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

Citation

Sumner, JG and Holland, BR and Jarvis, PD, The Algebra of the General Markov Model on Phylogenetic Trees and Networks, Bulletin of Mathematical Biology, 74, (4) pp. 858-880. ISSN 1522-9602 (2012) [Refereed Article]


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Copyright Statement

Copyright 2011 Society for Mathematical Biology

DOI: doi:10.1007/s11538-011-9691-z

Abstract

It is known that the Kimura 3ST model of sequence evolution on phylogenetic trees can be extended quite naturally to arbitrary split systems. However, this extension relies heavily on mathematical peculiarities of the associated Hadamard transformation, and providing an analogous augmentation of the general Markov model has thus far been elusive. In this paper, we rectify this shortcoming by showing how to extend the general Markov model on trees to include incompatible edges; and even further to more general network models. This is achieved by exploring the algebra of the generators of the continuous-time Markov chain together with the "splitting" operator that generates the branching process on phylogenetic trees. For simplicity, we proceed by discussing the two state case and then show that our results are easily extended to more states with little complication. Intriguingly, upon restriction of the two state general Markov model to the parameter space of the binary symmetric model, our extension is indistinguishable from the Hadamard approach only on trees; as soon as any incompatible splits are introduced the two approaches give rise to differing probability distributions with disparate structure. Through exploration of a simple example, we give an argument that our extension to more general networks has desirable properties that the previous approaches do not share. In particular, our construction allows for convergent evolution of previously divergent lineages; a property that is of significant interest for biological applications.

Item Details

Item Type:Refereed Article
Keywords:Split system Cluster system Markov process Maximum likelihood
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
Author:Sumner, JG (Dr Jeremy Sumner)
Author:Holland, BR (Associate Professor Barbara Holland)
Author:Jarvis, PD (Dr Peter Jarvis)
ID Code:75153
Year Published:2012 (online first 2011)
Funding Support:Australian Research Council (FT100100031)
Web of Science® Times Cited:8
Deposited By:Mathematics and Physics
Deposited On:2012-01-13
Last Modified:2014-10-14
Downloads:0

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