Phenotypic and Genotypic Values

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# Phenotypic and Genotypic Values
### Jinliang Yang
### March 10, 2022

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# Phenotype

Phenotype in a population can be characterized in terms of its __mean__ and __variance__.
  - Here we have a F2 population. 
  - The plant height ranges from 100-200 cm, with a mean of 150 cm.

A population can be characterized by its __allele__ and __genotype frequencies__.
 - Allele frequencies are `\(p = q = 0.5\)`.
 - Genotype frequencies are `\(p^2\)`, `\(2pq\)`, and `\(q^2\)`.

---

# A F2 population

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## Basic Phenotypic Model

`\begin{align*}
\mathbf{P} &= \mathbf{G} + \mathbf{E} \\
\end{align*}`

- `\(\mathbf{P}\)`: Phenotypic value
- `\(\mathbf{G}\)`: Genotypic value
- `\(\mathbf{E}\)`: Environmental deviation

### Genotypic Value

`\begin{align*}
\mathbf{G} &= \mathbf{A} + \mathbf{D} +  \mathbf{I} \\
\end{align*}`

- `\(\mathbf{A}\)`: Additive genetic value
- `\(\mathbf{D}\)`: Dominance value
- `\(\mathbf{I}\)`: Epistasis deviation

---

# Genotypic value

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At a given locus, there are three possible genotypes: `\(A_1A_1\)`, `\(A_1A_2\)`, and `\(A_2A_2\)`.

- Now we consider this locus has a large effect on plant height.

- Midpoint between `\(A_1A_1\)` and `\(A_2A_2\)` is commonly standardized to 0. Or, represented as `\(m\)`.

---

# Gene action (or mode of inheritance):

- Value of `\(A_1A_2\)` ( `\(d\)` ) indicates gene action at the locus
  - `\(d = 0\)`: __additive__ gene action
  - `\(d = a\)` or `\(-a\)`: __complete dominance__ or __complete recessive__
  - `\(d > a\)` or `\(d < -a\)`: __overdominance__ or __underdominance__
  - Any other values of `\(d\)` besides those above values are __partial dominance__

---

# Population Mean

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

What is the __Population mean ( `\(\mu\)` )__ or the expected genotypic value?

`\begin{align*}
\mu = \sum f_i x_i \\
\end{align*}`

where `\(f_i\)` is the frequency and `\(x_i\)` is the value of the event `\(i\)`.

`\begin{align*}
\mu = & p^2a + 2pqd - q^2a \\
    = & a(p-q) + 2pqd
\end{align*}`

where `\(\mu\)` is the deviation from the midpoint of the homozygotes.

- The first term comes from the homozygote values
- The second from the value of the heterozygote

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# Mutliple Loci

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When multiple loci are acting **independently** and contributing to overall value, the mean is:

`\begin{align*}
\mu =  \sum_{i=1}^k{a}_i(p_i - q_i) + 2\sum_{i=1}^kp_iq_id_i
\end{align*}`

- Summation over all the loci
- `\(k\)` is the number of loci affecting the genotypic value of a trait

---

# The average effect of `\(A_1\)`

- Consider the `\(A_1\)` allele. Under HWE, the probability an `\(A_1\)` allele combines with another `\(A_1\)` allele to form an `\(A_1A_1\)` genotype is `\(p\)`. The value of the `\(A_1A_1\)` genotype is `\(a\)`.

- The probability an `\(A_1\)` allele combines with an `\(A_2\)` allele to form a heterozygote, with value `\(d\)`, is `\(q\)`.

- Thus, the mean value of individuals that received `\(A_1\)` is `\(pa + qd\)`.

- And, the average effect of allele `\(A_1\)` is:

`\begin{align*}
\alpha_1 = & pa + dq - (a(p-q) + 2pqd) \\
  = & q(a + d(q - p)) \\
\end{align*}`

---

# Allele Substitution

Similarly,

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

What is effect of substituting an `\(A_1\)` allele for `\(A_2\)`?

This can be simply expressed as the __difference in average effect of the two alleles__:

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

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

---

# Genotypic values and Breeding values

### Genotypic value

Now, the value of a particular genotype can be partitioned into __the average effects of the alleles__ and any __residuals value__ that the average effects do not account for.

Specifically, the __genotypic value__ of `\(A_iA_j\)`
`\begin{align*}
G_{ij} = \mu + \alpha_i + \alpha_j + \delta_{ij}
\end{align*}`

- Where `\(\mu\)` is the population mean
- `\(\delta_{ij}\)` is the dominance deviation.

