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.     

Quartet MaxCut: A fast algorithm for amalgamating quartet trees

Accurate phylogenetic reconstruction methods are inherently computationally heavy and therefore are limited to relatively small numbers of taxa. Supertree construction is the task of amalgamating small trees over partial sets into a big tree over the complete taxa set. The need for fast and accurate supertree methods has become crucial due to the enormous number of new genomic sequences generated by modern technology and the desire to use them for classification purposes. In particular, the Assembling the Tree of Life (ATOL) program aims at constructing the evolutionary history of all living organisms on Earth. When dealing with unrooted trees, a quartet - an unrooted tree over four taxa - is the most basic piece of phylogenetic information. Therefore, quartet amalgamation stands at the heart of any supertree problem as it concerns combining many minimal pieces of information into a single, coherent, and more comprehensive piece of information.We have devised an extremely fast algorithm for quartet amalgamation and implemented it in a very efficient code. The new code can handle over a hundred millions of quartet trees over several hundreds of taxa with very high accuracy.     

Using max cut to enhance rooted trees consistency

Supertree methods are used to construct a large tree over a large set of taxa from a set of small trees over overlapping subsets of the complete taxa set. Since accurate reconstruction methods are currently limited to a maximum of a few dozen taxa, the use of a supertree method in order to construct the tree of life is inevitable. Supertree methods are broadly divided according to the input trees: When the input trees are unrooted, the basic reconstruction unit is a quartet tree. In this case, the basic decision problem of whether there exists a tree that agrees with all quartets is NP-complete. On the other hand, when the input trees are rooted, the basic reconstruction unit is a rooted triplet and the above decision problem has a polynomial time algorithm. However, when there is no tree which agrees with all triplets, it would be desirable to find the tree that agrees with the maximum number of triplets. However, this optimization problem was shown to be NP-hard. Current heuristic approaches perform min cut on a graph representing the triplets inconsistency and return a tree that is guaranteed to satisfy some required properties. In this work, we present a different heuristic approach that guarantees the properties provided by the current methods and give experimental evidence that it significantly outperforms currently used methods. This method is based on a divide and conquer approach, where the min cut in the divide step is replaced by a max cut in a variant of the same graph. The latter is achieved by a lightweight semidefinite programming-like heuristic that leads to very fast running times     

Root location in random trees: A polarity property of all sampling consistent phylogenetic models except one

Neutral macroevolutionary models, such as the Yule model, give rise to a probability distribution on the set of discrete rooted binary trees over a given leaf set. Such models can provide a signal as to the approximate location of the root when only the unrooted phylogenetic tree is known, and this signal becomes relatively more significant as the number of leaves grows. In this short note, we show that among models that treat all taxa equally, and are sampling consistent (i.e. the distribution on trees is not affected by taxa yet to be included), all such models, except one (the so-called PDA model), convey some information as to the location of the ancestral root in an unrooted tree.     

重建系统演化树的一种新方法--试错法

重建系统演化树是进化研究的一个极为重要的方面。系统树的构建依赖于一定的方法和数据。在分子系统演化研究中,所使用的数据大多是ＤＮＡ序列、氨基酸序列和分子标记。而就构树方法来说,ＮＪ法、ＭＬ法和ＭＰ法是三种最为普遍使用的方法。本文给出了一种新的建树方法,即试错法。该方法不但具有与ＮＪ法一样好的建树效果,而且不存在难以解释的负枝长问题。     

A genetic algorithm for maximum-likelihood phylogeny inference using nucleotide sequence data

Phylogeny reconstruction is a difficult computational problem, because the number of possible solutions increases with the number of included taxa. For example, for only 14 taxa, there are more than seven trillion possible unrooted phylogenetic trees. For this reason, phylogenetic inference methods commonly use clustering algorithms (e.g., the neighbor-joining method) or heuristic search strategies to minimize the amount of time spent evaluating nonoptimal trees. Even heuristic searches can be painfully slow, especially when computationally intensive optimality criteria such as maximum likelihood are used. I describe here a different approach to heuristic searching (using a genetic algorithm) that can tremendously reduce the time required for maximum-likelihood phylogenetic inference, especially for data sets involving large numbers of taxa. Genetic algorithms are simulations of natural selection in which individuals are encoded solutions to the problem of interest. Here, labeled phylogenetic trees are the individuals, and differential reproduction is effected by allowing the number of offspring produced by each individual to be proportional to that individual's rank likelihood score. Natural selection increases the average likelihood in the evolving population of phylogenetic trees, and the genetic algorithm is allowed to proceed until the likelihood of the best individual ceases to improve over time. An example is presented involving rbcL sequence data for 55 taxa of green plants. The genetic algorithm described here required only 6% of the computational effort required by a conventional heuristic search using tree bisection/reconnection (TBR) branch swapping to obtain the same maximum-likelihood topology.      

