Breeding value and dominance

class: center, middle, inverse, title-slide

.title[
# Breeding value and dominance
]
.author[
### Jinliang Yang
]
.date[
### Mar. 20, 2024
]

---

# Genotypic value

| Genotype  | Freq.   | Value   | Freq. `\(\times\)` Val.   | 
| :-------: |: ------- :| :-------: | :-------:  | 
| `\(A_1A_1\)`  | `\(p^2\)`   | `\(+a\)`    |  `\(p^2a\)`        |
| `\(A_1A_2\)`  | `\(2pq\)`   | `\(d\)`     | `\(2pqd\)`           |
| `\(A_2A_2\)`  | `\(q^2\)`   | `\(-a\)`    | `\(-q^2a\)`          |
|           |         | Sum =   | `\(a(p-q) + 2pqd\)`  |

### Parents pass on **alleles**, not genotypes.

### Average Effect of `\(A_1\)` and `\(A_2\)`

---

# Average Effect of `\(A_1\)` and `\(A_2\)`

The average effect of `\(A_1\)` is the **mean deviation** from the **population mean** of individuals which received that allele from one parent, and the alleles received from the other parents being at random.

According to Falconer & Mackey `Table 7.2`:

`\begin{align*}
\alpha_1 = q(a + d(q-p))
\end{align*}`

`\begin{align*}
\alpha_2 = -p(a + d(q-p))
\end{align*}`

### Allele substitution effect

`\begin{align*}
\alpha =  \alpha_1 - \alpha_2 =  a + d(q-p) \\
\end{align*}`

therefore,
`\begin{align*}
\alpha_1 = & q\alpha \\
\alpha_2 = & -p\alpha \\
\end{align*}`

---
# Breeding value (A)

Breeding value is the "value" of an individual as a parent. 
- The value of an individual, judged by the mean value of its progeny, is called the *breeding value* (__A__) of the individual.

- Note that breeding value is **population specific**.

### Theoretical definition

Sum of the average effects of the alleles an individual carries.

`\begin{align*}
BV =  \sum_{i=1}^k\sum_{j=1}^2\alpha_{ij}
\end{align*}`

Where summation occurs across the number of loci ( `\(k\)` ) and the two alleles present at each locus.

---
# Breeding value (A)

### Operational/Practical definition

The deviation of the value of an individual's offspring from the population mean.

.pull-left[
<div align="center">
<img src="bv.png" height=300>
</div>
]

.pull-right[
### To get the BV of sire X
1. mate it to many random dams

2. measure the performance of all the offspring

3. subtract population mean from the offspring mean and multiply by 2
]

---
# Breeding value (A)

Consider only one locus using the theoretical definition:

| Genotype  | Breeding Value  | 
| :-------: | :-------: | 
| `\(B_1B_1\)`  | `\(2\alpha_1\)` |  
| `\(B_1B_2\)`  | `\(\alpha_1 + \alpha_2\)` |
| `\(B_2B_2\)`  | `\(2\alpha_2\)`  |

### Booroola locus example

Returning to our two example populations and looking at breeding values of the three genotypes.

- Recall that `\(a=0.59\)` and `\(d=0.39\)`

- In __pop1__, freq of `\(B_1\)` allele is 0.25

- In **pop2**, freq of `\(B_1\)` allele is 0.85

---
# Breeding value (A)

Consider only one locus using the theoretical definition:

| Genotype  | Breeding Value  | BV of pop1 | BV of pop2 |
| :-------: | :-------: | | |
| `\(B_1B_1\)`  | `\(2\alpha_1\)` | | |
| `\(B_1B_2\)`  | `\(\alpha_1 + \alpha_2\)` | | |
| `\(B_2B_2\)`  | `\(2\alpha_2\)`  | | |

### Booroola locus example

Returning to our two example populations and looking at breeding values of the three genotypes.

