Lie Markov models.

Recent work has discussed the importance of multiplicative closure for the Markov models used in phylogenetics. For continuous-time Markov chains, a sufficient condition for multiplicative closure of a model class is ensured by demanding that the set of rate-matrices belonging to the model class form a Lie algebra. It is the case that some well-known Markov models do form Lie algebras and we refer to such models as "Lie Markov models". However it is also the case that some other well-known Markov models unequivocally do not form Lie algebras (GTR being the most conspicuous example). In this paper, we will discuss how to generate Lie Markov models by demanding that the models have certain symmetries under nucleotide permutations. We show that the Lie Markov models include, and hence provide a unifying concept for, "group-based" and "equivariant" models. For each of two and four character states, the full list of Lie Markov models with maximal symmetry is presented and shown to include interesting examples that are neither group-based nor equivariant. We also argue that our scheme is pleasing in the context of applied phylogenetics, as, for a given symmetry of nucleotide substitution, it provides a natural hierarchy of models with increasing number of parameters. We also note that our methods are applicable to any application of continuous-time Markov chains beyond the initial motivations we take from phylogenetics.

The algebra of the general Markov model on phylogenetic trees and networks.

It is known that the Kimura 3ST model of sequence evolution on phylogenetic trees can be extended quite naturally to arbitrary split systems. However, this extension relies heavily on mathematical peculiarities of the associated Hadamard transformation, and providing an analogous augmentation of the general Markov model has thus far been elusive. In this paper, we rectify this shortcoming by showing how to extend the general Markov model on trees to include incompatible edges; and even further to more general network models. This is achieved by exploring the algebra of the generators of the continuous-time Markov chain together with the "splitting" operator that generates the branching process on phylogenetic trees. For simplicity, we proceed by discussing the two state case and then show that our results are easily extended to more states with little complication. Intriguingly, upon restriction of the two state general Markov model to the parameter space of the binary symmetric model, our extension is indistinguishable from the Hadamard approach only on trees; as soon as any incompatible splits are introduced the two approaches give rise to differing probability distributions with disparate structure. Through exploration of a simple example, we give an argument that our extension to more general networks has desirable properties that the previous approaches do not share. In particular, our construction allows for convergent evolution of previously divergent lineages; a property that is of significant interest for biological applications.

Phylogenetic estimation with partial likelihood tensors.

We present an alternative method for calculating likelihoods in molecular phylogenetics. Our method is based on partial likelihood tensors, which are generalizations of partial likelihood vectors, as used in Felsenstein's approach. Exploiting a lexicographic sorting and partial likelihood tensors, it is possible to obtain significant computational savings. We show this on a range of simulated data by enumerating all numerical calculations that are required by our method and the standard approach.

Markov invariants and the isotropy subgroup of a quartet tree.

The purpose of this article is to show how the isotropy subgroup of leaf permutations on binary trees can be used to systematically identify tree-informative invariants relevant to models of phylogenetic evolution. In the quartet case, we give an explicit construction of the full set of representations and describe their properties. We apply these results directly to Markov invariants, thereby extending previous theoretical results by systematically identifying linear combinations that vanish for a given quartet. We also note that the theory is fully generalizable to arbitrary trees and is equally applicable to the related case of phylogenetic invariants. All results follow from elementary consideration of the representation theory of finite groups.

Markov invariants, plethysms, and phylogenetics.

We explore model-based techniques of phylogenetic tree inference exercising Markov invariants. Markov invariants are group invariant polynomials and are distinct from what is known in the literature as phylogenetic invariants, although we establish a commonality in some special cases. We show that the simplest Markov invariant forms the foundation of the Log-Det distance measure. We take as our primary tool group representation theory, and show that it provides a general framework for analyzing Markov processes on trees. From this algebraic perspective, the inherent symmetries of these processes become apparent, and focusing on plethysms, we are able to define Markov invariants and give existence proofs. We give an explicit technique for constructing the invariants, valid for any number of character states and taxa. For phylogenetic trees with three and four leaves, we demonstrate that the corresponding Markov invariants can be fruitfully exploited in applied phylogenetic studies.

Using the tangle: a consistent construction of phylogenetic distance matrices for quartets.

Distance based algorithms are a common technique in the construction of phylogenetic trees from taxonomic sequence data. The first step in the implementation of these algorithms is the calculation of a pairwise distance matrix to give a measure of the evolutionary change between any pair of the extant taxa. A standard technique is to use the log det formula to construct pairwise distances from aligned sequence data. We review a distance measure valid for the most general models, and show how the log det formula can be used as an estimator thereof. We then show that the foundation upon which the log det formula is constructed can be generalized to produce a previously unknown estimator which improves the consistency of the distance matrices constructed from the log det formula. This distance estimator provides a consistent technique for constructing quartets from phylogenetic sequence data under the assumption of the most general Markov model of sequence evolution.

Entanglement invariants and phylogenetic branching.

It is possible to consider stochastic models of sequence evolution in phylogenetics in the context of a dynamical tensor description inspired from physics. Approaching the problem in this framework allows for the well developed methods of mathematical physics to be exploited in the biological arena. We present the tensor description of the homogeneous continuous time Markov chain model of phylogenetics with branching events generated by dynamical operations. Standard results from phylogenetics are shown to be derivable from the tensor framework. We summarize a powerful approach to entanglement measures in quantum physics and present its relevance to phylogenetic analysis. Entanglement measures are found to give distance measures that are equivalent to, and expand upon, those already known in phylogenetics. In particular we make the connection between the group invariant functions of phylogenetic data and phylogenetic distance functions. We introduce a new distance measure valid for three taxa based on the group invariant function known in physics as the "tangle". All work is presented for the homogeneous continuous time Markov chain model with arbitrary rate matrices.