Pitfalls of heterogeneous processes for phylogenetic reconstruction

Different genes often have different phylogenetic histories. Even within regions having the same phylogenetic history, the mutation rates often vary. We investigate the prospects of phylogenetic reconstruction when all the characters are generated from the same tree topology, but the branch lengths vary (with possibly different tree shapes). Furthering work of Kolaczkowski and Thornton (2004, Nature 431: 980-984) and Chang (1996, Math. Biosci. 134: 189-216), we show examples where maximum likelihood (under a homogeneous model) is an inconsistent estimator of the tree. We then explore the prospects of phylogenetic inference under a heterogeneous model. In some models, there are examples where phylogenetic inference under any method is impossible - despite the fact that there is a common tree topology. In particular, there are nonidentifiable mixture distributions, i.e., multiple topologies generate identical mixture distributions. We address which evolutionary models have nonidentifiable mixture distributions and prove that the following duality theorem holds for most DNA substitution models. The model has either: (i) nonidentifiability - two different tree topologies can produce identical mixture distributions, and hence distinguishing between the two topologies is impossible; or (ii) linear tests - there exist linear tests which identify the common tree topology for character data generated by a mixture distribution. The theorem holds for models whose transition matrices can be parameterized by open sets, which includes most of the popular models, such as Tamura-Nei and Kimura's 2-parameter model. The duality theorem relies on our notion of linear tests, which are related to Lake's linear invariants.     

Phylogenetic mixtures on a single tree can mimic a tree of another topology

Phylogenetic mixtures model the inhomogeneous molecular evolution commonly observed in data. The performance of phylogenetic reconstruction methods where the underlying data are generated by a mixture model has stimulated considerable recent debate. Much of the controversy stems from simulations of mixture model data on a given tree topology for which reconstruction algorithms output a tree of a different topology; these findings were held up to show the shortcomings of particular tree reconstruction methods. In so doing, the underlying assumption was that mixture model data on one topology can be distinguished from data evolved on an unmixed tree of another topology given enough data and the "correct" method. Here we show that this assumption can be false. For biologists, our results imply that, for example, the combined data from two genes whose phylogenetic trees differ only in terms of branch lengths can perfectly fit a tree of a different topology.     

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.     

Permutation tests for random effects in linear mixed models

Inference regarding the inclusion or exclusion of random effects in linear mixed models is challenging because the variance components are located on the boundary of their parameter space under the usual null hypothesis. As a result, the asymptotic null distribution of the Wald, score, and likelihood ratio tests will not have the typical χ(2) distribution. Although it has been proved that the correct asymptotic distribution is a mixture of χ(2) distributions, the appropriate mixture distribution is rather cumbersome and nonintuitive when the null and alternative hypotheses differ by more than one random effect. As alternatives, we present two permutation tests, one that is based on the best linear unbiased predictors and one that is based on the restricted likelihood ratio test statistic. Both methods involve weighted residuals, with the weights determined by the among- and within-subject variance components. The null permutation distributions of our statistics are computed by permuting the residuals both within and among subjects and are valid both asymptotically and in small samples. We examine the size and power of our tests via simulation under a variety of settings and apply our test to a published data set of chronic myelogenous leukemia patients.     

A structural EM algorithm for phylogenetic inference.

A central task in the study of molecular evolution is the reconstruction of a phylogenetic tree from sequences of current-day taxa. The most established approach to tree reconstruction is maximum likelihood (ML) analysis. Unfortunately, searching for the maximum likelihood phylogenetic tree is computationally prohibitive for large data sets. In this paper, we describe a new algorithm that uses Structural Expectation Maximization (EM) for learning maximum likelihood phylogenetic trees. This algorithm is similar to the standard EM method for edge-length estimation, except that during iterations of the Structural EM algorithm the topology is improved as well as the edge length. Our algorithm performs iterations of two steps. In the E-step, we use the current tree topology and edge lengths to compute expected sufficient statistics, which summarize the data. In the M-Step, we search for a topology that maximizes the likelihood with respect to these expected sufficient statistics. We show that searching for better topologies inside the M-step can be done efficiently, as opposed to standard methods for topology search. We prove that each iteration of this procedure increases the likelihood of the topology, and thus the procedure must converge. This convergence point, however, can be a suboptimal one. To escape from such "local optima," we further enhance our basic EM procedure by incorporating moves in the flavor of simulated annealing. We evaluate these new algorithms on both synthetic and real sequence data and show that for protein sequences even our basic algorithm finds more plausible trees than existing methods for searching maximum likelihood phylogenies. Furthermore, our algorithms are dramatically faster than such methods, enabling, for the first time, phylogenetic analysis of large protein data sets in the maximum likelihood framework.     

