Phylogenetic Invariants: From the General Markov to Equivariant Models

In the last decade, some algebraic tools have been successfully applied to phylogenetic reconstruction. These tools are mainly based on the knowledge of equations describing algebraic varieties associated to Markov processes of molecular substitution on phylogenetic trees, the so called phylogenetic...

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Detalles Bibliográficos
Autores: Casanellas, Marta, Fernández-Sánchez, Jesús
Tipo de recurso: artículo
Fecha de publicación:2025
País:España
Institución:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:2072/489271
Acceso en línea:https://hdl.handle.net/2072/489271
Access Level:acceso abierto
Palabra clave:phylogenetic invariant
phylogenetic variety
evolutionary Markov process
models of nucleotide substitution
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spelling Phylogenetic Invariants: From the General Markov to Equivariant ModelsCasanellas, MartaFernández-Sánchez, Jesúsphylogenetic invariantphylogenetic varietyevolutionary Markov processmodels of nucleotide substitutionIn the last decade, some algebraic tools have been successfully applied to phylogenetic reconstruction. These tools are mainly based on the knowledge of equations describing algebraic varieties associated to Markov processes of molecular substitution on phylogenetic trees, the so called phylogenetic invariants. Although the theory involved allows for the explicit determination of these equations for all equivariant models (which include some of the most popular nucleotide substitution models), practical uses of these algebraic tools have been restricted to the case of the general Markov model. Arguably, one of the reasons for this restriction is that knowledge of linear representation theory is required before making these equations explicit. With the aim of enlarging the practical uses of algebraic phylogenetics, in this paper we prove that phylogenetic invariants for equivariant models can be derived from phylogenetic invariants for the general Markov model without the need of representation theory. Our main result states that the algebraic variety corresponding to an equivariant model on a phylogenetic tree T is an irreducible component of the variety corresponding to the general Markov model on T intersected with the linear space defined by the model. We also prove that for any equivariant model, those phylogenetic invariants that are relevant for practical uses (e.g., tree reconstruction) can be simply deduced from a single rank constraint on the matrices obtained by flattening the joint distribution at the leaves of the tree. This condition can be easily tested from singular values of the matrices, and it allows us to extend our results from trees to phylogenetic networks.Both authors were partially supported by the project reference PID2023-146936NB-I00 financed by the Spanish State Agency MCIN/AEI/10.13039/501100011033/FEDER, UE, by the Severo Ochoa and Maria de Maeztu Program for Centers and Units of Excellence in R&D (project CEX2020-001084-M) , and by the AGAUR project 2021 SGR 00603 Geometry of Manifolds and Applications, GEOMVAP.info:eu-repo/semantics/acceptedVersionSociety for Industrial and Applied Mathematics2025info:eu-repo/semantics/article34 p.application/pdfhttps://hdl.handle.net/2072/489271RECERCAT (Dipòsit de la Recerca de Catalunya)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésSIAM Journal on Applied Algebra and GeometryAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:2072/4892712026-05-29T05:05:01Z
dc.title.none.fl_str_mv Phylogenetic Invariants: From the General Markov to Equivariant Models
title Phylogenetic Invariants: From the General Markov to Equivariant Models
spellingShingle Phylogenetic Invariants: From the General Markov to Equivariant Models
Casanellas, Marta
phylogenetic invariant
phylogenetic variety
evolutionary Markov process
models of nucleotide substitution
title_short Phylogenetic Invariants: From the General Markov to Equivariant Models
title_full Phylogenetic Invariants: From the General Markov to Equivariant Models
title_fullStr Phylogenetic Invariants: From the General Markov to Equivariant Models
title_full_unstemmed Phylogenetic Invariants: From the General Markov to Equivariant Models
title_sort Phylogenetic Invariants: From the General Markov to Equivariant Models
dc.creator.none.fl_str_mv Casanellas, Marta
Fernández-Sánchez, Jesús
author Casanellas, Marta
author_facet Casanellas, Marta
Fernández-Sánchez, Jesús
author_role author
author2 Fernández-Sánchez, Jesús
author2_role author
dc.subject.none.fl_str_mv phylogenetic invariant
phylogenetic variety
evolutionary Markov process
models of nucleotide substitution
topic phylogenetic invariant
phylogenetic variety
evolutionary Markov process
models of nucleotide substitution
description In the last decade, some algebraic tools have been successfully applied to phylogenetic reconstruction. These tools are mainly based on the knowledge of equations describing algebraic varieties associated to Markov processes of molecular substitution on phylogenetic trees, the so called phylogenetic invariants. Although the theory involved allows for the explicit determination of these equations for all equivariant models (which include some of the most popular nucleotide substitution models), practical uses of these algebraic tools have been restricted to the case of the general Markov model. Arguably, one of the reasons for this restriction is that knowledge of linear representation theory is required before making these equations explicit. With the aim of enlarging the practical uses of algebraic phylogenetics, in this paper we prove that phylogenetic invariants for equivariant models can be derived from phylogenetic invariants for the general Markov model without the need of representation theory. Our main result states that the algebraic variety corresponding to an equivariant model on a phylogenetic tree T is an irreducible component of the variety corresponding to the general Markov model on T intersected with the linear space defined by the model. We also prove that for any equivariant model, those phylogenetic invariants that are relevant for practical uses (e.g., tree reconstruction) can be simply deduced from a single rank constraint on the matrices obtained by flattening the joint distribution at the leaves of the tree. This condition can be easily tested from singular values of the matrices, and it allows us to extend our results from trees to phylogenetic networks.
publishDate 2025
dc.date.none.fl_str_mv 2025
dc.type.none.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2072/489271
url https://hdl.handle.net/2072/489271
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv SIAM Journal on Applied Algebra and Geometry
dc.rights.none.fl_str_mv Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 34 p.
application/pdf
dc.publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
dc.source.none.fl_str_mv RECERCAT (Dipòsit de la Recerca de Catalunya)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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