Funnel landscape and mutational robustness as a result of evolution under thermal noise

TL;DR

This paper models biological evolution using a spin system to show how thermal noise influences the development of funnel landscapes and mutational robustness, revealing phase transitions and the emergence of robust, rapid expression dynamics.

Contribution

It introduces a statistical-mechanical spin model to analyze how thermal noise affects evolutionary phases, highlighting the emergence of funnel landscapes and robustness.

Findings

Evolved genotypes in an intermediate temperature phase exhibit rapid, funnel-like expression dynamics.

A phase transition occurs at a critical temperature T_S^{c1} leading to a spin-glass phase.

Mutational robustness and rapid expression are linked to the local Mattis state in the model.

Abstract

In biological systems, expression dynamics to shape a fitted phenotype for function has evolved through mutations to genes, as observed in the evolution of funnel landscape in protein. We study this evolutionary process with a statistical-mechanical model of interacting spins, where the fitted phenotype is represented by a configuration of a given set of "target spins" and interaction matrix J among spins is genotype evolving over generations. The expression dynamics is given by stochastic process with temperature T_S to decrease energy for a given set of J. The evolution of J is also stochastic with temperature T_J, following mutation in J and selection based on a fitness given by configurations of the target spins. Below a certain temperature T_S^{c2}, the highly adapted J evolves, whereasanother phase transition characterised by frustration occurs at T_S^{c1}<T_S^{c2}. At temperature lower than T_S^{c1}, the Hamiltonian exhibits a spin-glass like phase, where the dynamics requires long time steps to produce the fitted phenotype, and the fitness often decreases drastically by single mutation. In contrast, in the intermediate temperature phase between T_S^{c1} and T_S^{c2}, the evolved genotypes, that have no frustration around the target spins (we call "local Mattis state"), give a funnel-like rapid expression dynamics and are robust to mutation. These results imply that evolution under thermal noise beyond a certain level leads to funnel dynamics and mutational robustness. We will explain its mechanism with the statistical-mechanical method.