Heterosis and Inbreeding Depression

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.title[
# Heterosis and Inbreeding Depression
]
.author[
### Jinliang Yang
]
.date[
### April 22, 2024
]

---

# Heterosis

In the absence of selection, the heterosis on crossing a set of random lines is the same as the depression upon inbreeding.

.pull-left[
<div align="center">
<img src="figure14.2_het.png" height=380>
</div>
> Springer and Stupar, 2007
]

.pull-right[
<div align="center">
<img src="figure14.3_yield.png" height=300>
</div>
> Crow 1998
]

---

# Heterosis

.pull-left[
<div align="center">
<img src="figure14.4_m1.png" height=300>
</div>
> East, 1936; Shull, 1948; Crow, 1998; Charlesworth and Charlesworth, 1999

]

.pull-right[
<div align="center">
<img src="figure14.5_m2.png" height=300>
</div>
> Crow 1948; Gowen, 1952
]

----------

- __Pseudo-overdominance__: _Cockerham and Zeng, 1996_
- __Dosage__: _Birchler et al., 2003_

---

# Measurement of heterosis

The theoretical conclusion depends upon crossing a large number of random lines from the base population.

Here, let's turn from the average consequences of crossing to the specific crossing of two lines.

.pull-left[
<div align="center">
<img src="figure14.6_f1.png" height=250>
</div>
]

### High-parental heterosis
`\begin{align*}
& F_1 - Max(P_1, P_2) \\
\end{align*}`

### Mid-parental heterosis

`\begin{align*}
& F_1 - mean(P_1, P_2) \\
\end{align*}`

---

# Mid-parental heterosis

|  | Freq ( `\(A_1\)` )      | Freq ( `\(A_2\)` ) |
| :-------: | :-------: | :--------: |
| `\(P_1\)`  | `\(p\)` | `\(q\)` |
| `\(P_2\)`  | `\(p'\)` | `\(q'\)` |
| `\(P_1 - P_2\)`  | `\(y=p-p'\)` | `\(y=q'-q\)` |
| `\(P_2\)` re-written | `\(p'=p-y\)` | `\(q'=q+y\)` |

#### Population Mean

`\begin{align*}
& M = a(p-q) + 2dpq \\
\end{align*}`

Therefore,

`\begin{align*}
M_{P_1} &  = a(p-q) + 2dpq \\
M_{P_2} &  = a(p'-q') + 2dp'q' \\
& = a(p-y - q -y ) + 2d(p-y)(q+y) \\
\end{align*}`

The mid-parent genotypic value:

`\begin{align*}
 M_\bar{P} & = 1/2(M_{P_1} + M_{P_2}) \\
& = a(p-q-y) +d[2pq + y(p-q) -y^2] \\
\end{align*}`

---

# Genotypic value for F1

Frequencies:

|           | __pop1 ( `\(p\)` )__     |  __pop1 ( `\(q\)` )__ |
| :-------: | :-------: | :--------: |
| __pop2 ( `\(p-y\)` )__ | `\(p(p-y)\)`  | `\(q(p-y)\)` |
| __pop2 ( `\(q+y\)` )__ | `\(p(q+y)\)`  | `\(q(q+y)\)` |

Genotypic values:

|           | __A1__     |  __A2__ |
| :-------: | :-------: | :--------: |
| __A1__ | `\(A_1A_1=a\)`  | `\(A_1A_2=d\)` |
| __A2__ | `\(A_1A_2=d\)`  | `\(A_2A_2=-a\)` |

The mean genotypic value of the F1:

`\begin{align*}
M_{F_1} & = ap(p-y) + dq(p-y) + dp(q+y) -aq(q+y) \\
& = a(p-q-y) +d[2pq + y(p-q)] \\
\end{align*}`

---

# Mid-parental heterosis

`\begin{align*}
& H_{F_1} = M_{F_1} - M_\bar{P} \\
\end{align*}`

`\begin{align*}
& M_{F_1} = a(p-q-y) +d[2pq + y(p-q)] \\
& M_\bar{P}  =  a(p-q-y) +d[2pq + y(p-q) -y^2] \\
\end{align*}`

--------

`\begin{align*}
& H_{F_1} = M_{F_1} - M_\bar{P} = dy^2\\
\end{align*}`

Remember, `\(y\)` is the difference in allele frequency between populations.

### Extending across all loci

`\begin{align*}
& H_{F_1} = \sum_{i=1}^n dy^2\\
\end{align*}`

---

# About MPH

`\begin{align*}
& H_{F_1} = \sum_{i=1}^n dy^2\\
\end{align*}`

- `\(y\)` is the difference in allele frequency between populations.
- `\(n\)` is the number of loci in contributing to heterosis.

### Three conclusions:

- __Directional dominance__ is required for heterosis

- Amount of heterosis __specific to each individual cross__

- If lines with `\(F=1\)` crossed, then `\(y^2\)` can be only 1 or 0. If that is the case, then `\(H\)` is the summation of `\(d\)`s across loci with different alleles at each locus.
  - Recall that `\(F\)` here is the inbreeding coefficient.

---

# Heterosis in F2 generation

F2 being made by random mating among the individuals of the F1.

