Physical model of the genotype-to-phenotype map of proteins

Abstract: How DNA is mapped to functional proteins is a basic question of living matter. We introduce and study a physical model of protein evolution which suggests a mechanical basis for this map. Many proteins rely on large-scale motion to function. We therefore treat protein as learning amorphous matter that evolves towards such a mechanical function: Genes are binary sequences that encode the connectivity of the amino acid network that makes a protein. The gene is evolved until the network forms a shear band across the protein, which allows for long-range, soft modes required for protein function. The evolution reduces the high-dimensional sequence space to a low-dimensional space of mechanical modes, in accord with the observed dimensional reduction between genotype and phenotype of proteins. Spectral analysis of the space of $10^6$ solutions shows a strong correspondence between localization around the shear band of both mechanical modes and the sequence structure. Specifically, our model shows how mutations of the gene and their correlations occur at amino acids whose interactions determine the functional mode.