Asymmetry of responses

class: center, middle, inverse, title-slide

.title[
# Asymmetry of responses
]
.author[
### Jinliang Yang
]
.date[
### April 17, 2024
]

---

# Illinois long-term selection experiment

.pull-left[
- The experiment started in 1896 by C.G. Hopkins and is [still active](http://mooselab.cropsci.illinois.edu/longterm.html)!
- They are selecting lines for higher or lower concentration of protein or oil in the maize kernel.
- __Mass selection__ was used with a selection intensity of approximately 1 out of 5 for most of the experiment

> Selected on the basis of phenotype from a mixed population, their seeds are bulked and used to grow next generation.
]

.pull-right[

```r
d <- read.csv(file="https://jyanglab.com/AGRO-931/data/ILoil.csv", na.strings=".", header=TRUE)
par(bty="l", pty="m", mar=c(5, 4, 1, 1))
matplot(x=d$YR, y=d[, c("IHP", "ILP", "IHO", "ILO")], type="l", lty=1,
        xlab="Year", ylab="Concentration (%)", col=c("#8b2323","#8b2323","#458b74","#458b74"), lwd=4)
legend("topleft", c("Protein", "Oil"), lty=1, lwd=4, col=c("#8b2323","#458b74"), bty="n")
```

<img src="w13_c2_files/figure-html/unnamed-chunk-1-1.png" width="100%" style="display: block; margin: auto;" />
]

---

# Illinois long-term selection experiment

### Variability of the responses

.pull-left[
<img src="w13_c2_files/figure-html/unnamed-chunk-2-1.png" width="100%" style="display: block; margin: auto;" />
]

### Possible reasons?

.pull-right[

1. __Random drift__ due to the restricted number of parents

2. __Sampling error__ in estimating the generation mean

3. Variation of __selection differentials__

4. __Environmental factors__
]

---

# Illinois long-term selection experiment

### Asymmetry of responses
.pull-left[
<img src="w13_c2_files/figure-html/unnamed-chunk-3-1.png" width="100%" style="display: block; margin: auto;" />
]

### Discussion for possible reasons?

whether it is real (need replications)

.pull-right[

1. Random drift?
2. Selection differentials?
3. Inbreeding depression?
  - reduce the rate in the upward and increase it in the downward direction
4. Others?
]

---

# Asymmetry of responses

### Discussion for possible reasons?

- Genetic asymmetry
- Genes with large effects

#### Other reasons (F&M p211-215):

- Scalar asymmetry
  - uneven or non-proportional changes in different traits
  
- Indirect selection
 - For example, if a breeding program selects for increased grain yield in wheat and this trait is correlated with plant height, the response in plant height might show asymmetry.

- Maternal effects

---

# Consequences of selection

### 1. For __how long__ does the response continue?

### 2. By __how much__ the population mean can be changed?

### 3. What is the __genetic nature__ of the limit to further progress?

---

# Selection limits

## Long-term results

- __Selection limits__, or where response seems to stop, are usually unpredictable.

But they are very interesting from a biological point of view.

## Total response:

__Total response ( `$R_T$` )__ = selection limit in high direction - selection limit in low direction

---

# Total response

### Genetic architecture
- Total response depends on __number of loci__ contributing to a given amount of variation.

- For a given amount of variation, the number of loci is inversely related to __average size of effects__.

### Mutation
Long term results become unpredictable, because __mutation__ produces new variation whose nature we cannot predict.

---

# Theoretical maximum limit

### Two assumptions:

1. All the loci have the same magnitude of additive effect: `$a$`
2. All alleles start at the frequencies of 0.5.

Then the range between two homozygotes is 2a for one locus. For more than one loci:

`\begin{align*}
R_T & = \sum\limits_{i=1}^n2a = 2na\\
\end{align*}`

The additive variance, assuming all alleles start at a frequency of 0.50, for one locus

`\begin{align*}
\sigma_A^2 & = 2pq(a + d(q-p))^2 = \frac{1}{2}a^2 \\
\end{align*}`

For more than one loci:

`\begin{align*}
\sum\sigma_A^2 &= \sum\limits_{i=1}^n \frac{1}{2}a^2 = \frac{1}{2}na^2\\
\end{align*}`

---

# Theoretical maximum limit

### Two assumptions:
1. All the loci have the same magnitude of additive effect: `$a$`
2. All alleles start at the frequencies of 0.5.

