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Variance in Fst in the infinite island model


The most famous result in the study of structured populations come from Sewall Wright. He showed that in an island model, where each subpopulation is of size $N$ and the migration rate is $m$, then the pairwise $F_{ST}$ is

$$F_{ST} = frac{1}{4Nm+1}$$

This equation gives the expected $F_{ST}$. Because populations are finite in size ($N$), genetic drift yield this value to vary.

What is the variance in $F_{ST}$ in the infinite island model?


References

evolution in mendelian population is the original paper who derived this result from Sewall Wright.

Indirect measures of gene flow and migration: FST≠$frac{1}{4Nm+1}$ is an influential paper in the field.

GENE FLOW IN NATURAL POPULATIONS is a famous review as well.


From Lewontin and Krakauer 1973, the ratio

$$frac{F_{ST}(d-1)}{ar F_{ST}}$$

approximatively follows $chi^2$ distribution of degree $k=d-1$. Here $d$ is the number of demes (number of islands), $F_{ST}$ is the random variable of the $chi^2$ distribution and $ar F_{ST}$ is the average $F_{ST}$ that is $ar F_{ST} = frac{sum F_{ST}}{n}$, where $n$ is the number of $F_{ST}$ values.

The variance of a $chi^2$ distribution is $2k$, therefore

$$varleft(frac{F_{ST}(d-1)}{ar F_{ST}} ight) = 2d-2$$

Taking $frac{d-1}{ar F_{ST}}$ out of the ratio, the variance of $F_{ST}$ becomes

$$var(F_{ST})=left(frac{d-1}{ar F_{ST}} ight)^2(2d-2)$$

, which simplifies into

$$var(F_{ST}) = frac{2(d-1)^3}{ar F_{ST}^2}$$


The above expression is probably the most interesting result but one could go further and express the variance independently from the mean (by replacing $ar F_{ST}$ by Slatkin 1991 expectation for $ar F_{ST}$ in a finite island. It yields to

$$var(F_{ST}) = frac{2(d-1)^3}{left(frac{1}{1+4Nm(frac{d}{d-1})^2} ight)^2}$$

, which again "simplifies" into

$$var(F_{ST}) = frac{2 left(4 d^2 m N+d^2-2 d+1 ight)^2}{d-1}$$