Lie Markov models with purine/pyrimidine symmetry

Abstract

Continuous-time Markov chains are a standard tool in phylogenetic inference. If homogeneity is assumed, the chain is formulated by specifying time-independent rates of substitutions between states in the chain. In applications, there are usually extra constraints on the rates, depending on the situation. If a model is formulated in this way, it is possible to generalise it and allow for an inhomogeneous process, with time-dependent rates satisfying the same constraints. It is then useful to require that there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the definition of "Lie Markov models", which are precisely the class of models where such an average exists. These models form Lie algebras and hence concepts from Lie group theory are central to their derivation. In this paper, we concentrate on applications to phylogenetics and nucleotide evolution, and derive the complete hierarchy of Lie Markov models that respect the grouping of nucleotides into purines and pyrimidines -that is, models with purine/pyrimidine symmetry. We also discuss how to handle the subtleties of applying Lie group methods, most naturally defined over the complex field, to the stochastic case of a Markov process, where parameter values are restricted to be real and positive. In particular, we explore the geometric embedding of the cone of stochastic rate matrices within the ambient space of the associated complex Lie algebra.The whole list of Lie Markov models with purine/pyrimidine symmetry is available at

A New Hierarchy of Phylogenetic Models Consistent with Heterogeneous Substitution Rates

“…The ability to express the model with six nonnegative parameters is due to the set of stochastic rate matrices of this model forming a “polyhedral cone” having six “rays,” this being the origin of the “6” in the model name. Rays and polyhedral cones in this context are more fully explained in Fernández-Sánchez et al (2014).…”

“…For the purpose of easy comparison to the presentation given in Fernández-Sánchez et al (2014), note that we have renamed the basis matrices and added $A_2$ as an alternative to $A_1$. A table of the basis matrix renaming is in the Supplementary Material available on Dryad at http://dx.doi.org/10.5061/dryad.461g6.…”

“…The corresponding regions of stochasticity describe a geometric entity known as a convex polydreal cone. The interested reader is referred to Fernández-Sánchez et al (2014) for details.…”

“…The theoretical results of (Sumner et al 2012a) and (Fernández-Sánchez et al 2014) prove only that the Lie Markov models have “local multiplicative closure.” This means that the “average” rate matrix of a time varying process can be nonstochastic or even complex. Here, we see that “local” is really quite broad: phylogenies have to be quite deep before nonembeddability potentially becomes an issue, and very deep before the average $Q$ becomes complex.…”

“…This hierarchy was derived in Fernández-Sánchez et al (2014) and totals 37 models capable of distinguishing transitions from transversions. Here we present the models in a more accessible way; present some new results on model nesting, equilibrium frequencies, and parameterization; and test the models.…”