Population Values and Means

class: center, middle, inverse, title-slide

.title[
# Population Values and Means
]
.author[
### Jinliang Yang
]
.date[
### Mar. 18, 2024
]

---

# Basic Genetic Model

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

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

If we replicate a genotype in a large number of environments 
- Average environmental deviation would be __zero__ 
- And phenotypic average would be equal to the genotypic value of that genotype

### Genotypic Value (G)

When a single locus is under consideration,

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

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

---

# Genotypic value

- We need to assign arbitrary values to genotypes to dissect the genotypic value of an individual

- 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 `\(A_1A_2\)` 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

In a population in HWE:

| 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 ( `\(M\)` ) or the expected genotypic value?

According to *Formula (1)* in the STAT Note:
  
`\begin{align*}
E(f(X)) = & \sum\limits_{i=1}^kf(x_i)Pr(X = x_i) \\
M = & p^2a + 2pqd - q^2a \\
       = & a(p-q) + 2pqd
\end{align*}`

--
where `\(M\)` 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

---

# B Locus Example

Booroola (B) locus influences fecundity in Merino sheep

.pull-left[
| Genotype  | Mean litter size   | 
| :-------: |: ------- :| 
| `\(B_1B_1\)`  | `\(2.66\)`   |  
| `\(B_1B_2\)`  | `\(2.46\)`   |
| `\(B_2B_2\)`  | `\(1.48\)`   |

- Freq of `\(B_1 = 0.25\)`
]

.pull-right[
<div align="center">
<img src="sheep.png" height=150>
</div>
]

#### What is the M?

- First, determine midpoint:

- then, determine `\(a\)` and `\(d\)`:

- And finally calculate the population mean

---

# B Locus Example

Booroola (B) locus influences fecundity in Merino sheep

.pull-left[
| Genotype  | Mean litter size   | 
| :-------: |: ------- :| 
| `\(B_1B_1\)`  | `\(2.66\)`   |  
| `\(B_1B_2\)`  | `\(2.46\)`   |
| `\(B_2B_2\)`  | `\(1.48\)`   |

- Freq of `\(B_1 = 0.25\)`
]

.pull-right[
<div align="center">
<img src="sheep.png" height=150>
</div>
]

#### What is the M?

- First, determine midpoint:
  - `\((2.66 + 1.48)/2 = 2.07\)`
- then, determine `\(a\)` and `\(d\)`: 
  - `\(a=2.66 - 2.07 = 0.59\)`
  - `\(d=2.46 - 2.07 = 0.39\)`
- And finally calculate the population mean
  `\(M = a(p-q) + 2pqd = 0.59 \times (0.25 - 0.75) + 2 \times 0.25 \times 0.75 \times 0.39 = -0.149\)`

Note that this is the __deviation from the midpoint__, 2.07.

---
# Multiple Loci

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

`\begin{align*}
M =  \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

---
# Average effect of an allele

- Parents pass on **alleles**, not _genotypes_

- The average effect of an allele depends on 
  - the genotypic values ( `\(a\)`, `\(d\)` ) 
  - and allele frequency ( `\(p\)`, `\(q\)` )

- Thus, an average effect can be unique for each population because of differences in allele freq between populations.

--
### Average effect of `\(A_1\)`

- Mean deviation from the population mean of individuals who received that allele from one parent

- The other allele is received at random 
  - i.e., according to the allele freq, `\(q\)`, from the population

---

# The avarage 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 `\(q\)`.

- The upshot of this is that 
  - allele `\(A_1\)` forms __genotypes `\(A_1A_1\)`__  with value of `\(a\)` at a frequency of `\(p\)`, 
  - and forms __genotypes `\(A_1A_2\)`__ with a value of `\(d\)` at a frequency of `\(q\)`.

