Lecture 8: Types of Selection
September 14, 2015
Last Time
Inbreeding depression
Introduction to selection
Predicting allele frequency change in
response to selection
Today
Dominance and types of selection
Why do lethal recessives stick around?
Equilibrium under selection
Stable equilibrium: overdominance
Unstable equilibrium: underdominance
Lethal Recessives
A1A1
A1A2
A2A2
Relative Fitness (ω)
ω11
ω12
ω22
Relative Fitness (hs)
1
1-hs
1-s
For completely recessive case, h=0
For lethality, s=1
1
0.8
0.6
ω
0.4
0.2
0
A1AA11A1
A1A1
A1AA21A2
A1A2
A2AA22A2
A2A2
Lethal Recessive
Δq = -pqs[ph + q(1-h)]
1-2pqhs-q2s
=
-pq2
1-q2
=
For q<1
1+q
0.0
 h=0; s=1
 ω11=1; ω12=1-hs=1; ω22=1-s=0
-q2
-0.1
q
-0.2
-0.3
-0.4
 Δq more negative at large q
-0.5
0.0
0.4
0.6
0.8
1.0
q
 Population moves toward
maximum fitness
 Rate of change decreases at low
q
0.2
1.0
0.8

0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
q
0.8
1.0
Retention of Lethal Recessives
 As p approaches 1, rate of
change decreases
 Heterozygotes “hidden” from
selection (ω11=1; ω12=1-hs=1)
Heterozygotes:Homozygotes
 Very difficult to eliminate A2,
recessive deleterious allele from
population
 At low frequencies, most A2 are in
heterozygous state:
p
2pq
p
q
q
=
q
2q2
0.5
1
0.1
9
Ratio of A2 alleles in
0.01 99
heterozygotes versus
homozygotes
12
10
8
6
4
2
0
0.0
0.2
0.4
0.6
q
0.8
1.0
Time to reduce lethal recessives
1 1
t 
qt q0
See Hedrick 2011, p. 123 for derivation
It takes a very large number of generations to reduce
lethal recessive frequency once frequency gets low
Selection against Recessives
A1A1
ω
ω11
s
1
A1A2
A2A2
ω12
ω22
1-hs
1-s
For completely recessive case, h=0
For deleterious recessives, s<1
1
0.8
0.6
ω
0.4
0.2
0
AA
A1A
1A
1A1
1
1
AA
1AA
2A
1A2
1
2
AA
A2A
2A
2A2
2
2
Selection Against Recessives
Δq = -pqs[ph + q(1-h)]
1-2pqhs-q2s
=
-pq2s
1-q2s
 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?
=
-q2s(1-q)
1-q2s
0.00
s=0.2
s=0.2
s=0.2
-0.02
s=0.4
-0.04
q
s=1
s=0.4
-0.06
-0.08
-0.10
-0.12
0.0
0.2
0.4
0.6
q
Lethal recessive,
continues off chart
0.8
1.0
Modes of Selection on Single Loci
 Directional – One homozygous genotype
has the highest fitness
 Purifying selection AND
Darwinian/positive/adaptive selection
1
0.8
ω
0.6
0.4
0.2
0
 Depends on your perspective!
AAA
1A1
aa
AAa
1A2 A2A2
AAA
1A1
aa
AAa
1A2 A2A2
AAA
1A1
AAa1A2 A2aaA2
 0 ≤ h ≤ 1
1
 Overdominance – Heterozygous
genotype has the highest fitness
(balancing selection)
0.8
ω
0
 Underdominance – The heterozygous
h>1, (1-hs) < (1 – s) < 1 for s > 0
0.4
0.2
h<0, 1-hs > 1
genotypes has the lowest fitness
(diversifying selection)
0.6
1
0.8
ω
0.6
0.4
0.2
0
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
•Perturbations
from equilibrium
cause variable to
move toward
equilibrium
railslide.com
Unstable Equilibria
•Perturbations from
equilibrium cause
variable to move
away from equilibrium
Does selection always cause
average fitness to approach 1?
Under what conditions do we
reach an equilibrium while
polymorphisms still exist in the
population?
