Phylogenetic Mixture Model Research Articles

Dimensions of Group-Based Phylogenetic Mixtures.

Mixtures of group-based Markov models of evolution correspond to joins of toric varieties. In this paper, we establish a large number of cases for which these phylogenetic join varieties realize their expected dimension, meaning that they are nondefective. Nondefectiveness is not only interesting from a geometric point-of-view, but has been used to establish combinatorial identifiability for several classes of phylogenetic mixture models. Our focus is on group-based models where the equivalence classes of identified parameters are orbits of a subgroup of the automorphism group of the abelian group defining the model. In particular, we show that for these group-based models, the variety corresponding to the mixture of r trees with n leaves is nondefective when [Formula: see text]. We also give improved bounds for claw trees and give computational evidence that 2-tree and 3-tree mixtures are nondefective for smalln.

Classification of molecular sequence data using Bayesian phylogenetic mixture models

Rate variation among the sites of a molecular sequence is commonly found in applications of phylogenetic inference. Several approaches exist to account for this feature but they do not usually enable the investigator to pinpoint the sites that evolve under one or another rate of evolution in a straightforward manner. The focus is on Bayesian phylogenetic mixture models, augmented with allocation variables, as tools for site classification and quantification of classification uncertainty. The method does not rely on prior knowledge of site membership to classes or even the number of classes. Furthermore, it does not require correlated sites to be next to one another in the sequence alignment, unlike some phylogenetic hidden Markov or change-point models. In the approach presented, model selection on the number and type of mixture components is conducted ahead of both model estimation and site classification; the steppingstone sampler (SS) is used to select amongst competing mixture models. Example applications of simulated data and mitochondrial DNA of primates illustrate site classification via ‘augmented’ Bayesian phylogenetic mixtures. In both examples, all mixtures outperform commonly-used models of among-site rate variation and models that do not account for rate heterogeneity. The examples further demonstrate how site classification is readily available from the analysis output. The method is directly relevant to the choice of partitions in Bayesian phylogenetics, and its application may lead to the discovery of structure not otherwise recognised in a molecular sequence alignment. Computational aspects of Bayesian phylogenetic model estimation are discussed, including the use of simple Markov chain Monte Carlo (MCMC) moves that mix efficiently without tempering the chains. The contribution to the field of Bayesian phylogenetics is in (1) the use of mixture models augmented with allocation variables as tools for site classification and quantification of classification uncertainty, (2) the successful application of SS for selection of phylogenetic mixtures, and (3) the development of novel MCMC aspects of relevance to Bayesian phylogenetic models—whether mixtures or not.11The MCMC methods discussed in this paper have been coded in a C program; source files are available upon request. Supplementary material is available online (see Appendix A).

When Do Phylogenetic Mixture Models Mimic Other Phylogenetic Models?

Phylogenetic mixture models, in which the sites in sequences undergo different substitution processes along the same or different trees, allow the description of heterogeneous evolutionary processes. As data sets consisting of longer sequences become available, it is important to understand such models, for both theoretical insights and use in statistical analyses. Some recent articles have highlighted disturbing "mimicking" behavior in which a distribution from a mixture model is identical to one arising on a different tree or trees. Other works have indicated such problems are unlikely to occur in practice, as they require very special parameter choices. After surveying some of these works on mixture models, we give several new results. In general, if the number of components in a generating mixture is not too large and we disallow zero or infinite branch lengths, then it cannot mimic the behavior of a nonmixture on a different tree. On the other hand, if the mixture model is locally overparameterized, it is possible for a phylogenetic mixture model to mimic distributions of another tree model. Although theoretical questions remain, these sorts of results can serve as a guide to when the use of mixture models in either maximum likelihood or Bayesian frameworks is likely to lead to statistically consistent inference, and when mimicking due to heterogeneity should be considered a realistic possibility. [Phylogenetic mixture models; parameter identifiability; heterogeneous sequence evolution.].

