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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

     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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    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].

  • 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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    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.

  • 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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  • Lie Markov models

     Fernández Sánchez, Jesús
    Algebraic Statistics
    Presentation's date: 2012-06-12
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  • Phylogenetic mixtures for equivariant 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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  • GEOMETRÍA DE VARIEDADES ALGEBRAICAS Y APLICACIONES

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

  • GEOMETRIA DE VARIETATS I APLICACIONS

     Gonzalez Alonso, Victor; Alvarez Montaner, Josep; Plans Berenguer, Bernat; Barja Yañez, Miguel Angel; Berna Sepulveda, Isabel Silvana; Abío Roig, Ignasi; Elgueta Montó, Josep; Casanellas Rius, Marta; Fernández Sánchez, Jesús; Amoros Torrent, Jaume; Alberich Carramiñana, Maria
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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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  • 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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  • 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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  • Clases de equisingularidad de proyecciones birracionales de singularidades normales de superficie

     Fernández Sánchez, Jesús
    Approaching Mathematics through Algebra
    p. 145-159
    Presentation's date: 2007-06-25
    Presentation of work at congresses

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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
    Approaching Mathematics through Algebra
    p. 1
    Presentation of work at congresses

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  • Performance of a new invariants method for phylogenetic reconstruction

     Fernández Sánchez, Jesús
    Future Directions in Phylogenetic Methods and Models
    Presentation of work at congresses

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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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  • Geometría y Topología de Variedades Algebraicas y Simplécticas

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

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

     Alberich Carramiñana, Maria; Zuazua ., Enrique; Fernández Sánchez, Jesús
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  • RTACA - Red Temática de Álgebra Conmutativa y Aplicaciones

     Casanellas Rius, Marta; Elias ., Joan; 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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  • 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

     Alberich Carramiñana, Maria; Casas ., E.; Fernández Sánchez, Jesús
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  • Geometria Algebraica.

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

     Conte ., Alberto; Amoros Torrent, Jaume; Casanellas Rius, Marta; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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  • On polar germs of planet curves and related in variants

     Fernández Sánchez, Jesús
    Géométrie algébrique en Liberté VIII
    Presentation's date: 2000-03-28
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  • Geometria Algebraica.

     Casanellas Rius, Marta; Casas ., E.; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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  • Geometría local de sistemas lineales

     Casas ., E.; 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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  • AGE-Algebraic Geometry in Europe

     Casanellas Rius, Marta; Fernández Sánchez, Jesús
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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
    Vol. 24, num. 1, p. 288-293
    Date of publication: 2007-01
    Journal article

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  • Geometry of the Kimura 3-parameter model

     Casanellas Rius, Marta; Fernández Sánchez, Jesús
    Advances in applied mathematics
    Vol. 41, num. 3, p. 265-292
    Date of publication: 2008-03
    Journal article

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  • Equivalence of the Nash conjecture for primitive and sandwiched singularities

     Fernández Sánchez, Jesús
    Proceedings of the American Mathematical Society
    Vol. 133, num. 3, p. 677-679
    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
    Vol. 138, num. 1, p. 117-128
    Date of publication: 2005-01
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  • Factorization of complete ideals in normal birational extensions in dimension two

     Fernández Sánchez, Jesús
    Journal of algebra
    Vol. 314, num. 1, p. 344-361
    Date of publication: 2007-03
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  • Some results relating to adjacent ideals in dimension two

     Fernández Sánchez, Jesús
    Journal of pure and applied algebra
    Vol. 207, num. 2, p. 387-395
    Date of publication: 2006-08
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  • On sandwiched singularities and complete ideals

     Fernández Sánchez, Jesús
    Journal of pure and applied algebra
    Vol. 185, num. 1-3, p. 165-175
    Date of publication: 2003-12
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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
    Vol. 137A, num. 5, p. 913-921
    Date of publication: 2007-10
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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
    Vol. 312, p. 2461-2473
    Date of publication: 2009-01
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  • Equisingularity classes of birational projections of normal singularities to a plane

     Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
    Advances in mathematics
    Vol. 216, num. 2, p. 753-770
    Date of publication: 2007-12
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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
    Vol. 7, num. 1, p. 33-
    DOI: 10.1186/1748-7188-7-33
    Date of publication: 2012-11-28
    Journal article

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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.

  • Relevant phylogenetic invariants of evolutionary models

     Casanellas Rius, Marta; Fernández Sánchez, Jesús
    Journal de mathématiques pures et appliquées
    Vol. 96, num. 3, p. 207-229
    DOI: 10.1016/j.matpur.2010.11.002
    Date of publication: 2010-11-19
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  • Lie Markov models

     Sumner, Jeremy; Fernández Sánchez, Jesús; Jarvis, Peter
    Journal of theoretical biology
    Vol. 298, num. 7, p. 16-31
    DOI: 10.1016/j.jtbi.2011.12.017
    Date of publication: 2012-04-07
    Journal article

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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.

  • 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
    Vol. 61, num. 6, p. 1069-1074
    DOI: 10.1093/sysbio/sys042
    Date of publication: 2012-03-22
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