Evolutionary shaping of low-dimensional path facilitates robust and plastic switching between phenotypes

Abstract

Biological systems must be robust for stable functioning against perturbations, but robustness alone is insufficient. The ability to switch between appropriate states (phenotypes) in response to different conditions is essential for biological functions, as observed in allosteric enzymes and motor proteins. How are robustness and plasticity simultaneously acquired through evolution In an attempt to answer this question, we examine the evolution of genotypes that realize plastic switching between two endpoint phenotypes upon external inputs as well as stationary expressions of phenotypes. Here, we introduce a statistical physics model consisting of spins, with active sites and regulatory sites, which are distinct from each other. In our model, we represent the phenotype and genotype as spin configurations and spin-spin interactions, respectively. The fitness for selection is given by the spin configuration, whose behavior is governed by the genotypes. Specifically, the fitness for selection is given so that it takes a higher value as more active sites take two requested spin configurations depending on the states of the regulatory sites. The remaining spins do not directly affect the fitness, but they interact with other spins. We numerically evolve the matrices of spin-spin interactions (genotypes) by changing them with mutations and selection of those with higher fitness. Our numerical simulations show that characteristic genotypes with higher fitness evolve slightly above the phase transition temperature between replica-symmetric and replica-symmetry-breaking phase in spin-glass theory. These genotypes shape the two spin configurations separately depending on the regulation. Each phenotype is primarily represented by the first or second eigenmode of the genotypes. Smooth switching between the two phenotypes is achieved by following a one-dimensional quarter-circle that connects them. Upon changes in regulations, spin configurations are attracted to this path, which allows for robust and plastic switching between the two phenotypes. The statistical physics analysis based on the two eigenmodes shows that the free energy landscape has a valley along the one-dimensional quarter-circle switching path. Robust attraction to the path is achieved through the evolution of the interactions within nonactive and nonregulatory spin sites, which themselves do not contribute to fitness. Our findings indicate that the compatibility between robustness and plasticity is acquired by the evolution of low dimensionality in the phenotype space, which will be relevant to the understanding of the robust function of protein as well as material design.