The coalescent and $F$-statistics

Suppose we have a sample of alleles from a structured population. For alleles chosen randomly within populations let the average time to coalescence be $\bar t_0$. For alleles chosen randomly from different populations let the average time to coalescence be $\bar t_1$. If there are $k$ populations in our sample, the average time to coalescence for two alleles drawn at random without respect to population is⁷

$$\bar t = \frac{\left(\frac{k(k-1)}{2}\right)\bar t_1 + k\bar t_0} {\frac{k(k+1)}{2}} \quad .$$

Slatkin [4] pointed out that $F_{st}$ bears a simple relationship to average coalescence times within and among populations. Given these definitions of $\bar t$ and $\bar t_0$,

$$F_{st} = \frac{\bar t - \bar t_0}{\bar t} \quad .$$

So $F_{st}$ measures the proportional increase in time to coalescence that is due to populations being separate from one another. One way to think about that relationship is this: the longer it has been, on average, since alleles in different populations diverged from a common ancestor, the greater the chances that they have become different. An implication of this relationship is that $F$-statistics, by themselves, can't tell us much of anything about patterns of migration among populations.

A given pattern of among-population relationships might reflect a migration-drift equilibrium, a sequence of population splits followed by genetic isolation, or any combination of the two. If we are willing to assume that populations in our sample have been exchanging genes long enough to reach stationarity in the drift-migration process, then $F_{st}$ may tell us something about migration. If we are willing to assume that there's been no gene exchange among our populations, we can infer something about how recently they've diverged from one another. But unless we're willing to make one of those assumptions, we can't make any further progress.