Plant breeding

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# Plant breeding
### Jinliang Yang
### March 8, 2022

---

# From PopGen to QuantGen

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<img src="b4.0.png" height=450>
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> Wallace et al., 2018
]

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### Domestication 
  - Natural populations

### Conventional plant breeding
  - Breeding populations
]

---

# Plant breeding is an art and science

Plant breeding is the genetic improvement of plants for __human benefit__.

## As a science
- An objective basis for deciding which parents to cross
- Which selection methods to use
- Which progeny to keep
- Which cultivars to release

## As an art

- Requires subjective judgment in the design and implementation of a breeding program
- The intuition to determine one parent or one group of progeny or one cultivar is better than another.

---

# Changed definition on fitness

Plant breeding is the genetic improvement of plants for __human benefit__.

Breeders do not improve plants for the sake of the plants themselves.
  - For example, seed shattering, which allows a plant to propagate itself for the next generation, would be considered beneficial in cereal crops.

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<img src="sh1.png" height=200>
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> __Seed shattering phenotype in sorghum__. Seeds were scattered everywhere in the wild sorghum SV and firmly retained on the domesticated Tx430. (Lin et al., 2012.)

---

# Quantitative traits

__Quantitative traits__, relative to _qualitative traits_, characterized by a continuum of phenotypes.

Quantitative traits are studies with measures of central tendency (e.g., mean) and dispersion (e.g., variance)

Quantitative traits are controlled by the joint effect of many genes (i.e., __quantitative trait loci__).

---

# Breeding populations

Breeding populations are created by breeders to serve as a source of cultivars that meet specific breeding objectives.

- Genotype frequencies and allele frequencies to characterize them

-----------

Suppose a diploid breeding population is segregating at a locus with two alleles, `$A_1$` and `$A_2$`

| Genotype  | Number    | Freq.     | 
| :-------: |: ------- :| :-------: | 
| `$A_1A_1$`  | 240   | `$P_{11}=0.4$`  | 
| `$A_1A_2$`  | 240  | `$P_{12} =0.4$`  | 
| `$A_2A_2$`  | 120  | `$P_{22} =0.2$`  |
| Total     | 600  |

What is the allele frequence `$p$` and `$q$` for `$A_1$` and `$A_2$`?

`\begin{align*}
& p = P_{11} + 1/2P_{12} \\
& p + q =1
\end{align*}`

---

# Hardy-Weinberg Equilibrium

Suppose this diploid breeding population is mated at random.

The union of two gametes that have the `$A_1$` alleles leads to an individual with the `$A_1A_1$` genotype would be `$P_{11(RM)}=p^2=0.36$`

Similarly, `$P_{22(RM)}=q^2=0.16$`

With random mating, what is the probability of having an `$A_1A_2$` individual?

---

# Hardy-Weinberg Equilibrium

| Genotype  | Number    | Freq.     | After 1st RM |
| :-------: |: ------- :| :-------: | :-------: |
| `$A_1A_1$`  | 240   | `$P_{11}=0.4$`  | `$P_{11(RM)}=0.36$` |
| `$A_1A_2$`  | 240  | `$P_{12} =0.4$`  | `$P_{12(RM)}=0.48$` |
| `$A_2A_2$`  | 120  | `$P_{22} =0.2$`  | `$P_{22(RM)}=0.16$` |
| Total     | 600  | | |

Suppose this diploid breeding poulation is mated at random.

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

How about the allele frequencies after random mating?

`\begin{align*}
p &  = P_{11} + 1/2P_{12} \\
p_{(RM)} & = P_{11(RM)} + 1/2P_{12(RM)} \\
& = p^2 + 1/2 \times 2pq \\
& = p \times (p + q) \\
& = p
\end{align*}`

Random mating therefore __changes the genotype frequencies__ of the population, but it does not change the allele frequencies.

---

# Hardy-Weinberg Equilibrium

What happens after a second generation of random mating?

| Genotype  | Number    | Freq.     | After 1st RM | 2nd RM |
| :-------: |: ------- :| :-------: | :-------: | :-------: |
| `$A_1A_1$`  | 240   | `$P_{11}=0.4$`  | `$P_{11(RM)}=0.36$` | `$P_{11(RM)}=0.36$` |
| `$A_1A_2$`  | 240  | `$P_{12} =0.4$`  | `$P_{12(RM)}=0.48$` | `$P_{12(RM)}=0.48$` |
| `$A_2A_2$`  | 120  | `$P_{22} =0.2$`  | `$P_{22(RM)}=0.16$` | `$P_{22(RM)}=0.16$` |
| Total     | 600  | | |

### Three key features of HWE

1. The allele frequencies remain __constant__ from generation to generation.
2. The square of the array of allele frequencies is equal to the array of genotype frequencies.
`\begin{align*}
(p +q )^2 = p^2 + 2pq + q^2
\end{align*}`
3. If allele frequencies change due to external factors, __one generation of random mating__ will lead to equilibrium genotype frequencies.

