An efficient algorithm for approximating geodesic distances in tree space

The increasing use of phylogeny in biological studies is limited by the need to make available more efficient tools for computing distances between trees. The geodesic tree distance-introduced by Billera, Holmes, and Vogtmann-combines both the tree topology and edge lengths into a single metric. Despite the conceptual simplicity of the geodesic tree distance, algorithms to compute it don't scale well to large, real-world phylogenetic trees composed of hundred or even thousand leaves. In this paper, we propose the geodesic distance as an effective tool for exploring the likelihood profile in the space of phylogenetic trees, and we give a cubic time algorithm, GeoHeuristic, in order to compute an approximation of the distance. We compare it with the GTP algorithm, which calculates the exact distance, and the cone path length, which is another approximation, showing that GeoHeuristic achieves a quite good trade-off between accuracy (relative error always lower than 0.0001) and efficiency. We also prove the equivalence among GeoHeuristic, cone path, and Robinson-Foulds distances when assuming branch lengths equal to unity and we show empirically that, under this restriction, these distances are almost always equal to the actual geodesic.     

Genetic algorithm for large-scale maximum parsimony phylogenetic analysis of proteins

Inferring phylogeny is a difficult computational problem. For example, for only 13 taxa, there are more then 13 billion possible unrooted phylogenetic trees. Heuristics are necessary to minimize the time spent evaluating non-optimal trees. We describe here an approach for heuristic searching, using a genetic algorithm, that can reduce the time required for weighted maximum parsimony phylogenetic inference, especially for data sets involving a large number of taxa. It is the first implementation of a weighted maximum parsimony criterion using amino acid sequences. To validate the weighted criterion, we used an artificial data set and compared it to a number of other phylogenetic methods. Genetic algorithms mimic the natural selection's ability to solve complex problems. We have identified several parameters affecting the genetic algorithm. Methods were developed to validate these parameters, ensuring optimal performance. This approach allows the construction of phylogenetic trees with over 200 taxa in practical time on a regular PC.     

STEME: efficient EM to find motifs in large data sets

MEME and many other popular motif finders use the expectation-maximization (EM) algorithm to optimize their parameters. Unfortunately, the running time of EM is linear in the length of the input sequences. This can prohibit its application to data sets of the size commonly generated by high-throughput biological techniques. A suffix tree is a data structure that can efficiently index a set of sequences. We describe an algorithm, Suffix Tree EM for Motif Elicitation (STEME), that approximates EM using suffix trees. To the best of our knowledge, this is the first application of suffix trees to EM. We provide an analysis of the expected running time of the algorithm and demonstrate that STEME runs an order of magnitude more quickly than the implementation of EM used by MEME. We give theoretical bounds for the quality of the approximation and show that, in practice, the approximation has a negligible effect on the outcome. We provide an open source implementation of the algorithm that we hope will be used to speed up existing and future motif search algorithms.     

The average common substring approach to phylogenomic reconstruction.

We describe a novel method for efficient reconstruction of phylogenetic trees, based on sequences of whole genomes or proteomes, whose lengths may greatly vary. The core of our method is a new measure of pairwise distances between sequences. This measure is based on computing the average lengths of maximum common substrings, which is intrinsically related to information theoretic tools (Kullback-Leibler relative entropy). We present an algorithm for efficiently computing these distances. In principle, the distance of two l long sequences can be calculated in O(l) time. We implemented the algorithm using suffix arrays our implementation is fast enough to enable the construction of the proteome phylogenomic tree for hundreds of species and the genome phylogenomic forest for almost two thousand viruses. An initial analysis of the results exhibits a remarkable agreement with "acceptable phylogenetic and taxonomic truth." To assess our approach, our results were compared to the traditional (single-gene or protein-based) maximum likelihood method. The obtained trees were compared to implementations of a number of alternative approaches, including two that were previously published in the literature, and to the published results of a third approach. Comparing their outcome and running time to ours, using a "traditional" trees and a standard tree comparison method, our algorithm improved upon the "competition" by a substantial margin. The simplicity and speed of our method allows for a whole genome analysis with the greatest scope attempted so far. We describe here five different applications of the method, which not only show the validity of the method, but also suggest a number of novel phylogenetic insights.     