`\begin{align*}
R_T & = \sum\limits_{i=1}^n2a = 2na\\
\sum\sigma_A^2 &= \sum\limits_{i=1}^n \frac{1}{2}a^2 = \frac{1}{2}na^2\\
\end{align*}`

The relationship between the range of response and the additive variance is obtained as:

`\begin{align*}
\frac{R_T^2}{\sum\sigma_A^2} = \frac{4n^2a^2}{\frac{1}{2} na^2} = 8n \\
\end{align*}`

---
# For example:

| Observation | Exp1      | Exp2 | Exp3  | 
| :-------: | :-------: | :--------: | :-------: |
| `$R_T/\sum\sigma_A$`  | 100 | 50 | 10 |
| `$n$`  | ? | ? | ? |

---
# For example:

| Observation | Exp1      | Exp2 | Exp3  | 
| :-------: | :-------: | :--------: | :-------: |
| `$R_T/\sum\sigma_A$`  | 100 | 50 | 10 |
| `$n=R_T^2/8\sigma_A^2$`  | 1250 | 312.5 | 12.5 |

### Theory of limits:
The total response relative to the initial genetic variation, depends primarily on __the number of loci__.

Addressed the third question to some degree:
- What is the __genetic nature__ of the limit to further progress?

- i.e., the polygenicity --- how many genetic loci in determining the trait variation

---

# Theoretical maximum limit

`\begin{align*}
R_T & = \sum\limits_{i=1}^n2a = 2na\\
\end{align*}`

Only could be achieved if the effective population size being very large.

### In practice

Limited parents used => unfavorable alleles are fixed by genetic drift.

---

# Theoretical maximum limit

The chance of fixation of a favorable allele:

`\begin{align*}
& N_es \\
\end{align*}`

- `$s$`: selection coefficient. `$s = i(2a/\sigma_P)$` given by F&M eq. 11.8.
  - Selection coefficient refers to the reduced fitness of the genotype being selected against.
  
- `$N_e$`: effective population size

Therefore, the chance of fixation of a favorable is a function of __ `$N_ei$`__.

The total response should be greater:
1. with larger population size ( `$N_e$` )

2. with more intense selection ( `$i$` )

---

# Theoretical maximum limit

### Breeder's equation:

`\begin{align*}
& R = ih^2\sigma_P \\
\end{align*}`

This is the predicted response in one generation.

### The theoretical half-life of response:

Up to about `$2N_e$` generations. 
- `$N_e$` is the number of individuals that would give rise to the calculated sampling variance.

`\begin{align*}
& R_{max} = 2N_eih^2\sigma_P \\
\end{align*}`

> Robertson 1960

---

# The theoretical half-life of response:

`\begin{align*}
& R_{max} = 2N_eih^2\sigma_P \\
\end{align*}`

- The theoretical maximum response is the total response attained if the trait were controlled by an infinite number of loci.

- For maximum response to divergent selection (i.e. range between high and low lines)
 - selection intensity, `$i$`, is the sum of selection intensities in the two directions.

- This does no more than set an upper limit to what can be expected.

---

# Illinois long-term selection experiment

.pull-left[

### Predict the theoretical limit?

<img src="w13_c2_files/figure-html/unnamed-chunk-4-1.png" width="100%" style="display: block; margin: auto;" />
]

.pull-right[
#### From Moose et al., 2004 and Dudley and Lambert, 2004:

- In 1896, 163 ears from the open-pollinated variety Burr's White, 24 highest ears formed the Illinois high oil strain (IHO) (__Ne ~ 4-12__)
- Later, 300-500 kernels => __Ne raised to a maximum of 96__
- About 20% selected in each generation
- Realized `$h^2 = 0.25$`
- `$\sigma_P$` about 0.9

```r
ifun(0.2)
```

```
## [1] 1.39981
```
]

---

# Illinois long-term selection experiment

.pull-left[

### Predict the theoretical limit?

<img src="w13_c2_files/figure-html/unnamed-chunk-7-1.png" width="100%" style="display: block; margin: auto;" />
]

.pull-right[
`\begin{align*}
& R_{max} = 2N_eih^2\sigma_P \\
\end{align*}`

```r
2*12*ifun(0.2)*0.25*0.9
```

```
## [1] 7.558972
```

```r
2*96*ifun(0.2)*0.25*0.9
```

```
## [1] 60.47177
```

]

---

# Summary of the long-term selection

### 1. For __how long__ does the response continue?

`\begin{align*}
& 2N_e \\
\end{align*}`

### 2. By __how much__ the population mean can be changed?

`\begin{align*}
& R_{max} = 2N_eih^2\sigma_P \\
\end{align*}`

### 3. What is the __genetic nature__ of the limit to further progress?

`\begin{align*}
& n= \frac{R_T^2}{8\sigma_A^2}\\
\end{align*}`