Several studies have reported a negative correlation between estimates of the nonsynonymous to synonymous rate ratio (ω = dN/dS) and the sequence distance d in pairwise comparisons of the same gene from different species. That is, more divergent sequences produce smaller estimates of ω. Explanations for this negative correlation have included segregating nonsynonymous polymorphisms in closely related species and nonlinear dynamics of the ratio of two random variables. Here we study the statistical properties of the maximum-likelihood estimates of ω and d in pairwise alignments and explore the possibility that the negative correlation can be entirely explained by those properties. We show that the ω estimate is positively biased for small d and that the bias decreases with the increase of d. We also show that the estimates of ω and d are negatively correlated when ω < 1 and positively correlated when ω > 1. However, the bias in estimates of ω and the correlation between estimates of ω and d are not enough to explain the much stronger correlation observed in real data sets. We then explore the behavior of the estimates when the model is misspecified and suggest that the observed correlation may be due to protein-level selection that causes very different amino acids to be favored in different domains of the protein. Widely used models fail to account for such among-site heterogeneity and cause underestimation of the nonsynonymous rate and ω, with the bias being much stronger for distant sequences. We point out that tests of positive selection based on the ω ratio are invariant to the parameterization of the model and thus unaffected by bias in the ω estimates or the correlation between estimates of ω and d.
Keywords: evolutionary distance; nonsynonymous to synonymous rate ratio; pairwise alignment; selection.