### Breeding value

The __Breeding value__ associated with `\(A_iA_j\)` is defined as: the sum of `\(\alpha_i\)` and `\(\alpha_j\)`.
- Breeding value is the value of an individual as a parent!

---
# A breeding program for hybrid maize

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# Testcross effect of an allele

A common breeding procedure is to cross individuals or families that belong to one population with a __tester__ that belongs to a different population.

#### The population being testcrossed

The frequency of the `\(A_1\)` allele is `\(p\)` and the frequency of `\(A_2\)` allele is `\(q\)`.

#### The tester used

The frequency of the `\(A_1\)` allele is `\(p_T\)` and the frequency of `\(A_2\)` allele is `\(q_T\)`.

#### The genotype frequencies

The frequency of the 
- `\(A_1A_1\)` allele is `\(pp_T\)`,
- `\(A_2A_2\)` allele is `\(qq_T\)`,
- `\(A_1A_2\)` allele is `\(pq_T + p_Tq\)`.

---

# Testcross effect of an allele

#### The genotype frequencies

The frequency and the effect of the 
- `\(A_1A_1\)` allele is `\(pp_T\)` => `\(a\)`
- `\(A_2A_2\)` allele is `\(qq_T\)` => `\(-a\)`
- `\(A_1A_2\)` allele is `\(pq_T + p_Tq\)` => `\(d\)`

The testcross mean is the sum of the genotypic values multiplied by their frequencies:

`\begin{align*}
\mu_T = & a(pp_T - qq_T)  + d(pq_T + p_Tq) \\
\end{align*}`

---

# Average testcross effect of an allele

- Consider the `\(A_1\)` allele. Under HWE, the probability an `\(A_1\)` allele combines with __another `\(A_1\)` allele from the tester to form an `\(A_1A_1\)` genotype is `\(p_T\)`__. The value of the `\(A_1A_1\)` genotype is `\(a\)`.

- The probability an `\(A_1\)` allele combines with __an `\(A_2\)` allele  from the tester to form a heterozygote__, with value `\(d\)`, is `\(q_T\)`.

- Thus, the mean value of individuals that received `\(A_1\)` is `\(p_Ta + q_Td\)` and, the average test effect of allele `\(A_1\)` is:

`\begin{align*}
\alpha_1^T = & p_Ta + q_Td - (a(pp_T - qq_T)  + d(pq_T + p_Tq)) \\
  = & q(a + d(q_T - p_T))
\end{align*}`

---

# Average testcross effect of an allele

The average test effect of allele `\(A_1\)` is:
`\begin{align*}
\alpha_1^T =  q(a + d(q_T - p_T))
\end{align*}`

The average test effect of allele `\(A_2\)` is:
`\begin{align*}
\alpha_2^T =  -p(a + d(q_T - p_T))
\end{align*}`

The average testcross effect of an allele substitution is analogous to `\(\alpha\)` and is denoted by `\(\alpha^T\)`.

`\begin{align*}
\alpha^T & = \alpha_1^T - \alpha_2^T\\
 & = a + d(q_T - p_T) \\
\end{align*}`

- The average testcross effect of an allele substitution __does not depend on the allele frequencies in the population being testcrossed__.

- It underscores the dependence of testcross effects on the particular tester used.

---

# General and specific combining ability

Breeders of hybrid crops evaluate the performance of a set of inbreds from one population in crosses with a set of inbreds from a second population.

Consider two populations, `\(P1\)` and `\(P2\)`, each in Hardy-Weinberg equilibrium.

The genotypic value of `\(A_i^{P1}A_j^{P2}\)` is:

`\begin{align*}
G_{i^{P1}j^{P2}} = \mu_{P1 \times P2} + \alpha_i^{P1} + \alpha_j^{P2} + \delta_{ij}^{P1P2}
\end{align*}`

- Where `\(\mu_{P1 \times P2}\)` is the mean of the cross between the two populations
- `\(\alpha_i^{P1}\)` is the __average testcross effect__ of the `\(A_i^{P1}\)` allele
- `\(\alpha_j^{P2}\)` is the __average testcross effect__ of the `\(A_j^{P2}\)` allele
- `\(\delta_{ij}^{P1P2}\)` is the dominance deviation associated with the `\(A_i^{P1}A_j^{P2}\)` genotype.

-------

- Here, `\(\alpha_i^{P1}\)` and `\(\alpha_j^{P2}\)` represent the __general combining ability__ of the `\(A_i^{P1}\)` and `\(A_j^{P2}\)` alleles.
- `\(\delta_{ij}^{P1P2}\)` is the __specific combining ability__ between the `\(A_i^{P1}\)` and `\(A_j^{P2}\)` alleles.