Evolutionary Le Chatelier's Principle: Phenotypic Plasticity and Genetic Assimilation via Timescale Separation in the Price Equation

Abstract

Phenotypic plasticity and genetic assimilation play key roles in adaptive evolution, yet their underlying mechanism has lacked a unified physical description. A major theoretical difficulty lies in the fundamental difference in timescales, as phenotypic plasticity occurs rapidly within a generation whereas genetic changes accumulate slowly across generations. Here, we formalize these processes by bridging the continuous-time Price equation, a foundational equation of evolutionary dynamics, with the physical concept of timescale separation. A sudden environmental change induces a fast plastic displacement of the phenotype relative to the slow genotypic variable. Through genotype--phenotype coupling, this displacement generates an internal genetic stress. We demonstrate that genetic assimilation is a dynamical relaxation process in which the genotype evolves to resolve this self-generated stress. These evolutionary dynamics mathematically realize Le Chatelier's principle, where the slow genetic response naturally amplifies the initial plastic shift in the same direction. The theory predicts that a weaker restoring force, which can manifest as larger clonal phenotypic fluctuations, requires a longer evolutionary timescale for assimilation. In the ideal limit of cost-free, perfectly adaptive plasticity, the relaxation time diverges, so assimilation effectively stalls. This formulation provides a macroscopic physical mechanism for genetic assimilation, offering a universal response law for evolutionary systems in which rapid phenotypic responses precede slower genetic change.