Effectiveness of Selection

class: center, middle, inverse, title-slide

.title[
# Effectiveness of Selection
]
.author[
### Jinliang Yang
]
.date[
### Jan. 31, 2024
]

---

# Selection against the recessive allele

- Gametic contribution is the product of genotype freq and fitness

|                    | `\(A_1A_1\)`   | `\(A_1A_2\)`   | `\(A_2A_2\)`   |
| :-------: | : ------ : | :-------: | :-------: |
| Initial frq           | `\(p^2\)`    | `\(2pq\)`    | `\(q^2\)`     | 
| Coefficient of selection | `\(0\)`   | `\(0\)`      | `\(s\)`       |
| Fitness ( `\(w\)` )   |  `\(1\)`        |  `\(1\)`        |  `\(1-s\)`   | 
| Gametic contribution | `\(p^2\)`   | `\(2pq\)`    | `\(q^2(1-s)\)`    |

---

# Selection against the recessive allele

- Gametic contribution is the product of genotype freq and fitness

|                    | `\(A_1A_1\)`   | `\(A_1A_2\)`   | `\(A_2A_2\)`   | Total |
| :-------: | : ------ : | :-------: | :-------: | :-------: |
| Initial frq           | `\(p^2\)`    | `\(2pq\)`    | `\(q^2\)`     |  `\(1\)`        |
| Coefficient of selection | `\(0\)`   | `\(0\)`      | `\(s\)`       |             |
| Fitness ( `\(w\)` )   |  `\(1\)`        |  `\(1\)`        |  `\(1-s\)`   |            |
| Gametic contribution | `\(p^2\)`   | `\(2pq\)`    | `\(q^2(1-s)\)`    |  `\(1-sq^2\)` |

### After one generation of selection against recessive

Note that there has been a proportionate loss of `\(sq^2\)` due to the selection.

The `\(A_1\)` allele freq,

`\begin{align*}
p_1 & = \frac{(2p^2 + 2pq)/2}{p^2 + 2pq + q^2(1-s)} = \frac{p(p+q)}{1 -sq^2}\\
& = \frac{p}{1- sq^2} \\
\end{align*}`

---

# Selection

### After one generation of selection against recessive

`\begin{align*}
p_1  = \frac{p}{1- sq^2} \\
\end{align*}`

### Change in allele freq after one generation

`\begin{align*}
\Delta p & = p_1 - p_0  = \frac{p}{1- sq^2} - p\\
& = \frac{p-p(1-sq^2)}{1-sq^2} \\
& = \frac{spq^2}{1 - sq^2}
\end{align*}`

Then,

`\begin{align*}
\Delta q & = -\Delta p = -\frac{spq^2}{1 - sq^2}
\end{align*}`

---

# Simulation for selection against recessive

```r
deltap <- function(s, p){
  q <- 1-p
  return((s*p*q^2)/(1 - s*q^2))
}

p <- seq(0, 1, by=0.01)
plot(p, deltap(s=0.4, p), type="l", lwd=3, col="red", xlab="p", ylab="|p1 - p0|")
lines(p, deltap(s=0.2, p), lty=2, lwd=3, col="blue")
```

---

# Effectiveness of Selection

### Initial allele freq

- Most effective at intermediate freq
  
  - Inefficient when targeted recessive allele is rare

### Degree of dominance

- It will become more complex

---

# Degree of dominance

|                    | `\(A_1A_1\)`   | `\(A_1A_2\)`   | `\(A_2A_2\)`   | Total |
| :-------: | : ------ : | :-------: | :-------: | :-------: |
| Initial frq           | `\(p^2\)`    | `\(2pq\)`    | `\(q^2\)`     |  `\(1\)`        |
| Fitness ( `\(w\)` ): recessive  |  `\(1\)`        |  `\(1\)`        |  `\(1 -s\)`   |   
| Fitness ( `\(w\)` ): dominance   |  `\(w_{11}\)`        |  `\(w_{12}\)`        |  `\(w_{22}\)`   |            |
| Gametic contribution | `\(p^2\)`   | `\(2pq\)`    | `\(q^2(1-s)\)`    |  `\(1-sq^2\)` |

### Recessive:

`\begin{align*}
p_1 & = \frac{p^2 + pq}{p^2 + 2pq + q^2(1-s)} = \frac{p}{1- sq^2} \\
\end{align*}`

### Degree of dominance:

`\begin{align*}
p_1 & = \frac{p^2w_{11} + pqw_{12}}{p^2w_{11} + 2pqw_{12} + q^2w_{22}} \\
\end{align*}`

---

# Average fitness

`\begin{align*}
p_1 & = \frac{p^2w_{11} + pqw_{12}}{p^2w_{11} + 2pqw_{12} + q^2w_{22}} \\
\end{align*}`