---

# Hardy-Weinberg Equilibrium

To meet HWE, in addition to __random mating__ and the __absence of selection__, other conditions needed are:
- a large population
- absence of mutation and migration.

An `$F_2$` population from two inbred is in HWE at a single locus.

Breeders, however, routinely use procedures that cause deviations from HWE:
- Artificial selection
- Inbreeding during the development of progeny
- etc.

---

# Artificial selection

Selection, in general, refers to __a differential rate of reproduction__ among individuals that differ in their genotypes.
- In other words, selection occurs when some individuals __produce more progeny__ than others.

### Natural selection

Due to either differential fertility or differential viability.

### Artificial selection

- The nonselected plants do not contribute any progeny to the next generation.
- Whereas all the selected plants usually contribute the __same number of progeny__ to the next generation.
- The direct consequence of selection is _a change in allele frequencies_.

---

# Artificial selection

- Consider a single locus in a random-mating population.

- The __selection coefficient__ represents the severity of selection against a particular genotype.
  - `$s_{11}$` for `$A_1A_1$`, `$s_{12}$` for `$A_1A_2$`, `$s_{22}$` for `$A_2A_2$`

- The relative __fitness__ of each genotype is 
  - `$1- s_{11}$` for `$A_1A_1$`, `$1- s_{12}$` for `$A_1A_2$`, `$1- s_{22}$` for `$A_2A_2$`

- When selection occurs in an F2 population in HWE.

|                           | `$A_1A_1$`    |  `$A_1A_2$`     | `$A_2A_2$` |  Total |
| :-------: | :-------: | :-------: | :-------: | :-------: |
| Frequency                  | `$p^2$`             | `$2pq$`       | `$q^2$` |   1 |
| Relative fitness           |  `$1- s_{11}$`      | `$1- s_{12}$` | `$1- s_{22}$` | |
| Contribution               |  `$p^2(1- s_{11})$` | `$2pq(1- s_{12})$` | `$q^2 (1- s_{22})$` | T | 
| Frequency after selection  | `$\frac{p^2(1- s_{11})}{T}$` | `$\frac{2pq(1- s_{12})}{T}$` | `$\frac{q^2 (1- s_{22})}{T}$` | |

---

# Artificial selection

The frequency of the recessive allele after one generation of selection is:

`\begin{align*}
T & = p^2 - p^2s_{11} + 2pq - 2pqs_{12} + q^2 - q^2s_{22} \\
 & = 1 - (p^2s_{11} + 2pqs_{12} + q^2s_{22})
\end{align*}`

`\begin{align*}
q_1 & = \frac{pq(1-s_{12}) + q^2(1-s_{22})}{T} \\
 & = \frac{q(p -ps_{12} + q -qs_{22})}{1 - (p^2s_{11} + 2pqs_{12} + q^2s_{22})} \\
 & = \frac{q(1 -ps_{12} -qs_{22})}{1 - (p^2s_{11} + 2pqs_{12} + q^2s_{22})} \\
\end{align*}`

---

# Artificial selection

The change in allele frequency due to one generation of selection is

`\begin{align*}
\Delta_q & = q_1 -q \\
 & = \frac{pq[p(s_{11} - s_{12}) + q(s_{12} - s_{22})]}{1 - (p^2s_{11} + 2pqs_{12} + q^2s_{22})} \\
\end{align*}`

---

# Artificial selection

```r
Dq <- function(q, s11=0, s12=0, s22=0, n=10){
  
  out <- data.frame(n=0, q=q)
  # loop through n generations
  for (i in 1:n){
    p = 1 - q
    q <- (q*(1-p*s12 - q*s22))/(1- (p^2*s11 + 2*p*q*s12 + q^2*s22))
    tem <- data.frame(n=i, q= q)
    out <- rbind(out, tem)
  }
  return(out)
}
```

---

# Artificial selection

- We completely select __against__ the __recessive__ deleterious alleles: `$A_2A_2$` with `$s_{22}=1$`
- Selection coeffecients are `$s_{11} = s_{12} =0$` for `$A_1A_1$` and `$A_1A_2$` genotypes.

```r
df <- Dq(q=0.5, s11=0, s12=0, s22=1, n=10)
plot(df$n, df$q, type="l", xlab="Generations", main="F2 population",
     ylab="Freq of A2 allele", lwd=3, ylim=c(0, 0.5))
```

