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)