Funnel landscape and mutational robustness as a result of evolution under thermal noise
Abstract
In biological systems, expression dynamics to shape a fitted phenotype for function has evolved through mutations to genes, as observed in the evolution of funnel landscape in protein. We study this evolutionary process with a statistical-mechanical model of interacting spins, where the fitted phenotype is represented by a configuration of a given set of "target spins" and interaction matrix J among spins is genotype evolving over generations. The expression dynamics is given by stochastic process with temperature TS to decrease energy for a given set of J. The evolution of J is also stochastic with temperature TJ, following mutation in J and selection based on a fitness given by configurations of the target spins. Below a certain temperature TSc2, the highly adapted J evolves, whereasanother phase transition characterised by frustration occurs at TSc1<TSc2. At temperature lower than TSc1, the Hamiltonian exhibits a spin-glass like phase, where the dynamics requires long time steps to produce the fitted phenotype, and the fitness often decreases drastically by single mutation. In contrast, in the intermediate temperature phase between TSc1 and TSc2, the evolved genotypes, that have no frustration around the target spins (we call "local Mattis state"), give a funnel-like rapid expression dynamics and are robust to mutation. These results imply that evolution under thermal noise beyond a certain level leads to funnel dynamics and mutational robustness. We will explain its mechanism with the statistical-mechanical method.