The probability of a gene tree topology within a phylogenetic network with applications to hybridization detection

Gene tree topologies have proven a powerful data source for various tasks, including species tree inference and species delimitation. Consequently, methods for computing probabilities of gene trees within species trees have been developed and widely used in probabilistic inference frameworks. All these methods assume an underlying multispecies coalescent model. However, when reticulate evolutionary events such as hybridization occur, these methods are inadequate, as they do not account for such events. Methods that account for both hybridization and deep coalescence in computing the probability of a gene tree topology currently exist for very limited cases. However, no such methods exist for general cases, owing primarily to the fact that it is currently unknown how to compute the probability of a gene tree topology within the branches of a phylogenetic network. Here we present a novel method for computing the probability of gene tree topologies on phylogenetic networks and demonstrate its application to the inference of hybridization in the presence of incomplete lineage sorting. We reanalyze a Saccharomyces species data set for which multiple analyses had converged on a species tree candidate. Using our method, though, we show that an evolutionary hypothesis involving hybridization in this group has better support than one of strict divergence. A similar reanalysis on a group of three Drosophila species shows that the data is consistent with hybridization. Further, using extensive simulation studies, we demonstrate the power of gene tree topologies at obtaining accurate estimates of branch lengths and hybridization probabilities of a given phylogenetic network. Finally, we discuss identifiability issues with detecting hybridization, particularly in cases that involve extinction or incomplete sampling of taxa.     

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.     

A graphical method for detecting recombination in phylogenetic data sets

Current phylogenetic tree reconstruction methods assume that there is a single underlying tree topology for all sites along the sequence. The presence of mosaic sequences due to recombination violates this assumption and will cause phylogenetic methods to give misleading results due to the imposition of a single tree topology on all sites. The detection of mosaic sequences caused by recombination is therefore an important first step in phylogenetic analysis. A graphical method for the detection of recombination, based on the least squares method of phylogenetic estimation, is presented here. This method locates putative recombination breakpoints by moving a window along the sequence. The performance of the method is assessed by simulation and by its application to a real data set.      

Phylogeny of mixture models: robustness of maximum likelihood and non-identifiable distributions.

We address phylogenetic reconstruction when the data is generated from a mixture distribution. Such topics have gained considerable attention in the biological community with the clear evidence of heterogeneity of mutation rates. In our work we consider data coming from a mixture of trees which share a common topology, but differ in their edge weights (i.e., branch lengths). We first show the pitfalls of popular methods, including maximum likelihood and Markov chain Monte Carlo algorithms. We then determine in which evolutionary models, reconstructing the tree topology, under a mixture distribution, is (im)possible. We prove that every model whose transition matrices can be parameterized by an open set of multilinear polynomials, either has non-identifiable mixture distributions, in which case reconstruction is impossible in general, or there exist linear tests which identify the topology. This duality theorem, relies on our notion of linear tests and uses ideas from convex programming duality. Linear tests are closely related to linear invariants, which were first introduced by Lake, and are natural from an algebraic geometry perspective.     

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.     

The Phylogenetic Utility of the Codon-Degeneracy Model

The codon-degeneracy model (CDM) predicts relative frequencies of substitution for any set of homologous protein-coding DNA sequences based on patterns of nucleotide degeneracy, codon composition, and the assumption of selective neutrality. However, at present, the CDM is reliant on outside estimates of transition bias. A new method by which the power of the CDM can be used to find a synonymous transition bias that is optimal for any given phylogenetic tree topology is presented. An example is illustrated that utilizes optimized transition biases to generate CDM GF-scores for every possible phylogenetic tree for pocket gophers of the genus Orthogeomys. The resulting distribution of CDM GF-scores is compared and contrasted with the results of maximum parsimony and maximum likelihood methods. Although convergence on a single tree topology by the CDM and another method indicates greater support for that particular tree, the value of CDM GF-score as the sole optimality criterion for phylogeny reconstruction remains to be determined. It is clear, however, that the a priori estimation of an optimum transition bias from codon composition has a direct application to differentiating between alternative trees. Received: 13 October 1999 / Accepted: 28 April 2000     

