 
Summary: RESEARCH ARTICLES
Quantifying the Stationarity and Time Reversibility of the Nucleotide
Substitution Process
Federico Squartini and Peter F. Arndt
Department of Computational Molecular Biology, Max Planck Institute for Molecular Genetics, Berlin, Germany
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.
Introduction
When studying a natural phenomenon, it is a well
established and fruitful practice to disregard some of its
properties in order to get a simpler and neater mathematical
description. In the first stage, we can use physical and math
