eCite Digital Repository

Lie Markov models


Sumner, JG and Fernandez-Sanchez, J and Jarvis, PD, Lie Markov models, Journal of Theoretical Biology, 298 pp. 16-31. ISSN 1095-8541 (2012) [Refereed Article]

Restricted - Request a copy

Copyright Statement

Copyright 2011 Crown Copyright

DOI: doi:10.1016/j.jtbi.2011.12.017


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.

Item Details

Item Type:Refereed Article
Keywords:Phylogenetics; Lie algebras; Representation theory; Symmetry; Markov chains
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 mathematical sciences
UTAS Author:Sumner, JG (Associate Professor Jeremy Sumner)
UTAS Author:Jarvis, PD (Dr Peter Jarvis)
ID Code:76242
Year Published:2012 (online first 2011)
Funding Support:Australian Research Council (DP0877447)
Web of Science® Times Cited:40
Deposited By:Mathematics and Physics
Deposited On:2012-03-02
Last Modified:2015-06-23

Repository Staff Only: item control page