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Background

Quoting from Gillespie's book

The level of inbreeding is measured by the inbreeding coefficient, $F_I$, which is the probability that two alleles in an individual are identity by descent.

The two possible alleles are $A_1$ and $A_2$ and their frequencies are $p$ and $q=1-p$, respectively. Gillespie goes on and give the expected genotype frequencies given $p$ (and $q$) and $F_I$

$$ \begin{array}{r|c|c|c} Genotype & A_1A_1 & A_1A_2 & A_2A_2 \\ \hline Frequency & p^2(1-F_I)+pF_I & 2pq(1-F_I) & q^2(1-F_I)+qF_I \\ \end{array} $$

Question

Can you please help me to understand why those are the appropriate genotype frequencies?

Example of what is unclear to me

By the definition of $F_I$, I was expecting that the frequency of `$A_1A_2$ (and $A_2A_1$) would be $(1-F_I)$, that is the frequency (or probability) that two alleles in an individual are not identical by descent. I feel like the definition given above is wrong. Should it rather be something like the weight for sampling like individuals? That is if there were only two possible mates, one that is $A_1A_1$ and one that is either $A_1A_2$ or $A_2A_2$, then the probability for a $A_1A_1$ individual to mate with the other $A_1A_1$ is $F_I$.

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You're confused because you're failing to distinguish between 'identical' and 'identical by descent'. Some pairs of alleles would still be identical even in the absence of inbreeding.

We model the inbreeding by classifying allele pairs as IDB - always homozygous - or not IDB - distributed according to Hardy Weinberg. The frequency of a pair of alleles being A1,A2 (or A2,A1) is thus the frequency under H.W.E. (2pq) multiplied by the chance of them NOT being IBD (1-FI). And conversely, a pair can be homozygous and not IBD - at frequency p 2(1-FI), or because they are IBD - at frequency p F I.

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  • $\begingroup$ +1 Oh..ok! This is a little disturbing to me. The concept of IBD depends on an arbitrary time in the past beyond which if coalescence occurs, then we still call the two alleles IBS (Identical by state) and not IBD (I could cite Hartl and Clark I think but don't have the book with me today). Am I right? $\endgroup$ – Remi.b Apr 20 '16 at 20:21
  • $\begingroup$ Yes, but these models are just simple approximations. Concepts like time and coalescence don't even exist the perfect binomial world of Hardy Weinberg :) $\endgroup$ – Dermot Harnett Apr 20 '16 at 20:35
  • $\begingroup$ Ok. I don't feel 100% clear with the definitions of IBD and IBS so I just opened "Identity By Descent vs Identity By State". Thanks for your help $\endgroup$ – Remi.b Apr 20 '16 at 20:45

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