Performance of a New Invariants Method on Homogeneous and Nonhomogeneous Quartet Trees

The model-specific Markov embedding problem for symmetric group-based models

We study model embeddability, which is a variation of the famous embedding problem in probability theory, when apart from the requirement that the Markov matrix is the matrix exponential of a rate matrix, we additionally ask that the rate matrix follows the model structure. We provide a characterisation of model embeddable Markov matrices corresponding to symmetric group-based phylogenetic models. In particular, we provide necessary and sufficient conditions in terms of the eigenvalues of symmetric group-based matrices. To showcase our main result on model embeddability, we provide an application to hachimoji models, which are eight-state models for synthetic DNA. Moreover, our main result on model embeddability enables us to compute the volume of the set of model embeddable Markov matrices relative to the volume of other relevant sets of Markov matrices within the model.

Topological Metrizations of Trees, and New Quartet Methods of Tree Inference

Topological phylogenetic trees can be assigned edge weights in several natural ways, highlighting different aspects of the tree. Here, the rooted triple and quartet metrizations are introduced, and applied to formulate novel methods of inferring large trees from rooted triple and quartet data. These methods lead to new statistically consistent procedures for inference of a species tree from gene trees under the multispecies coalescent model.

Split Scores: A Tool to Quantify Phylogenetic Signal in Genome-Scale Data

Detecting variation in the evolutionary process along chromosomes is increasingly important as whole-genome data become more widely available. For example, factors such as incomplete lineage sorting, horizontal gene transfer, and chromosomal inversion are expected to result in changes in the underlying gene trees along a chromosome, while changes in selective pressure and mutational rates for different genomic regions may lead to shifts in the underlying mutational process. We propose the split score as a general method for quantifying support for a particular phylogenetic relationship within a genomic data set. Because the split score is based on algebraic properties of a matrix of site pattern frequencies, it can be rapidly computed, even for data sets that are large in the number of taxa and/or in the length of the alignment, providing an advantage over other methods (e.g., maximum likelihood) that are often used to assess such support. Using simulation, we explore the properties of the split score, including its dependence on sequence length, branch length, size of a split and its ability to detect true splits in the underlying tree. Using a sliding window analysis, we show that split scores can be used to detect changes in the underlying evolutionary process for genome-scale data from primates, mosquitoes, and viruses in a computationally efficient manner. Computation of the split score has been implemented in the software package SplitSup.

Dimensional Reduction for the General Markov Model on Phylogenetic Trees

We present a method of dimensional reduction for the general Markov model of sequence evolution on a phylogenetic tree. We show that taking certain linear combinations of the associated random variables (site pattern counts) reduces the dimensionality of the model from exponential in the number of extant taxa, to quadratic in the number of taxa, while retaining the ability to statistically identify phylogenetic divergence events. A key feature is the identification of an invariant subspace which depends only bilinearly on the model parameters, in contrast to the usual multi-linear dependence in the full space. We discuss potential applications including the computation of split (edge) weights on phylogenetic trees from observed sequence data.

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.

Invariant based quartet puzzling

Background

First proposed by Cavender and Felsenstein, and Lake, invariant based algorithms for phylogenetic reconstruction were widely dismissed by practicing biologists because invariants were perceived to have limited accuracy in constructing trees based on DNA sequences of reasonable length. Recent developments by algebraic geometers have led to the construction of lists of invariants which have been demonstrated to be more accurate on small sequences, but were limited in that they could only be used for trees with small numbers of taxa. We have developed and tested an invariant based quartet puzzling algorithm which is accurate and efficient for biologically reasonable data sets.

Results

We found that our algorithm outperforms Maximum Likelihood based quartet puzzling on data sets simulated with low to medium evolutionary rates. For faster rates of evolution, invariant based quartet puzzling is reasonable but less effective than maximum likelihood based puzzling.

Conclusions

This is a proof of concept algorithm which is not intended to replace existing reconstruction algorithms. Rather, the conclusion is that when seeking solutions to a new wave of phylogenetic problems (super tree algorithms, gene vs. species tree, mixture models), invariant based methods should be considered. This article demonstrates that invariants are a practical, reasonable and flexible source for reconstruction techniques.

Quartets MaxCut: a divide and conquer quartets algorithm

Accurate phylogenetic reconstruction methods are currently limited to a maximum of few dozens of taxa. Supertree methods construct a large tree over a large set of taxa, from a set of small trees over overlapping subsets of the complete taxa set. Hence, in order to construct the tree of life over a million and a half different species, the use of a supertree method over the product of accurate methods, is inevitable. Perhaps the simplest version of this task that is still widely applicable, yet quite challenging, is quartet-based reconstruction. This problem lies at the root of many tree reconstruction methods and theoretical as well as experimental results have been reported. Nevertheless, dealing with false, conflicting quartet trees remains problematic. In this paper, we describe an algorithm for constructing a tree from a set of input quartet trees even with a significant fraction of errors. We show empirically that conflicts in the inputs are handled satisfactorily and that it significantly outperforms and outraces the Matrix Representation with Parsimony (MRP) methods that have previously been most successful in dealing with supertrees. Our algorithm is based on a divide and conquer algorithm where our divide step uses a semidefinite programming (SDP) formulation of MaxCut. We remark that this builds on previous work of ours for piecing together trees from rooted triplet trees. The recursion for unrooted quartets, however, is more complicated in that even with completely consistent set of quartet trees the problem is NP-hard, as opposed to the problem for triples where there is a linear time algorithm. This complexity leads to several issues and some solutions of possible independent interest.

Hadamard conjugation for the Kimura 3ST model: combinatorial proof using path sets

Under a stochastic model of molecular sequence evolution the probability of each possible pattern of a characters is well defined. The Kimura's three-substitution-types (K3ST) model of evolution, allows analytical expression for these probabilities of by means of the Hadamard conjugation as a function of the phylogeny T and the substitution probabilities on each edge of TM . In this paper we produce a direct combinatorial proof of these results, using pathset distances which generalise pairwise distances between sequences. This interpretation provides us with tools that were proved useful in related problems in the mathematical analysis of sequence evolution.