Testing goodness of fit of a uniform truncation model

Several goodness-of-fit tests of a lifetime distribution have been suggested in the literature; many take into account censoring and/or truncation of event times. In some contexts, a goodness-of-fit test for the truncation distribution is of interest. In particular, better estimates of the lifetime distribution can be obtained when knowledge of the truncation law is exploited. In cross-sectional sampling, for example, there are theoretical justifications for the assumption of a uniform truncation distribution, and several studies have used it to improve the efficiency of their survival estimates. The duality of lifetime and truncation in the absence of censoring enables methods for testing goodness of fit of the lifetime distribution to be used for testing goodness of fit of the truncation distribution. However, under random censoring, this duality does not hold and different tests are required. In this article, we introduce several goodness-of-fit tests for the truncation distribution and investigate their performance in the presence of censored event times using simulation. We demonstrate the use of our tests on two data sets.     

Nonparametric bayes testing of changes in a response distribution with an ordinal predictor

Summary .   In certain biomedical studies, one may anticipate changes in the shape of a response distribution across the levels of an ordinal predictor. For instance, in toxicology studies, skewness and modality might change as dose increases. To address this issue, we propose a Bayesian nonparametric method for testing for distribution changes across an ordinal predictor. Using a dynamic mixture of Dirichlet processes, we allow the response distribution to change flexibly at each level of the predictor. In addition, by assigning mixture priors to the hyperparameters, we can obtain posterior probabilities of no effect of the predictor and identify the lowest dose level for which there is an appreciable change in distribution. The method also provides a natural framework for performing tests across multiple outcomes. We apply our method to data from a genotoxicity experiment.     

TESTING FOR UNEQUAL AMOUNTS OF EVOLUTION IN A CONTINUOUS CHARACTER ON DIFFERENT BRANCHES OF A PHYLOGENETIC TREE USING LINEAR AND SQUARED-CHANGE PARSIMONY: AN EXAMPLE USING LESSER ANTILLEAN ANOLIS LIZARDS

Although a large body of work investigating tests of correlated evolution of two continuous characters exists, hypotheses such as character displacement are really tests of whether substantial evolutionary change has occurred on a particular branch or branches of the phylogenetic tree. In this study, we present a methodology for testing such a hypothesis using ancestral character state reconstruction and simulation. Furthermore, we suggest how to investigate the robustness of the hypothesis test by varying the reconstruction methods or simulation parameters. As a case study, we tested a hypothesis of character displacement in body size of Caribbean Anolis lizards. We compared squared-change, weighted squared-change, and linear parsimony reconstruction methods, gradual Brownian motion and speciational models of evolution, and several resolution methods for linear parsimony. We used ancestor reconstruction methods to infer the amount of body size evolution, and tested whether evolutionary change in body size was greater on branches of the phylogenetic tree in which a transition from occupying a single-species island to a two-species island occurred. Simulations were used to generate null distributions of reconstructed body size change. The hypothesis of character displacement was tested using Wilcoxon Rank-Sums. When tested against simulated null distributions, all of the reconstruction methods resulted in more significant P-values than when standard statistical tables were used. These results confirm that P-values for tests using ancestor reconstruction methods should be assessed via simulation rather than from standard statistical tables. Linear parsimony can produce an infinite number of most parsimonious reconstructions in continuous characters. We present an example of assessing the robustness of our statistical test by exploring the sample space of possible resolutions. We compare ACCTRAN and DELTRAN resolutions of ambiguous character reconstructions in linear parsimony to the most and least conservative resolutions for our particular hypothesis.     