Maximum Parsimony and the Skewness Test: A Simulation Study of the Limits of Applicability

The maximum parsimony (MP) method for inferring phylogenies is widely used, but little is known about its limitations in non-asymptotic situations. This study employs large-scale computations with simulated phylogenetic data to estimate the probability that MP succeeds in finding the true phylogeny for up to twelve taxa and 256 characters. The set of candidate phylogenies are taken to be unrooted binary trees; for each simulated data set, the tree lengths of all (2n − 5)!! candidates are computed to evaluate quantities related to the performance of MP, such as the probability of finding the true phylogeny, the probability that the tree with the shortest length is unique, the probability that the true phylogeny has the shortest tree length, and the expected inverse of the number of trees sharing the shortest length. The tree length distributions are also used to evaluate and extend the skewness test of Hillis for distinguishing between random and phylogenetic data. The results indicate, for example, that the critical point after which MP achieves a success probability of at least 0.9 is roughly around 128 characters. The skewness test is found to perform well on simulated data and the study extends its scope to up to twelve taxa.     

Short quartet puzzling: a new quartet-based phylogeny reconstruction algorithm.

Quartet-based phylogeny reconstruction methods, such as Quartet Puzzling, were introduced in the hope that they might be competitive with maximum likelihood methods, without being as computationally intensive. However, despite the numerous quartet-based methods that have been developed, their performance in simulation has been disappointing. In particular, Ranwez and Gascuel, the developers of one of the best quartet methods, conjecture that quartet-based methods have inherent limitations that make them unable to produce trees as accurate as neighbor joining or maximum parsimony. In this paper, we present Short Quartet Puzzling, a new quartet-based phylogeny reconstruction algorithm, and we demonstrate the improved topological accuracy of the new method over maximum parsimony and neighbor joining, disproving the conjecture of Ranwez and Gascuel. We also show a dramatic improvement over Quartet Puzzling. Thus, while our new method is not compared to any ML method (as it is not expected to be as accurate as the best of these), this study shows that quartet methods are not as limited in performance as was previously conjectured, and opens the possibility to further improvements through new algorithmic designs.     

Unrooted trees discovered independently in philology and phylogenetics: a remarkable case of methodological convergence

We report on the only known case of independent discovery of unrooted trees in a historical science outside of biological systematics. The method of textual criticism (ecdotics, i.e., the building of text-version genealogies) created by French philologist Henri Quentin (1872–1935) proposes the use of a type of branching scheme equivalent to unrooted trees in phylogenetics. Because Quentin's method has never become the prevailing paradigm in philology, his insight into unrooted trees has not been noticed in previous studies comparing philology and phylogenetics. In fact, the modern use of unrooted trees in philology is seen as imported from phylogenetics. Quentin's procedure starts by building an unrooted tree (‘chain’) expressing the network of text versions (taxa) based on ‘variants’ (equivalent to unpolarized character states). Such undirected scheme is then rooted on the basis of extrinsic temporal information, thus resulting in a complete (rooted) hypothesis of relationships. Quentin asserts that the building of an unrooted tree precedes the determination of its orientation (rooting) and that the two procedures reflect distinct levels of structural organization, relying on different assumptions. Henri Quentin fully grasped the implications of time-reversible properties of unrooted trees and associated characters, in striking prescience of the same concepts developed in phylogenetics some 45 years later. The two versions of unrooted trees were developed entirely independently of each other and such convergence is testimony to the formal efficiency of approaching historical reconstruction in unrooted and rooted dimensions.     

Cytochrome b phylogeny of North American hares and jackrabbits (Lepus, lagomorpha) and the effects of saturation in outgroup taxa

Jackrabbits and hares, members of the genus Lepus, comprise over half of the species within the family Leporidae (Lagomorpha). Despite their ecological importance, potential economic impact, and worldwide distribution, the evolution of hares and jackrabbits has been poorly studied. We provide an initial phylogenetic framework for jackrabbits and hares so that explicit hypotheses about their evolution can be developed and tested. To this end, we have collected DNA sequence data from a 702-bp region of the mitochondrial cytochrome b gene and reconstructed the evolutionary history (via parsimony, neighbor joining, and maximum likelihood) of 11 species of Lepus, focusing on North American taxa. Due to problems of saturation, induced by multiple substitutions, at synonymous coding positions between the ingroup taxa and the outgroups (Oryctolagus and Sylvilagus), both rooted and unrooted trees were examined. Variation in tree topologies generated by different reconstruction methods was observed in analyses including the outgroups, but not in the analyses of unrooted ingroup networks. Apparently, substitutional saturation hindered the analyses when outgroups were considered. The trees based on the cytochrome b data indicate that the taxonomic status of some species needs to be reassessed and that species of Lepus within North America do not form a monophyletic entity.     

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.     

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.     

Phylogenetic identification of lateral genetic transfer events

Background  

Lateral genetic transfer can lead to disagreements among phylogenetic trees comprising sequences from the same set of taxa. Where topological discordance is thought to have arisen through genetic transfer events, tree comparisons can be used to identify the lineages that may have shared genetic information. An 'edit path' of one or more transfer events can be represented with a series of subtree prune and regraft (SPR) operations, but finding the optimal such set of operations is NP-hard for comparisons between rooted trees, and may be so for unrooted trees as well.     

