A general approximation for the dynamics of quantitative traits

TL;DR

This paper investigates the macroscopic dynamics of quantitative traits influenced by selection, mutation, and drift, extending maximum entropy methods to regimes with low mutation rates and multiple loci, showing high accuracy in predictions.

Contribution

It extends the maximum entropy approximation to low mutation regimes and multiple loci, improving understanding of trait dynamics without tracking allele frequencies.

Findings

Maximum entropy approximation remains accurate even with rapid changes.

Extension to multiple unlinked loci with unequal effects.

Effective in regimes with small mutation rates.

Abstract

Selection, mutation and random drift affect the dynamics of allele frequencies and consequently of quantitative traits. While the macroscopic dynamics of quantitative traits can be measured, the underlying allele frequencies are typically unobserved. Can we understand how the macroscopic observables evolve without following these microscopic processes? The problem has previously been studied by analogy with statistical mechanics: the allele frequency distribution at each time is approximated by the stationary form, which maximizes entropy. We explore the limitations of this method when mutation is small ($4N\!\mu<1$) so that populations are typically close to fixation and we extend the theory in this regime to account for changes in mutation strength. We consider a single diallelic locus under either directional selection, or with over-dominance, and then generalise to multiple unlinked biallelic loci with unequal effects. We find that the maximum entropy approximation is remarkably accurate, even when mutation and selection change rapidly.