- Recall that `\(a=0.59\)` and `\(d=0.39\)`

- In __pop1__, freq of `\(B_1\)` allele is 0.25

- In **pop2**, freq of `\(B_1\)` allele is 0.85

- `\(\alpha = a+d(q-p)\)`, 0.785 for pop1 and 0.317 for pop2

---
# Breeding value (A)

Consider only one locus using the theoretical definition:

| Genotype  | Breeding Value  | BV of pop1 | BV of pop2 |
| :-------: | :-------: | :----: | :----: |
| `\(B_1B_1\)`  | `\(2\alpha_1\)` | 1.18 | 0.10 |
| `\(B_1B_2\)`  | `\(\alpha_1 + \alpha_2\)` | 0.39 | `\(-0.22\)` |
| `\(B_2B_2\)`  | `\(2\alpha_2\)`  | `\(-0.39\)` | `\(-0.54\)` |

### Booroola locus example

Returning to our two example populations and looking at breeding values of the three genotypes.

- Recall that `\(a=0.59\)` and `\(d=0.39\)`
- In __pop1__, freq of `\(B_1\)` allele is 0.25
- In **pop2**, freq of `\(B_1\)` allele is 0.85
- `\(\alpha = a+d(q-p)\)`, 0.785 for pop1 and 0.317 for pop2

This demonstrates that the BV of an individual is dependent on the population of reference.
In pop1, where the freq of `\(B_1\)` is less, there is greater value of an `\(B_1B_1\)` individual as a parent.

---
# Breeding value (A)

Consider only one locus using the theoretical definition:

| Genotype  | Breeding Value  | Freq |
| :-------: | :-------: | :----: |
| `\(B_1B_1\)`  | `\(2\alpha_1=2q\alpha\)` | `\(p^2\)` | 
| `\(B_1B_2\)`  | `\(\alpha_1 + \alpha_2 = q\alpha - p\alpha\)` | `\(2pq\)` |
| `\(B_2B_2\)`  | `\(2\alpha_2 = -2p\alpha\)`  | `\(q^2\)` |

### What is the mean BV if the population is in HWE?

`\begin{align*}
BV = & 2q\alpha \times p^2 + (q\alpha - p\alpha) \times 2pq - 2p\alpha \times q^2 \\
   = & 2pq\alpha \times (p +q) - 2pq\alpha \times (p + q) \\
   = & 0
\end{align*}`

---

# BV of offspring

The expected breeding value of offspring (o) is `\(1/2\)` the sum of the sire and dam breeding values.

`\begin{align*}
E(A_o) = & 1/2(A_s + A_d)
\end{align*}`

Note here `\(o\)`, `\(s\)`, and `\(d\)` refer to offspring, sire (male), and dam (female) respectively.

- Different offspring of the same parents will differ in BV

- The expected BV is also the individual's expected phenotypic value

`\begin{align*}
E(P_o) = E(A_o) = & 1/2(A_s + A_d)
\end{align*}`

---

# Dominance deviation (D)

We can examine the failure of the additive value to reflect the genotypic value as a deviation --- **a dominance deviation**.

### G = A + D

- Population mean: `\(M = a(p-q) + 2dpq\)`

- Avg. allele substitution effect: `\(\alpha = a+d(q-p)\)`

| Genotype  | Genotypic Value | Value as deviated from `\(M\)`      | 
| :-------: | :-------: | :-----------: |
| `\(A_1A_1\)`  | `\(a\)`   |  ?   |
| `\(A_1A_2\)`  | `\(d\)`   |   ?  | 
| `\(A_2A_2\)`  | `\(-a\)`  |   ?  |

---

# Dominance deviation (D)

We can examine the failure of the additive value to reflect the genotypic value as a deviation --- **a dominance deviation**.

