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Biological systems must be robust for stable function against perturbations,but robustness alone is not sufficient. The ability to switch betweenappropriate states (phenotypes) in response to different conditions isessential for biological functions. How are robustness and plasticitysimultaneously acquired through evolution? We examine the evolution ofgenotypes that realize plastic switching between two phenotypes upon externalinputs, as well as stationary expressions of phenotypes. We introduce astatistical physics model consisting of spins, with active and regulatorysites, which are distinct from each other. We represent the phenotype andgenotype as spin configurations and the spin-spin interactions, respectively.The fitness for selection is given so that it takes a higher value as more ofthe active sites take two requested spin configurations depending on theregulation. We numerically evolve the interaction matrix by changing them withmutations and selection of those with higher fitness. Our numerical simulationsshow that characteristic genotypes evolve slightly above the transitiontemperature between replica symmetric and replica symmetry breaking phase.These genotypes shape two spin configurations separately depending on theregulation, where the two phenotypes are dominantly represented by thegenotypes' first and second eigenmodes, and smooth switching of two phenotypesare achieved by following a one-dimensional path connecting the two phenotypes.Upon changes in regulations, spin configurations are attracted to this path,which allows for robust and plastic switching between the two phenotypes. Thestatistical-physics analysis show that the free energy landscape has a valleyalong the switching path. Our finding indicates that the compatibility of therobustness and plasticity is acquired by the evolution of thelow-dimensionality in the phenotype space.

Evolutionary Shaping of Low-Dimensional Path Facilitates Robust and Plastic Switching Between Phenotypes