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In biological systems, expression dynamics that can provide fitted phenotypepatterns with respect to a specific function have evolved through mutations.This has been observed in the evolution of proteins for realizing foldingdynamics through which a target structure is shaped. We study this evolutionaryprocess by introducing a statistical-mechanical model of interacting spins,where a configuration of spins and their interactions $\bm{J}$ represent aphenotype and genotype, respectively. The phenotype dynamics are given by astochastic process with temperature $T_{S}$ under a Hamiltonian with $\bm{J}$.The evolution of $\bm{J}$ is also stochastic with temperature $T_{J}$ andfollows mutations introduced into $\bm{J}$ and selection based on a fitnessdefined for a configuration of a given set of target spins. Below a certaintemperature $T_{S}^{c2}$, the interactions $\bm{J}$ that achieve the targetpattern evolve, whereas another phase transition is observed at$T_{S}^{c1}<T_{S}^{c2}$. At low temperatures $T_{S}<T_{S}^{c1}$, theHamiltonian exhibits a spin-glass like phase, where the dynamics toward thetarget pattern require long time steps, and the fitness often decreasesdrastically as a result of a single mutation to $\bm{J}$. In theintermediate-temperature region, the dynamics to shape the target patternproceed rapidly and are robust to mutations of $\bm{J}$. The interactions inthis region have no frustration around the target pattern and results infunnel-type dynamics. We propose that the ubiquity of funnel-type dynamics, asobserved in protein folding, is a consequence of evolution subjected to thermalnoise beyond a certain level; this also leads to mutational robustness of thefitness.

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