For a single new variant allele subject to additive selection, it can be shown that the probability of fixation p, given a selection coefficient s in favor of the new variant in a population size N, is

         The equation is applicable for both positive and negative selection ( s < 0 or s > 0 ). The generalization here are specific for a small population, N = 100 with 2N = 200 alleles. As well, the numerical values cited below are mathematical expectations, where the actual result in any particular case is subject to chance. [The expectation of Heads or Tails is p(H,T) = 0.5. EXPAND].

[Left] Over a broad range of  s < or << 0.01, a deleterious allele (s < -0.01) is very unlikely to reach fixation. As s 0, there is an appreciable chance (~ 0.01) that a slightly deleterious allele can reach fixation, because of stochastic fluctuation in the small population. When s > 0.01, the probability of fixation increases linearly, with a slope of two, such that p = 2s. The shape of the graph indicates the probability of fixation of an advantageous novel allele increases rapidly as s increases.

[Middle] Over a narrow range of -0.01 < s < 0.01, the slope of the curve of p(s,100) increases. Notice three particular values. At (s = 0), a selectively neutral allele has p(0,N) = 1/(2N) = 1/200 = 0.005, the inverse of the number of alleles in the population. That is, all 2N alleles including the new one have an equal chance of fixation, in this case, s = 1/200 = 0.005. For a deleterious allele with s = -1/(2N) = -0.005, the probability of fixation is quite small, < 0.002, less than one in 500. For an advantageous allele with s = 1/(2N) = 0.005, the probability of fixation is > 0.01, about 1%. For s = 1/N = 0.01, the probability of fixation is about 2%. This curve shows the basis of the rule of thumb, that the relative importance of selection and drift shifts in the vicinity of s ~ 1/N.

[Right]. On a greatly expanded vertical scale of p(s,100), halving s from  -0.0250 to -0.0125 increases the relative probability of fixation of a deleterious allele more than twenty-fold, from <0.00001 to 0.00020.Selection remains weak. The shift is not apparent in the other two graphs because of the scale.

    If selection is strongly deleterious such that (2Ns < -1), the expectation of fixation is almost nil.

    Any new mutant allele B* necessarily arises in a single individual heterozygote B*A. If B* is recessive to A, it has essentially 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. If B* is dominant or semi-dominant, the heterozygote B*A will have a selection advantage, and drift will favor early formation of advantageous B*B* homozygotes. This is also the case if the population structure favors inbreeding, which also increase the expected frequency of B*B* homozygotes.

HOMEWORK: Use an Excel program to redraw the middle graph for N = 10. How does the curve of p(s,10) in a very small population compare with the results above? What does this say about the probability of fixation of new variants in very small vs moderate populations?