# 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.

• Could you post a reference or two in which that equation is derived?
– tel
Aug 12 '15 at 22:17
• @tel See my edit. Note also that I reduced the Fst formula making the standard assumption that $m>>\mu$ (, where $\mu$ is the mutation rate) just to make things easier. Thank you Aug 13 '15 at 0:39
• Say @Remi.b, if I had two populations, and for each population I knew the genotypes of each individual (e.g. Aa, AA, aa, etc.) but for multiple loci (e.g. AABb) do you know of any simple ways of calculating the gene flow between these two populations? I used some R packages in the past but they were a bit fiddly. If you can recommend me a R/Python package and/or the breakdown of the maths I would be much obliged. Aug 19 '15 at 11:12
• @hello_there_andy Is it good to hear from you again. You should make a new post for your question. I don't quite know the answer right now. Aug 19 '15 at 14:57

From Lewontin and Krakauer 1973, the ratio

$$\frac{F_{ST}(d-1)}{\bar F_{ST}}$$

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 $\bar F_{ST}$ is the average $F_{ST}$ that is $\bar F_{ST} = \frac{\sum F_{ST}}{n}$, where $n$ is the number of $F_{ST}$ values.

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

$$var\left(\frac{F_{ST}(d-1)}{\bar F_{ST}}\right) = 2d-2$$

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

$$var(F_{ST})=\left(\frac{d-1}{\bar F_{ST}}\right)^2(2d-2)$$

, which simplifies into

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

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

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

, which again "simplifies" into

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