NS 08-04

Probability of fixation u with variable s & N
Drift vs Selection in finite populations

For a new mutant allele A subject to Additive Selection, the probability of fixation u, given a selection coefficient s in a population size N, is

u(s,N) = (1 - e^-2s) / 1 - e^-4Ns)

as plotted in (A). The equation is true irrespective of positive or negative selection ( s < 0 or s > 0 ). At the extremes, u(0,N) = 1/(2N) for a neutral allele, and if s > 1/(2N)), fixation will likely occur irrespective of N. That is, the break point is when s and 2N are approximately reciprocals of each other.

    (B) [red box in (A)] If u > 2s, selection is strongly advantageous and the expectation of fixation of the new mutant is high irrespective of N.

    (C) [green box in (A)] If selection is weak ( -1 < 2Ns < 1 ), the expectation is closer to that for a neutral allele. If selection is strongly deleterious where s < 0 and ( 2Ns < -1 ), the probability of fixation is almost nil.

    NB: Any new mutant allele A* necessarily arises in a heterozygote A*A. If A* is recessive to A, it has close to a 50:50 chance of being lost in the next generation despite any selective advantage, because the two alleles in the heterozygous carrier are propagated with equal probability. The expectation of loss by chance remains high over the first few generations, unless inbreeding and (or) drift favor early formation of advantageous A*A* homozygotes.