The Disentangling Number for Phylogenetic Mixtures

We provide a logarithmic upper bound for the disentangling number on unordered lists of leaf labeled trees. This result is useful for analyzing phylogenetic mixture models. The proof depends on interpreting multisets of trees as high-dimensional contingency tables.

A phylogenetic mixture model for the identification of functionally divergent protein residues

To understand the evolution of molecular function within protein families, it is important to identify those amino acid residues responsible for functional divergence; i.e. those sites in a protein family that affect cofactor, protein or substrate binding preferences; affinity; catalysis; flexibility; or folding. Type I functional divergence (FD) results from changes in conservation (evolutionary rate) at a site between protein subfamilies, whereas type II FD occurs when there has been a shift in preferences for different amino acid chemical properties. A variety of methods have been developed for identifying both site types in protein subfamilies, both from phylogenetic and information-theoretic angles. However, evaluation of the performance of these methods has typically relied upon a handful of reasonably well-characterized biological datasets or analyses of a single biological example. While experimental validation of many truly functionally divergent sites (true positives) can be relatively straightforward, determining that particular sites do not contribute to functional divergence (i.e. false positives and true negatives) is much more difficult, resulting in noisy 'gold standard' examples. We describe a novel, phylogeny-based functional divergence classifier, FunDi. Unlike previous approaches, FunDi uses a unified mixture model-based approach to detect type I and type II FD. To assess FunDi's overall classification performance relative to other methods, we introduce two methods for simulating functionally divergent datasets. We find that the FunDi method performs better than several other predictors over a wide variety of simulation conditions. http://rogerlab.biochem.dal.ca/Software andrew.roger@dal.ca Supplementary data are available at Bioinformatics online.

Identifiability of Large Phylogenetic Mixture Models

Phylogenetic mixture models are statistical models of character evolution allowing for heterogeneity. Each of the classes in some unknown partition of the characters may evolve by different processes, or even along different trees. Such models are of increasing interest for data analysis, as they can capture the variety of evolutionary processes that may be occurring across long sequences of DNA or proteins. The fundamental question of whether parameters of such a model are identifiable is difficult to address, due to the complexity of the parameterization. Identifiability is, however, essential to their use for statistical inference.We analyze mixture models on large trees, with many mixture components, showing that both numerical and tree parameters are indeed identifiable in these models when all trees are the same. This provides a theoretical justification for some current empirical studies, and indicates that extensions to even more mixture components should be theoretically well behaved. We also extend our results to certain mixtures on different trees, using the same algebraic techniques.

On the artefactual parasitic eubacteria clan in conditioned logdet phylogenies: heterotachy and ortholog identification artefacts as explanations

BackgroundPhylogenetic reconstruction methods based on gene content often place all the parasitic and endosymbiotic eubacteria (parasites for short) together in a clan. Many other lines of evidence point to this parasites clan being an artefact. This artefact could be a consequence of the methods used to construct ortholog databases (due to some unknown bias), the methods used to estimate the phylogeny, or both.We test the idea that the parasites clan is an ortholog identification artefact by analyzing three different ortholog databases (COG, TRIBES, and OFAM), which were constructed using different methods, and are thus unlikely to share the same biases. In each case, we estimate a phylogeny using an improved version of the conditioned logdet distance method. If the parasites clan appears in trees from all three databases, it is unlikely to be an ortholog identification artefact.Accelerated loss of a subset of gene families in parasites (a form of heterotachy) may contribute to the difficulty of estimating a phylogeny from gene content data. We test the idea that heterotachy is the underlying reason for the estimation of an artefactual parasites clan by applying two different mixture models (phylogenetic and non-phylogenetic), in combination with conditioned logdet. In these models, there are two categories of gene families, one of which has accelerated loss in parasites. Distances are estimated separately from each category by conditioned logdet. This should reduce the tendency for tree estimation methods to group the parasites together, if heterotachy is the underlying reason for estimation of the parasites clan.ResultsThe parasites clan appears in conditioned logdet trees estimated from all three databases. This makes it less likely to be an artefact of database construction. The non-phylogenetic mixture model gives trees without a parasites clan. However, the phylogenetic mixture model still results in a tree with a parasites clan. Thus, it is not entirely clear whether heterotachy is the underlying reason for the estimation of a parasites clan. Simulation studies suggest that the phylogenetic mixture model approach may be unsuccessful because the model of gene family gain and loss it uses does not adequately describe the real data.ConclusionsThe most successful methods for estimating a reliable phylogenetic tree for parasitic and endosymbiotic eubacteria from gene content data are still ad-hoc approaches such as the SHOT distance method. however, the improved conditioned logdet method we developed here may be useful for non-parasites and can be accessed at http://www.liv.ac.uk/~cgrbios/cond_logdet.html

