A finite-population bound on the selection limit

R. Craig Stillwell 

Fisher’s fundamental theorem is an identity. Given current gene frequencies, the partial change in mean fitness, holding genotypic fitnesses at their additive predictions, equals the additive genetic variance in fitness divided by mean fitness. Price gave that reading and Ewens proved the discrete-generation form. Lessard and Grafen have argued over what the partial change represents. Akin and Shahshahani showed that selection on a frequency simplex is the gradient flow of mean fitness under the Fisher information metric. In Ewens’s variables, allele-frequency change under genic selection is half a natural-gradient step on mean fitness, and Fisher’s rate is the squared length of that step divided by twice mean fitness.
The identity uses whatever frequencies the population has. It does not give the distance from the additive ceiling once directional selection has finished, Robertson’s selection limit. Under the infinitesimal model I bound that gap for a Wright–Fisher population. Kimura’s diffusion approximation supplies a bias. As a fraction of the ceiling the bias is exponentially small once selection dominates drift at every locus, and in fitness units it grows with the ceiling. Hoeffding’s inequality bounds the spread of the fraction that is fixed. The number of loci is the count in that inequality, where a bound on a sample mean puts the sample size. Population size enters the bias through the fixation probability. The bound assumes N_e s≫1 at each locus. Where N_e s is of order one, the process is in Barton’s infinitesimal-selection regime.