Testing of the three multiplicatively closed (Lie Markov) model heirarchies which respect purine/pyrimidine, Watson-Crick, and amino/keto nucleotide groupings

We present three hierarchies of Lie Markov models of DNA sequence evolution. These models are (locally) “multiplicatively closed,” meaning that the composition of two Markov matrices in the model results, with some (rare) exceptions, in a third Markov matrix that is still in the model. Additionally, the models in each hierarchy respectively distinguish between (i) purines and pyrimadines (RY), (ii) Watson-Crick pairs (WS), and (iii) amino/keto pairs (MK), but otherwise treat the four nucleotides without distinction. The multiplicative closure property allows mathematically consistent modeling of time-inhomogeneous scenarios, unlike models that are not closed, such as the general time-reversible model (GTR) and many of its submodels. We derive the nesting relationships of the three model hierarchies and present software implementing the models. For a diverse range of biological data sets, we perform Bayesian information criterion model comparision analogous to that of the ModelTest framework. We find that our models outperform the GTR model in some (but not all) cases