Effective Pop Size

class: center, middle, inverse, title-slide

.title[
# Effective Pop Size
]
.author[
### Jinliang Yang
]
.date[
### Feb. 7, 2024
]

---

# Non-ideal population

Allele frequencies of populations often behave as if the population is smaller than it is.

### Census population size ( `\(N\)` or `\(N_c\)` )

- Just a count of how many individuals are there

### Effective population size ( `\(N_e\)` )

- Effective number of breeding individuals

- Describes the number of individuals that would __give rise to the observed sample variance__ _IF they bred in an ideal manner_

- In terms of inbreeding: `\(\Delta F = \frac{1}{2N}\)`
- In a real population: `\(\Delta F = \frac{1}{2N_e}\)`, and  `\(N_e= \frac{1}{2\Delta F}\)`

---

# Causes of `\(N_e\)`

Deviations in effective population size from the actual population size are caused by:

- #### Exclusion of closely related matings

- #### Different number of males and females

- #### Population bottleneck and expansion

- #### Minimize inbreeding

- #### Overlapping generations

---

# Exclusion of closely related matings

In general, exclusion of closely related matings has very minimal effect on `\(N_e\)`

### With self-fertilization excluded

`\begin{align*}
N_e \approx N + 1/2
\end{align*}`

### With sib-mating also excluded

`\begin{align*}
N_e \approx N + 2
\end{align*}`

---

# Different number of males and females

In domestic animals, the sexes are often unequally represented among the breeding individuals!

- No matter their relative numbers, males and females contribute equally to the alleles of the next generation

- Have to consider sampling variance of each sex

`\(N_e\)` is twice the harmonic mean and harmonic mean is

`\begin{align*}
\frac{1}{\frac{1}{2}(\frac{1}{N_m} + \frac{1}{N_f})}
\end{align*}`

Therefore,
`\begin{align*}
N_e & = 2 \times \frac{1}{\frac{1}{2}(\frac{1}{N_m} + \frac{1}{N_f})} \\
\frac{1}{N_e} & = \frac{1}{4N_m} +  \frac{1}{4N_f} \\
\end{align*}`

---

# The rate of inbreeding

Because
`\begin{align*}
\frac{1}{N_e} & = \frac{1}{4N_m} +  \frac{1}{4N_f} \\
\Delta F & = \frac{1}{2N_e} \\
\end{align*}`

So, the rate of inbreeding,
`\begin{align*}
V(p) = \Delta F = \frac{1}{2N_e} = \frac{1}{8N_m} +  \frac{1}{8N_f}
\end{align*}`

And `\(N_e\)` due to unequal contribution of sexes

`\begin{align*}
N_e = \frac{4N_mN_f}{N_m + N_f}
\end{align*}`

- If the `\(N_m = N_f = N/2\)`, simply, `\(N_e = N\)`

- If, for example, `\(4N_m = N_f = 4/5N\)`, now `\(N_e = \frac{16}{25}N\)`

- Therefore, `\(N_e\)` depends on the rarity of one sex

---

# Different number of males and females

The effective population size as a function of the proportion of males

```r
ne <- function(pnm=0.5, n=10){
  nm <- n*pnm # number of male
  nf <- n - nm # number of female
  ne <- 4*nm*nf/(nm + nf)
  return(ne)
}
p <- seq(from=0, to=1, by=0.01)
plot(p, ne(pnm=p, n=10), xlab="Nm/N", ylab="Ne", type="l", lwd=3, ylim=c(0,40))
lines(p, ne(pnm=p, n=20), type="l", lwd=3, col="red")
lines(p, ne(pnm=p, n=40), type="l", lwd=3, col="blue")
```

---

# Population bottleneck and expansion

Refer to change in number from generation to generation. 
  - It is another reason for `\(N_e\)` deviation from `\(N\)`

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

`\begin{align*}
\Delta F = \frac{1}{2N_i}
\end{align*}`

- Here, `\(N_i\)` is the population size at generation `\(i\)`.

### Average `\(N_e\)` across generations

`\begin{align*}
N_e = \frac{t}{\frac{1}{N_1} + \frac{1}{N_2} + ... + \frac{1}{N_t}}
\end{align*}`

- The smallest `\(N_i\)` has the most impact on `\(N_e\)`
  - Ex: pop size over 4 gens: 1000, 1000, 50, 1000
  - Average pop size = 762.5; and `\(N_e\)` = 174

---

# Variation in family size (fecundity)

Family size is the number of progeny of an individual parent or pair of parents that survive to become founders.

### In a stable population

- The family size ( `\(K\)` ) of a mating pair = 2 (on average, one male and one female) 
- The family size follows a Poisson distribution with __variance = mean = 2__ (or `\(K = V_k=2\)`).

`\begin{align*}
N_e \approx \frac{4N}{V_k  + 2}
\end{align*}`

- When variance `\(V_k =2\)`, the `\(N_e = N\)`

---

# Variation in family size (fecundity)

As usual, things are never this simple.

