Sequence entropy of folding and the absolute rate of amino acid substitutions

Adequate representations of protein evolution should consider how the acceptance of mutations depends on the sequence context in which they arise. However, epistatic interactions among sites in a protein result in time and spatial substitution rate heterogeneity beyond the capabilities of current models. Here, we exploit parallels between amino acid substitutions and chemical reaction kinetics to develop an improved theory of protein evolution. We constructed a mechanistic framework for modelling amino acid substitution rates that employs the formalisms of statistical mechanics, with population genetics principles underlying the analysis. Theoretical analyses and computer simulations of proteins under purifying selection for thermodynamic stability show that substitution rates and the stabilisation of resident amino acids (the ‘evolutionary Stokes shift’) can be predicted from biophysics and the effect of sequence entropy alone. Furthermore, we demonstrate that substitutions predominantly occur when epistatic interactions result in near neutrality; substitution rates are determined by how often epistasis results in such nearly neutral conditions. This theory provides a general framework for modelling protein sequence change under purifying selection, potentially explains patterns of convergence and mutation rates in real proteins that are incompatible with previous models, and provides a better null model for the detection of adaptive changes.