Accounting for uncertainty in the tree topology has little effect on the decision-theoretic approach to model selection in phylogeny estimation

Currently available methods for model selection used in phylogenetic analysis are based on an initial fixed-tree topology. Once a model is picked based on this topology, a rigorous search of the tree space is run under that model to find the maximum-likelihood estimate of the tree (topology and branch lengths) and the maximum-likelihood estimates of the model parameters. In this paper, we propose two extensions to the decision-theoretic (DT) approach that relax the fixed-topology restriction. We also relax the fixed-topology restriction for the Bayesian information criterion (BIC) and the Akaike information criterion (AIC) methods. We compare the performance of the different methods (the relaxed, restricted, and the likelihood-ratio test [LRT]) using simulated data. This comparison is done by evaluating the relative complexity of the models resulting from each method and by comparing the performance of the chosen models in estimating the true tree. We also compare the methods relative to one another by measuring the closeness of the estimated trees corresponding to the different chosen models under these methods. We show that varying the topology does not have a major impact on model choice. We also show that the outcome of the two proposed extensions is identical and is comparable to that of the BIC, Extended-BIC, and DT. Hence, using the simpler methods in choosing a model for analyzing the data is more computationally feasible, with results comparable to the more computationally intensive methods. Another outcome of this study is that earlier conclusions about the DT approach are reinforced. That is, LRT, Extended-AIC, and AIC result in more complicated models that do not contribute to the performance of the phylogenetic inference, yet cause a significant increase in the time required for data analysis.     

A Bayesian model for detecting past recombination events in DNA multiple alignments.

Most phylogenetic tree estimation methods assume that there is a single set of hierarchical relationships among sequences in a data set for all sites along an alignment. Mosaic sequences produced by past recombination events will violate this assumption and may lead to misleading results from a phylogenetic analysis due to the imposition of a single tree along the entire alignment. Therefore, the detection of past recombination is an important first step in an analysis. A Bayesian model for the changes in topology caused by recombination events is described here. This model relaxes the assumption of one topology for all sites in an alignment and uses the theory of Hidden Markov models to facilitate calculations, the hidden states being the underlying topologies at each site in the data set. Changes in topology along the multiple sequence alignment are estimated by means of the maximum a posteriori (MAP) estimate. The performance of the MAP estimate is assessed by application of the model to data sets of four sequences, both simulated and real.     

Target-capture phylogenomics provide insights on gene and species tree discordances in Old World treefrogs (Anura: Rhacophoridae)

Genome-scale data have greatly facilitated the resolution of recalcitrant nodes that Sanger-based datasets have been unable to resolve. However, phylogenomic studies continue to use traditional methods such as bootstrapping to estimate branch support; and high bootstrap values are still interpreted as providing strong support for the correct topology. Furthermore, relatively little attention has been given to assessing discordances between gene and species trees, and the underlying processes that produce phylogenetic conflict. We generated novel genomic datasets to characterize and determine the causes of discordance in Old World treefrogs (Family: Rhacophoridae)—a group that is fraught with conflicting and poorly supported topologies among major clades. Additionally, a suite of data filtering strategies and analytical methods were applied to assess their impact on phylogenetic inference. We showed that incomplete lineage sorting was detected at all nodes that exhibited high levels of discordance. Those nodes were also associated with extremely short internal branches. We also clearly demonstrate that bootstrap values do not reflect uncertainty or confidence for the correct topology and, hence, should not be used as a measure of branch support in phylogenomic datasets. Overall, we showed that phylogenetic discordances in Old World treefrogs resulted from incomplete lineage sorting and that species tree inference can be improved using a multi-faceted, total-evidence approach, which uses the most amount of data and considers results from different analytical methods and datasets.     

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.     