- Because of the random mating, allele freq in F2 will be the HW freq of the F1
  
  - Allele freq in F1, being the mean of the allele freq in the two parental popl.s `\(=\frac{p + p'}{2}=(p -\frac{1}{2}y)\)`
  
  - Similarly, another allele freq `\(=(q + \frac{1}{2}y)\)`
  
  - Plug these values in to the function to calculate population mean `\(M = a(p-q) + 2dpq\)`

### Population Mean and heterosis

`\begin{align*}
& M_{F_2} = a(p-q-y) + d[2pq + y(p-q) - \frac{1}{2} y^2] \\
\end{align*}`

`\begin{align*}
H_{F_2} & = M_{F_2} - M_{\bar{P}} \\
& = \frac{1}{2}dy^2 = \frac{1}{2}H_{F_1} \\
\end{align*}`

---
# Inbreeding decreases population mean

The change of mean from F1 to F2 may therefore be regarded as __inbreeding depression__

.pull-left[
Reduction of the __mean phenotypic value__ displayed by characters associated with reproductive capacity or physiological efficiency.
]

.pull-right[
<div align="center">
<img src="table14.1.png" height=350>
</div>
]

---

# Inbreeding decreases population mean

The change of mean from F1 to F2 may therefore be regarded as __inbreeding depression__

.pull-left[
Reduction of the __mean phenotypic value__ displayed by characters associated with reproductive capacity or physiological efficiency.
- Tends to affect traits related to __fitness and viability__ more than other traits.
]

.pull-right[
<div align="center">
<img src="table14.1.png" height=350>
</div>
]

---

# Inbreeding coefficient (F)
 
.pull-left[
- Average allele frequency across entire population does not change, they only change within each line.

- Any change in mean value must be caused by changes in __genotype frequencies__.
- As inbreeding coefficient (F) increases:
 - frequency of __heterozygotes goes down__ 
 - frequency of __homozygotes goes up__
]

.pull-right[
<div align="center">
<img src="figure14.1_basepop.png" height=300>
</div>
]

What we want to know is __the change in mean value__ across all lines derived from a base population.

---

# Inbreeding depression

From Chapter 3 that the genotype frequencies after inbreeding are

| Genotype | Freq      | Value | 
| :-------: | :-------: | :--------: | 
| `\(A_1A_1\)`  | `\(\bar{p}^2 + \bar{p}\bar{q}F\)` | `\(a\)` | 
| `\(A_1A_2\)`  | `\(2\bar{pq} - 2\bar{p}\bar{q}F\)` | `\(d\)` | 
| `\(A_2A_2\)`  | `\(\bar{q}^2 + \bar{p}\bar{q}F\)` | `\(-a\)` |

where, `\(\bar{p}\)` and `\(\bar{q}\)` are the allele frequencies in the whole population.

---

# Inbreeding depression

From Chapter 3 that the genotype frequencies after inbreeding are

| Genotype | Freq      | Value |  Freq `\(\times\)` value |
| :-------: | :-------: | :--------: | :------------:|
| `\(A_1A_1\)`  | `\(\bar{p}^2 + \bar{p}\bar{q}F\)` | `\(a\)` | `\(\bar{p}^2a + \bar{p}\bar{q}Fa\)` |
| `\(A_1A_2\)`  | `\(2\bar{pq} - 2\bar{p}\bar{q}F\)` | `\(d\)` | `\(2\bar{pq}d - 2\bar{p}\bar{q}Fd\)` |
| `\(A_2A_2\)`  | `\(\bar{q}^2 + \bar{p}\bar{q}F\)` | `\(-a\)` | `\(-\bar{q}^2a - \bar{p}\bar{q}Fa\)` |

where, `\(\bar{p}\)` and `\(\bar{q}\)` are the allele frequencies in the whole population.

### The population mean after inbreeding is then

`\begin{align*}
M_F &  =  \bar{p}^2a + \bar{p}\bar{q}Fa + 2\bar{pq}d - 2\bar{p}\bar{q}Fd -\bar{q}^2a - \bar{p}\bar{q}Fa  \\
& = a(\bar{p}^2 - \bar{q}^2) + 2d\bar{p}\bar{q}(1-F) \\
& = a(\bar{p} - \bar{q}) + 2d\bar{p}\bar{q}(1-F) \\
\end{align*}`

Note that `\(M_0\)` is the population mean before inbreeding.
`\begin{align*}
M_0  = a(\bar{p} - \bar{q}) + 2d\bar{p}\bar{q} \\
\end{align*}`

---

# The population mean after inbreeding

`\begin{align*}
& M_F   =  a(\bar{p} - \bar{q}) + 2d\bar{p}\bar{q}(1-F) \\
& M_0  = a(\bar{p} - \bar{q}) + 2d\bar{p}\bar{q} \\
\end{align*}`

Therefore,

`\begin{align*}
M_F  = M_0 - 2d\bar{p}\bar{q}F \\
\end{align*}`

where `\(M_0\)` here is the population mean before inbreeding.

### Equation about inbreeding depression
This equation brings two important points:

- Change in population mean __only occurs when `\(d \neq 0\)`__.

- A locus contributes to inbreeding depression most __when `\(p = q = 0.5\)`__.

---

# Combined effects of all loci

`\begin{align*}
M_F & = \sum a(\bar{p} - \bar{q}) + 2\sum d\bar{p}\bar{q}(1-F) \\
 & = \sum a(\bar{p} - \bar{q}) + 2\sum d\bar{p}\bar{q} - 2F\sum d\bar{p}\bar{q} \\
\end{align*}`

### Equation interpretation

- This means that a preponderance of dominance in one direction is needed.
 - e.g., the alleles that increase the trait value need to be dominant more than recessive.

- In the absence of epistasis, `\(M_F\)` declines linearly with increasing `\(F\)`.