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.     

Reconstructing phylogeny by quadratically approximated maximum likelihood

Maximum likelihood (ML) for phylogenetic inference from sequence data remains a method of choice, but has computational limitations. In particular, it cannot be applied for a global search through all potential trees when the number of taxa is large, and hence a heuristic restriction in the search space is required. In this paper, we derive a quadratic approximation, QAML, to the likelihood function whose maximum is easily determined for a given tree. The derivation depends on Hadamard conjugation, and hence is limited to the simple symmetric models of Kimura and of Jukes and Cantor. Preliminary testing has demonstrated the accuracy of QAML is close to that of ML.     

Genetic Algorithm-Based Maximum-Likelihood Analysis for Molecular Phylogeny

A heuristic approach to search for the maximum-likelihood (ML) phylogenetic tree based on a genetic algorithm (GA) has been developed. It outputs the best tree as well as multiple alternative trees that are not significantly worse than the best one on the basis of the likelihood criterion. These near-optimum trees are subjected to further statistical tests. This approach enables ones to infer phylogenetic trees of over 20 taxa taking account of the rate heterogeneity among sites on practical time scales on a PC cluster. Computer simulations were conducted to compare the efficiency of the present approach with that of several likelihood-based methods and distance-based methods, using amino acid sequence data of relatively large (5–24) taxa. The superiority of the ML method over distance-based methods increases as the condition of simulations becomes more realistic (an incorrect model is assumed or many taxa are involved). This approach was applied to the inference of the universal tree based on the concatenated amino acid sequences of vertically descendent genes that are shared among all genomes whose complete sequences have been reported. The inferred tree strongly supports that Archaea is paraphyletic and Eukarya is specifically related to Crenarchaeota. Apart from the paraphyly of Archaea and some minor disagreements, the universal tree based on these genes is largely consistent with the universal tree based on SSU rRNA. Received: 4 January 2001 / Accepted: 16 May 2001     

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).     

Efficiencies of fast algorithms of phylogenetic inference under the criteria of maximum parsimony, minimum evolution, and maximum likelihood when a large number of sequences are used

In phylogenetic inference by maximum-parsimony (MP), minimum-evolution (ME), and maximum-likelihood (ML) methods, it is customary to conduct extensive heuristic searches of MP, ME, and ML trees, examining a large number of different topologies. However, these extensive searches tend to give incorrect tree topologies. Here we show by extensive computer simulation that when the number of nucleotide sequences (m) is large and the number of nucleotides used (n) is relatively small, the simple MP or ML tree search algorithms such as the stepwise addition (SA) plus nearest neighbor interchange (NNI) search and the SA plus subtree pruning regrafting (SPR) search are as efficient as the extensive search algorithms such as the SA plus tree bisection-reconnection (TBR) search in inferring the true tree. In the case of ME methods, the simple neighbor-joining (NJ) algorithm is as efficient as or more efficient than the extensive NJ+TBR search. We show that when ME methods are used, the simple p distance generally gives better results in phylogenetic inference than more complicated distance measures such as the Hasegawa-Kishino-Yano (HKY) distance, even when nucleotide substitution follows the HKY model. When ML methods are used, the simple Jukes-Cantor (JC) model of phylogenetic inference generally shows a better performance than the HKY model even if the likelihood value for the HKY model is much higher than that for the JC model. This indicates that at least in the present case, selecting of a substitution model by using the likelihood ratio test or the AIC index is not appropriate. When n is small relative to m and the extent of sequence divergence is high, the NJ method with p distance often shows a better performance than ML methods with the JC model. However, when the level of sequence divergence is low, this is not the case.     

