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

TL;DR

This paper analyzes the algebraic structure of the strand symmetric phylogenetic substitution model, classifies Markov invariants, and provides explicit polynomial forms, with practical implications for estimating phylogenetic distances.

Contribution

It introduces a representation theoretic approach to classify and enumerate Markov invariants in the model, providing explicit forms and counting formulas for quadratic and cubic cases.

Findings

Precisely 1/3(3^L+(-1)^L) quadratic invariants

Exactly 6^{L-1} cubic invariants

Quadratic invariants estimate phylogenetic distances

Abstract

We consider the continuous-time presentation of the strand symmetric phylogenetic substitution model (in which rate parameters are unchanged under nucleotide permutations given by Watson-Crick base conjugation). Algebraic analysis of the model's underlying structure as a matrix group leads to a change of basis where the rate generator matrix is given by a two-part block decomposition. We apply representation theoretic techniques and, for any (fixed) number of phylogenetic taxa $L$ and polynomial degree $D$ of interest, provide the means to classify and enumerate the associated Markov invariants. In particular, in the quadratic and cubic cases we prove there are precisely 1/3$(3^L+(-1)^L)$ and $6^{L-1}$ linearly independent Markov invariants, respectively. Additionally, we give the explicit polynomial forms of the Markov invariants for (i) the quadratic case with any number of taxa $L$, and (ii) the cubic case in the special case of a three-taxa phylogenetic tree. We close by showing our results are of practical interest since the quadratic Markov invariants provide independent estimates of phylogenetic distances based on (i) substitution rates within Watson-Crick conjugate pairs, and (ii) substitution rates across conjugate base pairs.