# What is the conversion between r and FST?

From Schonman (2013):

...allele A can only invade under Hamilton’s condition R=$F_{ST}$ > C/B.

From Harpending (2002):

The best general definition of the coefficient of relation $R_{XY}$ between individuals X and Y is (Bulmer, 1994) $R_{XY}$ = $F_{XY}$/$F_{XX}$, where $F_{XY}$ is the kinship between X and Y and $F_{XX}$ is the kinship of X with himself [..]

[..] $F_{ST}$ is just the coefficient of kinship between members of the same deme [..]

[..] $F_{Self} = \frac{(1 + F_{ST})}{2}$ [..]

This means, unless I am mistaken, that if I take Harpending's equation and substitute $F_{ST}$ for $F_{XY}$ and substitute 1/2(1+$F_{ST}$) for $F_{XX}$, I can calculate within-deme relatedness: 2*$F_{ST}$/(1+$F_{ST}$).

Yet Schonman says R=$F_{ST}$?

What am I doing wrong here?

• The low attention your post received is likely related to its format. I've improved the format for the quotation and the references. You probably want to improve the format for the equations. For example, "Fst" should be "$F_{ST}$" (by typing $F_{ST}$). Here is a tutorial for formatting math equations. – Remi.b May 7 '18 at 1:12
• @Remi.b Is this question too difficult or too trivial? Do you know the answer? – sterid May 7 '18 at 1:56
• It is not an easy question. My first thoughts were that $F_{ST}$ depends upon the genetic diversity while $r$ (or $R$ as it is apparently called here) is not. So equating the two sounds hard. You first quote comes from the social evolution literature and Nash's work into the discussion just makes things unnecessarily complicated. – Remi.b May 7 '18 at 2:23
• Sentences like FST is just the coefficient of kinship between members of the same deme sounds wrong to me because $F_{ST}$ depends upon the genetic diversity in the total population but it might be hard to give an opinion on an out-of-context quotation (I have not read the paper). Also, I am not used to work with he coefficient of relatedness (or of kinship; to me these two concepts are the same but I might be mistaken) and might gets a bit lost here. – Remi.b May 7 '18 at 2:23
• @Remi.b What do you mean by Nash's work? You're not used to using coefficient of relatedness (r)? Isn't that measure very commonly used? Could you read the paper? It is rather short and I think you would be able to understand it because you are familiar with math formulations in evolutionary biology. – sterid May 7 '18 at 2:26

I am very unsure but here are my thoughts...

We can work out from

$$R_{XY} = F_{XY}/F_{XX}$$

I do not really understand the concept of coefficient of kinship and the concept of coefficient of kinship with oneself but you say

According to the paper, kinship with himself is 0.5 in a sexual population

So, let's assume

$$R_{XY} = 2F_{XY}$$

Let's assume that the individual Y is drawn from anywhere in the total population. Again, I don't really understand the concept of coefficient of kinship but let's assume that $F_{XY}$ is the same as the probability of identity by descent between these two individuals, then in Nei (1973) terms, $F_{XY} = J_T = 1 - H_T$. Hence, $R_{XY} = 2\left(1 - H_T\right)$ or $H_T = 1-\frac{R_{XY}}{2}$

Given that

$$F_{ST} = 1 - \frac{H_S}{H_T}$$

it results that

$$H_T = \frac{H_S}{1-F_{ST}}$$

and therefore,

$$1-\frac{R_{XY}}{2} = \frac{H_S}{1-F_{ST}}$$

solving into

$$R_{XY} = \frac{ 1 - \frac{H_S}{1-F_{ST}} }{2} = \frac{1-F_{ST} - H_T}{2(1-F_{ST})}$$

I would recommend you making some more reading on the coefficient of kinship to ensure that my interpretation here is correct.

• Oh Remi, I apologize. The thing is, in the paper, Harpending says that kinship with oneself is 0.5. But later on in the paper, Harpending gives a different equation for kinship with oneself in a subdivided population. That is the one I thought should be used, since $F_{ST}$ implies a subdivided population, right? So, that is the one I used in phrasing my question above. – sterid May 7 '18 at 3:20
• Ok.... see new answer then! I am not sure it is of any help though. – Remi.b May 7 '18 at 3:35

I am not very at ease with the concepts of coefficient of kinship but

if $F_{self} = F_{XX}$, then

$$R_{XY} = 2F_{XY}(1 + F_{ST})$$

Assuming that Y is a random individual from the total population and if $F_{XY}$ is the probability of identity by descent of haplotypes X and Y (which I am unsure because I don't really ), then $F_{XY} = H_T$ and hence

$$R_{XY} = 2H_T(1 + F_{ST})$$

But really I am very unsure about all that. Note also that the small calculation is based upon the equation $F_{Self}$ = 1/2(1 + $F_{ST}$, which I don't really understand! I would recommend you making some more reading on the coefficient of kinship to ensure that my interpretation here is correct.

• $R_{XY}$=2$F_{XY}$/(1+$F_{ST}$), correct? – sterid May 7 '18 at 3:43
• Is the original expression $F_{Self} = \frac{1}{2(1 +$F_{ST})}$or$F_{Self} = \frac{1 + F_{ST}}{2}\$? It is unclear from the format you are currently using in the question. I might have assumed the wrong one. – Remi.b May 7 '18 at 13:51
• The original is the latter. – sterid May 7 '18 at 19:21
• I tried to modify the format but I don’t think it worked. I’m on my phone and won’t have a computer for the rest of the day. – sterid May 7 '18 at 19:32
• The format is corrected now. – sterid May 9 '18 at 1:16