A rapid heuristic algorithm for finding minimum evolution trees

The minimum sum of branch lengths (S), or the minimum evolution (ME) principle, has been shown to be a good optimization criterion in phylogenetic inference. Unfortunately, the number of topologies to be analyzed is computationally prohibitive when a large number of taxa are involved. Therefore, simplified, heuristic methods, such as the neighbor-joining (NJ) method, are usually employed instead. The NJ method analyzes only a small number of trees (compared with the size of the entire search space); so, the tree obtained may not be the ME tree (for which the S value is minimum over the entire search space). Different compromises between very restrictive and exhaustive search spaces have been proposed recently. In particular, the "stepwise algorithm" (SA) utilizes what is known in computer science as the "beam search," whereas the NJ method employs a "greedy search." SA is virtually guaranteed to find the ME trees while being much faster than exhaustive search algorithms. In this study we propose an even faster method for finding the ME tree. The new algorithm adjusts its search exhaustiveness (from greedy to complete) according to the statistical reliability of the tree node being reconstructed. It is also virtually guaranteed to find the ME tree. The performances and computational efficiencies of ME, SA, NJ, and our new method were compared in extensive simulation studies. The new algorithm was found to perform practically as well as the SA (and, therefore, ME) methods and slightly better than the NJ method. For searching for the globally optimal ME tree, the new algorithm is significantly faster than existing ones, thus making it relatively practical for obtaining all trees with an S value equal to or smaller than that of the NJ tree, even when a large number of taxa is involved.     

Accelerated likelihood surface exploration: the likelihood ratchet

The existence of multiple likelihood maxima necessitates algorithms that explore a large part of the tree space. However, because of computational constraints, stepwise addition-based tree-searching methods do not allow for this exploration in reasonable time. Here, I present an algorithm that increases the speed at which the likelihood landscape can be explored. The iterative algorithm combines the computational speed of distance-based tree construction methods to arrive at approximations of the global optimum with the accuracy of optimality criterion based branch-swapping methods to improve on the result of the starting tree. The algorithm moves between local optima by iteratively perturbing the tree landscape through a process of reweighting randomly drawn samples of the underlying sequence data set. Tests on simulated and real data sets demonstrated that the optimal solution obtained using stepwise addition-based heuristic searches was found faster using the algorithm presented here. Tests on a previously published data set that established the presence of tree islands under maximum likelihood demonstrated that the algorithm identifies the same tree islands in a shorter amount of time than that needed using stepwise addition. The algorithm can be readily applied using standard software for phylogenetic inference.     

Reconstruction of ancestral genomic sequences using likelihood.

A challenging task in computational biology is the reconstruction of genomic sequences of extinct ancestors, given the phylogenetic tree and the sequences at the leafs. This task is best solved by calculating the most likely estimate of the ancestral sequences, along with the most likely edge lengths. We deal with this problem and also the variant in which the phylogenetic tree in addition to the ancestral sequences need to be estimated. The latter problem is known to be NP-hard, while the computational complexity of the former is unknown. Currently, all algorithms for solving these problems are heuristics without performance guarantees. The biological importance of these problems calls for developing better algorithms with guarantees of finding either optimal or approximate solutions.We develop approximation, fix parameter tractable (FPT), and fast heuristic algorithms for two variants of the problem; when the phylogenetic tree is known and when it is unknown. The approximation algorithm guarantees a solution with a log-likelihood ratio of 2 relative to the optimal solution. The FPT has a running time which is polynomial in the length of the sequences and exponential in the number of taxa. This makes it useful for calculating the optimal solution for small trees. Moreover, we combine the approximation algorithm and the FPT into an algorithm with arbitrary good approximation guarantee (PTAS). We tested our algorithms on both synthetic and biological data. In particular, we used the FPT for computing the most likely ancestral mitochondrial genomes of hominidae (the great apes), thereby answering an interesting biological question. Moreover, we show how the approximation algorithms find good solutions for reconstructing the ancestral genomes for a set of lentiviruses (relatives of HIV). Supplementary material of this work is available at www.nada.kth.se/~isaac/publications/aml/aml.html.     

