Gaussian selection on a polygenic trait from the gene's-eye view.

Abstract

We study a polygenic trait under Gaussian selection at statistical equilibrium, where allelic effect, mutation rate and mutational bias are heterogeneous across loci. The model assumes a large number $L$ of biallelic sites subject to reversible mutations, each allele described by its frequency in the population. Using a diffusion approximation, a mean-field approximation and neglecting linkage disequilibrium, we predict consistent phenomena across several regimes of selection: (1) a small deviation $\Delta^*$ of the trait mean from its optimal value appears and persists due to genetic mutations not aligned with selection; (2) while this deviation is often undetectable at the trait level, it leaves a substantial signature at the locus level by favoring alleles reducing it, resulting in genic selection with mean coefficient $s^*$ proportional to $-\Delta^*$ acting pervasively; (3) with stronger selection on the trait, (3a) the value of $\Delta^*$ is decreased but the intensity of genic selection is increased in inverse proportion, resulting in an essentially constant, non negligible value of $s^*$. We show how the stationary distribution of allelic frequencies can be obtained from $\Delta^*$. The latter can then be characterized semi-explicitly as the solution to a fixed-point equation. Finally, we find tractable expressions for several macroscopic observables of interest (genetic variance, description of the fluctuations of the trait mean as an Ornstein-Uhlenbeck process). The orders of magnitude of the macroscopic observables can be derived on a wide region of the parameter space. The model shows good fit and can straightforwardly be extended to accommodate pleiotropy, dominance, and some forms of epistasis. We also discuss the different types of breakdown which may occur (Bulmer effect, Hill-Robertson effect, breakdown of the Ornstein-Uhlenbeck approximation for the dynamics of the trait mean, depletion of genetic variability due to low mutation rates).