Abstract
The measure of synonymous and non-synonymous substitution rates (dS and dN) is useful for assessing selection operating on protein sequences or for investigating mutational processes affecting genomes. In particular, the ratio 
is expected to be a good proxy of ω, the probability of fixation of non-synonymous mutations relative to that of neutral mutations. Standard methods for estimating dN, dS or ω rely on the assumption that the base composition of sequences is at the equilibrium of the evolutionary process. In many clades, this assumption of stationarity is in fact incorrect, and we show here through simulations and through analyses of empirical data that non-stationarity biases the estimate of dN, dS and ω. We show that the bias in the estimate of ω can be fixed by explicitly considering non-stationarity in the modeling of codon evolution, in a maximum likelihood framework. Moreover, we propose an exact method of estimate of dN and dS on branches, based on stochastic mapping, that can take into account non-stationarity. This method can be directly applied to any kind of model of evolution of codons, as long as neutrality is clearly parameterized.