A Phylogenetic Mixture Model for Gene Family Loss in Parasitic Bacteria

A Phylogenetic Mixture Model for Gene Family Loss in Parasitic Bacteria

A Phylogenetic Mixture Model for Gene Family Loss in Parasitic Bacteria

Gene families are frequently gained and lost from prokaryotic genomes. It is widely believed that the rate of loss was accelerated for some but not all gene families in lineages that became parasites or endosymbionts. This leads to a form of heterotachy that may be responsible for the poor performance of phylogeny estimation based on gene content. We describe a mixture model that accounts for this heterotachy. We show that this model fits data on the distribution of gene families across bacteria from the COG database much better than previous models. However, it still favors an artifactual tree topology in which parasites form a clade over the more plausible 16S topology. In contrast to a previous model of genome dynamics, our model suggests that the ancestral bacterium had a small genome. We suggest that models of gene family gain and loss are likely to be more useful for understanding genome dynamics than for estimating phylogenetic trees.

Networks, trees, and treeshrews: assessing support and identifying conflict with multiple loci and a problematic root.

Multiple unlinked genetic loci often provide a more comprehensive picture of evolutionary history than any single gene can, but analyzing multigene data presents particular challenges. Differing rates and patterns of nucleotide substitution, combined with the limited information available in any data set, can make it difficult to specify a model of evolution. In addition, conflict among loci can be the result of real differences in evolutionary process or of stochastic variance and errors in reconstruction. We used 6 presumably unlinked nuclear loci to investigate relationships within the mammalian family Tupaiidae (Scandentia), containing all but one of the extant tupaiid genera. We used a phylogenetic mixture model to analyze the concatenated data and compared this with results using partitioned models. We found that more complex models were not necessarily preferred under tests using Bayes factors and that model complexity affected both tree length and parameter variance. We also compared the results of single-gene and multigene analyses and used splits networks to analyze the source and degree of conflict among genes. Networks can show specific relationships that are inconsistent with each other; these conflicting and minority relationships, which are implicitly ignored or collapsed by traditional consensus methods, can be useful in identifying the underlying causes of topological uncertainty. In our data, conflict is concentrated around particular relationships, not widespread throughout the tree. This pattern is further clarified by considering conflict surrounding the root separately from conflict within the ingroup. Uncertainty in rooting may be because of the apparent evolutionary distance separating these genera and our outgroup, the tupaiid genus Dendrogale. Unlike a previous mitochondrial study, these nuclear data strongly suggest that the genus Tupaia is not monophyletic with respect to the monotypic Urogale, even when uncertainty about rooting is taken into account. These data concur with mitochondrial DNA on other relationships, including the close affinity of Tupaia tana with the enigmatic Tupaia splendidula and of Tupaia belangeri with Tupaia glis. We also discuss the taxonomic and biogeographic implications of these results.

