Correlation between the substitution rate and rate variation among sites in protein evolution

It is well known that the rate of amino acid substitution varies among different proteins and among different sites of a protein. It is, however, unclear whether the extent of rate variation among sites of a protein and the mean substitution rate of the protein are correlated. We used two approaches to analyze orthologous protein sequences of 51 nuclear genes of vertebrates and 13 mitochondrial genes of mammals. In the first approach, no assumptions of the distribution of the rate variation among sites were made, and in the second approach, the gamma distribution was assumed. Through both approaches, we found a negative correlation between the extent of among-site rate variation and the average substitution rate of a protein. That is, slowly evolving proteins tend to have a high level of rate variation among sites, and vice versa. We found this observation consistent with a simple model of the neutral theory where most sites are either invariable or neutral. We conclude that the correlation is a general feature of protein evolution and discuss its implications in statistical tests of positive Darwinian selection and molecular time estimation of deep divergences.