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.

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).

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.

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.

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.

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.

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.

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.