Phylogenomics reveal a robust fungal tree of life

Our understanding of the tree of life (TOL) is still fragmentary. Until recently, molecular phylogeneticists have built trees based on ribosomal RNA sequences and selected protein sequences, which, however, usually suffered from lack of support for the deeper branches and inconsistencies probably due to limited subsampling of the entire genome. Now, phylogenetic hypotheses can be based on the analysis of full genomes. We used available complete genome data as well as the eukaryote orthologous group (KOG) proteins to reconstruct with confidence basal branches of the fungal TOL. Phylogenetic analysis of a core of 531 KOGs shared among 21 fungal genomes, three animal genomes and one plant genome showed a single tree with high support resulting from four different methods of phylogenetic reconstruction. The single tree that we inferred from our dataset showed excellent nodal support for each branch, suggesting that it reflects the true phylogenetic relationships of the species involved.     

A polynomial time algorithm for calculating the probability of a ranked gene tree given a species tree

ABSTRACT: BACKGROUND: The ancestries of genes form gene trees which do not necessarily have the same topology as the species tree due to incomplete lineage sorting. Available algorithms determining the probability of a gene tree given a species tree require exponential computational runtime. RESULTS: In this paper, we provide a polynomial time algorithm to calculate the probability of a ranked gene tree topology for a given species tree, where a ranked tree topology is a tree topology with the internal vertices being ordered. The probability of a gene tree topology can thus be calculated in polynomial time if the number of orderings of the internal vertices is a polynomial number. However, the complexity of calculating the probability of a gene tree topology with an exponential number of rankings for a given species tree remains unknown. CONCLUSIONS: Polynomial algorithms for calculating ranked gene tree probabilities may become useful in developing methodology to infer species trees based on a collection of gene trees, leading to a more accurate reconstruction of ancestral species relationships.     

Effects of taxon sampling and tree reconstruction methods on phylodiversity metrics

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Experimental design criteria in phylogenetics: where to add taxa

Accurate phylogenetic inference is a topic of intensive research and debate and has been studied in response to many different factors: for example, differences in the method of reconstruction, the shape of the underlying tree, the substitution model, and varying quantities and types of data. Investigating whether the conditions used might lead to inaccurate inference has been attempted through elaborate data exploration but less attention has been given to creating a unified methodology to enable experimental designs in phylogenetic analysis to be improved and so avoid suboptimal conditions. Experimental design has been part of the field of statistics since the seminal work of Fisher in the early 20th century and a large body of literature exists on how to design optimum experiments. Here we investigate the use of the Fisher information matrix to decide between candidate positions for adding a taxon to a fixed topology, and introduce a parameter transformation that permits comparison of these different designs. This extension to Goldman (1998. Proc. R. Soc. Lond. B. 265: 1779-1786) thus allows investigation of "where to add taxa" in a phylogeny. We compare three different measures of the total information for selecting the position to add a taxon to a tree. Our methods are illustrated by investigating the behavior of the three criteria when adding a branch to model trees, and by applying the different criteria to two biological examples: a simplified taxon-sampling problem in the balsaminoid Ericales and the phylogeny of seed plants.     

Guided tree topology proposals for Bayesian phylogenetic inference

Increasingly, large data sets pose a challenge for computationally intensive phylogenetic methods such as Bayesian Markov chain Monte Carlo (MCMC). Here, we investigate the performance of common MCMC proposal distributions in terms of median and variance of run time to convergence on 11 data sets. We introduce two new Metropolized Gibbs Samplers for moving through "tree space." MCMC simulation using these new proposals shows faster average run time and dramatically improved predictability in performance, with a 20-fold reduction in the variance of the time to estimate the posterior distribution to a given accuracy. We also introduce conditional clade probabilities and demonstrate that they provide a superior means of approximating tree topology posterior probabilities from samples recorded during MCMC.     

The identifiability of tree topology for phylogenetic models, including covarion and mixture models.

