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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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  • Modelos de Lie Markov con simetría fijada

     Fernández Sánchez, Jesús
    Congreso de Jóvenes Investigadores Real Sociedad Matemática Española
    Presentation's date: 2013-09-16
    Presentation of work at congresses

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  • 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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  • 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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  • 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
    Journal article

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  • Lie Markov models

     Sumner, Jeremy; Fernández Sánchez, Jesús; Jarvis, Peter
    Journal of theoretical biology
    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.

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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
    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.

  • Algebraic Tools in Phylogenomics.  Open access

     Kedzierska, Anna Magdalena
    Defense's date: 2012-03-16
    Department of Applied Mathematics I, Universitat Politècnica de Catalunya
    Theses

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    En aquesta tesi interdisciplinar desenvolupem eines algebraiques per a problemes en filogenètica i genòmica. Per estudiar l'evolució molecular de les espècies sovint s'usen models evolutius estocàstics. L'evolució es representa en un arbre (anomenat filogenètic) on les espècies actuals corresponen a fulles de l'arbre i els nodes interiors corresponen a ancestres comuns a elles. La longitud d'una branca de l'arbre representa la quantitat de mutacions que han ocorregut entre les dues espècies adjacents a la branca. Llavors l'evolució de seqüències d'ADN en aquestes espècies es modelitza amb un procés Markov ocult al llarg de l'arbre. Si el procés de Markov se suposa a temps continu, normalment s'assumeix que també és homogeni i, en tal cas, els paràmetres del model són les entrades d'una raó de mutació instantània i les longituds de les branques. Si el procés de Markov és a temps discret, llavors els paràmetres del model són les probabilitats condicionades de substitució de nucleòtids al llarg de l'arbre i no hi ha cap hipòtesi d'homogeneïtat. Aquests últims són els tipus de models que considerem en aquesta tesi i són, per tant, més generals que els de temps continu. Des d'aquesta perspectiva s'estudien els problemes més bàsics de la filogenètica: donat un conjunt de seqüències d'ADN, com decidim quin és el model evolutiu més adequat? com inferim de forma eficient els paràmetres del model? I fins i tot, tal i com també hem provat en aquesta tesi, és possible que les espècies no hagin evolucionat seguint un sol arbre sinó una mescla d'arbres i llavors cal abordar aquestes preguntes en aquest cas més general. Per a models evolutius a temps continu i homogenis, s'ha proposat solucions diverses a aquestes preguntes al llarg de les últimes dècades. En aquesta tesi resolem aquests dos problemes per a models evolutius a temps discret usant tècniques algebraiques provinents d'àlgebra lineal, teoria de grups, geometria algebraica i estadística algebraica. A més a més, la nostra solució per al primer problema és vàlida també per a mescles filogenètiques. Hem fet tests dels mètodes proposats en aquesta tesi sobre dades simulades i dades reals del projectes ENCODE (Encyclopedia Of DNA Elements). Per tal de provar els nostres mètodes hem donat algoritmes per a generar seqüències evolucionant sota un model a temps discret amb un nombre esperat de mutacions prefixat. I així mateix, hem demostrat que aquests algorismes generen totes les seqüències possibles (per la majoria de models). Els tests sobre dades simulades mostren que els mètodes proposats són molt acurats i els resultats sobre dades reals permeten corroborar hipòtesis prèviament formulades. Tots els mètodes proposats en aquesta tesi han estat implementats per a un nombre arbitrari d'espècies i estan disponibles públicament.

    In this thesis we develop interdisciplinary algebraic tools for genomic and phylogenetic problems. To study the molecular evolution of species one often uses stochastic evolutionary models. The evolution is represented in a tree (called phylogenetic tree) whose leaves represent current species and whose internal nodes correspond to their common ancestors. The length of a branch of the tree represents the number of mutations that have occurred between the two species adjacent to the branch. Then ,the evolution of DNA sequences in these species is modeled with a hidden Markov process along the tree. If the Markov process is assumed to be continuous in time, it is usually assumed homogeneous as well and, if so, the model parameters are the instantaneous rate of mutation and the lengths of the branches. If the Markov process is discrete in time, then the model parameters are the conditional probabilities of nucleotide substitution along the tree and there is no assumption of homogeneity. The latter are the types of models we consider in this thesis and are therefore more general than the homogeneous continuous ones. From this perspective we study the basic problems of phylogenetics: Given a set of DNA sequences, what is the evolutionary model that best fits the data? how can we efficiently infer the model parameters? Also, as we also checked in this thesis, it is possible that species have not evolved along a single tree but a mixture of trees so that we need to address these questions in this more general case. For continuous-time, homogeneous, evolutionary models, several solutions to these questions have been proposed during the last decades. In this thesis we solve these two problems for discrete-time evolutionary models, using algebraic techniques from linear algebra, group theory, algebraic geometry and algebraic statistics. In addition, our solution to the first problem is also valid for phylogenetic mixtures. We have made tests of the methods proposed in this thesis on simulated and real data from ENCODE Project (Encyclopedia Of DNA Elements). To test our methods, we also provide algorithms to generate sequences evolving under discrete-time models with a given expected number of mutations. Even more, we have proved that these algorithms generate all possible sequences (for most models). Tests on simulated data show that the methods are very accurate and our results on real data confirm hypotheses previously formulated. All the methods in this thesis have been implemented for an arbitrary number of species and are publicly available.

  • 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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  • Phylogenetic reconstruction using algebraic geometry

     Casanellas Rius, Marta; Fernández Sánchez, Jesús
    Arbor (Madrid)
    Date of publication: 2010-12-31
    Journal article

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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
    Journal article

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  • GEOMETRÍA DE VARIEDADES ALGEBRAICAS Y APLICACIONES

     Alberich Carramiñana, Maria; Barja Yañez, Miguel Angel; Elgueta Montó, Josep; Amoros Torrent, Jaume; Fernández Sánchez, Jesús; 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
    Report

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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.

  • Principality of curves on sandwiched singularities

     Fernández Sánchez, Jesús
    Mathematical research letters
    Date of publication: 2010
    Journal article

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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
    Journal article

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  • GEOMETRIA DE VARIETATS I APLICACIONS

     Berna Sepulveda, Isabel Silvana; Barja Yañez, Miguel Angel; Casanellas Rius, Marta; Plans Berenguer, Bernat; Alvarez Montaner, Josep; Gonzalez Alonso, Victor; Elgueta Montó, Josep; Abío Roig, Ignasi; Amoros Torrent, Jaume; 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
    Journal article

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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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  • Classication of the birational projections of a sandwiched surface singularity to a plane

     Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
    Date: 2008-09
    Report

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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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  • 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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  • Factorization of complete ideals in normal birational extensions in dimension two

     Fernández Sánchez, Jesús
    Journal of algebra
    Date of publication: 2007-03
    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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  • 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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  • 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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  • 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
    Journal article

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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
    Journal article

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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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  • Geometría y Topología de Variedades Algebraicas y Simplécticas

     Casanellas Rius, Marta; Barja Yañez, Miguel Angel; Fernández Sánchez, Jesús
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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
    Journal article

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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
    Journal article

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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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  • RTACA - Red Temática de Álgebra Conmutativa y Aplicaciones

     Elias ., Joan; Casanellas Rius, Marta; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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  • Geometría y topología de las variedades algebraicas

     Barja Yañez, Miguel Angel; Roig Marti, Agustin; 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
    Journal article

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  • Geometria Algebraica.

     Casanellas Rius, Marta; 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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  • 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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  • 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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  • Geometría local de sistemas lineales

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

     Casanellas Rius, Marta; Casas ., E.; 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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