Two stationary nonhomogeneous Markov models of nucleotide sequence evolution

The general Markov model (GMM) of nucleotide substitution does not assume the evolutionary process to be stationary, reversible, or homogeneous. The GMM can be simplified by assuming the evolutionary process to be stationary. A stationary GMM is appropriate for analyses of phylogenetic data sets that are compositionally homogeneous; a data set is considered to be compositionally homogeneous if a statistical test does not detect significant differences in the marginal distributions of the sequences. Though the general time-reversible (GTR) model assumes stationarity, it also assumes reversibility and homogeneity. We propose two new stationary and nonhomogeneous models - one constrains the GMM to be reversible, whereas the other does not. The two models, coupled with the GTR model, comprise a set of nested models that can be used to test the assumptions of reversibility and homogeneity for stationary processes. The two models are extended to incorporate invariable sites and used to analyze a seven-taxon hominoid data set that displays compositional homogeneity. We show that within the class of stationary models, a nonhomogeneous model fits the hominoid data better than the GTR model. We note that if one considers a wider set of models that are not constrained to be stationary, then an even better fit can be obtained for the hominoid data. However, the methods for reducing model complexity from an extremely large set of nonstationary models are yet to be developed.