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.

A field theoretic approach to non-equilibrium population genetics in the strong selection regime

“…From a modeling standpoint, this class of demographies provides a convenient example for the application of PGFT to a relatively complex demography that can be defined using a relatively simple deterministic function for the population size. Cyclical demographies can be parameterized in a number of ways (see, e.g., [37] for a recent example), which do not all describe equivalent demographies, as the shape of the population size trajectory over the course of each oscillation may differ in different models. Here, I will analyze one class of cyclical demographies as an example, which are defined by the following three-parameter function representing the time-dependence of the diploid population size N ( t ).…”

“…Towards this end, PGFT is not a priori necessary; there have been many attempts at describing non-equilibrium dynamics in a number of ways, ranging from a focus on fixation probability in a changing population size [38, 39] or changing environment [40] (or both simultaneously [37]), integral solutions to neutral non-equilibrium dynamics of the allele frequency spectrum [41], exact solutions for the transition density in the form of an infinite series (for constant population size [42] and variable population size [43]), a system of ODEs for the moments of the allele frequency distribution [44, 46], numerical solutions to the time-dependent diffusion equations [44–46], analysis of the stochastic dynamics of traveling waves of fitness [52], amongst many others. Several approaches involve the use of path integrals [47–51, 65], which is a natural way to characterize the trajectory that an allele takes through a population over time, also employed in the present manuscript.…”

“…However, the majority of the mathematics developed to model population genetics is, by construction, restricted to equilibrium or pseudo-equilibrium regimes. This is for both analytic and intuitive simplicity, but, additionally, forays far beyond equilibrium population genetics are notoriously difficult to describe mathematically, particularly when more than one parameter can temporally vary (see, e.g., [37] for a recent paper on co-varying parameters). There are, of course, a number of exceptions (i.e., non-equilibrium models [27, 37–52]), some of which are highlighted in the Discussion.…”

“…This is for both analytic and intuitive simplicity, but, additionally, forays far beyond equilibrium population genetics are notoriously difficult to describe mathematically, particularly when more than one parameter can temporally vary (see, e.g., [37] for a recent paper on co-varying parameters). There are, of course, a number of exceptions (i.e., non-equilibrium models [27, 37–52]), some of which are highlighted in the Discussion. Non-equilibrium descriptions in the literature are often constructed using ad hoc techniques suited to model a specific demography or phenomena of interest (e.g., models presented in [27, 37]).…”

“…There are, of course, a number of exceptions (i.e., non-equilibrium models [27, 37–52]), some of which are highlighted in the Discussion. Non-equilibrium descriptions in the literature are often constructed using ad hoc techniques suited to model a specific demography or phenomena of interest (e.g., models presented in [27, 37]). Instead, this manuscript aims to describe a distinct and generalizable formalism that is flexible enough to characterize a wide range of behavior in non-equilibrium populations in a unified framework.…”