A statistical-mechanical study of evolution of robustness: An approach from two-temperature models

Abstract

Recently we proposed a simple statistical-mechanical model of interacting spins for studying evolutional process in biological system. In our model, 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 fixed Hamiltonian with J. Meanwhile, the evolution of J is also stochastic with temperature TJ and follows mutations introduced into J and selection based on a fitness given by equilibrium spin configurations of a given set of target spins. Using Monte Carlo simulations, it is found that frustration around the target spins is strongly suppressed for the interactions J evolved in the intermediate TS temperature region. The evolved J's give the funnel-like dynamics and show robustness to mutations. This model can be regarded as an extension of partial annealing model, in which the interactions between spins are dynamical variables with characteristic time scale widely separated from that of the spins. We also discuss the partially annealed Sherrington-Kirkpatrick model by particularly paying attention to the frustration.