If males can mate with more than one female, `\(V_k\)` is likely to be different for males and females ( `\(V_{km} \neq V_{kf}\)` )

`\begin{align*}
N_e & \approx \frac{4N}{\frac{V_{km} + V_{kf}}{2}  + 2} \\
& \approx \frac{8N}{V_{km} + V_{kf}  + 4}
\end{align*}`

Variation of family size, `\(V_k\)`, is one of the most important causes of `\(N_e\)` being less than `\(N\)`.

---

# Increase `\(N_e\)` and minimize inbreeding

`\begin{align*}
N_e = \frac{1}{2\Delta F} &  \approx \frac{4N}{V_k  + 2} \\
& \approx 2N \\
\end{align*}`

In many experiments, you want to minimize inbreeding

- By choosing individuals a parents, `\(V_k\)` can be reduced below its random amount.

- Can choose equal number of individuals from each family for the next generation, so `\(V_k=0\)`

What if unequal number of males and females?

- Can eliminate variance by choosing one male from each sire and one female from each dam

`\begin{align*}
N_e & \approx \frac{16N_mN_f}{N_m + 3N_f} \\
\end{align*}`

> Gowe, Robertson, and Latter, 1959

---

# Swine example

- A swine breeding program has 15 males and 45 females
- Select to breed 1 son from each sire and one daughter of each dam
- What is the expected inbreeding coefficients after one generation and 23 generations?

`\begin{align*}
N_e & \approx \frac{16N_mN_f}{N_m + 3N_f} \\
\end{align*}`

```r
ne <- function(nm, nf){
  return(16*nm*nf/(nm + 3*nf))
}
ne(nm=15, nf=45)
```

```
## [1] 72
```

`\begin{align*}
\Delta F = \frac{1}{2N_e} = \frac{1}{2 \times 72} = 0.0069
\end{align*}`

`\begin{align*}
F_t & = 1 - (1 - \Delta F)^t \\
F_{23} & = 1 - (1 - 0.0069)^{23} \\
& = 0.15
\end{align*}`

---

# Swine example

- A swine breeding program has 15 males and 45 females
- ~~Select to breed 1 son from each sire and one daughter of each dam~~
- __What if they had selected individuals and mated them at random__
- What is the expected inbreeding coefficients after one generation and 23 generations?

`\begin{align*}
N_e = \frac{4N_mN_f}{N_m + N_f} \\
\end{align*}`

```r
ne1 <- function(nm, nf){
  return(4*nm*nf/(nm + nf))
}
ne1(nm=15, nf=45)
```

```
## [1] 45
```

`\begin{align*}
\Delta F = \frac{1}{2N_e} = \frac{1}{2 \times 45} = 0.011
\end{align*}`

`\begin{align*}
F_t & = 1 - (1 - \Delta F)^t \\
F_{23} & = 1 - (1 - 0.011)^{23} \\
& = 0.22
\end{align*}`

---
# Overlapping generations

Differences in lifetime add to differences of fertility in increasing the variance of family size.

- This often confounds things
  - i.e., longer lived individuals can contribute more progeny

`\begin{align*}
N_e \approx \frac{4N_cL}{V_k  + 2}
\end{align*}`

- `\(N_c\)`, number born in corhort defined by a time interval

- `\(L\)`, length of generation (avg age of the parents at the birth of offspring)

- `\(V_k\)`, variance in family size

---

# Other considerations for `\(N_e\)`

> "Effective popopulation size and patterns of molecular evolution and variation" by [Charlesworth 2019](https://www.nature.com/articles/nrg2526)

- #### Formulas given only useful when population history is known

- #### Drift randomly affects allele frequencies, and also creates LD between alleles

- #### The amount of drift is proportional to `\(N_e\)`

- #### LD breaks up with recombination

---

# `\(N_e\)` from LD

Expected LD as a function of recombination rate is:

`\begin{align*}
E(r^2) \approx \frac{1}{1 + 4N_ec}
\end{align*}`

- `\(r^2\)`, the expected LD
- `\(c\)`, the recombination rate

In this equation, as `\(N_ec\)` approaches 0, `\(E(r^2)\)` approaches 1.

As `\(N_e\)` and `\(c\)` increase, `\(E(r^2)\)` approaches 0.

---

# A real-world study

> "Extent of linkage disequilibrium and effective population size in Finnish Landrace and Finnish Yorkshire pig breeds" [Uimari and Tapio, 2011](https://academic.oup.com/jas/article/89/3/609/4764206?login=true)

### Inbreeding from pedigree data

- 554,000 Yorkshire and 608,000 Landrace
- Used equation `\(\Delta F = 1/2N_e\)` 
- Pedigree-based `\(N_e= 91\)` and `\(61\)` for Landrace and Yorkshire

### Inbreeding from SNP data

- Genotype data (60K SNP) for 86 Landrace and 32 Yorkshire boars
- Estimated `\(r^2\)` based upon haplotype frequencies
- Estimated `\(c\)` assuming 1 cm = 1 Mb
- Genotype-based `\(N_e = 80\)` and `\(55\)` for Landrace and Yorkshire