Heterozygote Advantage
(Overdominance)
1
0.8
ω
0.6
0.4
0.2
0
 New notation for simplicity (hopefully):
Fitness
Fitness in terms of si
1 2 p0 q0w12 q 2w 22 p0 q0w12 + q02w 22
q1 =
+
=
2
w
w
w
AA
A1
A1
Aa
A1
A2
aa
A2
A2
Genotype
A1A1 A1A2 A2A2
ω11
ω12
ω22
1 – s1
1
1 – s2
p0 q0 + q02 (1- s2 )
q1 = 2
p0 (1- s1 ) + 2 p0 q0 + q02 (1- s2 )
Equilibrium under Overdominance
 Equilibrium occurs under
three conditions: q=0,
q=1 (trivial), and
s1p – s2q = 0
s1 peq  s2 qeq  0
s2 qeq  s1 (1  qeq )
s2 qeq  s1qeq  s1
qeq ( s1  s2 )  s1
s1
qeq 
s1  s2
Equilibrium under Overdominance
 Allele frequency always
approaches same value of
q when perturbed away
from equilibrium value
 Stable equilibrium
 Allele frequency change
moves population toward
maximum average fitness
s1
qeq 
s1  s2
Heterozygote Disadvantage
(Underdominance)
1
0.8
ω
0.6
0.4
0.2
0
AAA
1A1
Fitness
Fitness in terms of s1 and s2
A1A1
ω11
1 + s1
AAa1A2 A2aaA2
Genotype
A1A2
A2A2
ω12
ω22
1
1 + s2
s1
qeq 
s1  s2
Heterozygote Disadvantage (Underdominance)
Genotype
A1A1
A1A2
A2A2
Fitness
ω11
ω12
ω22
Fitness in terms of s1 and s2
11+s
+ 1s
1
11+s
+ t2
s1 = 0.1
s2 = 0.1
Equilibrium under Underdominance
 Allele frequency moves
away from equilibrium
point and to extremes
when perturbed
 Unstable equilibrium
 Equilibrium point is at
local minimum for average
fitness
 Population approaches
trivial equilibria: fixation
of one allele
Where are equilibrium points?
ω11 =1.1 ω12 = 1 ω22 = 1.1
Underdominance Revisited
Fitness
Fitness in terms of s1 and s2
Fitness in terms of s and h
h 1
s1  hs
Genotype
A1A1 A1A2
ω11
ω12
1 + s1
1
1
1-hs
s1
hs
ω
s
s2
s2  hs  s
 s (h  1)
A2A2
ω22
1 + s2
1-s
A1A1
A1A2 A2A2
Why does “nontrivial” equilibrium
occur with underdominance?
 Why doesn’t A1 allele
always go to fixation if
A1A1 is most fit
genotype?
ω
Proportion of A1 alleles in
heterozygous state:
pq
(pq+p2)
= q
A1A1
A1A2 A2A2
What determines the equilibrium
point with underdominance?
s1=0.2; s2=0.2
ω11=1; ω12=0.8; ω22=1
ω
s1=0.2; s2=0.05
ω11=0.85; ω12=0.8; ω22=1
Allele Frequency (q)
 Why does equilibrium
point of A1 allele
frequency increase
when selection
coefficient decreases?
A 1A 1
A1A2
A2A2
s1
qeq =
s1 + s2
s1 peq  s2 qeq
Example: Kuru in Fore Tribespeople
 Prion disease in Fore tribesmen
 Transmitted by cannibalism of
relatives by women/children
 Cannibalism stopped in 1950’s
 Older people exposed to selection,
younger are ‘controls’
 Identified locus that causes
susceptibility: Prion Protein
Gene, PRNP
 MM and VV are susceptible, MV are
resistant
Mead et al. 2003 Science 300:640
http://learn.genetics.utah.edu/features/prions/kuru.cfm
Kuru and Heterozygote Advantage
1 v
(s 
)
Selection coefficient
2 (only females)
0.403
0.2985
0.373
sMM
qeq 
 0.483
sMM  sVV
 Tremendous selective advantage in favor of
heterozygotes
 Balancing selection maintains polymorphism in human
populations