Phylogenetic mixture models for proteins

Standard protein substitution models use a single amino acid replacement rate matrix that summarizes the biological, chemical and physical properties of amino acids. However, site evolution is highly heterogeneous and depends on many factors: genetic code; solvent exposure; secondary and tertiary structure; protein function; etc. These impact the substitution pattern and, in most cases, a single replacement matrix is not enough to represent all the complexity of the evolutionary processes. This paper explores in maximum-likelihood framework phylogenetic mixture models that combine several amino acid replacement matrices to better fit protein evolution.We learn these mixture models from a large alignment database extracted from HSSP, and test the performance using independent alignments from TREEBASE.We compare unsupervised learning approaches, where the site categories are unknown, to supervised ones, where in estimations we use the known category of each site, based on its exposure or its secondary structure. All our models are combined with gamma-distributed rates across sites. Results show that highly significant likelihood gains are obtained when using mixture models compared with the best available single replacement matrices. Mixtures of matrices also improve over mixtures of profiles in the manner of the CAT model. The unsupervised approach tends to be better than the supervised one, but it appears difficult to implement and highly sensitive to the starting values of the parameters, meaning that the supervised approach is still of interest for initialization and model comparison. Using an unsupervised model involving three matrices, the average AIC gain per site with TREEBASE test alignments is 0.31, 0.49 and 0.61 compared with LG (named after Le & Gascuel 2008 Mol. Biol. Evol. 25, 1307-1320), WAG and JTT, respectively. This three-matrix model is significantly better than LG for 34 alignments (among 57), and significantly worse for 1 alignment only. Moreover, tree topologies inferred with our mixture models frequently differ from those obtained with single matrices, indicating that using these mixtures impacts not only the likelihood value but also the output tree. All our models and a PhyML implementation are available from http://atgc.lirmm.fr/mixtures.

Modelling heterotachy in phylogenetic inference by reversible-jump Markov chain Monte Carlo

The rate at which a given site in a gene sequence alignment evolves over time may vary. This phenomenon--known as heterotachy--can bias or distort phylogenetic trees inferred from models of sequence evolution that assume rates of evolution are constant. Here, we describe a phylogenetic mixture model designed to accommodate heterotachy. The method sums the likelihood of the data at each site over more than one set of branch lengths on the same tree topology. A branch-length set that is best for one site may differ from the branch-length set that is best for some other site, thereby allowing different sites to have different rates of change throughout the tree. Because rate variation may not be present in all branches, we use a reversible-jump Markov chain Monte Carlo algorithm to identify those branches in which reliable amounts of heterotachy occur. We implement the method in combination with our 'pattern-heterogeneity' mixture model, applying it to simulated data and five published datasets. We find that complex evolutionary signals of heterotachy are routinely present over and above variation in the rate or pattern of evolution across sites, that the reversible-jump method requires far fewer parameters than conventional mixture models to describe it, and serves to identify the regions of the tree in which heterotachy is most pronounced. The reversible-jump procedure also removes the need for a posteriori tests of 'significance' such as the Akaike or Bayesian information criterion tests, or Bayes factors. Heterotachy has important consequences for the correct reconstruction of phylogenies as well as for tests of hypotheses that rely on accurate branch-length information. These include molecular clocks, analyses of tempo and mode of evolution, comparative studies and ancestral state reconstruction. The model is available from the authors' website, and can be used for the analysis of both nucleotide and morphological data.

Phylogenetic Mixture Models Can Reduce Node-Density Artifacts

We investigate the performance of phylogenetic mixture models in reducing a well-known and pervasive artifact of phylogenetic inference known as the node-density effect, comparing them to partitioned analyses of the same data. The node-density effect refers to the tendency for the amount of evolutionary change in longer branches of phylogenies to be underestimated compared to that in regions of the tree where there are more nodes and thus branches are typically shorter. Mixture models allow more than one model of sequence evolution to describe the sites in an alignment without prior knowledge of the evolutionary processes that characterize the data or how they correspond to different sites. If multiple evolutionary patterns are common in sequence evolution, mixture models may be capable of reducing node-density effects by characterizing the evolutionary processes more accurately. In gene-sequence alignments simulated to have heterogeneous patterns of evolution, we find that mixture models can reduce node-density effects to negligible levels or remove them altogether, performing as well as partitioned analyses based on the known simulated patterns. The mixture models achieve this without knowledge of the patterns that generated the data and even in some cases without specifying the full or true model of sequence evolution known to underlie the data. The latter result is especially important in real applications, as the true model of evolution is seldom known. We find the same patterns of results for two real data sets with evidence of complex patterns of sequence evolution: mixture models substantially reduced node-density effects and returned better likelihoods compared to partitioning models specifically fitted to these data. We suggest that the presence of more than one pattern of evolution in the data is a common source of error in phylogenetic inference and that mixture models can often detect these patterns even without prior knowledge of their presence in the data. Routine use of mixture models alongside other approaches to phylogenetic inference may often reveal hidden or unexpected patterns of sequence evolution and can improve phylogenetic inference.

