Statistical-mechanical study of evolution of robustness in noisy environments.

In biological systems, expression dynamics that can provide fitted phenotype patterns with respect to a specific function have evolved through mutations. This has been observed in the evolution of proteins for realizing folding dynamics through which a target structure is shaped. We study this evolutionary process by introducing a statistical-mechanical model of interacting spins, where a configuration of spins and their interactions J represent a phenotype and genotype, respectively. The phenotype dynamics are given by a stochastic process with temperature TS under a Hamiltonian with J. The evolution of J is also stochastic with temperature TJ and follows mutations introduced into J and selection based on a fitness defined for a configuration of a given set of target spins. Below a certain temperature TS(c2), the interactions J that achieve the target pattern evolve, whereas another phase transition is observed at TS(c1)<TS(c2). At low temperatures TS<TS(c1), the Hamiltonian exhibits a spin-glass-like phase, where the dynamics toward the target pattern require long time steps, and the fitness often decreases drastically as a result of a single mutation to J. In the intermediate-temperature region, the dynamics to shape the target pattern proceed rapidly and are robust to mutations of J. The interactions in this region have no frustration around the target pattern and results in funnel-type dynamics. We propose that the ubiquity of funnel-type dynamics, as observed in protein folding, is a consequence of evolution subjected to thermal noise beyond a certain level; this also leads to mutational robustness of the fitness.