The space of phylogenetic mixtures for equivariant models

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.Comment: 28 pages, 1 figure; to appear in Algorithms for Molecular Biolog

Algebraic tools in phylogenomics.

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.Postprint (published version

Invariant versus classical quartet inference when evolution is heterogeneous across sites and lineages

One reason why classical phylogenetic reconstruction methods fail to correctly infer the underlying topology is because they assume oversimplified models. In this paper we propose a topology reconstruction method consistent with the most general Markov model of nucleotide substitution, which can also deal with data coming from mixtures on the same topology. It is based on an idea of Eriksson on using phylogenetic invariants and provides a system of weights that can be used as input of quartet-based methods. We study its performance on real data and on a wide range of simulated 4-taxon data (both time-homogeneous and nonhomogeneous, with or without among-site rate heterogeneity, and with different branch length settings). We compare it to the classical methods of neighbor-joining (with paralinear distance), maximum likelihood (with different underlying models), and maximum parsimony. Our results show that this method is accurate and robust, has a similar performance to ML when data satisfies the assumptions of both methods, and outperforms all methods when these are based on inappropriate substitution models or when both long and short branches are present. If alignments are long enough, then it also outperforms other methods when some of its assumptions are violated.Comment: 32 pages; 9 figure

Toric geometry of the 3-Kimura model for any tree

In this paper we present geometric features of group based models. We focus on the 3-Kimura model. We present a precise geometric description of the variety associated to any tree on a Zariski open set. In particular this set contains all biologically meaningful points. Our motivation is a conjecture of Sturmfels and Sullivant on the degree in which the ideal associated to 3-Kimura model is generated

A novel algebraic approach to time-reversible evolutionary models

In the last years algebraic tools have been proven to be useful in phylogenetic reconstruction and model selection by means of the study of phylogenetic invariants. However, up to now, the models studied from an algebraic viewpoint are either too general or too restrictive (as group-based models with a uniform stationary distribution) to be used in practice. In this paper we provide a new framework to work with time-reversible models, which are the most widely used by biologists. In our approach we consider algebraic time-reversible models on phylogenetic trees (as defined by Allman and Rhodes) and introduce a new inner product to make all transition matrices of the process diagonalizable through the same orthogonal eigenbasis. This framework generalizes the Fourier transform widely used to work with group-based models and recovers some of the well known results. As illustration, we exploit the combination of our technique with algebraic geometry tools to provide relevant phylogenetic invariants for trees evolving under the Tamura-Nei model of nucleotide substitution

Phylogenetic mixtures and linear invariants for equal input models

The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely long sequences. However, when the underlying Markov process supports linear phylogenetic invariants, then provided these are sufficiently informative, the identifiability of the tree topology can be restored. In this paper, we investigate a class of processes that support linear invariants once the stationary distribution is fixed, the ‘equal input model’. This model generalizes the ‘Felsenstein 1981’ model (and thereby the Jukes–Cantor model) from four states to an arbitrary number of states (finite or infinite), and it can also be described by a ‘random cluster’ process. We describe the structure and dimension of the vector spaces of phylogenetic mixtures and of linear invariants for any fixed phylogenetic tree (and for all trees—the so called ‘model invariants’), on any number n of leaves. We also provide a precise description of the space of mixtures and linear invariants for the special case of n=4 leaves. By combining techniques from discrete random processes and (multi-) linear algebra, our results build on a classic result that was first established by James Lake (Mol Biol Evol 4:167–191, 1987).Peer ReviewedPostprint (author's final draft

Local equations for equivariant evolutionary models

Phylogenetic varieties related to equivariant substitution models have been studied largely in the last years. One of the main objectives has been finding a set of generators of the ideal of these varieties, but this has not yet been achieved in some cases (for example, for the general Markov model this involves the open “salmon conjecture”, see [2]) and it is not clear how to use all generators in practice. Motivated by applications in biology, we tackle the problem from another point of view. The elements of the ideal that could be useful for applications in phylogenetics only need to describe the variety around certain points of no evolution (see [13]). We produce a collection of explicit equations that describe the variety on a Zariski open neighborhood of these points (see Theorem 5.4). Namely, for any tree T on any number of leaves (and any degrees at the interior nodes) and for any equivariant model on any set of states ¿, we compute the codimension of the corresponding phylogenetic variety. We prove that this variety is smooth at general points of no evolution and, if a mild technical condition is satisfied (“d-claw tree hypothesis”), we provide an algorithm to produce a complete intersection that describes the variety around these points.Peer ReviewedPostprint (author's final draft

Invariant versus classical quartet inference when evolution is heterogeneous across sites and lineages

One reason why classical phylogenetic reconstruction methods fail to correctly infer the underlying topology is because they assume oversimplified models. In this article, we propose a quartet reconstruction method consistent with the most general Markov model of nucleotide substitution, which can also deal with data coming from mixtures on the same topology. Our proposed method uses phylogenetic invariants and provides a system of weights that can be used as input for quartet-based methods. We study its performance on real data and on a wide range of simulated 4-taxon data (both time-homogeneous and nonhomogeneous, with or without among-site rate heterogeneity, and with different branch length settings). We compare it to the classical methods of neighbor-joining (with paralinear distance), maximum likelihood (with different underlying models), and maximum parsimony. Our results show that this method is accurate and robust, has a similar performance to maximum likelihood when data satisfies the assumptions of both methods, and outperform the other methods when these are based on inappropriate substitution models. If alignments are long enough, then it also outperforms other methods when some of its assumptions are violatedPeer ReviewedPostprint (author's final draft