Robustness of maximum likelihood tree estimation against different patterns of base substitutions

Summary In the maximum likelihood (ML) method for estimating a molecular phylogenetic tree, the pattern of nucleotide substitutions for computing likelihood values is assumed to be simpler than that of the actual evolutionary process, simply because the process, considered to be quite devious, is unknown. The problem, however, is that there has been no guarantee to endorse the simplification.To study this problem, we first evaluated the robustness of the ML method in the estimation of molecular trees against different nucleotide substitution patterns, including Jukes and Cantor's, the simplest ever proposed. Namely, we conducted computer simulations in which we could set up various evolutionary models of a hypothetical gene, and define a true tree to which an estimated tree by the ML method was to be compared. The results show that topology estimation by the ML method is considerably robust against different ratios of transitions to transversions and different GC contents, but branch length estimation is not so. The ML tree estimation based on Jukes and Cantor's model is also revealed to be resistant to GC content, but rather sensitive to the ratio of transitions to transversions.We then applied the ML method with different substitution patterns to nucleotide sequence data ontax gene from T-cell leukemia viruses whose evolutionary process must have been more complicated than that of the hypothetical gene. The results are in accordance with those from the simulation study, showing that Jukes and Cantor's model is as useful as a more complicated one for making inferences about molecular phylogeny of the viruses.     

Treelength optimization for phylogeny estimation

The standard approach to phylogeny estimation uses two phases, in which the first phase produces an alignment on a set of homologous sequences, and the second phase estimates a tree on the multiple sequence alignment. POY, a method which seeks a tree/alignment pair minimizing the total treelength, is the most widely used alternative to this two-phase approach. The topological accuracy of trees computed under treelength optimization is, however, controversial. In particular, one study showed that treelength optimization using simple gap penalties produced poor trees and alignments, and suggested the possibility that if POY were used with an affine gap penalty, it might be able to be competitive with the best two-phase methods. In this paper we report on a study addressing this possibility. We present a new heuristic for treelength, called BeeTLe (Better Treelength), that is guaranteed to produce trees at least as short as POY. We then use this heuristic to analyze a large number of simulated and biological datasets, and compare the resultant trees and alignments to those produced using POY and also maximum likelihood (ML) and maximum parsimony (MP) trees computed on a number of alignments. In general, we find that trees produced by BeeTLe are shorter and more topologically accurate than POY trees, but that neither POY nor BeeTLe produces trees as topologically accurate as ML trees produced on standard alignments. These findings, taken as a whole, suggest that treelength optimization is not as good an approach to phylogenetic tree estimation as maximum likelihood based upon good alignment methods.     

SATe-II: very fast and accurate simultaneous estimation of multiple sequence alignments and phylogenetic trees

Highly accurate estimation of phylogenetic trees for large data sets is difficult, in part because multiple sequence alignments must be accurate for phylogeny estimation methods to be accurate. Coestimation of alignments and trees has been attempted but currently only SATé estimates reasonably accurate trees and alignments for large data sets in practical time frames (Liu K., Raghavan S., Nelesen S., Linder C.R., Warnow T. 2009b. Rapid and accurate large-scale coestimation of sequence alignments and phylogenetic trees. Science. 324:1561-1564). Here, we present a modification to the original SATé algorithm that improves upon SATé (which we now call SATé-I) in terms of speed and of phylogenetic and alignment accuracy. SATé-II uses a different divide-and-conquer strategy than SATé-I and so produces smaller more closely related subsets than SATé-I; as a result, SATé-II produces more accurate alignments and trees, can analyze larger data sets, and runs more efficiently than SATé-I. Generally, SATé is a metamethod that takes an existing multiple sequence alignment method as an input parameter and boosts the quality of that alignment method. SATé-II-boosted alignment methods are significantly more accurate than their unboosted versions, and trees based upon these improved alignments are more accurate than trees based upon the original alignments. Because SATé-I used maximum likelihood (ML) methods that treat gaps as missing data to estimate trees and because we found a correlation between the quality of tree/alignment pairs and ML scores, we explored the degree to which SATé's performance depends on using ML with gaps treated as missing data to determine the best tree/alignment pair. We present two lines of evidence that using ML with gaps treated as missing data to optimize the alignment and tree produces very poor results. First, we show that the optimization problem where a set of unaligned DNA sequences is given and the output is the tree and alignment of those sequences that maximize likelihood under the Jukes-Cantor model is uninformative in the worst possible sense. For all inputs, all trees optimize the likelihood score. Second, we show that a greedy heuristic that uses GTR+Gamma ML to optimize the alignment and the tree can produce very poor alignments and trees. Therefore, the excellent performance of SATé-II and SATé-I is not because ML is used as an optimization criterion for choosing the best tree/alignment pair but rather due to the particular divide-and-conquer realignment techniques employed.     

