Derivation of the Neutral Equation

In an ideal population with effective size Ne, the inbreeding coefficient is simply

F = 1 / 2Ne

This population then comprises
2Ne alleles at any locus: formation of diploid individuals is a random draw & replacement exercise from this gene pool. For any individual, the probability that the first allele drawn is the same as itself is 1 [think about it]: the probability of drawing the same allele again is simply the reciprocal of the gene pool size, thus 1 / 2Ne.

In a finite population with effective size Ne AND some degree of inbreeding, the inbreeding coefficient F at time t (Ft) is related to that in the previous generation Ft-1 by a recursion equation,

Ft = (1/2Ne)(1) + (1 - 1/2Ne)(Ft-1)
.
The first term applies to inbred individuals, the second to non-inbred individuals, as follows. [Notice that this term is the same as the first equation, above]. The expectation of drawing
any particular allele once is 1 / 2Ne. If that allele occurs in an inbred individual, then the expectation that the second allele is identical by descent with the first is necessarily 1. Otherwise, the expectation that the second allele chosen is not the same as the first is (1 - 1/2Ne). In this case, the probability that the second allele drawn would be identical with the first would by definition be zero, except that some fraction of the population was inbred in the previous generation Ft-1, so the probability of identity by descent is modified. The recursion equation then continues to similar terms for t-2, t-3, and so on. Considering the whole series, the inbreeding coefficient at equilibrium Fe is Ft = Ft-1 = Fe .    [The form of the equation is identical with that previously used to estimate the expectations of f(AA) & f(BB) under inbreeding].

Now consider how mutation changes the probability of identity by descent. The expectation that any single allele mutates is
µ. Among individuals with alleles identical by descent, the expectation that neither allele mutates (and thus remain identical by descent with each other) is (1 - µ)2. At equilibrium Fe,

Fe = [ (1/2Ne)(1) + (1 - 1/2Ne)(Fe) ] [ 1 - µ ]2

Expanding, and neglecting µ2 terms

F
e =
(1 - 2
µ) / (4Neµ + 1 - 2
µ)

and if 2
µ << 1

Fe = (1) / (4Neµ + 1)

and because H
e = 1 - F
e

He
= 1 - (1) / (4Neµ + 1) = (4Neµ + 1 - 1) / (4Neµ + 1) = (4Neµ) / (4Neµ + 1)

This is an extremely important equation in evolutionary population genetics, because it enables estimation of population size from observed genetic data, or vice versa.

Figure & text ©2022 Steven M Carr