### G = A + D

- Population mean: `\(M = a(p-q) + 2dpq\)`

- Avg. allele substitution effect: `\(\alpha = a+d(q-p)\)`

| Genotype  | Genotypic Value | Value as deviated from `\(M\)`      | 
| :-------: | :-------: | :-----------: |
| `\(A_1A_1\)`  | `\(a\)`   |  `\(a -M = 2q(a-pd) = 2q(\alpha - qd)\)`   |
| `\(A_1A_2\)`  | `\(d\)`   |   `\(d- M = a(q-p) + d(1-2pq) = (q-p)\alpha + 2pqd\)`  | 
| `\(A_2A_2\)`  | `\(-a\)`  |   `\(-a-M = -2p(a+qd) = -2p(\alpha + pd)\)`  |

---

# Dominance deviation (D)

We can examine the failure of the additive value to reflect the genotypic value as a deviation --- **a dominance deviation**.

### G = A + D

| Genotype  |  Value as deviated from `\(M\)`      | Breeding Value  | Dominance Deviation  | 
| :-------: |  :-----------: | :-----------: | :-------: | :-------: | 
| `\(A_1A_1\)`  |   `\(2q(\alpha - qd)\)`   | `\(2q\alpha\)`    |   ?  |  
| `\(A_1A_2\)`  |   `\((q-p)\alpha + 2pqd\)`  | `\((q-p)\alpha\)` |    ?   |  
| `\(A_2A_2\)`  |   `\(-2p(\alpha + pd)\)`  | `\(-2p\alpha\)`   |  ?   |

---

# Dominance deviation (D)

We can examine the failure of the additive value to reflect the genotypic value as a deviation --- **a dominance deviation**.

### G = A + D

| Genotype  |  Value as deviated from `\(M\)`      | Breeding Value  | Dominance Deviation  | 
| :-------: | :-------: | :-----------: | :-----------: | :-------: | :-------: | 
| `\(A_1A_1\)`  |   `\(2q(\alpha - qd)\)`   | `\(2q\alpha\)`    |   `\(-2q^2d\)`   |  
| `\(A_1A_2\)`  |   `\((q-p)\alpha + 2pqd\)`  | `\((q-p)\alpha\)` |    `\(2pqd\)`   |  
| `\(A_2A_2\)`  |   `\(-2p(\alpha + pd)\)`  | `\(-2p\alpha\)`   |  `\(-2p^2d\)`   |

- Note that in the absence of dominance, breeding values and genotypic values are the same.

---

# Dominance deviation (D)

- It follows that the mean dominance deviation is zero.

The mean dominance deviation:

`\begin{align*}
 = & p^2 \times (-2q^2d) + 2pq \times 2pqd + q^2 \times (-2p^2d) \\
 = & - 2p^2q^2d + 4p^2q^2d - 2p^2q^2d \\
 = & 0
\end{align*}`

---
# Interaction deviation (epistasis)

- Epistasis = interaction of alleles at **different** loci

- Also causes deviation of genotypic value from additive genetic value (or BV)

Imagine a trait controlled by two loci, A and B. Then

`\begin{align*}
 G = & G_A + G_B + I_{AB} \\
   = & (\alpha_i + \alpha_j + \delta_{ij}) + (\alpha_k + \alpha_l + \delta_{kl}) + I_{AB}
\end{align*}`

Where

`\begin{align*}
 I_{AB} = & \alpha_i\alpha_k + \alpha_i\alpha_l + \alpha_j\alpha_k + \alpha_j\alpha_l \\
    & + \alpha_i\delta_{kl} + \alpha_j\delta_{kl} + \alpha_k\delta_{ij} + \alpha_l\delta_{ij} \\
    & + \delta_{ij}\delta_{kl}
\end{align*}`

---
# Interaction deviation (epistasis)

- Interaction effect can arise through interaction
  - Between the average allele effects at each locus ( __A__ `\(\times\)` __A__ )
  - Between average effect and dominance deviation ( __A__ `\(\times\)` __D__ )
  - And between dominance deviations ( __D__ `\(\times\)` __D__ )

- As # of loci increases, so does # of interactions (exponentially)

- Summed to contribute to the epistatic deviation ( `\(I\)` )

`\begin{align*}
G = \sum_{i=1}^k(A_i + D_i) + I
\end{align*}`