`\begin{align*}
\bar{w} = p^2w_{11} + 2pqw_{12} + q^2w_{22} \\
\end{align*}`

- `\(w_{11}\)`: genotype specific fitness for `\(A_1A_1\)`
- `\(w_{12}\)`: genotype specific fitness for `\(A_1A_2\)`
- `\(w_{22}\)`: genotype specific fitness for `\(A_2A_2\)`

`\begin{align*}
\Delta p & = \frac{p^2w_{11} + pqw_{12}}{\bar{w}} - p \\
& = \frac{pq(p(w_{11} - w_{12}) + q(w_{12} - w_{22}))}{\bar{w}}
\end{align*}`

---

# Practice Problem

In a population of fireflies, allele `\(A_1\)` results in a brighter light. We wish to select for the `\(A_1\)` allele to save money on electricity.

36% of the population is `\(A_1A_1\)`. We spray all the dimly lit `\(A_2A_2\)` individuals so they are sterile but estimate that we missed 10%, which randomly mate with the others.

### Questions:

- What is the fitness of each genotype?

- What type of dominance model is this?

- What is the proportion of fireflies _carrying_ the unwanted, `\(A_2\)` allele after one generation of selection?

---

# Average fitness

- Define each of your variables ( `\(A_1\)`, `\(A_2\)`, `\(p\)`, `\(q\)` )

- Identify % survial/contribution of each genotype

- Calculate the relative fitness ( `\(w\)` ) for each genotype

- Use the general equation with `\(\bar{w}\)`

`\begin{align*}
p_1 & = \frac{p^2w_{11} + pqw_{12}}{p^2w_{11} + 2pqw_{12} + q^2w_{22}} \\
\end{align*}`

- When you get your answer does it make sense?

- For example, for the favorable allele, i.e., `\(A_1\)` allele, does `\(p_1\)` increase?

---

# Allele freq change over time

Selection for generations to achieve target frequency:

- `\(p_0\)` = current frequency
- `\(p_t\)` = frequency at future time
- `\(t\)` = number of generations

--
----------

### Selection for Dominant allele ( `\(s=1\)` )

It is the same as selection against recessive genotype.

`\begin{align*}
p_1 = & \frac{p}{1 - sq^2} \\
= & \frac{1-q}{(1+q)(1-q)} = \frac{1}{1+q}
\end{align*}`

`\begin{align*}
p_2 = & \frac{p_1}{1 - sq^2_1} \\
= & \frac{1-q_1}{(1-q_1^2)} = \frac{1 - (1-p_1)}{1 - (1-p_1)^2} = \frac{\frac{1}{1+q}}{1 - (\frac{q}{1+q})^2}
= \frac{1+q}{1+2q}
\end{align*}`

---

# Allele freq change over time

`\begin{align*}
p_1  & = \frac{1}{1+q} \\
p_2  & = \frac{1+q}{1+2q} \\
p_3  & = \frac{1+2q}{1+3q} \\
\end{align*}`

A pattern emerges ...

`\begin{align*}
p_t  & = \frac{1+(t-1)q_0}{1+tq_0} \\
q_t  & = 1- p_t = \frac{q_0}{1+tq_0} \\
\end{align*}`

The generations required to move from `\(p_0\)` to `\(p_t\)` is the same as to move from `\(q_0\)` to `\(q_t\)`

`\begin{align*}
1 + tq_0 & = \frac{q_0}{q_t} \\
t & = \frac{q_0 - q_t}{q_tq_0} = \frac{1}{q_t} - \frac{1}{q_0}
\end{align*}`

---

# Simulation for selection against recessive

`\begin{align*}
p_t  & = \frac{1+(t-1)q_0}{1+tq_0} \\
q_t  & = 1- p_t = \frac{q_0}{1+tq_0} \\
\end{align*}`

```r
get_pt <- function(p0, t){
  q0 <- 1 - p0
  return(1- q0/(1+t*q0))
  }
```

`\begin{align*}
t & = \frac{q_0 - q_t}{q_tq_0} = \frac{1}{q_t} - \frac{1}{q_0}
\end{align*}`

```r
get_t <- function(p0, pt){
  q0 = 1- p0
  qt = 1 - pt
  return(1/qt - 1/q0)
  }
```

---

# Simulation for selection against recessive

```r
t <- 1:50
get_t(p0=0.6, pt=0.9) #7.5
```

```
## [1] 7.5
```

```r
plot(t, get_pt(0.05, t), type="l", lwd=3, col="red", 
     xlab="generations", ylab="p", ylim=c(0, 1))
points(x=get_t(p0=0.05, pt=0.6), y=0.6, col="blue", cex=3, pch=16)
points(x=get_t(p0=0.05, pt=0.9), y=0.9, col="blue", cex=3, pch=16)
```