Lipid-dependent Generation of Dual Topology for a Membrane Protein

The mechanism by which membrane proteins exhibit structural and functional duality in the same membrane or different membranes is unknown. We posit that such duality is determined by both the protein sequence and the membrane lipid composition wherein a spatial or temporal change in the latter can result in a post-assembly change in protein structure and function. To investigate whether co-existence of multiple topological conformers is dependent on the membrane lipid composition, we determined the topological organization of lactose permease in an Escherichia coli model cell system in which phosphatidylethanolamine membrane content can be systematically varied. At intermediate levels of phosphatidylethanolamine a mixture of native and topologically mis-oriented conformers co-existed. There was no threshold level of phosphatidylethanolamine determining a sharp transition from one conformer to the other. Co-existing conformers were not in rapid equilibrium at a static lipid composition indicating that duality of topology is established during an early folding step. Depletion of intermediate levels of phosphatidylethanolamine after final protein assembly resulted in complete mis-orientation of the native conformer. Combined with previous results, such topological dynamics are reversible in both directions. We propose a thermodynamically based model for how lipid-protein interactions can result in a mixed topological organization and how changes in lipid composition can result in changes in the ratio of topologically distinct conformers of proteins. These observations demonstrate a potential lipid-dependent biological switch for generating dynamic structural and functional heterogeneity for a protein within the same membrane or between different membranes in more complex eukaryotic cells.     

Inferring gene regulatory networks using differential evolution with local search heuristics

We present a memetic algorithm for evolving the structure of biomolecular interactions and inferring the effective kinetic parameters from the time series data of gene expression using the decoupled Ssystem formalism. We propose an Information Criteria based fitness evaluation for gene network model selection instead of the conventional Mean Squared Error (MSE) based fitness evaluation. A hill-climbing local-search method has been incorporated in our evolutionary algorithm for efficiently attaining the skeletal architecture which is most frequently observed in biological networks. The suitability of the method is tested in gene circuit reconstruction experiments, varying the network dimension and/or characteristics, the amount of gene expression data used for inference and the noise level present in expression profiles. The reconstruction method inferred the network topology and the regulatory parameters with high accuracy. Nevertheless, the performance is limited to the amount of expression data used and the noise level present in the data. The proposed fitness function has been found more suitable for identifying correct network topology and for estimating the accurate parameter values compared to the existing ones. Finally, we applied the methodology for analyzing the cell-cycle gene expression data of budding yeast and reconstructed the network of some key regulators.     

Horned lizard (Phrynosoma) phylogeny inferred from mitochondrial genes and morphological characters: understanding conflicts using multiple approaches

The genus Phrynosoma includes 13 species of North American lizards characterized by unique and highly derived morphologies and ecologies. Understanding interspecific relationships within this genus is essential for testing hypotheses about character evolution in this group. We analyzed mitochondrial ND4 and cytochrome b gene sequence data from all species of Phrynosoma in conjunction with a previously published dataset including 12S and 16S rRNA gene sequences and morphological characters. We used multiple phylogenetic methods and diagnostic tests for data combinability and taxonomic congruence to investigate the data in separate and combined analyses. Separate data partitions resulted in several well-supported lineages, but taxonomic congruence was lacking between topologies from separate and combined analyses. Partitioned Bremer support analyses also reveals conflict between data partitions in certain tree regions. When taxa associated with well-supported clades were removed from analyses, phylogenetic signal was lost. Combined, our results initially suggest conflict between data partitions, but further tests show the data are only appropriate for phylogenetic reconstruction of those parts of the topology that were well resolved. Nonetheless, our data analyses reveal five well-supported clades: (1) Phrynosoma ditmarsi and Phrynosoma hernandesi, (2) P. ditmarsi, P. hernandesi, and Phrynosoma douglasii, (3) P. ditmarsi, P. hernandesi, P. douglasii, and Phrynosoma orbiculare, (4) Phrynosoma mcallii and Phrynosoma platyrhinos, and (5) Phrynosoma braconnieri and Phrynosoma taurus.     