---

# Artificial selection

- We completely select against the __heterozygote `$A_1A_2$` with `$s_{12}=1$`__
- Selection coeffecients are `$s_{11} = s_{22} =0$` for `$A_1A_1$` and `$A_2A_2$` genotypes.

```r
df <- Dq(q=0.5, s11=0, s12=1, s22=0, n=10)
#df <- Dq(q=0.4, s11=0, s12=1, s22=0, n=10)
#df <- Dq(q=0.6, s11=0, s12=1, s22=0, n=10)
plot(df$n, df$q, type="l", xlab="Generations", main="F2 population",
     ylab="Freq of A2 allele", lwd=3, ylim=c(0, 1))
```

---

# Artificial selection

The change in `$q$` with complete selection against the heterozygote depends on the initial `$q$` value.

- `$\Delta_q$` is __zero__ if `$q$` is = 0.50
- `$\Delta_q$` is __positive__ if `$q$` is > 0.50
- `$\Delta_q$` is __negative__ if `$q$` is < 0.50

Changes in allele frequency lead to changes in the population mean of a given phenotype.

An allele becomes __fixed__ when all the other alleles originally present in a population become lost.
- Join effect of __selection__ and __inbreeding__.

---

# Inbreeding and relatedness

__Inbreeding__ results when two __related__ individuals are mated.
  - Two individuals are related if they have at least one ancestor in common.
  - But if the common ancestor is too remote, its effect on inbreeding is negligible.

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Two alleles are __identical by state__ if they physically represent the same allele, e.g., in P1.

In contrast, alleles are __identical by descent__ if they are copies of the same allele present in a common ancestor.
- The copies of the A1 allele in P2 and P3 are not only IBS but are also IBD.
]

---

# Coefficient of coancestry

The probability that two alleles are identical by descent can be deduced from the system of mating or from the pedigree structure.

The __coefficient of coancestry ( `$f_{XY}$` )__ between individuals X and Y is the probability that, at a single locus, a random allele from X and Y are IBD.

- `$f_{XY}=0$` indicates two individuals have no relationship.

- `$f_{XY}=1$` at a given locus indicates that the two individuals are homozygous for copies of the same allele found in an ancestor.

- `$f_{XY}=1$` _across the genome_ indicates that the two individuals are __fully inbred and genetically identical__.

---

# Coefficient of inbreeding

The coefficient of inbreeding ( `$F$` ) is the probability that, at a single locus, the two alleles in the __same individual__ are IBD.

- The coefficient of inbreeding measures IBD within an individual ( `$F=f_{XX}$` )

- Whereas the coefficient of coancestry measures IBD between two individuals ( `$f_{XY}$` )

- Similarly, `$F=0$` indicates no inbreeding and `$F=1$` indicates complete inbreeding.

---

# Selfing

The increase in the `$F$` coefficient is __halved__ upon each generation of selfing.

In a cross between inbreds, the coefficient of inbreeding among individual plants in the `$F_n$` or `$S_{n-2}$` generation is:

`\begin{align*}
F = 1 - (\frac{1}{2})^{n-2}
\end{align*}`

Plant breeders work with individual plants or with families.

The `$F$` coefficient __among families (e.g., `$F_3$`)__ is equal to the `$F$` coefficient __among the individual plant (e.g., `$F_2$`)__ that were selfed to form the families (Cockerham, 1961).

|      Plants    | Families        |  Freq of heterozygotes     | `$F$` | 
| :-------:      | :-------:       | :-------: | :-------: | :-------: |
| `$F_2$` or `$S_0$` | `$F_3$` or `$S_1$`  | `$P_{12}$`    | `$0$` |  
| `$F_3$` or `$S_1$` | `$F_4$` or `$S_2$`  | `$0.5P_{12}$` | `$0.5$` | 
| `$F_4$` or `$S_2$` | `$F_5$` or `$S_3$`  | `$0.25P_{12}$` | `$0.75$` |  
| `$F_5$` or `$S_3$` | `$F_6$` or `$S_4$`  | `$0.125P_{12}$` | `$0.875$` |

---

# A F2 population

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Another way to define __coefficient of inbreeding ( `$F$` )__ is equal to:

- The proportion by which heterozygosity is reduced upon inbreeding, relative to a population in Hardy-Weinberg equilibrium.

`\begin{align*}
F &= 1 - \frac{P_{12(F)}}{P_{12}} \\
\end{align*}`

We assume the `$F2$` population is non-inbred.
- This assumption is made so that genetic parameters, such as the __population mean__ and __variance__, are defined for a non-inbred base population.