Jesús Fernández-Sánchez

Is the General Time-Reversible Model Bad for Molecular Phylogenetics?

The general time-reversible (GTR) model (Tavare 1986) has been the workhorse of molecular phylogenetics for the last decade. GTR sits at the top of the ModelTest hierarchy of models (Posada and Crandall 1998) and, usually with the addition of invariant sites and a gamma distribution of rates across sites, is currently by far the most commonly selected model for phylogenetic inference (see Table 1). However, a recent publication (Sumner et al. 2012) shows that GTR, along with several other commonly used models, has an undesirable mathematical property that may be a cause of concern for the thoughtful phylogeneticist. In mathematical terms, the problem is simple: matrix multiplication of two GTR substitution matrices does not return another GTR matrix. It is the purpose of this article to give examples that demonstrate why this lack of closure may pose a problem for phylogenetic analysis and thus add GTR to the growing list of factors that are known to cause model misspecification in phylogenetics.

On sandwiched singularities and complete ideals

Given a complete ideal I in a two-dimensional normal complete local C-algebra having a rational singularity, we prove that there is a bijection between the set of complete ideals of codimension one contained in I and the set of points in the exceptional locus of the surface X=BlI(R). As a consequence, in the case the ring R is regular, we are able to read from the Enriques diagram of the cluster of base points of I the number of singularities on X as well as their fundamental cycles and multiplicities.

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 violated.

Equivalence of the Nash conjecture for primitive and sandwiched singularities

We show that in order to prove the Nash Conjecture for sandwiched singularities it is enough to prove it for primitive singularities.

A New Hierarchy of Phylogenetic Models Consistent with Heterogeneous Substitution Rates

When the process underlying DNA substitutions varies across evolutionary history, some standard Markov models underlying 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, nonhomogeneous Lie Markov models have been identified as the class of models that are consistent in the face of a changing process of DNA substitutions regardless of taxon sampling. 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 well, with the best performing models having 8–10 parameters and the ability to recognize the distinction between purines and pyrimidines.

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, under some time restrictions, there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the definition of “Lie Markov models” which, as we will show, 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 space of phylogenetic mixtures for equivariant models

BackgroundThe selection of an evolutionary model to best fit given molecular data is usually a heuristic choice. In his seminal 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 an open question, however, whether these equations were sufficient to fully characterize the evolutionary model under consideration.ResultsHere we prove that, for most equivariant models of evolution, the space of distributions satisfying these linear equations coincides with the space of distributions arising from mixtures of trees. In other words, we prove that the evolution of an observed multiple sequence alignment can be modeled by a mixture of phylogenetic trees under an equivariant evolutionary model if and only if the distribution of patterns at its columns satisfies the linear equations mentioned above. Moreover, we provide a set of linearly independent equations defining this space of phylogenetic mixtures for each equivariant model and for any number of taxa. Lastly, we use these results to perform a study of identifiability of phylogenetic mixtures.ConclusionsThe space of phylogenetic mixtures under equivariant models is a linear space that fully characterizes the evolutionary model. We provide an explicit algorithm to obtain the equations defining these spaces for a number of models and taxa. Its implementation has proved to be a powerful tool for model selection.

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.

On adjacent complete ideals above a given complete ideal

Given a complete m-primary ideal J in a local regular two-dimensional ring ( R ,m), we describe every adjacent complete ideal above J as the integral closure of some ideal ( f , g ) for suitable f , g associated to J . We also provide a geometrical procedure that gives its base points, thus determining its equisingularity class. We decompose the set I J of these adjacent ideals in terms of the Rees valuations of J . As a consequence, we obtain a geometrical characterization of the finiteness of I J .

Reconstrucción filogenética usando geometría algebraica

Una nueva aproximacion a la reconstruccion filogenetica basada en la geometria algebraica esta ganando fuerza en los ultimos anos. Fijado un modelo evolutivo para un conjunto de especies, las distribuciones teoricas de los nucleotidos de estas especies satisfacen ciertas relaciones algebraicas que llamamos invariantes. Estos invariantes son de interes teorico y practico dado que se pueden utilizar para inferir filogenias. En este articulo, explicamos como usar los invariantes para implementar algoritmos de reconstruccion filogenetica y mostramos como el uso de tecnicas y resultados teoricos procedentes del algebra conmutativa y la geometria algebraica puede contribuir en la mejora en la eficacia y la eficiencia de estos algoritmos.