Performance of flip supertree construction with a heuristic algorithm

Supertree methods are used to assemble separate phylogenetic trees with shared taxa into larger trees (supertrees) in an effort to construct more comprehensive phylogenetic hypotheses. In spite of much recent interest in supertrees, there are still few methods for supertree construction. The flip supertree problem is an error correction approach that seeks to find a minimum number of changes (flips) to the matrix representation of the set of input trees to resolve their incompatibilities. A previous flip supertree algorithm was limited to finding exact solutions and was only feasible for small input trees. We developed a heuristic algorithm for the flip supertree problem suitable for much larger input trees. We used a series of 48- and 96-taxon simulations to compare supertrees constructed with the flip supertree heuristic algorithm with supertrees constructed using other approaches, including MinCut (MC), modified MC (MMC), and matrix representation with parsimony (MRP). Flip supertrees are generally far more accurate than supertrees constructed using MC or MMC algorithms and are at least as accurate as supertrees built with MRP. The flip supertree method is therefore a viable alternative to other supertree methods when the number of taxa is large.     

Coalescent-based species tree inference from gene tree topologies under incomplete lineage sorting by maximum likelihood

Incomplete lineage sorting can cause incongruence between the phylogenetic history of genes (the gene tree) and that of the species (the species tree), which can complicate the inference of phylogenies. In this article, I present a new coalescent-based algorithm for species tree inference with maximum likelihood. I first describe an improved method for computing the probability of a gene tree topology given a species tree, which is much faster than an existing algorithm by Degnan and Salter (2005). Based on this method, I develop a practical algorithm that takes a set of gene tree topologies and infers species trees with maximum likelihood. This algorithm searches for the best species tree by starting from initial species trees and performing heuristic search to obtain better trees with higher likelihood. This algorithm, called STELLS (which stands for Species Tree InfErence with Likelihood for Lineage Sorting), has been implemented in a program that is downloadable from the author's web page. The simulation results show that the STELLS algorithm is more accurate than an existing maximum likelihood method for many datasets, especially when there is noise in gene trees. I also show that the STELLS algorithm is efficient and can be applied to real biological datasets.     

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.     

Genetic algorithms and parallel processing in maximum-likelihood phylogeny inference

We investigated the usefulness of a parallel genetic algorithm for phylogenetic inference under the maximum-likelihood (ML) optimality criterion. Parallelization was accomplished by assigning each "individual" in the genetic algorithm "population" to a separate processor so that the number of processors used was equal to the size of the evolving population (plus one additional processor for the control of operations). The genetic algorithm incorporated branch-length and topological mutation, recombination, selection on the ML score, and (in some cases) migration and recombination among subpopulations. We tested this parallel genetic algorithm with large (228 taxa) data sets of both empirically observed DNA sequence data (for angiosperms) as well as simulated DNA sequence data. For both observed and simulated data, search-time improvement was nearly linear with respect to the number of processors, so the parallelization strategy appears to be highly effective at improving computation time for large phylogenetic problems using the genetic algorithm. We also explored various ways of optimizing and tuning the parameters of the genetic algorithm. Under the conditions of our analyses, we did not find the best-known solution using the genetic algorithm approach before terminating each run. We discuss some possible limitations of the current implementation of this genetic algorithm as well as of avenues for its future improvement.     

RAxML-III: a fast program for maximum likelihood-based inference of large phylogenetic trees

MOTIVATION: The computation of large phylogenetic trees with statistical models such as maximum likelihood or bayesian inference is computationally extremely intensive. It has repeatedly been demonstrated that these models are able to recover the true tree or a tree which is topologically closer to the true tree more frequently than less elaborate methods such as parsimony or neighbor joining. Due to the combinatorial and computational complexity the size of trees which can be computed on a Biologist's PC workstation within reasonable time is limited to trees containing approximately 100 taxa. RESULTS: In this paper we present the latest release of our program RAxML-III for rapid maximum likelihood-based inference of large evolutionary trees which allows for computation of 1.000-taxon trees in less than 24 hours on a single PC processor. We compare RAxML-III to the currently fastest implementations for maximum likelihood and bayesian inference: PHYML and MrBayes. Whereas RAxML-III performs worse than PHYML and MrBayes on synthetic data it clearly outperforms both programs on all real data alignments used in terms of speed and final likelihood values. Availability SUPPLEMENTARY INFORMATION: RAxML-III including all alignments and final trees mentioned in this paper is freely available as open source code at http://wwwbode.cs.tum/~stamatak CONTACT: stamatak@cs.tum.edu.     

