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Today Dominance and types of selection
Why do lethal recessives stick around?
Equilibrium under selection
Stable equilibrium: overdominance
Unstable equilibrium: underdominance
Lethal Recessives
For completely recessive case, h=0
For lethality, s=1
ω
A1A1 A1A2 A2A2
0
0.2
0.4
0.6
0.8
1
A1A1 A1A2 A2A2A1A1 A1A2 A2A2
A1A1 A1A2
A2A2
Relative Fitness (ω) ω11 ω12
ω22
Relative Fitness (hs) 1 1-hs 1-s
Lethal Recessive
For q<1
h=0; s=1
ω11=1; ω12=1-hs=1; ω22=1-s=0
Δq more negative at large q
Population moves toward maximum fitness
Rate of change decreases at low q
Δq = -pqs[ph + q(1-h)]
1-2pqhs-q2s
-pq2
1-q2=
-q2
1+q=
q
0.0 0.2 0.4 0.6 0.8 1.0-0.5
-0.4
-0.3
-0.2
-0.1
0.0
q
q
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Retention of Lethal Recessives As p approaches 1, rate of change decreases
Very difficult to eliminate A2, recessive deleterious allele from population
Heterozygotes “hidden” from selection (ω11=1; ω12=1-hs=1)
At low frequencies, most A2 are in heterozygous state:
q
p
2q2
2pq=
q
pq
0.50.1
0.01
1999
q
0.0 0.2 0.4 0.6 0.8 1.0Heterozygotes:Homozygotes
0
2
4
6
8
10
12
Ratio of A2 alleles in heterozygotes versus homozygotes
Time to reduce lethal recessives
It takes a very large number of generations to reduce lethal recessive frequency once frequency gets low
0
11
qqt
t
See Hedrick 2011, p. 123 for derivation
Selection against Recessives
For completely recessive case, h=0
For deleterious recessives, s<1
A1A1 A1A2 A2A2
ω ω11 ω12 ω22
s 1 1-hs 1-s
ω
A1A1 A1A2 A2A2
0
0.2
0.4
0.6
0.8
1
A1A1 A1A2 A2A2A1A1 A1A2 A2A2
Selection Against Recessives
h=0; 0<s<1
Maximum rate of change at intermediate allele frequencies
Location of maximum depends on s: q ≈ 2/3 for small s
Where is maximum rate of change in q for lethal recessive?
What is final value of q?
What is final average fitness of population?
Δq = -pqs[ph + q(1-h)]
1-2pqhs-q2s
-pq2s
1-q2s=
-q2s(1-q)
1-q2s=
q
0.0 0.2 0.4 0.6 0.8 1.0-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
q
s=0.2
q
0.0 0.2 0.4 0.6 0.8 1.0-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
q s=0.4
s=0.2
q
0.0 0.2 0.4 0.6 0.8 1.0-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
q
s=0.2
s=0.4
s=1
Lethal recessive, continues off chart
Modes of Selection on Single Loci Directional – One homozygous
genotype has the highest fitness
Purifying selection AND Darwinian/positive/adaptive selection
Depends on your perspective!
0 ≤ h ≤ 1
Overdominance – Heterozygous genotype has the highest fitness (balancing selection)
h<0, 1-hs > 1
Underdominance – The heterozygous genotypes has the lowest fitness (diversifying selection)
h>1, (1-hs) < (1 – s) < 1 for s > 0
0
0.2
0.4
0.6
0.8
1
AA Aa aa
ω
A1A1 A1A2 A2A2
0
0.2
0.4
0.6
0.8
1
AA Aa aa
ω
A1A1 A1A2 A2A2
0
0.2
0.4
0.6
0.8
1
AA Aa aa
ω
A1A1 A1A2 A2A2
Equilibrium The point at which allele frequencies become constant through time
Two types of equilibria
Stable
Unstable
The question: stable or unstable?
What happens if I move q a little bit away from equilibrium?
Stable Equilibria
railslide.com
•Perturbations from equilibrium cause variable to move toward equilibrium
Heterozygote Advantage (Overdominance)
New notation for simplicity (hopefully):
Genotype
A1A1 A1A2 A2A2
Fitness ω11 ω12 ω22
Fitness in terms of s and h 1 – s1 1 1 – s2
0
0.2
0.4
0.6
0.8
1
AA Aa aa
ω
A1A1
A1A2
A2A2
Equilibrium under Overdominance
Equilibrium occurs under three conditions: q=0, q=1 (trivial), and
s1p – s2q = 0
021 eqeq qsps
)1(12 eqeq qsqs
112 sqsqs eqeq
121 )( sssqeq
21
1
ss
sqeq
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