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

Abstract

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 WatsonCrick 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.

Systematics and symmetry in molecular phylogenetic modelling: perspectives from physics

“…Consequently, the all-ones left null eigenvector (responsible for probability conservation of the Markov transition matrix) is also a right eigenvector (so that they are doubly stochastic); moreover, these models have the uniform distribution as stationary state-that is, with each character having probability 1 4 . A commonly used model which allows for data with non-uniform base frequencies, in the most parsimonious parametrization, is the Felsenstein (1981) model 28 [25],…”

“…As can be seen from figure 5 however, there is a large hierarchy of Lie-Markov models with this S 2 S 2 pairing symmetry-in fact, except for degenerate cases, there is a threefold multiplicity of such pairing groups-one for purine-pyrimidine R = {A, G}, Y = {C, T}, one for weak-strong W = {A, T}, S = {C, G}, and one for amino-keto M = {A, C}, K = {T, G} pairings (see [20]). In the so-called strand symmetric model [27,28] the symmetry is with respect to the strong-weak pairings (CG), (AT), and (CG)(AT), and only these permutations are used in restoring the action of arbitrary permutations of the rate matrix 32 .…”

Adventures in Invariant Theory

“…There are many more gems to be examined amongst Markov invariants for different models and subgroups [16,17], with potential practical and theoretical interest. As one instance of as yet unexplored terrain for K = 3, we have evidence [28,29] at degree eight for stochastic tangle (stangle) invariants with mixed weight, since it turns out that g (8) (51 3 ),(2 4 ),(2 4 ) = 1 (≡ g (8) (2 4 ),(51 3 ),(2 4 ) ≡ g (8) (2 4 ),(2 4 ),(51 3 ) ).…”

Full reconstruction of non-stationary strand-symmetric models on rooted phylogenies

“…Strand-symmetric substitution models have also been approached from a theoretical perspective (Casanellas and Sullivant 2005;Jarvis and Sumner 2013) in the context of reversible models. Jarvis and Sumner (2013) make the observation that strand-symmetric models enjoy the property of closure; that a model which is strandsymmetric on two adjacent phylogenetic branches is strand-symmetric across the two branches as well.…”