There was a conversation on twitter about Haldane’s 2s approximation to the fixation probability of an allele, and how it related to the diffusion approximation of the same quantity. This followed from a blog post by Adam Eyre-Walker. I thought I’d write a couple of notes up on it. This post could likely do with more thought/editing but I thought it would be useful to put it out there.
The 2s result is the correct (ignoring terms of s^2 and higher) answer for the probability of a mutation never being lost in an infinite population with Poisson number of offspring with mean 1+s. The reason why this is “never being lost” instead of fixed, is that the population is infinite. So to persist indefinitely the allele has to escape loss permanently, by never being absorbed by the zero state.
This disagrees with the fixation probability from the diffusion, which is given by (1-exp(-4Nes/(2N))/(1-exp(-4Nes))) ~ 2s (Ne/N)/(1-exp(-4Nes))). Note the various roles played by Ne (eff. pop. size) and N in both equations (or the lack thereof).
Haldane’s result is not quite “right” for “real” populations (e.g. as modeled by Wright Fisher, and its diffusion limit) for 2 reasons.
The first is that population size is finite, so to fix we only need to reach a size 2N individuals (and then we will never be lost). Weakly beneficial mutations (Ns~1) are slightly more likely to fix than the 2s probability, as they only have to reach 2N to never be lost. Similarly deleterious mutations will never escape loss in infinite population, but can in finite pop. by reach 2N individuals. This is captured by the denominator of the fixation probability under the diffusion model, which that this increases the fixation prob. of alleles with |Ns|~1. The absorption of alleles at 2N copies can also be modeled in finite individual models (i.e. not the diffusion limit), I seem to remember that Rick Durrett’s book has a section on this.
The second issue with the 2s result is that it assumes that the individuals have Poisson distributed number offspring with variance 1 (actually our selected type has mean and var 1+s, but we ignore the s). However, in practice that isn’t quite true as our number of offspring (in Wright Fisher) is binomial with p=1/2N (actually it not quite this due to s, but we can ignore that). That also drops the dependance on Ne out of the equation, this can be factored back in as the branching process escaping loss probability is easily modified for non-Poisson variance. Where for an allele, with mean offspring 1+s and variance in offspring V, the probability the branching process escapes loss is ~2s/V.