0
p=1
WAa −Waa
p=
2WAa −WAA −Waa
Equilibria
p=0
p=1
WAa −Waa
p=
2WAa −WAA −Waa
The last equilibrium is only valid (between 0 and 1)
when WAA < WAa > Waa or WAA > WAa < Waa.
The last equilibrium is only valid (between 0 and 1)
when WAA < WAa > Waa or WAA > WAa < Waa.
With directional selection, there is no polymorphic
equilibrium (frequencies will always go to 0 or 1)
What are the polymorphic equilibria for the examples
described earlier? (Try this at home)
One-locus diploid model
One-locus diploid model
Equilibria
p=0
p=1
WAa −Waa
p=
2WAa −WAA −Waa
Examples: Sickle-cell anemia
p will approach the polymorphic equilibrium when
WAA < WAa > Waa (heterozygote advantage), but it will
be repelled from it when WAA > WAa < Waa
(heterozygote disadvantage).
Sickle-cell anemia is a human disease affecting the shape and
flexibility of red blood cells.
It is caused by a single mutation in the sixth amino acid of the
chain of hemoglobin.
The mutant form of hemoglobin (S) tends to crystalize and form
chains, causing distortions in red blood cells.
One-locus diploid model
One-locus diploid model
Examples: Sickle-cell anemia
Examples: Sickle-cell anemia
Homozygous carriers of S experience the greatest degree of
sickling and tend to suffer severe anemic attacks.
Heterozygous carriers of S also suffer from sickling of red blood
cells, but to a lesser degree.
However, heterozygous carriers are less likely to die from
malaria, since red blood cells infected with the parasites causing
malaria tend to sickle and be destroyed.
The fitnesses of each genotype can be estimated from
the extent to which genotypic frequencies among
adults depart from Hardy-Weinberg.
For the Nigerian population studied:
WAA = 0.88
WAS = 1
WSS = 0.14
What is the expected equilibrium frequency of the
non-mutant allele (A)?
One-locus diploid model
One-locus diploid model
Examples: Sickle-cell anemia
Examples: Decline of a recessive allele
The fitnesses of each genotype can be estimated from
the extent to which genotypic frequencies among
adults depart from Hardy-Weinberg.
For the Nigerian population studied:
WAA = 0.88
WAS = 1
WSS = 0.14
The frequency of the non-mutant allele (A) in this
population was 0.877, which matches extremely well!
Dawson (1970) studied the decline of a recessive lethal
mutation (Waa = 0) in a population of flour beetles.
He started a laboratory population of beetles with an
initial frequency of the mutation of 1/2.
Changes were consistent
with a recessive or nearly
recessive mutation.
WAa = 1
WAa = 0.9