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    Testing of the three multiplicatively closed (Lie Markov) model heirarchies which respect purine/pyrimidine, Watson-Crick, and amino/keto nucleotide groupings  Open access

     Woodhams, Michael D.; Fernández Sánchez, Jesús; Sumner, Jeremy
    Date: 2014-03
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    We present three hierarchies of Lie Markov models of DNA sequence evolution. These models are (locally) ¿multiplicatively closed,¿ meaning that the composition of two Markov matrices in the model results, with some (rare) exceptions, in a third Markov matrix that is still in the model. Additionally, the models in each hierarchy respectively distinguish between (i) purines and pyrimadines (RY), (ii) Watson-Crick pairs (WS), and (iii) amino/keto pairs (MK), but otherwise treat the four nucleotides without distinction. The multiplicative closure property allows mathematically consistent modeling of time-inhomogeneous scenarios, unlike models that are not closed, such as the general time-reversible model (GTR) and many of its submodels. We derive the nesting relationships of the three model hierarchies and present software implementing the models. For a diverse range of biological data sets, we perform Bayesian information criterion model comparision analogous to that of the ModelTest framework. We find that our models outperform the GTR model in some (but not all) cases.

    We present three hierarchies of Lie Markov models of DNA sequence evolution. These models are (locally) “multiplicatively closed,” meaning that the composition of two Markov matrices in the model results, with some (rare) exceptions, in a third Markov matrix that is still in the model. Additionally, the models in each hierarchy respectively distinguish between (i) purines and pyrimadines (RY), (ii) Watson-Crick pairs (WS), and (iii) amino/keto pairs (MK), but otherwise treat the four nucleotides without distinction. The multiplicative closure property allows mathematically consistent modeling of time-inhomogeneous scenarios, unlike models that are not closed, such as the general time-reversible model (GTR) and many of its submodels. We derive the nesting relationships of the three model hierarchies and present software implementing the models. For a diverse range of biological data sets, we perform Bayesian information criterion model comparision analogous to that of the ModelTest framework. We find that our models outperform the GTR model in some (but not all) cases.

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    Lie Markov models with purine/pyrimidine symmetry  Open access

     Fernández Sánchez, Jesús; Sumner, Jeremy; Jarvis, Peter; Woodhams, Michael D.
    Date: 2013-06
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    Continuous-time Markov chains are a standard tool in phylogenetic inference. If homo- geneity is assumed, the chain is formulated by specifying time-independent rates of substitu- tions 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 gen- eralise it and allow for an inhomogeneous process, with time-dependent rates satisfying the same constraints. It is then useful to require that there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the denition of \Lie Markov models", which 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 dened over the complex eld, 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.

    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 there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the definition of "Lie Markov models", which 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.

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    Local description of phylogenetic group-based models  Open access

     Casanellas Rius, Marta; Fernández Sánchez, Jesús; Michalek, Mateusz
    Date: 2013-12
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    Motivated by phylogenetics, our aim is to obtain a system of equations that dene a phylogenetic variety on an open set containing the biologically meaningful points. In this paper we consider phylogenetic varieties dened via group-based models. For any nite abelian group G, we provide an explicit construction of codimX phylogenetic invariants (polynomial equations) of degree at most jGj that dene the variety X on a Zariski open set U. The set U contains all biologically meaningful points when G is the group of the Kimura 3-parameter model. In particular, our main result conrms [Mic12, Conjecture 7.9] and, on the set U, Conjectures 29 and 30 of [SS05].

    Motivated by phylogenetics, our aim is to obtain a system of equations that de ne a phylogenetic variety on an open set containing the biologically meaningful points. In this paper we consider phylogenetic varieties de ned via group-based models. For any nite abelian group G, we provide an explicit construction of codimX phylogenetic invariants (polynomial equations) of degree at most jGj that de ne the variety X on a Zariski open set U. The set U contains all biologically meaningful points when G is the group of the Kimura 3-parameter model. In particular, our main result con rms [Mic12, Conjecture 7.9] and, on the set U, Conjectures 29 and 30 of [SS05].

