The statistical mechanics of a polygenic characterunder stabilizing selection, mutation and drift

TL;DR

This paper develops a statistical mechanics framework to efficiently model the evolution of polygenic traits under stabilizing selection, mutation, and drift, accurately capturing key dynamics with only four macroscopic variables.

Contribution

It introduces a novel macroscopic variable approach derived from entropy maximization, simplifying the analysis of polygenic trait evolution compared to previous methods.

Findings

Accurately describes trait mean and variance dynamics with four macroscopic variables.

Shows genetic variance is stable under directional selection when drift is included.

Reveals hysteresis effects indicating memory of genetic states in trait evolution.

Abstract

By exploiting an analogy between population genetics and statistical mechanics, we study the evolution of a polygenic trait under stabilizing selection, mutation, and genetic drift. This requires us to track only four macroscopic variables, instead of the distribution of all the allele frequencies that influence the trait. These macroscopic variables are the expectations of: the trait mean and its square, the genetic variance, and of a measure of heterozygosity, and are derived from a generating function that is in turn derived by maximizing an entropy measure. These four macroscopics are enough to accurately describe the dynamics of the trait mean and of its genetic variance (and in principle of any other quantity). Unlike previous approaches that were based on an infinite series of moments or cumulants, which had to be truncated arbitrarily, our calculations provide a well-defined approximation procedure. We apply the framework to abrupt and gradual changes in the optimum, as well as to changes in the strength of stabilizing selection. Our approximations are surprisingly accurate, even for systems with as few as 5 loci. We find that when the effects of drift are included, the expected genetic variance is hardly altered by directional selection, even though it fluctuates in any particular instance. We also find hysteresis, showing that even after averaging over the microscopic variables, the macroscopic trajectories retain a memory of the underlying genetic states.