Neutral Theory

Derivation of the Neutral Equation

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

        F = 1 / 2N_e

This population then comprises 2N_ealleles 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 / 2N_e.

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

        F_t = (1/2N_e)(1) + (1 - 1/2N_e)(F_t-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 - 1/2N_e). 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 F_t-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 F_e is F_t = F_t-1 = F_e .    [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 F_e,

          F_e = [ (1/2N_e)(1) + (1 - 1/2N_e)(F_e) ] [ 1 - µ ]²^{Expanding, and
neglecting}^{µ² terms

        F_e
= (1 - 2}^{µ) / (4N}^_e^{µ
+}^{1 -}²^µ)^{and if 2}^{µ << 1}^{^{F_e = (1}^{) / (4N}^_e^{µ +}¹⁾}^{and because H}e^{= 1 - F}^_e        H_e = 1 - (1) / (4N_eµ + 1) = (4N_eµ + 1 - 1) / (4N_eµ + 1) = (4N_eµ) / (4N_eµ + 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.