Fast and accurate phylogeny reconstruction algorithms based on the minimum-evolution principle.

The Minimum Evolution (ME) approach to phylogeny estimation has been shown to be statistically consistent when it is used in conjunction with ordinary least-squares (OLS) fitting of a metric to a tree structure. The traditional approach to using ME has been to start with the Neighbor Joining (NJ) topology for a given matrix and then do a topological search from that starting point. The first stage requires O(n(3)) time, where n is the number of taxa, while the current implementations of the second are in O(p n(3)) or more, where p is the number of swaps performed by the program. In this paper, we examine a greedy approach to minimum evolution which produces a starting topology in O(n(2)) time. Moreover, we provide an algorithm that searches for the best topology using nearest neighbor interchanges (NNIs), where the cost of doing p NNIs is O(n(2) + p n), i.e., O(n(2)) in practice because p is always much smaller than n. The Greedy Minimum Evolution (GME) algorithm, when used in combination with NNIs, produces trees which are fairly close to NJ trees in terms of topological accuracy. We also examine ME under a balanced weighting scheme, where sibling subtrees have equal weight, as opposed to the standard "unweighted" OLS, where all taxa have the same weight so that the weight of a subtree is equal to the number of its taxa. The balanced minimum evolution scheme (BME) runs slower than the OLS version, requiring O(n(2) x diam(T)) operations to build the starting tree and O(p n x diam(T)) to perform the NNIs, where diam(T) is the topological diameter of the output tree. In the usual Yule-Harding distribution on phylogenetic trees, the diameter expectation is in log(n), so our algorithms are in practice faster that NJ. Moreover, this BME scheme yields a very significant improvement over NJ and other distance-based algorithms, especially with large trees, in terms of topological accuracy.     

Breakpoint medians and breakpoint phylogenies: a fixed-parameter approach

With breakpoint distance, the genome rearrangement field delivered one of the currently most popular measures in phylogenetic studies for related species. Here, BREAKPOINT MEDIAN, which is NP-complete already for three given species (whose genomes are represented as signed orderings), is the core basic problem. For the important special case of three species, approximation (ratio 7/6) and exact heuristic algorithms were developed. Here, we provide an exact, fixed-parameter algorithm with provable performance bounds. For instance, a breakpoint median for three signed orderings over nelements that causes at most d breakpoints can be computed in time O((2.15)(d).n). We show the algorithm's practical usefulness through experimental studies. In particular, we demonstrate that a simple implementation of our algorithm combined with a new tree construction heuristic allows for a new approach to breakpoint phylogeny, yielding evolutionary trees that are competitive in comparison with known results developed in a recent series of papers that use clever algorithm engineering methods.     

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.     

Neighbor joining algorithms for inferring phylogenies via LCA distances.

Reconstructing phylogenetic trees efficiently and accurately from distance estimates is an ongoing challenge in computational biology from both practical and theoretical considerations. We study algorithms which are based on a characterization of edge-weighted trees by distances to LCAs (Least Common Ancestors). This characterization enables a direct application of ultrametric reconstruction techniques to trees which are not necessarily ultrametric. A simple and natural neighbor joining criterion based on this observation is used to provide a family of efficient neighbor-joining algorithms. These algorithms are shown to reconstruct a refinement of the Buneman tree, which implies optimal robustness to noise under criteria defined by Atteson. In this sense, they outperform many popular algorithms such as Saitou and Nei's NJ. One member of this family is used to provide a new simple version of the 3-approximation algorithm for the closest additive metric under the iota (infinity) norm. A byproduct of our work is a novel technique which yields a time optimal O (n (2)) implementation of common clustering algorithms such as UPGMA.     