ImOSM: intermittent evolution and robustness of phylogenetic methods

Among the criteria to evaluate the performance of a phylogenetic method, robustness to model violation is of particular practical importance as complete a priori knowledge of evolutionary processes is typically unavailable. For studies of robustness in phylogenetic inference, a utility to add well-defined model violations to the simulated data would be helpful. We therefore introduce ImOSM, a tool to imbed intermittent evolution as model violation into an alignment. Intermittent evolution refers to extra substitutions occurring randomly on branches of a tree, thus changing alignment site patterns. This means that the extra substitutions are placed on the tree after the typical process of sequence evolution is completed. We then study the robustness of widely used phylogenetic methods: maximum likelihood (ML), maximum parsimony (MP), and a distance-based method (BIONJ) to various scenarios of model violation. Violation of rates across sites (RaS) heterogeneity and simultaneous violation of RaS and the transition/transversion ratio on two nonadjacent external branches hinder all the methods recovery of the true topology for a four-taxon tree. For an eight-taxon balanced tree, the violations cause each of the three methods to infer a different topology. Both ML and MP fail, whereas BIONJ, which calculates the distances based on the ML estimated parameters, reconstructs the true tree. Finally, we report that a test of model homogeneity and goodness of fit tests have enough power to detect such model violations. The outcome of the tests can help to actually gain confidence in the inferred trees. Therefore, we recommend using these tests in practical phylogenetic analyses.     

Multiple maxima of likelihood in phylogenetic trees: an analytic approach

Maximum likelihood (ML) is a widely used criterion for selecting optimal evolutionary trees. However, the nature of the likelihood surface for trees is still not sufficiently understood, especially with regard to the frequency of multiple optima. Here, we initiate an analytic study for identifying sequences that generate multiple optima. We concentrate on the problem of optimizing edge weights for a given tree or trees (as opposed to searching through the space of all trees). We report a new approach to computing ML directly, which we have used to find large families of sequences that have multiple optima, including sequences with a continuum of optimal points. Such data sets are best supported by different (two or more) phylogenies that vary significantly in their timings of evolutionary events. Some standard biological processes can lead to data with multiple optima, and consequently the field needs further investigation. Our results imply that hill-climbing techniques as currently implemented in various software packages cannot guarantee that one will find the global ML point, even if it is unique.     

Reanalysis of Murphy et al.'s data gives various mammalian phylogenies and suggests overcredibility of Bayesian trees

Murphy and colleagues reported that the mammalian phylogeny was resolved by Bayesian phylogenetics. However, the DNA sequences they used had many alignment gaps and undetermined nucleotide sites. We therefore reanalyzed their data by minimizing unshared nucleotide sites and retaining as many species as possible (13 species). In constructing phylogenetic trees, we used the Bayesian, maximum likelihood (ML), maximum parsimony (MP), and neighbor-joining (NJ) methods with different substitution models. These trees were constructed by using both protein and DNA sequences. The results showed that the posterior probabilities for Bayesian trees were generally much higher than the bootstrap values for ML, MP, and NJ trees. Two different Bayesian topologies for the same set of species were sometimes supported by high posterior probabilities, implying that two different topologies can be judged to be correct by Bayesian phylogenetics. This suggests that the posterior probability in Bayesian analysis can be excessively high as an indication of statistical confidence and therefore Murphy et al.'s tree, which largely depends on Bayesian posterior probability, may not be correct.     

Empirical problems of the hierarchical likelihood ratio test for model selection

Advocates of maximum likelihood (ML) approaches to phylogenetics commonly cite as one of their primary advantages the use of objective statistical criteria for model selection. Currently, a particular implementation of the likelihood ratio test (LRT) is the most commonly used model-selection criterion in phylogenetics. This approach requires the choice of a starting point and a parameter addition (or removal) sequence that can affect all ML inferences (i.e., topology, model, and all evolutionary parameters). Here, several alternative starting points and parameter sequences are tested in empirical data sets to assess their influence on model selection and optimal topology. In the studied data sets, varying model-selection protocols leads to selection of different models that, in some cases, lead to different ML trees. Given the sensitivity of the LRT, some possible solutions to model selection (within the hypothesis testing approach) are outlined, and alternative model-selection criteria are discussed. Some of the suggested alternatives seem to lack these problems, although their behavior and adequacy for phylogenetics needs to be further explored.     

