# $F_{ST}$ when considering a multi-allelic locus

Sewall Wright defined the $F_{ST}$ in a metapopulation as being:

$$F_{ST} = \frac{\text{Var}(p)}{\bar p (1-\bar p)}$$

, where $p$ is a vector of frequencies of a given allele and $\bar p$ and $\text{Var}(p)$ are the mean and variance of this vector.

For example, consider a metapopulation made of 4 subpopulations. The allele frequencies in these 4 subpopulations are p=[0.2, 0.5, 0.8, 0.3]. $\bar p$ is the mean of $p$ ($\bar p = 0.45$) and $\text{Var}(p)$ is the variance of $p$ ($\text{Var}(p )=0.07 \space$).

Question

Can we define $F_{ST}$ for a multiple allele locus? Or should $F_{ST}$ be defined for each allele independently?

• Can you provide a cite for the idea? The form of these things varies with the source. $F_{ST}$ is a symbol. What is it supposed to stand for, in English? May 30, 2014 at 15:36
• @daniel $F_{ST}$ is also called fixation index. It is an index used in population genetics in order to describe the population structure (non-random mating throughout the population) of a metapopulation. Here is the wiki page. I think there is only one definition of $F_{ST}$ but several ways to equivalently understand it. There are a bunch of ways to measure and approximate the $F_{ST}$ but this is the matter of another discussion. May 30, 2014 at 15:46
• @daniel You may also want to have a look to the table and to the slide that follows this table on this video at 31'44''. It gives an explanation of where the definition I wrote in my post comes from. I am personally not really able to give you a good intuitive explanation of what $F_{ST}$ is. I am only able to give the mathematical definition of it. Here is a related question I recently asked that might help a bit! May 30, 2014 at 15:53

Short answer: yes, people have formulated ways to estimate $$F_{ST}$$ for multiallelic loci, e.g. microsatellites. For a review, see here.
Specifically, Nei could define $$F_{ST}$$ for multiple alleles as
$$F_{ST} = \frac{(H_t - H_s)}{H_t}$$,
There exist alternate statistics such as $$G_{ST}$$ and $$R_{ST}$$ (links under each) which are devised specifically for the multiallelic problem, see also the review for further discussion. Basically, microsatellites have high mutation rates such that they deflate the statistic; this will likely be the case for most multiallelic loci.