Quartets MaxCut: a divide and conquer quartets algorithm

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

Bayesian phylogeny analysis via stochastic approximation Monte Carlo

Monte Carlo methods have received much attention in the recent literature of phylogeny analysis. However, the conventional Markov chain Monte Carlo algorithms, such as the Metropolis–Hastings algorithm, tend to get trapped in a local mode in simulating from the posterior distribution of phylogenetic trees, rendering the inference ineffective. In this paper, we apply an advanced Monte Carlo algorithm, the stochastic approximation Monte Carlo algorithm, to Bayesian phylogeny analysis. Our method is compared with two popular Bayesian phylogeny software, BAMBE and MrBayes, on simulated and real datasets. The numerical results indicate that our method outperforms BAMBE and MrBayes. Among the three methods, SAMC produces the consensus trees which have the highest similarity to the true trees, and the model parameter estimates which have the smallest mean square errors, but costs the least CPU time.     

On Nakhleh's Metric for Reduced Phylogenetic Networks

We prove that Nakhleh's metric for reduced phylogenetic networks is also a metric on the classes of tree-child phylogenetic networks, semibinary tree-sibling time consistent phylogenetic networks, and multilabeled phylogenetic trees. We also prove that it separates distinguishable phylogenetic networks. In this way, it becomes the strongest dissimilarity measure for phylogenetic networks available so far. Furthermore, we propose a generalization of that metric that separates arbitrary phylogenetic networks.     

Reconstructing reticulate evolution in species-theory and practice.

We present new methods for reconstructing reticulate evolution of species due to events such as horizontal transfer or hybrid speciation; both methods are based upon extensions of Wayne Maddison's approach in his seminal 1997 paper. Our first method is a polynomial time algorithm for constructing phylogenetic networks from two gene trees contained inside the network.We allow the network to have an arbitrary number of reticulations, but we limit the reticulation in the network so that the cycles in the network are node-disjoint ("galled"). Our second method is a polynomial time algorithm for constructing networks with one reticulation, where we allow for errors in the estimated gene trees. Using simulations, we demonstrate improved performance of this method over both NeighborNet and Maddison's method.     

Generalized Verlet Algorithm for Efficient Molecular Dynamics Simulations with Long-range Interactions

For the purpose of molecular dynamics simulations of large biopolymers we have developed a new method to accelerate the calculation of long-range pair interactions (e.g. Coulomb interaction). The algorithm introduces distance classes to schedule updates of non-bonding interactions and to avoid unnecessary computations of interactions between particles which are far apart. To minimize the error caused by the updating schedule, the Verlet integration scheme has been modified. The results of the method are compared to those of other approximation schemes as well as to results obtained by numerical integration without approximation. For simulation of a protein with 12 637 atoms our approximation scheme yields a reduction of computer time by a factor of seven. The approximation suggested can be implemented on sequential as well as on parallel computers. We describe an implementation on a (Transputer-based) MIMD machine with a systolic ring architecture.     

Approximate Maximum Parsimony and Ancestral Maximum Likelihood

We explore the maximum parsimony (MP) and ancestral maximum likelihood (AML) criteria in phylogenetic tree reconstruction. Both problems are NP-hard, so we seek approximate solutions. We formulate the two problems as Steiner tree problems under appropriate distances. The gist of our approach is the succinct characterization of Steiner trees for a small number of leaves for the two distances. This enables the use of known Steiner tree approximation algorithms. The approach leads to a 16/9 approximation ratio for AML and asymptotically to a 1.55 approximation ratio for MP.     

Distorted metrics on trees and phylogenetic forests

We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree T on n leaves with a path metric d, consider the pairwise distances {d(u,v)} between leaves. It is well known that these determine the tree and the d length of all edges. Here, we consider distortions d of d such that, for all leaves u and v, it holds that |d(u,v)-dmacr(u,v)|1.....T₀ such that the true tree T may be obtained from that forest by adding alpha-1 edges and alpha-1les2^-Omega(M/g)n. Our distorted metric result implies a reconstruction algorithm of phylogenetic forests with a small number of trees from sequences of length logarithmic in the number of species. The reconstruction algorithm is applicable for the general Markov model. Both the distorted metric result and its applications to phylogeny are almost tight     

Computing the Distribution of a Tree Metric

The Robinson-Foulds (RF) distance is by far the most widely used measure of dissimilarity between trees. Although the distribution of these distances has been investigated for 20 years, an algorithm that is explicitly polynomial time has yet to be described for computing the distribution for trees around a given tree. In this paper, we derive a polynomial-time algorithm for this distribution. We show how the distribution can be approximated by a Poisson distribution determined by the proportion of leaves that lie in “cherries” of the given tree. We also describe how our results can be used to derive normalization constants that are required in a recently proposed maximum likelihood approach to supertree construction.     

