Lecture 8: Types of Selection

February 5, 2014

Last Time 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

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

Unstable Equilibria

•Perturbations from equilibrium cause variable to move away from 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

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

21

1

ss

sqeq