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     

Closure operations in phylogenetics

Closure operations are a useful device in both the theory and practice of tree reconstruction in biology and other areas of classification. These operations take a collection of trees (rooted or unrooted) that classify overlapping sets of objects at their leaves, and infer further tree-like relationships. In this paper we investigate closure operations on phylogenetic trees; both rooted and unrooted; as well as on X-splits, and in a general abstract setting. We derive a number of new results, particularly concerning the completeness (and incompleteness) and complexity of various types of closure rules.     

GeneTRACE-reconstruction of gene content of ancestral species

While current computational methods allow the reconstruction of individual ancestral protein sequences, reconstruction of complete gene content of ancestral species is not yet an established task. In this paper, we describe GENETRACE, an efficient linear-time algorithm that allows the reconstruction of evolutionary history of individual protein families as well as the complete gene content of ancestral species. The performance of the method was validated with a simulated evolution program called SimulEv. Our results indicate that given a set of correct phylogenetic profiles and a correct species tree, ancestral gene content can be reconstructed with sensitivity and selectivity of more than 90%. SimulEv simulations were also used to evaluate performance of the reconstruction of gene content-based phylogenetic trees, suggesting that these trees may be accurate at the terminal branches but suffer from long branch attraction near the root of the tree.     

A fully resolved consensus between fully resolved phylogenetic trees

Nowadays, there are many phylogeny reconstruction methods, each with advantages and disadvantages. We explored the advantages of each method, putting together the common parts of trees constructed by several methods, by means of a consensus computation. A number of phylogenetic consensus methods are already known. Unfortunately, there is also a taboo concerning consensus methods, because most biologists see them mainly as comparators and not as phylogenetic tree constructors. We challenged this taboo by defining a consensus method that builds a fully resolved phylogenetic tree based on the most common parts of fully resolved trees in a given collection. We also generated results showing that this consensus is in a way a kind of "median" of the input trees; as such it can be closer to the correct tree in many situations.     

Optimization over a class of tree shape statistics

Tree shape statistics quantify some aspect of the shape of a phylogenetic tree. They are commonly used to compare reconstructed trees to evolutionary models and to find evidence of tree reconstruction bias. Historically, to find a useful tree shape statistic, formulas have been invented by hand and then evaluated for utility. This paper presents the first method which is capable of optimizing over a class of tree shape statistics, called binary recursive tree shape statistics (BRTSS). After defining the BRTSS class, a set of algebraic expressions is defined which can be used in the recursions. The set of tree shape statistics definable using these expressions in the BRTSS is very general and includes many of the statistics with which phylogenetic researchers are already familiar. We then present a practical genetic algorithm which is capable of performing optimization over BRTSS given any objective function. The chapter concludes with a successful application of the methods to find a new statistic which indicates a significant difference between two distributions on trees which were previously postulated to have similar properties.     

Optimal algorithms for local vertex quartet cleaning

MOTIVATION: Reconstructing evolutionary trees is an important problem in biology. A response to the computational intractability of most of the traditional criteria for inferring evolutionary trees has been a focus on new criteria, particularly quartet-based methods that seek to merge trees derived on subsets of four species from a given species-set into a tree for that entire set. Unfortunately, most of these methods are very sensitive to errors in the reconstruction of the trees for individual quartets of species. A recently developed technique called quartet cleaning can alleviate this difficulty in certain cases by using redundant information in the complete set of quartet topologies for a given species-set to correct such errors. RESULTS: In this paper, we describe two new local vertex quartet cleaning algorithms which have optimal time complexity and error-correction bound, respectively. These are the first known local vertex quartet cleaning algorithms that are optimal with respect to either of these attributes.     

Inferring phylogeny from whole genomes

MOTIVATION: Inferring species phylogenies with a history of gene losses and duplications is a challenging and an important task in computational biology. This problem can be solved by duplication-loss models in which the primary step is to reconcile a rooted gene tree with a rooted species tree. Most modern methods of phylogenetic reconstruction (from sequences) produce unrooted gene trees. This limitation leads to the problem of transforming unrooted gene tree into a rooted tree, and then reconciling rooted trees. The main questions are 'What about biological interpretation of choosing rooting?', 'Can we find efficiently the optimal rootings?', 'Is the optimal rooting unique?'. RESULTS: In this paper we present a model of reconciling unrooted gene tree with a rooted species tree, which is based on a concept of choosing rooting which has minimal reconciliation cost. Our analysis leads to the surprising property that all the minimal rootings have identical distributions of gene duplications and gene losses in the species tree. It implies, in our opinion, that the concept of an optimal rooting is very robust, and thus biologically meaningful. Also, it has nice computational properties. We present a linear time and space algorithm for computing optimal rooting(s). This algorithm was used in two different ways to reconstruct the optimal species phylogeny of five known yeast genomes from approximately 4700 gene trees. Moreover, we determined locations (history) of all gene duplications and gene losses in the final species tree. It is interesting to notice that the top five species trees are the same for both methods. AVAILABILITY: Software and documentation are freely available from http://bioputer.mimuw.edu.pl/~gorecki/urec