Extending Classical Population Genetics for Real-World Complexity — Epoche C1
The Wright-Fisher model states that each new generation of a population of $N$ diploid individuals is formed by drawing $2N$ gene copies at random, with replacement, from the $2N$ copies present in the previous generation. Almost every complaint that classical population genetics is too idealised to describe real populations is a complaint about one of the assumptions packed into that single sentence: that mating is random, that population size is constant, that generations do not overlap, that there is no spatial structure, and that every individual has the same expected reproductive success independently of every other. This essay takes four of those assumptions in turn, shows the specific modification that relaxes each one, and — this is the point of the exercise — quantifies what the relaxation costs. In two of the four cases the classical model survives with one parameter rewritten. In the fourth it does not, and knowing which case is which is the substance of the claim that these frameworks are extendable. One quantity recurs throughout and must be defined before it is used. The effective population size $N_e$ is not a headcount. It is the size of an idealised Wright-Fisher population that would show the same rate of genetic drift — the same random wandering of gene frequencies from generation to generation — as the real population under study. Most of the extensions below work by leaving the mathematics of the idealised model intact and substituting a suitable $N_e$ for the census size $N$. That strategy is the workhorse of the field, and its eventual failure in the fourth case is the most instructive result here. Non-random mating: the inbreeding coefficient and what it costs The first assumption to go is panmixia, the assumption that any individual is as likely to mate with any other. Under panmixia and no selection, a locus with two alleles $A_1$ and $A_2$ at frequencies $p$ and $q = 1-p$ settles in one generation at the Hardy-Weinberg genotype frequencies $p^2$, $2pq$ and $q^2$. Wright's inbreeding coefficient $F$ measures the departure from this: it is the probability that the two alleles in an individual at a given locus are copies of the same ancestral allele, inherited through both parents from a shared ancestor. Genotype frequencies become $$P(A_1A_1) = p^2 + Fpq, \qquad P(A_1A_2) = 2pq(1-F), \qquad P(A_2A_2) = q^2 + Fpq,$$ which is the correction the original version of this essay gave for the homozygote, now completed for all three genotypes. It is worth checking that the three add to 1, since this is where the structure of the correction is visible: the terms $+Fpq$, $-2Fpq$ and $+Fpq$ cancel exactly, so $F$ moves probability out of the heterozygote class and splits it evenly between the two homozygote classes without altering the allele frequency $p$ at all. Inbreeding rearranges genotypes; it does not by itself change gene frequencies. The consequence for drift can be stated exactly. For a population with inbreeding coefficient $F$, the effective size is approximately $$N_e \approx \frac{N}{1+F}.$$ The reason is that inbreeding makes the two gene copies within an individual partially redundant as independent samples of the gene pool: at $F=1$, complete selfing, an individual carries what is effectively one allele in duplicate, and a population of $N$ such individuals drifts like $N$ gene copies rather than $2N$ — hence $N_e = N/2$. So the cost of this first extension is a factor of at most two in $N_e$, which is small compared with the effects considered below. Selection: the diffusion approximation, and a factor that was missing The second assumption to relax is neutrality. The classical device for combining selection with drift is the diffusion approximation: instead of tracking the exact discrete Markov chain of allele counts, one approximates the frequency $p$ by a continuous process, valid when selection is weak and the population large. The result is a stochastic differential equation with a deterministic term for selection and a random term for drift. The version given in the original text was $$\frac{dp}{dt} = p(1-p)\frac{d\bar{w}}{dp} + \sqrt{\frac{p(1-p)}{2N_e}}\,\xi(t),$$ where $\bar{w}$ is the mean fitness of the population and $\xi(t)$ is white noise. The noise term is correct; the deterministic term is not, and the error is a factor of two together with a missing normalisation. The correct expression, due to Wright and derived in Crow and Kimura (1970), is $$\Delta p = \frac{p(1-p)}{2}\frac{d\ln\bar{w}}{dp} = \frac{p(1-p)}{2\bar{w}}\frac{d\bar{w}}{dp}.$$ The derivation shows where the two comes from and is short enough to give. Let the three genotypes have fitnesses $w_{11}$, $w_{12}$, $w_{22}$, so that $\bar{w} = p^2w_{11}+2pqw_{12}+q^2w_{22}$. Write $w_1 = pw_{11}+qw_{12}$ for the marginal fitness of allele $A_1$ — its average fitness over the genotypes it finds itself in — and $w_2 = pw_{12}+qw_{22}$ likewise. Then $\bar{w} = pw_1+qw_2$, and after selection the frequency is $p' = pw_1/\bar{w}$, so $\Delta p = p(w_1-\bar{w})/\bar{w} = pq(w_1-w_2)/\bar{w}$. Differentiating $\bar{w}$ with respect to $p$ gives $2(w_1-w_2)$, and the factor of two appears there: $\bar{w}$ is quadratic in $p$ because each individual carries two gene copies, so differentiating counts both. Substituting yields Wright's formula. Dividing by $\bar{w}$ matters whenever selection is strong, because $\bar{w}$ is the normalising constant that keeps the frequencies summing to one; under weak selection $\bar{w}\approx 1$ and it can be dropped, but the factor of two cannot. The noise term has an equally concrete origin. In the Wright-Fisher model the number of $A_1$ copies in the next generation is a binomial draw of $2N$ trials with success probability $p$, whose variance is $2Np(1-p)$; dividing by $(2N)^2$ to convert counts into a frequency gives $\mathrm{Var}(p') = p(1-p)/(2N)$. The $2N$ in the denominator is the number of gene copies, not of individuals. Replacing $N$ by $N_e$ giv