A Mixture Model for Quantum Dot Images of Kinesin Motor Assays

Summary We introduce a nearly automatic procedure to locate and count the quantum dots in images of kinesin motor assays. Our procedure employs an approximate likelihood estimator based on a two‐component mixture model for the image data; the first component has a normal distribution, and the other component is distributed as a normal random variable plus an exponential random variable. The normal component has an unknown variance, which we model as a function of the mean. We use B‐splines to estimate the variance function during a training run on a suitable image, and the estimate is used to process subsequent images. Parameter estimates are generated for each image along with estimates of standard errors, and the number of dots in the image is determined using an information criterion and likelihood ratio tests. Realistic simulations show that our procedure is robust and that it leads to accurate estimates, both of parameters and of standard errors.     

Phylogenetic relationships and heterogeneous evolutionary processes among phrynosomatine sand lizards (Squamata, Iguanidae) revisited

Phylogenetic analyses of DNA sequences were conducted to evaluate four alternative hypotheses of phrynosomatine sand lizard relationships. Sequences comprising 2871 aligned base pair positions representing the regions spanning ND1-COI and cyt b-tRNA(Thr) of the mitochondrial genome from all recognized sand lizard species were analyzed using unpartitioned parsimony and likelihood methods, likelihood methods with assumed partitions, Bayesian methods with assumed partitions, and Bayesian mixture models. The topology (Uma, (Callisaurus, (Cophosaurus, Holbrookia))) and thus monophyly of the "earless" taxa, Cophosaurus and Holbrookia, is supported by all analyses. Previously proposed topologies in which Uma and Callisaurus are sister taxa and those in which Holbrookia is the sister group to all other sand lizard taxa are rejected using both parsimony and likelihood-based significance tests with the combined, unparitioned data set. Bayesian hypothesis tests also reject those topologies using six assumed partitioning strategies, and the two partitioning strategies presumably associated with the most powerful tests also reject a third previously proposed topology, in which Callisaurus and Cophosaurus are sister taxa. For both maximum likelihood and Bayesian methods with assumed partitions, those partitions defined by codon position and tRNA stem and nonstems explained the data better than other strategies examined. Bayes factor estimates comparing results of assumed partitions versus mixture models suggest that mixture models perform better than assumed partitions when the latter were not based on functional characteristics of the data, such as codon position and tRNA stem and nonstems. However, assumed partitions performed better than mixture models when functional differences were incorporated. We reiterate the importance of accounting for heterogeneous evolutionary processes in the analysis of complex data sets and emphasize the importance of implementing mixed model likelihood methods.     

Bayesian estimation of concordance among gene trees

Multigene sequence data have great potential for elucidating important and interesting evolutionary processes, but statistical methods for extracting information from such data remain limited. Although various biological processes may cause different genes to have different genealogical histories (and hence different tree topologies), we also may expect that the number of distinct topologies among a set of genes is relatively small compared with the number of possible topologies. Therefore evidence about the tree topology for one gene should influence our inferences of the tree topology on a different gene, but to what extent? In this paper, we present a new approach for modeling and estimating concordance among a set of gene trees given aligned molecular sequence data. Our approach introduces a one-parameter probability distribution to describe the prior distribution of concordance among gene trees. We describe a novel 2-stage Markov chain Monte Carlo (MCMC) method that first obtains independent Bayesian posterior probability distributions for individual genes using standard methods. These posterior distributions are then used as input for a second MCMC procedure that estimates a posterior distribution of gene-to-tree maps (GTMs). The posterior distribution of GTMs can then be summarized to provide revised posterior probability distributions for each gene (taking account of concordance) and to allow estimation of the proportion of the sampled genes for which any given clade is true (the sample-wide concordance factor). Further, under the assumption that the sampled genes are drawn randomly from a genome of known size, we show how one can obtain an estimate, with credibility intervals, on the proportion of the entire genome for which a clade is true (the genome-wide concordance factor). We demonstrate the method on a set of 106 genes from 8 yeast species.     