  • GEOMATRIA ALGEBRAICA, SIMPLECTICA, ARITMETICA Y APLICACIONES

     Alberich Carramiñana, Maria; Amoros Torrent, Jaume; Barja Yañez, Miguel Angel; Elgueta Montó, Josep; Fernández Sánchez, Jesús; Ventura González Alonso, Daniel; Barbieri Solha, Romero; Miranda Galceran, Eva; Pascual Gainza, Pedro; Roig Marti, Agustin; Roig Maranges, Abdo; Feliu Trijueque, Elsienda; Gálvez Carrillo, Maria Immaculada; Casanellas Rius, Marta
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  • Modelos de Lie Markov con simetría fijada

     Fernández Sánchez, Jesús
    Congreso de Jóvenes Investigadores Real Sociedad Matemática Española
    Presentation's date: 2013-09-16
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  • Lie Markov models with prescribed symmetry

     Fernández Sánchez, Jesús
    Applied Algebraic Geometry
    Presentation's date: 2013-08-03
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    The space of phylogenetic mixtures for equivariant models  Open access

     Casanellas Rius, Marta; Fernández Sánchez, Jesús; Kedzierska, A.M.
    Algorithms for Molecular Biology
    Date of publication: 2012-11-28
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    The selection of the most suitable evolutionary model to analyze the given molecular data is usually left to biologist's choice. In his famous book, J Felsenstein suggested that certain linear equations satisfied by the expected probabilities of patterns observed at the leaves of a phylogenetic tree could be used for model selection. It remained open the question regarding whether these equations were enough for characterizing the evolutionary model. Here we prove that, for equivariant models of evolution, the space of distributions satisfying these linear equations coincides with the space of distributions arising from mixtures of trees on a set of taxa. In other words, we prove that an alignment is produced from a mixture of phylogenetic trees under an equivariant evolutionary model if and only if its distribution of column patterns satisfies the linear equations mentioned above. Moreover, for each equivariant model and for any number of taxa, we provide a set of linearly independent equations defining this space of phylogenetic mixtures. This is a powerful tool that has already been successfully used in model selection. We also use the results obtained to study identifiability issues for phylogenetic mixtures.

  • Is the general time-reversible model bad for molecular phylogenetics?

     Sumner, Jeremy; Jarvis, Peter; Fernández Sánchez, Jesús; Kaine, Bodie; Woodhams, Michael; Holland, Barbara
    Systematic biology
    Date of publication: 2012-03-22
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  • Lie Markov models

     Sumner, Jeremy; Fernández Sánchez, Jesús; Jarvis, Peter
    Journal of theoretical biology
    Date of publication: 2012-04-07
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    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.

  • Phylogenetic mixtures for equivariant models

     Fernández Sánchez, Jesús
    Algebraic Statistics
    Presentation's date: 2012-06-12
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  • Lie Markov models

     Fernández Sánchez, Jesús
    Algebraic Statistics
    Presentation's date: 2012-06-12
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  • An approach to phylogenetic inference by means of group representation theory

     Fernández Sánchez, Jesús
    Congreso de Jóvenes Investigadores Real Sociedad Matemática Española
    Presentation's date: 2011-09-06
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  • On adjacent complete ideals above a given complete ideal

     Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
    Proceedings of the Royal Society of Edinburgh. Section A, mathematics
    Date of publication: 2010
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  • Phylogenetic reconstruction using algebraic geometry

     Casanellas Rius, Marta; Fernández Sánchez, Jesús
    Arbor (Madrid)
    Date of publication: 2010-12-31
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    A new approach to phylogenetic reconstruction has been emerging in the last years. Given an evolutionary model, the joint probability distribution of the nucleotides for these species satisfy some algebraic constraints called invariants. These invariants have theoretical and practical interest, since they can be used to infer phylogenies. In this paper, we explain how to use these invariants to design algorithms for phylogenetic reconstruction and we show how the application of tools and theoretical results coming from commutative algebra and algebraic geometry can improve the performance and the efficiency of these algorithms.

