Natural Selection
M. Drew LaMar
October 31, 2016
Model Design
Selection
| Genotype | \( A_{1}A_{1} \) | \( A_{1}A_{2} \) | \( A_{2}A_{2} \) |
|---|---|---|---|
| Fitness | \( w_{11} \) | \( w_{12} \) | \( w_{22} \) |
| Frequency\( _{t} \) | \( p_{t}^2 \) | \( 2p_{t}q_{t} \) | \( q_{t}^2 \) |
| Count\( _{t} \) | \( p_{t}^2N_{t} \) | \( 2p_{t}q_{t}N_{t} \) | \( q_{t}^2N_{t} \) |
| Count\( _{t+1} \) | \( p_{t}^2N_{t}w_{11} \) | \( 2p_{t}q_{t}N_{t}w_{12} \) | \( q_{t}^2N_{t}w_{22} \) |
\[ \begin{align} N_{2,t+1} & = 2p_{t}q_{t}N_{t}w_{12} + 2q_{t}^2N_{t}w_{22} \\ N_{t+1} & = 2\bigl(p_{t}^2N_{t}w_{11} + 2p_{t}q_{t}N_{t}w_{12} + q_{t}^2N_{t}w_{22}\bigr) \end{align} \]
Question: So what is the frequency of allele \( A_{2} \) in generation \( t+1 \)?
Model Design
\[ \begin{align} N_{2,t+1} & = 2p_{t}q_{t}N_{t}w_{12} + 2q_{t}^2N_{t}w_{22} \\ N_{t+1} & = 2\bigl(p_{t}^2N_{t}w_{11} + 2p_{t}q_{t}N_{t}w_{12} + q_{t}^2N_{t}w_{22}\bigr) \end{align} \]
Answer: \[ \begin{align} q_{t+1} & = \frac{N_{2,t+1}}{N_{t+1}} \\ & = \frac{2p_{t}q_{t}N_{t}w_{12} + 2q_{t}^2N_{t}w_{22}}{2\bigl(p_{t}^2N_{t}w_{11} + 2p_{t}q_{t}N_{t}w_{12} + q_{t}^2N_{t}w_{22}\bigr)} \\ & = \frac{p_{t}q_{t}w_{12} + q_{t}^2w_{22}}{p_{t}^2w_{11} + 2p_{t}q_{t}w_{12} + q_{t}^2w_{22}} \end{align} \]
Model Design
Replace \( p_{t} = 1-q_{t} \) to arrive at
\[ q_{t+1} = \frac{(1-q_{t})q_{t}w_{12} + q_{t}^2w_{22}}{(1-q_{t})^2w_{11} + 2(1-q_{t})q_{t}w_{12} + q_{t}^2w_{22}}. \]
Since we are concerned with relative fitness, we have
| Genotype | \( A_{1}A_{1} \) | \( A_{1}A_{2} \) | \( A_{2}A_{2} \) |
|---|---|---|---|
| Fitness | \( 1 \) | \( 1+f(s,t) \) | \( 1+g(s,t) \) |
Model Design
We consider 5 common types of selection:
- Codominance (genic selection):
| Genotype | \( A_{1}A_{1} \) | \( A_{1}A_{2} \) | \( A_{2}A_{2} \) |
|---|---|---|---|
| Fitness | \( 1 \) | \( 1+s \) | \( 1+2s \) |
- Complete dominance (\( A_{2} \) is dominant):
| Genotype | \( A_{1}A_{1} \) | \( A_{1}A_{2} \) | \( A_{2}A_{2} \) |
|---|---|---|---|
| Fitness | \( 1 \) | \( 1+s \) | \( 1+s \) |
Model Design
- Complete recessiveness (\( A_{2} \) is recessive):
| Genotype | \( A_{1}A_{1} \) | \( A_{1}A_{2} \) | \( A_{2}A_{2} \) |
|---|---|---|---|
| Fitness | \( 1 \) | \( 1 \) | \( 1+s \) |
- Overdominance (\( s > 0 \), \( s > t \), heterozygote highest fitness):
| Genotype | \( A_{1}A_{1} \) | \( A_{1}A_{2} \) | \( A_{2}A_{2} \) |
|---|---|---|---|
| Fitness | \( 1 \) | \( 1+s \) | \( 1+t \) |
- Underdominance (\( s < 0 \), \( s < t \), heterozygote lowest fitness):
| Genotype | \( A_{1}A_{1} \) | \( A_{1}A_{2} \) | \( A_{2}A_{2} \) |
|---|---|---|---|
| Fitness | \( 1 \) | \( 1+s \) | \( 1+t \) |
Example: Codominance
Codominance (genic selection):
| Genotype | \( A_{1}A_{1} \) | \( A_{1}A_{2} \) | \( A_{2}A_{2} \) |
|---|---|---|---|
| Fitness | \( 1 \) | \( 1+s \) | \( 1+2s \) |
\[ \begin{align} q_{t+1} & = \frac{(1-q_{t})q_{t}(1+s) + q_{t}^2(1+2s)}{(1-q_{t})^2 + 2(1-q_{t})q_{t}(1+s) + q_{t}^2(1+2s)} \\ & = F(q_{t},s) \end{align} \]
Example: Codominance
Simulate: \( s = 0.01 \), \( q_{0} = 0.04 \)
Example: Codominance
Simulate: \( s = 0.01 \), \( q_{0} = 0.04 \)
N <- 3000
s <- 0.01
q <- rep(0.04,N)
for (i in 1:(N-1)) {
qi <- q[i] # For readability
q[i+1] <- ((1-qi)*qi*(1+s) + qi*qi*(1+2*s))/((1-qi)*(1-qi) + 2*(1-qi)*qi*(1+s) + qi*qi*(1+2*s))
}
plot(1:N,q)
Directional selection
Codominance (left); Dominant (center); Recessive (right)
Recessive deleterious alleles
What happens when \( A_{2} \) is recessive and deleterious? Suppose \( s = -0.01 \).
The proportion of \( A_{2} \) in the population after 3000 generations is 0.0184496!!! That's only a drop of 54%!
Balancing selection
Definition [Wikipedia]:
Balancing selection refers to a number of selective processes by which multiple alleles (different versions of a gene) are actively maintained in the gene pool of a population at frequencies longer than expected from genetic drift alone.
Overdominance (\( s > 0 \), \( s > t \), heterozygote highest fitness):
| Genotype | \( A_{1}A_{1} \) | \( A_{1}A_{2} \) | \( A_{2}A_{2} \) |
|---|---|---|---|
| Fitness | \( 1 \) | \( 1+s \) | \( 1+t \) |
Overdominance
Simulate: \( s = 0.04 \), \( t = 0.02 \)
Underdominance
Underdominance (\( s < 0 \), \( s < t \), heterozygote lowest fitness):
| Genotype | \( A_{1}A_{1} \) | \( A_{1}A_{2} \) | \( A_{2}A_{2} \) |
|---|---|---|---|
| Fitness | \( 1 \) | \( 1+s \) | \( 1+t \) |
Simulate: \( s = -0.02 \), \( t = -0.01 \)
