A statistical-mechanical study of evolution of robustness in noisy environment

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 $\bm{J}$ represent a phenotype and genotype, respectively. The phenotype dynamics are given by a stochastic process with temperature $T_{S}$ under a Hamiltonian with $\bm{J}$. The evolution of $\bm{J}$ is also stochastic with temperature $T_{J}$ and follows mutations introduced into $\bm{J}$ and selection based on a fitness defined for a configuration of a given set of target spins. Below a certain temperature $T_{S}^{c2}$, the interactions $\bm{J}$ that achieve the target pattern evolve, whereas another phase transition is observed at $T_{S}^{c1}<T_{S}^{c2}$. At low temperatures $T_{S}<T_{S}^{c1}$, 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 $\bm{J}$. In the intermediate-temperature region, the dynamics to shape the target pattern proceed rapidly and are robust to mutations of $\bm{J}$. 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.