arXiv subjects

Jesús Fernández-Sánchez

Publications and source records attributed to Jesús Fernández-Sánchez.

13 recordsLinked to original sources

arXiv2025

Phylogenetic invariants: straightforward 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 phylogenetic trees evolving under Markov processes of molecular substitution, the so called phylogenetic invariants. Although the theory involved allows to explicitly obtain 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 trees evolving under 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 a phylogenetic tree evolving under an equivariant model is an irreducible component of the variety corresponding to the same tree under the general Markov model cut 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 extends our results from trees to phylogenetic networks.

q-bio.PE↗

arXiv2022

Designing weights for quartet-based methods when data is heterogeneous across lineages

Homogeneity across lineages is a common assumption in phylogenetics according to which nucleotide substitution rates remain constant in time and do not depend on lineages. This is a simplifying hypothesis which is often adopted to make the process of sequence evolution more tractable. However, its validity has been explored and put into question in several papers. On the other hand, dealing successfully with the general case (heterogeneity across lineages) is one of the key features of phylogenetic reconstruction methods based on algebraic tools. The goal of this paper is twofold. First, we present a new weighting system for quartets (ASAQ) based on algebraic and semi-algebraic tools, thus specially indicated to deal with data evolving under heterogeneus rates. This method combines the weights two previous methods by means of a test based on the positivity of the branch length estimated with the paralinear distance. ASAQ is statistically consistent when applied to GM data, considers rate and base composition heterogeneity among lineages and does not assume stationarity nor time-reversibility. Second, we test and compare the performance of several quartet-based methods for phylogenetic tree reconstruction (namely, Quartet Puzzling, Weight Optimization and Wilson's method) in combination with ASAQ weights and other weights based on algebraic and semi-algebraic methods or on the paralinear distance. These tests are applied to both simulated and real data and support Weight Optimization with ASAQ weights as a reliable and successful reconstruction method.

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arXiv2021

The embedding problem for Markov matrices

Characterizing whether a Markov process of discrete random variables has an homogeneous continuous-time realization is a hard problem. In practice, this problem reduces to deciding when a given Markov matrix can be written as the exponential of some rate matrix (a Markov generator). This is an old question known in the literature as the embedding problem (Elfving37), which has been only solved for matrices of size $2\times 2$ or $3\times 3$. In this paper, we address this problem and related questions and obtain results in two different lines. First, for matrices of any size, we give a bound on the number of Markov generators in terms of the spectrum of the Markov matrix. Based on this, we establish a criterion for deciding whether a generic Markov matrix (different eigenvalues) is embeddable and propose an algorithm that lists all its Markov generators. Then, motivated and inspired by recent results on substitution models of DNA, we focus in the $4\times 4$ case and completely solve the embedding problem for any Markov matrix. The solution in this case is more concise as the embeddability is given in terms of a single condition.

math.PR↗

arXiv2020

SAQ: semi-algebraic quartet reconstruction method

We present the phylogenetic quartet reconstruction method SAQ (Semi-algebraic quartet reconstruction). SAQ is consistent with the most general Markov model of nucleotide substitution and, in particular, it allows for rate heterogeneity across lineages. Based on the algebraic and semi-algebraic description of distributions that arise from the general Markov model on a quartet, the method outputs normalized weights for the three trivalent quartets (which can be used as input of quartet-base methods). We show that SAQ is a highly competitive method that outperforms most of the well known reconstruction methods on data simulated under the general Markov model on 4-taxon trees. Moreover, it also achieves a high performance on data that violates the underlying assumptions.

q-bio.PE↗

arXiv2020

Distance to the stochastic part of phylogenetic varieties

Modelling the substitution of nucleotides along a phylogenetic tree is usually done by a hidden Markov process. This allows to define a distribution of characters at the leaves of the trees and one might be able to obtain polynomial relationships among the probabilities of different characters. The study of these polynomials and the geometry of the algebraic varieties defined by them can be used to reconstruct phylogenetic trees. However, not all points in these algebraic varieties have biological sense. In this paper, we explore the extent to which adding semi-algebraic conditions arising from the restriction to parameters with statistical meaning can improve existing methods of phylogenetic reconstruction. To this end, our aim is to compute the distance of data points to algebraic varieties and to the stochastic part of these varieties. Computing these distances involves optimization by nonlinear programming algorithms. We use analytical methods to find some of these distances for quartet trees evolving under the Kimura 3-parameter or the Jukes-Cantor models. Numerical algebraic geometry and computational algebra play also a fundamental role in this paper.

