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.
Answer
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
$$var(F_{ST}) = \frac{2 \left(4 d^2 m N+d^2-2 d+1\right)^2}{d-1}$$
No comments:
Post a Comment