Phylogenetic relationships and divergence time estimate of African anguilliform catfish (Siluriformes: Clariidae) inferred from ribosomal gene and spacer sequences

The catfish family Clariidae comprises species in which the body shape ranges from fusiform to anguilliform. Recent studies have shown that this body elongation is the result of convergent evolution. This paper aims to study the evolution towards anguilliformity in a phylogenetic framework. Sequences of 29 taxa were analyzed using the neighbor-joining, maximum-likelihood, maximum-parsimony, and Bayesian inference algorithms and the parsimony algorithm in POY. The study yields phylogenetic hypotheses showing well-supported clades. Anguilliformity appears to have arisen at least four times, each time having a sister group relation with a fusiform Clarias-like ancestor. Divergence time estimation indicates that the African Clariidae started radiating between 123 and 56 My ago.     

Performance, accuracy, and Web server for evolutionary placement of short sequence reads under maximum likelihood

We present an evolutionary placement algorithm (EPA) and a Web server for the rapid assignment of sequence fragments (short reads) to edges of a given phylogenetic tree under the maximum-likelihood model. The accuracy of the algorithm is evaluated on several real-world data sets and compared with placement by pair-wise sequence comparison, using edit distances and BLAST. We introduce a slow and accurate as well as a fast and less accurate placement algorithm. For the slow algorithm, we develop additional heuristic techniques that yield almost the same run times as the fast version with only a small loss of accuracy. When those additional heuristics are employed, the run time of the more accurate algorithm is comparable with that of a simple BLAST search for data sets with a high number of short query sequences. Moreover, the accuracy of the EPA is significantly higher, in particular when the sample of taxa in the reference topology is sparse or inadequate. Our algorithm, which has been integrated into RAxML, therefore provides an equally fast but more accurate alternative to BLAST for tree-based inference of the evolutionary origin and composition of short sequence reads. We are also actively developing a Web server that offers a freely available service for computing read placements on trees using the EPA.     

Stochastic search strategy for estimation of maximum likelihood phylogenetic trees

The maximum likelihood (ML) method of phylogenetic tree construction is not as widely used as other tree construction methods (e.g., parsimony, neighbor-joining) because of the prohibitive amount of time required to find the ML tree when the number of sequences under consideration is large. To overcome this difficulty, we propose a stochastic search strategy for estimation of the ML tree that is based on a simulated annealing algorithm. The algorithm works by moving through tree space by way of a "local rearrangement" strategy so that topologies that improve the likelihood are always accepted, whereas those that decrease the likelihood are accepted with a probability that is related to the proportionate decrease in likelihood. Besides greatly reducing the time required to estimate the ML tree, the stochastic search strategy is less likely to become trapped in local optima than are existing algorithms for ML tree estimation. We demonstrate the success of the modified simulated annealing algorithm by comparing it with two existing algorithms (Swofford's PAUP* and Felsenstein's DNAMLK) for several theoretical and real data examples.     

Analytic solutions of maximum likelihood on forks of four taxa

This work deals with symbolic mathematical solutions to maximum likelihood on small phylogenetic trees. Maximum likelihood (ML) is increasingly used as an optimality criterion for selecting evolutionary trees, but finding the global optimum is a hard computational task. In this work, we give general analytic solutions for a family of trees with four taxa, two state characters, under a molecular clock. Previously, analytical solutions were known only for three taxa trees. The change from three to four taxa incurs a major increase in the complexity of the underlying algebraic system, and requires novel techniques and approaches. Despite the simplicity of our model, solving ML analytically in it is close to the limit of today's tractability. Four taxa rooted trees have two topologies--the fork (two subtrees with two leaves each) and the comb (one subtree with three leaves, the other with a single leaf). Combining the properties of molecular clock fork trees with the Hadamard conjugation, and employing the symbolic algebra software Maple, we derive a number of topology dependent identities. Using these identities, we substantially simplify the system of polynomial equations for the fork. We finally employ the symbolic algebra software to obtain closed form analytic solutions (expressed parametrically in the input data).     

