Joint effect of changing selection and demography on the site frequency spectrum

Abstract

The site frequency spectrum (SFS) is an important statistic that summarizes the molecular variation in a population, and used to estimate population-genetic parameters and detect natural selection. While the equilibrium SFS in a constant environment is quite well studied, recent research has focused on nonequilibrium SFS to elucidate the role of demography when selection is constant in time and of fluctuating selection in a population of constant size. However, the joint effect of time-dependent selection and population size has not been investigated so far. Here, we study the SFS in a randomly mating, diploid population in which both the population size and selection coefficient vary periodically with time using a diffusion theory approach, and derive simple analytical expressions for the time-averaged SFS in slowly and rapidly changing environments. We show that for strong selection and in slowly changing environments, the time-averaged SFS differs significantly from the equilibrium SFS when the population experiences both positive and negative cycles of the selection coefficient. The deviation depends on the time spent by the population in the deleterious part of the selection cycle and the phase difference between the selection coefficient and population size. In particular, we find that the time-averaged SFS in slowly to moderately fast varying, on-average neutral environment has the same qualitative shape as the equilibrium SFS for positively selected mutant but differs quantitatively from it which can be captured by an effective population size.