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    Invariants, una aproximació a la filogenètica des de l'àlgebra  Open access

     Fernández Sánchez, Jesús
    Butlletí de la Societat Catalana de Matemàtiques
    Date of publication: 2010
    Journal article

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    En els darrers anys, una nova aproximació a la reconstrucció filogenètica basada en tècniques provinents de l’àlgebra s’ha estat consolidant. Fixat un model evolutiu per a un conjunt donat d’espècies, els invariants són relacions algebraiques satisfetes per les distribucions teòriques de nucleòtids d’aquestes espècies. En aquest article s’exposa com es poden fer servir els invariants per implementar algoritmes de reconstrucció filogenètica i s’explica com l’eficiència d’aquests algoritmes es veu beneficiada per resultats teòrics provinents de la geometria algebraica i la representació de grups.

  • Relevant phylogenetic invariants of evolutionary models

     Casanellas Rius, Marta; Fernández Sánchez, Jesús
    Journal de mathématiques pures et appliquées
    Date of publication: 2010-11-19
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  • Principality of curves on sandwiched singularities

     Fernández Sánchez, Jesús
    Mathematical research letters
    Date of publication: 2010
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    Relevant phylogenetic invariants of evolutionary models  Open access

     Casanellas Rius, Marta; Fernández Sánchez, Jesús
    Date: 2010-01
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    Recently there have been several attempts to provide a whole set of generators of the ideal of the algebraic variety associated to a phylogenetic tree evolving under an algebraic model. These algebraic varieties have been proven to be useful in phylogenetics. In this paper we prove that, for phylogenetic reconstruction purposes, it is enough to consider generators coming from the edges of the tree, the so-called edge invariants. This is the algebraic analogous to Buneman's Splits Equivalence Theorem. The interest of this result relies on its potential applications in phylogenetics for the widely used evolutionary models such as Jukes-Cantor, Kimura 2 and 3 parameters, and General Markov models.

  • GEOMETRÍA DE VARIEDADES ALGEBRAICAS Y APLICACIONES

     Amoros Torrent, Jaume; Barja Yañez, Miguel Angel; Alberich Carramiñana, Maria; Elgueta Montó, Josep; Fernández Sánchez, Jesús; Gonzalez Alonso, Victor; Berna Sepulveda, Isabel Silvana; Ferragut Amengual, Antoni Manel; Casanellas Rius, Marta
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  • GEOMETRIA DE VARIETATS I APLICACIONS

     Casanellas Rius, Marta; Berna Sepulveda, Isabel Silvana; Barja Yañez, Miguel Angel; Gonzalez Alonso, Victor; Abío Roig, Ignasi; Plans Berenguer, Bernat; Alvarez Montaner, Josep; Amoros Torrent, Jaume; Elgueta Montó, Josep; Fernández Sánchez, Jesús; Alberich Carramiñana, Maria
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  • On the exceptional locus of the birational projections of a normal surface singularity into a plane

     Fernández Sánchez, Jesús
    Journal of algebra
    Date of publication: 2009-01
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  • Classification of the birational projections of a sandwiched surface singularity to a plane

     Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
    Seminar on Singularities: algebraic methods
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  • Performance of a new invariants method for phylogenetic reconstruction

     Fernández Sánchez, Jesús
    Society for Molecular Biology and Evolution. Meeting
    Presentation's date: 2008-01-01
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  • Classication of the birational projections of a sandwiched surface singularity to a plane

     Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
    Date: 2008-09
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  • Geometry of the Kimura 3-parameter model

     Casanellas Rius, Marta; Fernández Sánchez, Jesús
    Advances in applied mathematics
    Date of publication: 2008-03
    Journal article

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  • Geometry of the Kimura 3-parameter model

