Population Genetics - Effective Population Size

Previously we were only looking at populations that satisfy H-W populations with random mating, non-overlapping generations and constant size, except for the fact that their size is not infinite.
The Wright-Fisher model accounts for this finite size, but still relies on other assumptions that are almost never true in nature
This model is still useful if we convert the actual population into a population of "effective" individuals which do satisfy W-F conditions.
$N_{e}$ is the size of an ideal population having the same amount of drift as an observed population of size $N$ - it is a measurement of the strength of genetic drift.
Amount of drift given by sampling variance in W-F model:

Var {p}_{exp} = \frac{p q}{2 N}

Effective population size is:

N_{exp} = \frac{p q}{2 Var {p}_{obs}}

In most cases, $N_{e} < N$ because:

Populations change over time
Number of males and females in the effective population are not equal - for example fewer males contribute to reproduction than females
some individuals have more reproductive success than others
variance in reproductive number is greater than expected by chance (i.e. not Binomial)

Mutation-drift balance

\theta = 4N_e \mu$$where: - $\mu$ is the per-nucleotide mutation rate - $\theta$ is per-nucleotide heterozygosity at neutral sites. - Meaning the probability of heterozygosity on all neutral mutation gene loci divided by the number of nucleotides. - Therefore, we can calculate $N_e$ if we know $\theta$ and $\mu$ for our population. ## Why is $N_e \lt N$? ### Variation in population size over time - [Harmonic mean](https://en.wikipedia.org/wiki/Harmonic_mean) population size where population size changes over time $t$: $$N_e ={t \over \sum_{i=1}^t N_i^{-1}}

Harmonic mean gives more weight to small numbers
We use this mean since small populations lead to large reductions in diversity (bottleneck)

N_{e} = \frac{4 N_{m} N_{f}}{N_{m} + N_{f}}

N_{e} = \frac{4 N - 2}{{σ_{k}}^{2} + 2}

${σ_{k}}^{2}$ is the variance in the number of offspring per adult
If ${σ_{k}}^{2}$ = the variance of a Binomial random process $2 (N - 1) / N$ , then $N_{e} = N$ .
If ${σ_{k}}^{2}$ is larger then expected under random processes, then $N_{e} < N$ .
- If the number of offpsring per adult varies more than what is expectedby chance, effective population size is smaller than census size of population.
If it is smaller than expected, we get $N_{e} > N$ .
- So, in a population where every individual has the same number of offpsring, the effective population is higher?
- What does this actually mean?

Genetic drift causes fluctuations in allele frequencies, even if they are under selection
It's possible that the effect of genetic drift is strong enough to dominate the evolutionary trajectory of allele frequency in a population.
Genetic drift sets a limit to natural selection. So, an allele is effectively neutral when:

| s | < \frac{1}{2 N_{e}}

So, because of genetic drift, in a small enough population beneficial mutations can become purged and deleterious mutations can become fixed.
Inbreeding coefficient $F$ can be estimated by the amount of heterozygosity in a population, which is lower in small populations:

F = \frac{H_{expected} - H_{observed}}{H_{expected}}, with H_{expected} = 2 p q

$F > 0$ expresses the excess of homozygosity relative to H-W expectation - meaning there is lots of inbreeding and many alleles are fixed