Comparative performance of supertree algorithms in large data sets using the soapberry family (Sapindaceae) as a case study

For the last 2 decades, supertree reconstruction has been an active field of research and has seen the development of a large number of major algorithms. Because of the growing popularity of the supertree methods, it has become necessary to evaluate the performance of these algorithms to determine which are the best options (especially with regard to the supermatrix approach that is widely used). In this study, seven of the most commonly used supertree methods are investigated by using a large empirical data set (in terms of number of taxa and molecular markers) from the worldwide flowering plant family Sapindaceae. Supertree methods were evaluated using several criteria: similarity of the supertrees with the input trees, similarity between the supertrees and the total evidence tree, level of resolution of the supertree and computational time required by the algorithm. Additional analyses were also conducted on a reduced data set to test if the performance levels were affected by the heuristic searches rather than the algorithms themselves. Based on our results, two main groups of supertree methods were identified: on one hand, the matrix representation with parsimony (MRP), MinFlip, and MinCut methods performed well according to our criteria, whereas the average consensus, split fit, and most similar supertree methods showed a poorer performance or at least did not behave the same way as the total evidence tree. Results for the super distance matrix, that is, the most recent approach tested here, were promising with at least one derived method performing as well as MRP, MinFlip, and MinCut. The output of each method was only slightly improved when applied to the reduced data set, suggesting a correct behavior of the heuristic searches and a relatively low sensitivity of the algorithms to data set sizes and missing data. Results also showed that the MRP analyses could reach a high level of quality even when using a simple heuristic search strategy, with the exception of MRP with Purvis coding scheme and reversible parsimony. The future of supertrees lies in the implementation of a standardized heuristic search for all methods and the increase in computing power to handle large data sets. The latter would prove to be particularly useful for promising approaches such as the maximum quartet fit method that yet requires substantial computing power.     

Molecular systematics and biogeography of the southern South american freshwater "crabs" Aegla (decapoda: Anomura: Aeglidae) using multiple heuristic tree search approaches

Recently new heuristic genetic algorithms such as Treefinder and MetaGA have been developed to search for optimal trees in a maximum likelihood (ML) framework. In this study we combined these methods with other standard heuristic approaches such as ML and maximum parsimony hill-climbing searches and Bayesian inference coupled with Markov chain Monte Carlo techniques under homogeneous and mixed models of evolution to conduct an extensive phylogenetic analysis of the most abundant and widely distributed southern South American freshwater"crab,"the Aegla(Anomura: Aeglidae). A total of 167 samples representing 64 Aegla species and subspecies were sequenced for one nuclear (28S rDNA) and four mitochondrial (12S and 16S rDNA, COI, and COII) genes (5352 bp total). Additionally, six other anomuran species from the genera Munida,Pachycheles, and Uroptychus(Galatheoidea), Lithodes(Paguroidea), and Lomis(Lomisoidea) and the nuclear 18S rDNA gene (1964 bp) were included in preliminary analyses for rooting the Aegla tree. Nonsignificantly different phylogenetic hypotheses resulted from all the different heuristic methods used here, although the best scored topologies found under the ML hill-climbing, Bayesian, and MetaGA approaches showed considerably better likelihood scores (Delta> 54) than those found under the MP and Treefinder approaches. Our trees provided strong support for most of the recognized Aegla species except for A. cholchol,A. jarai,A. parana,A. marginata, A. platensis, and A. franciscana, which may actually represent multiple species. Geographically, the Aegla group was divided into a basal western clade (21 species and subspecies) composed of two subclades with overlapping distributions, and a more recent central-eastern clade (43 species) composed of three subclades with fairly well-recognized distributions. This result supports the Pacific-Origin Hypothesis postulated for the group; alternative hypotheses of Atlantic or multiple origins were significantly rejected by our analyses. Finally, we combined our phylogenetic results with previous hypotheses of South American paleodrainages since the Jurassic to propose a biogeographical framework of the Aegla radiation.