Statistical Tests for Associations between Two Directed Acyclic Graphs

Biological data, and particularly annotation data, are increasingly being represented in directed acyclic graphs (DAGs). However, while relevant biological information is implicit in the links between multiple domains, annotations from these different domains are usually represented in distinct, unconnected DAGs, making links between the domains represented difficult to determine. We develop a novel family of general statistical tests for the discovery of strong associations between two directed acyclic graphs. Our method takes the topology of the input graphs and the specificity and relevance of associations between nodes into consideration. We apply our method to the extraction of associations between biomedical ontologies in an extensive use-case. Through a manual and an automatic evaluation, we show that our tests discover biologically relevant relations. The suite of statistical tests we develop for this purpose is implemented and freely available for download.     

Hierarchical phylogenetic models for analyzing multipartite sequence data

Debate exists over how to incorporate information from multipartite sequence data in phylogenetic analyses. Strict combined-data approaches argue for concatenation of all partitions and estimation of one evolutionary history, maximizing the explanatory power of the data. Consensus/independence approaches endorse a two-step procedure where partitions are analyzed independently and then a consensus is determined from the multiple results. Mixtures across the model space of a strict combined-data approach and a priori independent parameters are popular methods to integrate these methods. We propose an alternative middle ground by constructing a Bayesian hierarchical phylogenetic model. Our hierarchical framework enables researchers to pool information across data partitions to improve estimate precision in individual partitions while permitting estimation and testing of tendencies in across-partition quantities. Such across-partition quantities include the distribution from which individual topologies relating the sequences within a partition are drawn. We propose standard hierarchical priors on continuous evolutionary parameters across partitions, while the structure on topologies varies depending on the research problem. We illustrate our model with three examples. We first explore the evolutionary history of the guinea pig (Cavia porcellus) using alignments of 13 mitochondrial genes. The hierarchical model returns substantially more precise continuous parameter estimates than an independent parameter approach without losing the salient features of the data. Second, we analyze the frequency of horizontal gene transfer using 50 prokaryotic genes. We assume an unknown species-level topology and allow individual gene topologies to differ from this with a small estimable probability. Simultaneously inferring the species and individual gene topologies returns a transfer frequency of 17%. We also examine HIV sequences longitudinally sampled from HIV+ patients. We ask whether posttreatment development of CCR5 coreceptor virus represents concerted evolution from middisease CXCR4 virus or reemergence of initial infecting CCR5 virus. The hierarchical model pools partitions from multiple unrelated patients by assuming that the topology for each patient is drawn from a multinomial distribution with unknown probabilities. Preliminary results suggest evolution and not reemergence.     

Structural and mechanistic analyses of endo-glycoceramidase II, a membrane-associated family 5 glycosidase in the Apo and GM3 ganglioside-bound forms

endo-Glycoceramidase, a membrane-associated family 5 glycosidase, deviates from the typical polysaccharide substrate specificity of other soluble members of the family, preferentially hydrolyzing glycosidic linkages between the oligosaccharide and ceramide moieties of gangliosides. Here we report the first x-ray crystal structures of an endo-glycoceramidase from Rhodococcus sp., in the apo form, in complex with the ganglioside G(M3) (Svennerholm ganglioside nomenclature (Svennerholm, L. (1964) J. Lipid Res. 5, 145-155)), and trapped as a glycosyl-enzyme intermediate. These snapshots provide the first molecular insight into enzyme recognition and association with gangliosides, revealing the structural adaptations necessary for glycosidase-catalyzed hydrolysis and detailing a novel ganglioside binding topology. Consistent with the chemical duality of the substrate, the active site of endo-glycoceramidase is split into a wide, polar cavity to bind the polyhydroxylated oligosaccharide moiety and a narrow, hydrophobic tunnel to bind the ceramide lipid chains. The specific interactions with the ceramide polar head group manifest a surprising aglycone specificity, an observation substantiated by our kinetic analyses. Collectively, the reported structural and kinetic data provide insight toward rational redesign of the synthetic glycosynthase mutant of endo-glycoceramidase to enable facile synthesis of nonnatural, therapeutically useful gangliosides.