Phylogenetic Inference with Weighted Codon Evolutionary Distances

We develop a new approach to estimate a matrix of pairwise evolutionary distances from a codon-based alignment based on a codon evolutionary model. The method first computes a standard distance matrix for each of the three codon positions. Then these three distance matrices are weighted according to an estimate of the global evolutionary rate of each codon position and averaged into a unique distance matrix. Using a large set of both real and simulated codon-based alignments of nucleotide sequences, we show that this approach leads to distance matrices that have a significantly better treelikeness compared to those obtained by standard nucleotide evolutionary distances. We also propose an alternative weighting to eliminate the part of the noise often associated with some codon positions, particularly the third position, which is known to induce a fast evolutionary rate. Simulation results show that fast distance-based tree reconstruction algorithms on distance matrices based on this codon position weighting can lead to phylogenetic trees that are at least as accurate as, if not better, than those inferred by maximum likelihood. Finally, a well-known multigene dataset composed of eight yeast species and 106 codon-based alignments is reanalyzed and shows that our codon evolutionary distances allow building a phylogenetic tree which is similar to those obtained by non-distance-based methods (e.g., maximum parsimony and maximum likelihood) and also significantly improved compared to standard nucleotide evolutionary distance estimates.     

Identifying the rooted species tree from the distribution of unrooted gene trees under the coalescent

Gene trees are evolutionary trees representing the ancestry of genes sampled from multiple populations. Species trees represent populations of individuals—each with many genes—splitting into new populations or species. The coalescent process, which models ancestry of gene copies within populations, is often used to model the probability distribution of gene trees given a fixed species tree. This multispecies coalescent model provides a framework for phylogeneticists to infer species trees from gene trees using maximum likelihood or Bayesian approaches. Because the coalescent models a branching process over time, all trees are typically assumed to be rooted in this setting. Often, however, gene trees inferred by traditional phylogenetic methods are unrooted. We investigate probabilities of unrooted gene trees under the multispecies coalescent model. We show that when there are four species with one gene sampled per species, the distribution of unrooted gene tree topologies identifies the unrooted species tree topology and some, but not all, information in the species tree edges (branch lengths). The location of the root on the species tree is not identifiable in this situation. However, for 5 or more species with one gene sampled per species, we show that the distribution of unrooted gene tree topologies identifies the rooted species tree topology and all its internal branch lengths. The length of any pendant branch leading to a leaf of the species tree is also identifiable for any species from which more than one gene is sampled.     

A heuristic approach of maximum likelihood method for inferring phylogenetic tree and an application to the mammalian SOX-3 origin of the testis-determining gene SRY

Applying the tree bisection and reconnection (TBR) algorithm, we have developed a heuristic method (maximum likelihood (ML)-TBR) for inferring the ML tree based on tree topology search. For initial trees from which iterative processes start in ML-TBR, two cases were considered: one is 100 neighbor-joining (NJ) trees based on the bootstrap resampling and the other is 100 randomly generated trees. The same ML tree was obtained in both cases. All different iterative processes started from 100 independent initial trees ultimately converged on one optimum tree with the largest log-likelihood value, suggesting that a limited number of initial trees will be quite enough in ML-TBR. This also suggests that the optimum tree corresponds to the global optimum in tree topology space and thus probably coincides with the ML tree inferred by intact ML analysis. This method has been applied to the inference of phylogenetic tree of the SOX family members. The mammalian testis-determining gene SRY is believed to have evolved from SOX-3, a member of the SOX family, based on several lines of evidence, including their sequence similarity, the location of SOX-3 on the X chromosome and some aspects of their expression. This model should be supported directly from the phylogenetic tree of the SOX family, but no evidence has been provided to date. A recently published NJ tree shows implausibly remote origin of SRY, suggesting that a more sophisticated method is required for understanding this problem. The ML tree inferred by the present method showed that the SRYs of marsupial and placental mammals form a monophyletic cluster which had diverged from the mammalian SOX-3 in the early evolution of mammals.     