- Thus, the mean value of individuals that received `\(A_1\)` is `\(pa + qd\)` 
  - and, the average effect of allele `\(A_1\)` is (recall definition above):

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

---
# Allele Substitution

--
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*}`

---

# Revisit the Booroola locus example

| Genotype  | Mean litter size   | 
| :-------: |: ------- :| 
| `\(B_1B_1\)`  | `\(2.66\)`   |  
| `\(B_1B_2\)`  | `\(2.46\)`   |
| `\(B_2B_2\)`  | `\(1.48\)`   |

- Freq of `\(B_1 = 0.25\)`, `\(p = 1- q = 0.25\)` 
- Recall that `\(a= 0.59\)`, `\(d = 0.39\)`, and `\(M=-0.149\)`

#### What is the average effect of allele `\(B_1\)`?

- Mean of individuals receiving a `\(B_1\)` allele (others coming from random) is:  `\(pa + qd\)`
  - `\(0.25 \times 0.59 + 0.75 \times 0.39 = 0.44\)`
  
- Averge effect of `\(B_1\)`:
  - `\(\alpha_1 = 0.44 - M = 0.44 - (-0.149) = 0.589\)`

---

# Revisit the Booroola locus example

| Genotype  | Mean litter size   | 
| :-------: |: ------- :| 
| `\(B_1B_1\)`  | `\(2.66\)`   |  
| `\(B_1B_2\)`  | `\(2.46\)`   |
| `\(B_2B_2\)`  | `\(1.48\)`   |

- Freq of `\(B_1 = 0.25\)`, `\(p = 1- q = 0.25\)`
- Recall that `\(a= 0.59\)`, `\(d = 0.39\)`, and `\(M=-0.149\)`

#### What is the average effect of allele substitution `\(\alpha\)`?

`\begin{align*}
\alpha = & a + d(q - p) \\
 = & 0.59 + 0.39 \times (0.75 - 0.25) \\
 = & 0.785
\end{align*}`

#### The average effect of allele `\(B_2\)` or `\(\alpha_2\)`?

`\begin{align*}
\alpha = & \alpha_1 - \alpha_2 \\
\alpha_2 = & \alpha_1 - \alpha \\
 = & 0.589 - 0.785 = - 0.196 \\
\end{align*}`

---

# Revisit the Booroola locus example

| Genotype  | Mean litter size   | 
| :-------: |: ------- :| 
| `\(B_1B_1\)`  | `\(2.66\)`   |  
| `\(B_1B_2\)`  | `\(2.46\)`   |
| `\(B_2B_2\)`  | `\(1.48\)`   |

- Freq of `\(B_1 = 0.25\)`, `\(p = 1- q = 0.25\)`
- Recall that `\(a= 0.59\)`, `\(d = 0.39\)`, and `\(M=-0.149\)`

#### What is the `\(M\)` and `\(\alpha\)` values if __ `\(B_1 = 0.85\)`__?

`\begin{align*}
M = & a(p-q) + 2pqd \\
 = & 0.59 \times (0.85-0.15) + 2 \times 0.85 \times 0.15 \times 0.39 \\
 = & 0.512 \\
\alpha = & a + d(q - p) \\
 = & 0.59 + 0.39 \times (0.15 - 0.85) \\
 = & 0.317
\end{align*}`

--
As population mean increases, average effect of the allele substitution decreases.

---

# Avg effect of A1 vs. allele freq

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

When d=0, 
`\begin{align*}
\alpha_1 = & q(a + d(q - p))  = (1-p)a = a - pa\\
\end{align*}`

Average effect of an allele plotted against its frequency in the population.

---

# Avg effect  vs. allele freq

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

- If dominant is absent (i.e. d=0), `\(A_1A_2\)` is right in the middle in any case, so `\(\alpha\)` is just equal to `\(a\)`.

- If dominant is present, randomly change `\(A_2\)` alleles to `\(A_1\)` alleles will have a greater effect when `\(A_1\)` alleles are less frequent.

`\begin{align*}
\alpha = & a + d(q - p) \\
= & a + d(1-p-p) \\
= & (a +d) - 2d \times p \\
\end{align*}`