Quantifying the stationarity and time reversibility of the nucleotide substitution process

Markov models describing the evolution of the nucleotide substitution process, widely used in phylogeny reconstruction, usually assume the hypotheses of stationarity and time reversibility. Although these models give meaningful results when applied to biological data, it is not clear if the 2 assumptions mentioned above hold and, if not, how much sequence evolution processes deviate from them. To this aim, we introduce 2 sets of indices that can be calculated from the nucleotide distribution and the substitution rates. The stationarity indices (STIs) can be used to test the validity of the equilibrium assumption. The irreversibility indices (IRIs) are derived from the Kolmogorov cycle conditions for time reversibility and quantify the degree of nontime reversibility of a process. We have computed STIs and IRIs for the evolutionary process of 2 lineages, Drosophila simulans and Homo sapiens. In the latter case, we use a modified form of the indices that takes into account the CpG decay process. In both cases, we find statistically significant deviations from the ideal case of a process that has reached stationarity and is time reversible.