Lie Markov models with purine/pyrimidine symmetry

Continuous-time Markov chains are a standard tool in phylogenetic inference. If homogeneity is assumed, the chain is formulated by specifying time-independent rates of substitutions between states in the chain. In applications, there are usually extra constraints on the rates, depending on the situa...

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Vydáno v:	Journal of mathematical biology Ročník 70; číslo 4; s. 855 - 891
Hlavní autoři:	Fernández-Sánchez, Jesús, Sumner, Jeremy G., Jarvis, Peter D., Woodhams, Michael D.
Médium:	Journal Article Publikace
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2015 Springer Nature B.V
Témata:	15 Linear and multilinear algebra; matrix theory 22 Topological groups, lie groups 22E Lie groups 52 Convex and discrete geometry 52B Polytopes and polyhedra 62 Statistics 62P Applications Animals Applications of Mathematics Classificació AMS DNA - genetics Evolution, Molecular Humans Lie algebras Lie, Àlgebres de Markov Chains Markov model Matemàtiques i estadística Mathematical and Computational Biology Mathematical Concepts Mathematics Mathematics and Statistics Models, Genetic Permutation groups Phylogenetics Phylogeny Purine Nucleotides - genetics Pyrimidine Nucleotides - genetics Representation theory sequences Stochastic Processes Àrees temàtiques de la UPC 15A30 62P10 Permutation groups Lie algebras 22E60 52B99 Phylogenetics Representation theory Markov model
ISSN:	0303-6812, 1432-1416, 1432-1416
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Popis
Shrnutí:	Continuous-time Markov chains are a standard tool in phylogenetic inference. If homogeneity is assumed, the chain is formulated by specifying time-independent rates of substitutions between states in the chain. In applications, there are usually extra constraints on the rates, depending on the situation. If a model is formulated in this way, it is possible to generalise it and allow for an inhomogeneous process, with time-dependent rates satisfying the same constraints. It is then useful to require that, under some time restrictions, there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the definition of “Lie Markov models” which, as we will show, are precisely the class of models where such an average exists. These models form Lie algebras and hence concepts from Lie group theory are central to their derivation. In this paper, we concentrate on applications to phylogenetics and nucleotide evolution, and derive the complete hierarchy of Lie Markov models that respect the grouping of nucleotides into purines and pyrimidines—that is, models with purine/pyrimidine symmetry. We also discuss how to handle the subtleties of applying Lie group methods, most naturally defined over the complex field, to the stochastic case of a Markov process, where parameter values are restricted to be real and positive. In particular, we explore the geometric embedding of the cone of stochastic rate matrices within the ambient space of the associated complex Lie algebra.
Bibliografie:	SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-1 ObjectType-Feature-2 content type line 23
ISSN:	0303-6812 1432-1416 1432-1416
DOI:	10.1007/s00285-014-0773-z