     Casanellas Rius, Marta; Fernández Sánchez, Jesús
    Future Directions in Phylogenetic Methods and Models
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  • Equisingularity classes of birational projections of normal singularities to a plane

     Fernández Sánchez, Jesús; Alberich Carramiñana, Maria
    I Congreso Hispano-Francés de Matemàticas
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  • Equisingularity classes of birational projections of normal singularities to a plane

     Fernández Sánchez, Jesús
    I Congreso Hispano-Francés de Matemàticas
    Presentation's date: 2007-07-09
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  • Equisingularity classes of birational projections of normal singularities to a plane

     Fernández Sánchez, Jesús
    Approaching Mathematics through Algebra
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  • Performance of a new invariants method for phylogenetic reconstruction

     Fernández Sánchez, Jesús
    Future Directions in Phylogenetic Methods and Models
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  • Performance of a new invariants method on homogeneous and nonhomogeneous quartet trees

     Casanellas Rius, Marta; Fernández Sánchez, Jesús
    Molecular biology and evolution
    Date of publication: 2007-01
    Journal article

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  • Equisingularity classes of birational projections of normal singularities to a plane

     Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
    Advances in mathematics
    Date of publication: 2007-12
    Journal article

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  • Factorization of adjacent complete ideals in dimension 2

     Fernández Sánchez, Jesús
    Proceedings of the Royal Society of Edinburgh. Section A, mathematics
    Date of publication: 2007-10
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  • Factorization of complete ideals in normal birational extensions in dimension two

     Fernández Sánchez, Jesús
    Journal of algebra
    Date of publication: 2007-03
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  • Clases de equisingularidad de proyecciones birracionales de singularidades normales de superficie

     Fernández Sánchez, Jesús
    Approaching Mathematics through Algebra
    Presentation's date: 2007-06-25
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  • Geometría y Topología de Variedades Algebraicas y Simplécticas

     Casanellas Rius, Marta; Barja Yañez, Miguel Angel; Fernández Sánchez, Jesús
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  • Singularidades de morfismos y variedades II

     Alberich Carramiñana, Maria; Casas ., E.; Fernández Sánchez, Jesús
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  • Ingenio Mathematica (i-MATH)

     Zuazua ., Enrique; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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  • Some results relating to adjacent ideals in dimension two

     Fernández Sánchez, Jesús
    Journal of pure and applied algebra
    Date of publication: 2006-08
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  • RTACA - Red Temática de Álgebra Conmutativa y Aplicaciones

     Elias ., Joan; Casanellas Rius, Marta; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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  • Factoritación de ideales completos y singularidades sandwiched

     Fernández Sánchez, Jesús
    Congreso Conjunto de Matemáticas RSME-SCM-SEIO-SEMA
    Presentation's date: 2005-02-02
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  • Equivalence of the Nash conjecture for primitive and sandwiched singularities

     Fernández Sánchez, Jesús
    Proceedings of the American Mathematical Society
    Date of publication: 2005-03
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  • Nash families of smooth arcs on a sandwiched singularity

     Fernández Sánchez, Jesús
    Mathematical proceedings of the Cambridge Philosophical Society
    Date of publication: 2005-01
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  • Geometría y topología de las variedades algebraicas

     Roig Marti, Agustin; Barja Yañez, Miguel Angel; Pascual Gainza, Pedro; Elgueta Montó, Josep; Abío Roig, Ignasi; Amoros Torrent, Jaume; Casanellas Rius, Marta; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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  • Singularidades de morfismos y variedades

     Casas ., E.; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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  • On sandwiched singularities and complete ideals

     Fernández Sánchez, Jesús
    Journal of pure and applied algebra
    Date of publication: 2003-12
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  • Geometria Algebraica.

     Casanellas Rius, Marta; Casas ., E.; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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  • EAGER-European Algebraic Geometry Research training network

     Amoros Torrent, Jaume; Conte ., Alberto; Casanellas Rius, Marta; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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  • Geometria Algebraica.

     Casas ., E.; Casanellas Rius, Marta; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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