For a model of molecular evolution to be useful for phylogenetic inference, the topology of evolutionary trees must be identifiable. That is, from a joint distribution the model predicts, it must be possible to recover the tree parameter. We establish tree identifiability for a number of phylogenetic models, including a covarion model and a variety of mixture models with a limited number of classes. The proof is based on the introduction of a more general model, allowing more states at internal nodes of the tree than at leaves, and the study of the algebraic variety formed by the joint distributions to which it gives rise. Tree identifiability is first established for this general model through the use of certain phylogenetic invariants.     

Orthonormal transform to decompose the variance of a life-history trait across a phylogenetic tree

In recent years, there has been an increased interest in studying the variability of a quantitative life-history trait across a set of species sharing a common phylogeny. However, such studies have suffered from an insufficient development of statistical methods aimed at decomposing the trait variance with respect to the topological structure of the tree. Here we propose a new and generic approach that expresses the topological properties of the phylogenetic tree via an orthonormal basis, which is further used to decompose the trait variance. Such a decomposition provides a structure function, referred to as an "orthogram," which is relevant to characterize in both graphical and statistical aspects the dependence of trait values on the topology of the tree ("phylogenetic dependence"). We also propose four complementary test statistics to be computed from orthogram values that help to diagnose both the intensity and the nature of phylogenetic dependence. The relevance of the method is illustrated by the analysis of three phylogenetic data sets, drawn from the literature and typifying contrasted levels and aspects of phylogenetic dependence. Freely available routines which have been programmed in the R framework are also proposed.     

cBrother: relaxing parental tree assumptions for Bayesian recombination detection

Bayesian multiple change-point models accurately detect recombination in molecular sequence data. Previous Java-based implementations assume a fixed topology for the representative parental data. cBrother is a novel C language implementation that capitalizes on reduced computational time to relax the fixed tree assumption. We show that cBrother is 19 times faster than its predecessor and the fixed tree assumption can influence estimates of recombination in a medically-relevant dataset. Availability: cBrother can be freely downloaded from http://www.biomath.org/dormanks/ and can be compiled on Linux, Macintosh and Windows operating systems. Online documentation and a tutorial are also available at the site.     

Phylogenomics and the flowering plant tree of life

The advances accelerated by next-generation sequencing and long-read sequencing technologies continue to provide an impetus for plant phylogenetic study.In the past decade,a large number of phylogenetic studies adopting hundreds to thousands of genes across a wealth of clades have emerged and ushered plant phylogenetics and evolution into a new era.In the meantime,a roadmap for researchers when making decisions across different approaches for their phylogenomic research design is imminent.This r...     

Species trees from gene trees: reconstructing Bayesian posterior distributions of a species phylogeny using estimated gene tree distributions

The desire to infer the evolutionary history of a group of species should be more viable now that a considerable amount of multilocus molecular data is available. However, the current molecular phylogenetic paradigm still reconstructs gene trees to represent the species tree. Further, commonly used methods of combining data, such as the concatenation method, are known to be inconsistent in some circumstances. In this paper, we propose a Bayesian hierarchical model to estimate the phylogeny of a group of species using multiple estimated gene tree distributions, such as those that arise in a Bayesian analysis of DNA sequence data. Our model employs substitution models used in traditional phylogenetics but also uses coalescent theory to explain genealogical signals from species trees to gene trees and from gene trees to sequence data, thereby forming a complete stochastic model to estimate gene trees, species trees, ancestral population sizes, and species divergence times simultaneously. Our model is founded on the assumption that gene trees, even of unlinked loci, are correlated due to being derived from a single species tree and therefore should be estimated jointly. We apply the method to two multilocus data sets of DNA sequences. The estimates of the species tree topology and divergence times appear to be robust to the prior of the population size, whereas the estimates of effective population sizes are sensitive to the prior used in the analysis. These analyses also suggest that the model is superior to the concatenation method in fitting these data sets and thus provides a more realistic assessment of the variability in the distribution of the species tree that may have produced the molecular information at hand. Future improvements of our model and algorithm should include consideration of other factors that can cause discordance of gene trees and species trees, such as horizontal transfer or gene duplication.