Empirical determination of effective gap penalties for sequence comparison

MOTIVATION: No general theory guides the selection of gap penalties for local sequence alignment. We empirically determined the most effective gap penalties for protein sequence similarity searches with substitution matrices over a range of target evolutionary distances from 20 to 200 Point Accepted Mutations (PAMs). RESULTS: We embedded real and simulated homologs of protein sequences into a database and searched the database to determine the gap penalties that produced the best statistical significance for the distant homologs. The most effective penalty for the first residue in a gap (q+r) changes as a function of evolutionary distance, while the gap extension penalty for additional residues (r) does not. For these data, the optimal gap penalties for a given matrix scaled in 1/3 bit units (e.g. BLOSUM50, PAM200) are q=25-0.1 * (target PAM distance), r=5. Our results provide an empirical basis for selection of gap penalties and demonstrate how optimal gap penalties behave as a function of the target evolutionary distance of the substitution matrix. These gap penalties can improve expectation values by at least one order of magnitude when searching with short sequences, and improve the alignment of proteins containing short sequences repeated in tandem.     

A transition probability model for amino acid substitutions from blocks.

Substitution matrices have been useful for sequence alignment and protein sequence comparisons. The BLOSUM series of matrices, which had been derived from a database of alignments of protein blocks, improved the accuracy of alignments previously obtained from the PAM-type matrices estimated from only closely related sequences. Although BLOSUM matrices are scoring matrices now widely used for protein sequence alignments, they do not describe an evolutionary model. BLOSUM matrices do not permit the estimation of the actual number of amino acid substitutions between sequences by correcting for multiple hits. The method presented here uses the Blocks database of protein alignments, along with the additivity of evolutionary distances, to approximate the amino acid substitution probabilities as a function of actual evolutionary distance. The PMB (Probability Matrix from Blocks) defines a new evolutionary model for protein evolution that can be used for evolutionary analyses of protein sequences. Our model is directly derived from, and thus compatible with, the BLOSUM matrices. The model has the additional advantage of being easily implemented.     

Detection of recombination in DNA multiple alignments with hidden Markov models.

Conventional phylogenetic tree estimation methods assume that all sites in a DNA multiple alignment have the same evolutionary history. This assumption is violated in data sets from certain bacteria and viruses due to recombination, a process that leads to the creation of mosaic sequences from different strains and, if undetected, causes systematic errors in phylogenetic tree estimation. In the current work, a hidden Markov model (HMM) is employed to detect recombination events in multiple alignments of DNA sequences. The emission probabilities in a given state are determined by the branching order (topology) and the branch lengths of the respective phylogenetic tree, while the transition probabilities depend on the global recombination probability. The present study improves on an earlier heuristic parameter optimization scheme and shows how the branch lengths and the recombination probability can be optimized in a maximum likelihood sense by applying the expectation maximization (EM) algorithm. The novel algorithm is tested on a synthetic benchmark problem and is found to clearly outperform the earlier heuristic approach. The paper concludes with an application of this scheme to a DNA sequence alignment of the argF gene from four Neisseria strains, where a likely recombination event is clearly detected.     

An improved general amino acid replacement matrix

Amino acid replacement matrices are an essential basis of protein phylogenetics. They are used to compute substitution probabilities along phylogeny branches and thus the likelihood of the data. They are also essential in protein alignment. A number of replacement matrices and methods to estimate these matrices from protein alignments have been proposed since the seminal work of Dayhoff et al. (1972). An important advance was achieved by Whelan and Goldman (2001) and their WAG matrix, thanks to an efficient maximum likelihood estimation approach that accounts for the phylogenies of sequences within each training alignment. We further refine this method by incorporating the variability of evolutionary rates across sites in the matrix estimation and using a much larger and diverse database than BRKALN, which was used to estimate WAG. To estimate our new matrix (called LG after the authors), we use an adaptation of the XRATE software and 3,912 alignments from Pfam, comprising approximately 50,000 sequences and approximately 6.5 million residues overall. To evaluate the LG performance, we use an independent sample consisting of 59 alignments from TreeBase and randomly divide Pfam alignments into 3,412 training and 500 test alignments. The comparison with WAG and JTT shows a clear likelihood improvement. With TreeBase, we find that 1) the average Akaike information criterion gain per site is 0.25 and 0.42, when compared with WAG and JTT, respectively; 2) LG is significantly better than WAG for 38 alignments (among 59), and significantly worse with 2 alignments only; and 3) tree topologies inferred with LG, WAG, and JTT frequently differ, indicating that using LG impacts not only the likelihood value but also the output tree. Results with the test alignments from Pfam are analogous. LG and a PHYML implementation can be downloaded from http://atgc.lirmm.fr/LG.     

