Chapter 6, Population Genetics by Graham Coop

Neutral Diversity and Population Structure

Presented by Sei Chang

The chapter discusses how genetic differentiation build up between closely related population. Using \(F_{ST}\) statistic, Coop provided quantitative formulation to understand the genetic divergence between two sub-population, for example, after migration.

Population split model

To understand the effect of genetic drift of population structure, imagine a population of constant size of \(N_e\) diploid individuals that \(T\) generations in the past split into two sub-populations each of size \(N_e\) individuals, which do not subsequently exchange migrants. In the current day we sample an equal number of alleles from both subpopulations.

Consider a pair of alleles sampled within one of our sub-populations and think about their per site heterozygosity. These alleles have experienced a population of size \(N_e\) and so the probability that they differ is \(H_S \approx 4N_e \mu\) (assuming that \(N_e \mu \ll 1\), using the equation for heterozygosity within a population).

Assume equal sampling from both sub-populations, total heterozygosity is given by,

\[ H_T = 0.5 H_S + 0.5 H_B \] where \(H_B\) is the probability that a pair of alleles drawn from the two different sub-populations differ from each other. A pair of alleles from different sub-populations cannot find a common ancestor with each other for at least \(T\) generations into the past as they are in distinct populations (not connected by migration). Once these alleles find themselves back in the combined ancestral population it takes them on average \(2N\) generations to coalesce. So the total opportunity for mutation between the pair of alleles sampled from different populations is \(2 (T + 2N )\) generations of meioses, such that the probability that the pairs of alleles is different is \[ H_B \approx 2\mu ( T + 2 N) %\left( 1-(1-\mu)^{2T} \right) + (1-\mu)^{2T} %\frac{\theta}{\theta+1} \]

\[ F_{ST} \approx \frac{ \mu T}{\mu T + 4N_e\mu } = \frac{ T}{ T + 4N_e} \]

Other concepts

Some other concepts that we discussed include a simple model of migration, incomplete lineage sorting, and testing for gene flow. Summary (from the book):

• Only a small number of migrants between populations per generation is sufficient to prevent the build up of neutral allele frequency differentiation.
• Incomplete lineage sorting of ancestral variation is one source of disagreement between population/species-trees and gene trees. It occurs when the split times between populations are in quick enough succession that lineages do not have time coalese between more closely related populations.
• Gene flow can also lead to patterns similar to incomplete lineage sorting. We can test between a model of incomplete lineage sorting and gene flow using tests such as ABBA-BABA.