Accurate prediction of substitution rates at protein sites with a mutation-selection model

The pattern of substitutions at sites in proteins provides invaluable information about their biophysical and functional importance and what selection pressures are acting at individual sites. Amino acid site rates are typically estimated using phenomenological models where sequence variability is described by rate factors that scale the overall substitution rate in a protein to sites. In this study, we demonstrate that site rates can be calculated accurately from amino acid sequences from a multiple sequence alignment using a mutation-selection model in combination with a simple nucleotide substitution model. The method performs better than the standard phylogenetic approach on sequences generated by structure-based evolutionary dynamics simulations, robustly estimates rates for shallow multiple sequence alignments, and can be rapidly calculated also on larger sequence alignments. On natural sequences, site rates from the mutation-selection model are strongly correlated with rates calculated with the empirical Bayes methods. The model complements other work in providing a link between amino acid substitution rates and equilibrium frequency distributions at sites in proteins. We show how an ensemble of equilibrium frequency vectors can be used to represent the rate variation encoded in empirical amino acid substitution matrices. This study demonstrates that a rapid and simple method can be developed from the mutation-selection model to predict substitution rates from amino acid data, complementing the standard phylogenetic approach.