Estimation of evolutionary parameters with phylogenetic trees

An important issue in the phylogenetic analysis of nucleotide sequence data using the maximum likelihood (ML) method is the underlying evolutionary model employed. We consider the problem of simultaneously estimating the tree topology and the parameters in the underlying substitution model and of obtaining estimates of the standard errors of these parameter estimates. Given a fixed tree topology and corresponding set of branch lengths, the ML estimates of standard evolutionary model parameters are asymptotically efficient, in the sense that their joint distribution is asymptotically normal with the variance–covariance matrix given by the inverse of the Fisher information matrix. We propose a new estimate of this conditional variance based on estimation of the expected information using a Monte Carlo sampling (MCS) method. Simulations are used to compare this conditional variance estimate to the standard technique of using the observed information under a variety of experimental conditions. In the case in which one wishes to estimate simultaneously the tree and parameters, we provide a bootstrapping approach that can be used in conjunction with the MCS method to estimate the unconditional standard error. The methods developed are applied to a real data set consisting of 30 papillomavirus sequences. This overall method is easily incorporated into standard bootstrapping procedures to allow for proper variance estimation.     

An evolutionary model for maximum likelihood alignment of DNA sequences

Summary Most algorithms for the alignment of biological sequences are not derived from an evolutionary model. Consequently, these alignment algorithms lack a strong statistical basis. A maximum likelihood method for the alignment of two DNA sequences is presented. This method is based upon a statistical model of DNA sequence evolution for which we have obtained explicit transition probabilities. The evolutionary model can also be used as the basis of procedures that estimate the evolutionary parameters relevant to a pair of unaligned DNA sequences. A parameter-estimation approach which takes into account all possible alignments between two sequences is introduced; the danger of estimating evolutionary parameters from a single alignment is discussed.     

MEGA5: molecular evolutionary genetics analysis using maximum likelihood, evolutionary distance, and maximum parsimony methods

Comparative analysis of molecular sequence data is essential for reconstructing the evolutionary histories of species and inferring the nature and extent of selective forces shaping the evolution of genes and species. Here, we announce the release of Molecular Evolutionary Genetics Analysis version 5 (MEGA5), which is a user-friendly software for mining online databases, building sequence alignments and phylogenetic trees, and using methods of evolutionary bioinformatics in basic biology, biomedicine, and evolution. The newest addition in MEGA5 is a collection of maximum likelihood (ML) analyses for inferring evolutionary trees, selecting best-fit substitution models (nucleotide or amino acid), inferring ancestral states and sequences (along with probabilities), and estimating evolutionary rates site-by-site. In computer simulation analyses, ML tree inference algorithms in MEGA5 compared favorably with other software packages in terms of computational efficiency and the accuracy of the estimates of phylogenetic trees, substitution parameters, and rate variation among sites. The MEGA user interface has now been enhanced to be activity driven to make it easier for the use of both beginners and experienced scientists. This version of MEGA is intended for the Windows platform, and it has been configured for effective use on Mac OS X and Linux desktops. It is available free of charge from http://www.megasoftware.net.     

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.     

Sequence alignment with an appropriate substitution matrix.

A widely used algorithm for computing an optimal local alignment between two sequences requires a parameter set with a substitution matrix and gap penalties. It is recognized that a proper parameter set should be selected to suit the level of conservation between sequences. We describe an algorithm for selecting an appropriate substitution matrix at given gap penalties for computing an optimal local alignment between two sequences. In the algorithm, a substitution matrix that leads to the maximum alignment similarity score is selected among substitution matrices at various evolutionary distances. The evolutionary distance of the selected substitution matrix is defined as the distance of the computed alignment. To show the effects of gap penalties on alignments and their distances and help select appropriate gap penalties, alignments and their distances are computed at various gap penalties. The algorithm has been implemented as a computer program named SimDist. The SimDist program was compared with an existing local alignment program named SIM for finding reciprocally best-matching pairs (RBPs) of sequences in each of 100 protein families, where RBPs are commonly used as an operational definition of orthologous sequences. SimDist produced more accurate results than SIM on 50 of the 100 families, whereas both programs produced the same results on the other 50 families. SimDist was also used to compare three types of substitution matrices in scoring 444,461 pairs of homologous sequences from the 100 families.