The polymorphism frequency spectrum of finitely many sites under selection

The distribution of genetic polymorphisms in a population contains information about evolutionary processes. The Poisson random field (PRF) model uses the polymorphism frequency spectrum to infer the mutation rate and the strength of directional selection. The PRF model relies on an infinite-sites approximation that is reasonable for most eukaryotic populations, but that becomes problematic when is large ( greater, similar 0.05). Here, we show that at large mutation rates characteristic of microbes and viruses the infinite-sites approximation of the PRF model induces systematic biases that lead it to underestimate negative selection pressures and mutation rates and erroneously infer positive selection. We introduce two new methods that extend our ability to infer selection pressures and mutation rates at large : a finite-site modification of the PRF model and a new technique based on diffusion theory. Our methods can be used to infer not only a "weighted average" of selection pressures acting on a gene sequence, but also the distribution of selection pressures across sites. We evaluate the accuracy of our methods, as well that of the original PRF approach, by comparison with Wright-Fisher simulations.