A novel use of equilibrium frequencies in models of sequence evolution

Current mathematical models of amino acid sequence evolution are often applied in variants that match their expected amino acid frequencies to those observed in a data set under analysis. This has been achieved by setting the instantaneous rate of replacement of a residue i by another residue j proportional to the observed frequency of the resulting residue j. We describe a more general method that maintains the match between expected and observed frequencies but permits replacement rates to be proportional to the frequencies of both the replaced and resulting residues, raised to powers other than 1. Analysis of a database of amino acid alignments shows that the description of the evolutionary process in a majority (approximately 70% of 182 alignments) is significantly improved by use of the new method, and a variety of analyses indicate that parameter estimation with the new method is well-behaved. Improved evolutionary models increase our understanding of the process of molecular evolution and are often expected to lead to improved phylogenetic inferences, and so it seems justified to consider our new variants of existing standard models when performing evolutionary analyses of amino acid sequences. Similar methods can be used with nucleotide substitution models, but we have not found these to give corresponding significant improvements to our ability to describe the processes of nucleotide sequence evolution.