Medusozoan Phylogeny and Character Evolution Clarified by New Large and Small Subunit rDNA Data and an Assessment of the Utility of Phylogenetic Mixture Models

A newly compiled data set of nearly complete sequences of the large subunit of the nuclear ribosome (LSU or 28S) sampled from 31 diverse medusozoans greatly clarifies the phylogenetic history of Cnidaria. These data have substantial power to discern among many of the competing hypotheses of relationship derived from prior work. Moreover, LSU data provide strong support at key nodes that were equivocal based on other molecular markers. Combining LSU sequences with those of the small subunit of the nuclear ribosome (SSU or 18S), we present a detailed working hypothesis of medusozoan relationships and discuss character evolution within this diverse clade. Stauromedusae, comprising the benthic, so-called stalked jellyfish, appears to be the sister group of all other medusozoans, implying that the free-swimming medusa stage, the motor nerve net, and statocysts of ecto-endodermal origin are features derived within Medusozoa. Cubozoans, which have had uncertain phylogenetic affinities since the elucidation of their life cycles, form a clade-named Acraspeda-with the scyphozoan groups Coronatae, Rhizostomeae, and Semaeostomeae. The polyps of both cubozoans and hydrozoans appear to be secondarily simplified. Hydrozoa is comprised by two well-supported clades, Trachylina and Hydroidolina. The position of Limnomedusae within Trachylina indicates that the ancestral hydrozoan had a biphasic life cycle and that the medusa was formed via an entocodon. Recently hypothesized homologies between the entocodon and bilaterian mesoderm are therefore suspect. Laingiomedusae, which has often been viewed as a close ally of the trachyline group Narcomedusae, is instead shown to be unambiguously a member of Hydroidolina. The important model organisms of the Hydra species complex are part of a clade, Aplanulata, with other hydrozoans possessing direct development not involving a ciliated planula stage. Finally, applying phylogenetic mixture models to our data proved to be of little additional value over a more traditional phylogenetic approach involving explicit hypothesis testing and bootstrap analyses under multiple optimality criteria. [18S; 28S; Cubozoa; Hydrozoa; medusa; molecular systematics; polyp; Scyphozoa; Staurozoa.].

A phylogenetic mixture model for detecting pattern-heterogeneity in gene sequence or character-state data.

We describe a general likelihood-based 'mixture model' for inferring phylogenetic trees from gene-sequence or other character-state data. The model accommodates cases in which different sites in the alignment evolve in qualitatively distinct ways, but does not require prior knowledge of these patterns or partitioning of the data. We call this qualitative variability in the pattern of evolution across sites "pattern-heterogeneity" to distinguish it from both a homogenous process of evolution and from one characterized principally by differences in rates of evolution. We present studies to show that the model correctly retrieves the signals of pattern-heterogeneity from simulated gene-sequence data, and we apply the method to protein-coding genes and to a ribosomal 12S data set. The mixture model outperforms conventional partitioning in both these data sets. We implement the mixture model such that it can simultaneously detect rate- and pattern-heterogeneity. The model simplifies to a homogeneous model or a rate-variability model as special cases, and therefore always performs at least as well as these two approaches, and often considerably improves upon them. We make the model available within a Bayesian Markov-chain Monte Carlo framework for phylogenetic inference, as an easy-to-use computer program.