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

Citation

Jarvis, PD and Sumner, JG, Matrix group structure and Markov invariants in the strand symmetric phylogenetic substitution model, Journal of Mathematical Biology, 73 pp. 259-282. ISSN 0303-6812 (2016) [Refereed Article]

Copyright Statement

Copyright 2015 Springer-Verlag Berlin Heidelberg

DOI: doi:10.1007/s00285-015-0951-7

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 13(3L + (−1)L) and 6L−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-taxon 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.

Item Details

Item Type:Refereed Article
Keywords:phylogenetics, group theory, representation theory, Lie algebras, Markov invariants
Research Division:Mathematical Sciences
Research Group:Applied Mathematics
Research Field:Biological Mathematics
Objective Division:Expanding Knowledge
Objective Group:Expanding Knowledge
Objective Field:Expanding Knowledge in the Physical Sciences
Author:Jarvis, PD (Dr Peter Jarvis)
Author:Sumner, JG (Dr Jeremy Sumner)
ID Code:107772
Year Published:2016
Funding Support:Australian Research Council (DE130100423)
Web of Science® Times Cited:2
Deposited By:Mathematics and Physics
Deposited On:2016-03-23
Last Modified:2017-10-30
Downloads:0

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