Jeremy G. Sumner

Is the General Time-Reversible Model Bad for Molecular Phylogenetics?

The general time-reversible (GTR) model (Tavare 1986) has been the workhorse of molecular phylogenetics for the last decade. GTR sits at the top of the ModelTest hierarchy of models (Posada and Crandall 1998) and, usually with the addition of invariant sites and a gamma distribution of rates across sites, is currently by far the most commonly selected model for phylogenetic inference (see Table 1). However, a recent publication (Sumner et al. 2012) shows that GTR, along with several other commonly used models, has an undesirable mathematical property that may be a cause of concern for the thoughtful phylogeneticist. In mathematical terms, the problem is simple: matrix multiplication of two GTR substitution matrices does not return another GTR matrix. It is the purpose of this article to give examples that demonstrate why this lack of closure may pose a problem for phylogenetic analysis and thus add GTR to the growing list of factors that are known to cause model misspecification in phylogenetics.

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.

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.

Tensor Rank, Invariants, Inequalities, and Applications

Though algebraic geometry over

The Algebra of the General Markov Model on Phylogenetic Trees and Networks

\mathbb C

Path integral formulation and Feynman rules for phylogenetic branching models

is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the

Matrix group structure and Markov invariants in the strand symmetric phylogenetic substitution model

n\times n\times n

A New Hierarchy of Phylogenetic Models Consistent with Heterogeneous Substitution Rates

tensors of rank

Lie Markov models with purine/pyrimidine symmetry

n

Maximum likelihood estimates of pairwise rearrangement distances

over