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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Authors: Fernández Sánchez, Jesús|||0000-0002-7062-8042, Sumner, Jeremy, Jarvis, Peter, Woodhams, Michael D.
Format: article
Publication Date:2015
Country:España
Institution:Universitat Politècnica de Catalunya (UPC)
Repository:UPCommons. Portal del coneixement obert de la UPC
Language:English
OAI Identifier:oai:upcommons.upc.edu:2117/76817
Online Access:https://hdl.handle.net/2117/76817
https://dx.doi.org/10.1007/s00285-014-0773-z
Access Level:Open access
Keyword:Lie algebras
Phylogenetics
Markov model
Representation theory
Permutation groups
sequences
Lie, Àlgebres de
Classificació AMS::15 Linear and multilinear algebra
matrix theory
Classificació AMS::22 Topological groups, lie groups::22E Lie groups
Classificació AMS::52 Convex and discrete geometry::52B Polytopes and polyhedra
Classificació AMS::62 Statistics::62P Applications
Àrees temàtiques de la UPC::Matemàtiques i estadística
id ES_83e9d414ec3b6f16bcb4301c6edf0bd2
oai_identifier_str oai:upcommons.upc.edu:2117/76817
network_acronym_str ES
network_name_str España
repository_id_str
spelling Lie Markov models with purine/pyrimidine symmetryFernández Sánchez, Jesús|||0000-0002-7062-8042Sumner, JeremyJarvis, PeterWoodhams, Michael D.Lie algebrasPhylogeneticsMarkov modelRepresentation theoryPermutation groupsLie algebrassequencesLie, Àlgebres deClassificació AMS::15 Linear and multilinear algebramatrix theoryClassificació AMS::22 Topological groups, lie groups::22E Lie groupsClassificació AMS::52 Convex and discrete geometry::52B Polytopes and polyhedraClassificació AMS::62 Statistics::62P ApplicationsÀrees temàtiques de la UPC::Matemàtiques i estadísticaContinuous-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.20152015-03-0120152015-09-15journal articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/76817https://dx.doi.org/10.1007/s00285-014-0773-zreponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2http://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/768172026-05-27T15:37:01Z
dc.title.none.fl_str_mv Lie Markov models with purine/pyrimidine symmetry
title Lie Markov models with purine/pyrimidine symmetry
spellingShingle Lie Markov models with purine/pyrimidine symmetry
Fernández Sánchez, Jesús|||0000-0002-7062-8042
Lie algebras
Phylogenetics
Markov model
Representation theory
Permutation groups
Lie algebras
sequences
Lie, Àlgebres de
Classificació AMS::15 Linear and multilinear algebra
matrix theory
Classificació AMS::22 Topological groups, lie groups::22E Lie groups
Classificació AMS::52 Convex and discrete geometry::52B Polytopes and polyhedra
Classificació AMS::62 Statistics::62P Applications
Àrees temàtiques de la UPC::Matemàtiques i estadística
title_short Lie Markov models with purine/pyrimidine symmetry
title_full Lie Markov models with purine/pyrimidine symmetry
title_fullStr Lie Markov models with purine/pyrimidine symmetry
title_full_unstemmed Lie Markov models with purine/pyrimidine symmetry
title_sort Lie Markov models with purine/pyrimidine symmetry
dc.creator.none.fl_str_mv Fernández Sánchez, Jesús|||0000-0002-7062-8042
Sumner, Jeremy
Jarvis, Peter
Woodhams, Michael D.
author Fernández Sánchez, Jesús|||0000-0002-7062-8042
author_facet Fernández Sánchez, Jesús|||0000-0002-7062-8042
Sumner, Jeremy
Jarvis, Peter
Woodhams, Michael D.
author_role author
author2 Sumner, Jeremy
Jarvis, Peter
Woodhams, Michael D.
author2_role author
author
author
dc.subject.none.fl_str_mv Lie algebras
Phylogenetics
Markov model
Representation theory
Permutation groups
Lie algebras
sequences
Lie, Àlgebres de
Classificació AMS::15 Linear and multilinear algebra
matrix theory
Classificació AMS::22 Topological groups, lie groups::22E Lie groups
Classificació AMS::52 Convex and discrete geometry::52B Polytopes and polyhedra
Classificació AMS::62 Statistics::62P Applications
Àrees temàtiques de la UPC::Matemàtiques i estadística
topic Lie algebras
Phylogenetics
Markov model
Representation theory
Permutation groups
Lie algebras
sequences
Lie, Àlgebres de
Classificació AMS::15 Linear and multilinear algebra
matrix theory
Classificació AMS::22 Topological groups, lie groups::22E Lie groups
Classificació AMS::52 Convex and discrete geometry::52B Polytopes and polyhedra
Classificació AMS::62 Statistics::62P Applications
Àrees temàtiques de la UPC::Matemàtiques i estadística
description 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.
publishDate 2015
dc.date.none.fl_str_mv 2015
2015-03-01
2015
2015-09-15
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/76817
https://dx.doi.org/10.1007/s00285-014-0773-z
url https://hdl.handle.net/2117/76817
https://dx.doi.org/10.1007/s00285-014-0773-z
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2

http://creativecommons.org/licenses/by-nc-nd/3.0/es/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2

http://creativecommons.org/licenses/by-nc-nd/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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