Fast local fragment chaining using sum-of-pair gap costs

Background

When inferring phylogenetic trees different algorithms may give different trees. To study such effects a measure for the distance between two trees is useful. Quartet distance is one such measure, and is the number of quartet topologies that differ between two trees.

Results

We have derived a new algorithm for computing the quartet distance between a pair of general trees, i.e. trees where inner nodes can have any degree ≥ 3. The time and space complexity of our algorithm is sub-cubic in the number of leaves and does not depend on the degree of the inner nodes. This makes it the fastest algorithm so far for computing the quartet distance between general trees independent of the degree of the inner nodes.

Conclusions

We have implemented our algorithm and two of the best competitors. Our new algorithm is significantly faster than the competition and seems to run in close to quadratic time in practice.     

Correlation between strand asymmetry and phylogeny in mitochondrial DNA

An evolutionary distance is introduced in order to propose an efficient and feasible procedure for phylogeny studies. Our analysis are based on the strand asymmetry property of mitochondrial DNA, but can be applied to other genomes. Comparison of our results with those reported in conventional phylogenetic trees, gives confidence about our approximation. Our findings support the hypotheses about the origin of the skew and its dependence upon evolutionary pressures, and improves previous efforts on using the strand asymmetry property of genomes for phylogeny inference. For the evolutionary distance introduced here, we observe that the more adequate technique for tree reconstructions correspond to an average link method which employs a sequential clustering algorithm.     

NJML: a hybrid algorithm for the neighbor-joining and maximum-likelihood methods

In the reconstruction of a large phylogenetic tree, the most difficult part is usually the problem of how to explore the topology space to find the optimal topology. We have developed a "divide-and-conquer" heuristic algorithm in which an initial neighbor-joining (NJ) tree is divided into subtrees at internal branches having bootstrap values higher than a threshold. The topology search is then conducted by using the maximum-likelihood method to reevaluate all branches with a bootstrap value lower than the threshold while keeping the other branches intact. Extensive simulation showed that our simple method, the neighbor-joining maximum-likelihood (NJML) method, is highly efficient in improving NJ trees. Furthermore, the performance of the NJML method is nearly equal to or better than existing time-consuming heuristic maximum-likelihood methods. Our method is suitable for reconstructing relatively large molecular phylogenetic trees (number of taxa >/= 16).     

A simple algorithm to infer gene duplication and speciation events on a gene tree

MOTIVATION: When analyzing protein sequences using sequence similarity searches, orthologous sequences (that diverged by speciation) are more reliable predictors of a new protein's function than paralogous sequences (that diverged by gene duplication), because duplication enables functional diversification. The utility of phylogenetic information in high-throughput genome annotation ('phylogenomics') is widely recognized, but existing approaches are either manual or indirect (e.g. not based on phylogenetic trees). Our goal is to automate phylogenomics using explicit phylogenetic inference. A necessary component is an algorithm to infer speciation and duplication events in a given gene tree. RESULTS: We give an algorithm to infer speciation and duplication events on a gene tree by comparison to a trusted species tree. This algorithm has a worst-case running time of O(n(2)) which is inferior to two previous algorithms that are approximately O(n) for a gene tree of sequences. However, our algorithm is extremely simple, and its asymptotic worst case behavior is only realized on pathological data sets. We show empirically, using 1750 gene trees constructed from the Pfam protein family database, that it appears to be a practical (and often superior) algorithm for analyzing real gene trees. AVAILABILITY: http://www.genetics.wustl.edu/eddy/forester.