Fixation Probability with Stochastic Differential Equations — Epoche C1
The quantity in question A single new copy of a beneficial mutation, arising once in a population of ten thousand diploid individuals and conferring a one-in-ten-thousand advantage, has about a one-in-five-thousand chance of eventually replacing every other copy at its locus — and roughly a four-in-five chance of disappearing within the first few generations, before selection has had any opportunity to act. That number, the fixation probability , is what this essay derives. It is the probability that a mutant allele — one of the alternative forms of a gene at a given position on the chromosome — reaches frequency one, meaning every gene copy in the population descends from that original mutation. Three symbols carry the argument, and each needs its convention fixed, because the literature uses several. The selection coefficient $s$ will denote the advantage conferred by a single copy of the mutant allele: an individual carrying one copy leaves on average a fraction $1+s$ of the offspring of a non-carrier, and an individual carrying two copies a fraction $1+2s$. The effective population size $N_e$ is the size of an idealised population — random mating, constant size, Poisson-distributed offspring numbers — that would lose genetic variation by chance at the same rate as the real one; for a real population with unequal sex ratios, family-size variation or fluctuating numbers it is usually smaller than the census size $N$. And $x$ or $X_t$ denotes the frequency of the mutant allele, a number between zero and one. Because each of the $N$ diploid individuals carries two gene copies, a single new mutation starts at frequency $1/(2N)$. The claim this essay examines is that fixation probability is essentially a function of $s$ alone, drift being background noise. That claim is half right in a way that is worth making precise: $s$ does set a ceiling, but which side of the ceiling a population sits on is decided by the product $4N_e s$, and the ceiling itself is reached only from a direction that is easy to get backwards. Where the factor of two comes from Before the diffusion machinery, the classical answer deserves its derivation, because the derivation shows immediately that the famous coefficient is not a constant of nature. Fisher in 1922 and Haldane in 1927 treated the early fate of a single mutant copy as a branching process: while the mutant is rare it sits almost always in heterozygotes, it competes with nobody like itself, and its lineage grows or dies independently of the rest of the population. Let each carrier leave a Poisson-distributed number of offspring with mean $1+s$, and let $q$ be the probability that the lineage eventually dies out. Conditioning on the first generation's offspring number and using the fact that the probability generating function of a Poisson variable with mean $\lambda$ is $e^{\lambda(z-1)}$, $$q = e^{-(1+s)(1-q)}.$$ Write $\pi = 1 - q$ for the survival probability. Taking logarithms gives $\ln(1-\pi) = -(1+s)\pi$; expanding the left side as $-\pi - \pi^2/2 - \pi^3/3 - \cdots$ and dividing through by $\pi$ leaves $1 + \pi/2 + O(\pi^2) = 1+s$, so that $\pi \approx 2s$. The factor of two is therefore not fundamental: it is the reciprocal of the $1/2$ produced by the quadratic term in the logarithm, and that term traces back to the variance of the offspring distribution. Repeating the calculation with an offspring distribution of mean $1+s$ and variance $\sigma^2$ gives $\pi \approx 2s/\sigma^2$. For the Poisson case $\sigma^2 = 1$, which recovers $2s$; for organisms with high variance in reproductive success — many marine invertebrates, for instance — $\sigma^2$ can exceed one substantially and the fixation probability falls in proportion. Patwa and Wahl (2008) survey the range of life-history models for which this correction has been computed. The lesson is that $2s$ is a result about idealised reproduction, not a universal law, and the essay's classical formula should be read accordingly. The diffusion, written as a stochastic differential equation The branching-process argument covers only the rare phase and says nothing about what happens once the mutant is common. The diffusion approximation covers the whole trajectory, and it is obtained by asking what the frequency change per generation looks like when the population is large and the per-generation change is small. Two quantities suffice: the mean change and its variance. The mean change comes from selection. With the fitness convention above, the standard one-locus recursion gives, to first order in $s$, a change per generation of $M(x) = s\,x(1-x)$. The product $x(1-x)$ appears because selection can only act on variation: if the allele is absent ($x=0$) or already fixed ($x=1$) there is nothing to change. The variance of the change comes from random sampling of gametes. The next generation's $2N_e$ gene copies are drawn as if binomially from the current frequency, and a binomial proportion based on $n$ draws has variance $x(1-x)/n$, so with $n = 2N_e$ copies $$V(x) = \frac{x(1-x)}{2N_e}.$$ The $2$ is there because each of $N_e$ diploid individuals carries two copies; halving the number of individuals doubles the sampling noise. Putting the two together and letting the generation become an infinitesimal time step gives the stochastic differential equation $$dX_t = s\,X_t(1-X_t)\,dt + \sqrt{\frac{X_t(1-X_t)}{2N_e}}\;dW_t,$$ where $W_t$ is standard Brownian motion — a continuous random path whose increment over an interval of length $dt$ has mean zero and variance $dt$. The first term is the deterministic push from selection; the second is genetic drift, and it is largest at intermediate frequencies for the same reason the selection term is. From a martingale to Kimura's formula This section obtains the fixation probability from that equation, starting from the neutral case, which a reader can check in one line, and then taking the step that selection requires. Set $s = 0$. The equation then has no drift