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 in the current generation 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. The expectation of drawing any particular allele once is 1 / (2N_e). As above, 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, for non-inbred individuals, the expectation that the second allele chosen is not the same as the first is (1 - 1/(2N_e)). However, some fraction of the population at time t was inbred in the previous generation. Call this F_t-1, and the probability of identity by descent is increased by that factor. The recursion equation then continues to similar terms for F_t-2, F_t-3, and so on. Considering the whole series, the inbreeding coefficient at equilibrium F_eq is F_t = F_t-1 = F_eq .    [The form of the recursion 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 observing two alleles identical by descent, but not by allelic state. 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_eq,

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

        F_eq
= (1 - 2}^{µ) / (4N}^_e^{µ
+}^{1 -}²^µ)^{and if 2}^{µ << 1 while N_e
>> 1}^{^{F_eq = (1}^{) / (4N}^_e^{µ
+}¹⁾}^{and because H}exp^{= 1 - F_eq}        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 given a constant mutation rate µ, it enables estimation of population size from observed genetic data. For example, in the semi-log plot above, for a typical µ = 10^-6, a measured H_obs = 0.30 predicts N_e ~ 100,000.

HOMEWORK: For µ = 10^-5and H_obs = 0.01, 0.02, 0.04, 0.08, 0.16, & 0.32, calculate the expected N_e. [HINT: what is the equation for N_e ?]

       If N_e = 1/ µ , what is the expected value of H_exp? Does the value of µ make a difference to the calculation? Why or why not?