q-bio.PE↗

arXiv2020

An open set of $4\times4$ embeddable matrices whose principal logarithm is not a Markov generator

A Markov matrix is embeddable if it can represent a homogeneous continuous-time Markov process. It is well known that if a Markov matrix has real and pairwise-different eigenvalues, then the embeddability can be determined by checking whether its principal logarithm is a rate matrix or not. The same holds for Markov matrices close enough to the identity matrix or that rule a Markov process subjected to certain restrictions. In this paper we prove that this criterion cannot be generalized and we provide open sets of Markov matrices that are embeddable and whose principal logarithm is a not a rate matrix.

math.PR↗

arXiv2020

Rank conditions on phylogenetic networks

Less rigid than phylogenetic trees, phylogenetic networks allow the description of a wider range of evolutionary events. In this note, we explain how to extend the rank invariants from phylogenetic trees to phylogenetic networks evolving under the general Markov model and the equivariant models.

q-bio.PE↗

arXiv2019

Embeddability and rate identifiability of Kimura 2-parameter matrices

Deciding whether a Markov matrix is embeddable (i.e. can be written as the exponential of a rate matrix) is an open problem even for $4\times 4$ matrices. We study the embedding problem and rate identifiability for the K80 model of nucleotide substitution. For these $4\times 4$ matrices, we fully characterize the set of embeddable K80 Markov matrices and the set of embeddable matrices for which rates are identifiable. In particular, we describe an open subset of embeddable matrices with non-identifiable rates. This set contains matrices with positive eigenvalues and also diagonal largest in column matrices, which might lead to consequences in parameter estimation in phylogenetics. Finally, we compute the relative volumes of embeddable K80 matrices and of embeddable matrices with identifiable rates. This study concludes the embedding problem for the more general model K81 and its submodels, which had been initiated by the last two authors in a separate work.

q-bio.PE↗

arXiv2019

Embeddability of Kimura 3ST Markov matrices

In this note, we characterize the embeddability of generic Kimura 3ST Markov matrices in terms of their eigenvalues. As a consequence, we are able to compute the volume of such matrices relative to the volume of all Markov matrices within the model. We also provide examples showing that, in general, mutation rates are not identifiable from substitution probabilities. These examples also illustrate that symmetries between mutation probabilities do not necessarily arise from symmetries between the corresponding mutation rates.

q-bio.PE↗

arXiv2014

A new hierarchy of phylogenetic models consistent with heterogeneous substitution rates

When the process underlying DNA substitutions varies across evolutionary history, the standard Markov models underlying standard phylogenetic methods are mathematically inconsistent. The most prominent example is the general time reversible model (GTR) together with some, but not all, of its submodels. To rectify this deficiency, Lie Markov models have been developed as the class of models that are consistent in the face of a changing process of DNA substitutions. Some well-known models in popular use are within this class, but are either overly simplistic (e.g. the Kimura two-parameter model) or overly complex (the general Markov model). On a diverse set of biological data sets, we test a hierarchy of Lie Markov models spanning the full range of parameter richness. Compared against the benchmark of the ever-popular GTR model, we find that as a whole the Lie Markov models perform remarkably well, with the best performing models having eight parameters and the ability to recognise the distinction between purines and pyrimidines.

q-bio.PE↗

arXiv2014

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.

q-bio.PE↗

arXiv2014

Local description of phylogenetic group-based models

Motivated by phylogenetics, our aim is to obtain a system of equations that define a phylogenetic variety on an open set containing the biologically meaningful points. In this paper we consider phylogenetic varieties defined via group-based models. For any finite abelian group $G$, we provide an explicit construction of $codim X$ phylogenetic invariants (polynomial equations) of degree at most $|G|$ that define 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 confirms a conjecture by the third author and, on the set $U$, a couple of conjectures by Bernd Sturmfels and Seth Sullivant.

math.AG↗

arXiv2013

Lie Markov models with purine/pyrimidine symmetry

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. The whole list of Lie Markov models with purine/pyrimidine symmetry is available at http://www.pagines.ma1.upc.edu/~jfernandez/LMNR.pdf.

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