Designing Weights for Quartet-Based Methods When Data are Heterogeneous Across Lineages.

Homogeneity across lineages is a general assumption in phylogenetics according to which nucleotide substitution rates are common to all lineages. Many phylogenetic methods relax this hypothesis but keep a simple enough model to make the process of sequence evolution more tractable. On the other hand, dealing successfully with the general case (heterogeneity of rates 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 especially indicated to deal with data evolving under heterogeneous rates. This method combines the weights of two previous methods by means of a test based on the positivity of the branch lengths estimated with the paralinear distance. ASAQ is statistically consistent when applied to data generated under the general Markov model, 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 QFM, wQFM, quartet puzzling, weight optimization and Willson's method) in combination with several systems of weights, including 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 that improves upon the accuracy of global methods (such as neighbor-joining or maximum likelihood) in the presence of long branches or on mixtures of distributions on trees.

Embeddability and rate identifiability of Kimura 2-parameter matrices.

Deciding whether a substitution matrix is embeddable (i.e. the corresponding Markov process has a continuous-time realization) is an open problem even for [Formula: see text] matrices. We study the embedding problem and rate identifiability for the K80 model of nucleotide substitution. For these [Formula: see text] 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.

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.

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.

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.

SAQ: Semi-Algebraic Quartet Reconstruction.

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-based 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.

The space of phylogenetic mixtures for equivariant models.

BACKGROUND: The 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. RESULTS: Here 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. CONCLUSIONS: The 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.

Avaliaçäo radiográfica da reabsorçäo radicular, consecutiva ao tratamento ortodôntico (cinco anos após a remoçäo dos aparelhos) / Radiographic evaluation of root resorption five years after orthodontic treatment

Encetou-se uma pesquisa radiográfica, cinco anos após a remoçäo dos aparelhos ortodônticos, para avaliar o comportamento das reabsorçöes radiculares, consecutivas à terapia ortodôntica. Ao final do estudo, verificou-se que a reabsorçäo radicular observada ao término do tratamento ortodôntico, cessa com a remoçäo do aparelho, na maioria dos dentes observados, e nos poucos que apresentaram alguma progressäo, näo foi contínua, ocorrendo apenas uma remodelaçäo apical. Aconselha-se, entretanto, o ajuste oclusal, após seis meses do término do tratamento e o controle por 2 ou 3 anos, das reabsorçöes mais evidentes

Incisivo central superior dilacerado: tratamento clínico integrado / Dilacerated upper central incisor

Os autores apresentam discussão das indicações e técnicas de tratamento para casos de dilaceração radicular, com apresentação de caso clínico que mereceu abordagem multidisciplinar, ortodôntico-cirúrgico-protética