A fast-flowing river scours its banks, yet part of the water slips away into a side channel. Evolution works the same way: even when selection pushes an allele hard toward fixation, randomness can steer it into the dead end of loss. For a long time researchers watched only the main channel, the deterministic trajectory along which the allele frequency climbs. The side channel, where the allele vanishes, was written off as irrelevant. That was a big mistake: for the variance of the frequency, the error reached 100%.
The author of the study proposes watching both channels. He writes the statistic of the allele frequency as the sum of two conditional expectations: one conditioned on the allele fixing, the other on it being lost. Each of them is rewritten in terms of a modified process with altered selection. Then comes an expansion in a small parameter, the inverse of the selection strength. The result is explicit expressions for the mean and the variance of the frequency at any point in time.
A numerical comparison with exact results from the Wright-Fisher model shows that the new method sharply reduces errors. At selection strength R=50 and an initial frequency of 0.001, the error in the mean drops from 10.6% to 3.2%, and the error in the variance from 12.6% to 4.9%. The fluctuation contribution matters most at early times, when the deterministic variance is nearly zero.
Researchers now have a tool for seeing both channels of the river. The method applies to the analysis of time series of allele frequencies in experimental evolution, viral populations, and artificial selection programs. It helps estimate selection strength more accurately and predict the fate of beneficial alleles in small populations. The river of evolution has become more transparent.
Even under strong selection, when the deterministic force is large, the variance of the allele frequency can be almost entirely determined by random fluctuations for a long time. For example, at R=50 and initial frequency y=0.001, the deterministic contribution to the variance is essentially zero up to time τ≈5, whereas the fluctuation contribution accurately describes the actual variance.