Designing an A* algorithm for calculating edit distance between rooted-unordered trees.

Tree structures are useful for describing and analyzing biological objects and processes. Consequently, there is a need to design metrics and algorithms to compare trees. A natural comparison metric is the "Tree Edit Distance," the number of simple edit (insert/delete) operations needed to transform one tree into the other. Rooted-ordered trees, where the order between the siblings is significant, can be compared in polynomial time. Rooted-unordered trees are used to describe processes or objects where the topology, rather than the order or the identity of each node, is important. For example, in immunology, rooted-unordered trees describe the process of immunoglobulin (antibody) gene diversification in the germinal center over time. Comparing such trees has been proven to be a difficult computational problem that belongs to the set of NP-Complete problems. Comparing two trees can be viewed as a search problem in graphs. A* is a search algorithm that explores the search space in an efficient order. Using a good lower bound estimation of the degree of difference between the two trees, A* can reduce search time dramatically. We have designed and implemented a variant of the A* search algorithm suitable for calculating tree edit distance. We show here that A* is able to perform an edit distance measurement in reasonable time for trees with dozens of nodes.     

treegraph: automated drawing of complex tree figures using an extensible tree description format

treegraph assists in producing complex ready‐to‐publish figures of phylogenetic trees. The TGF format used by the program automates formatting of several different statistical support value types (confidence estimates) per tree node. Moreover, internal text and graphical labels are automatically arranged at the nodes as are annotations for clades or groups of terminals. treegraph imports nexus trees and related file formats. Beyond common tree edit operations, simultaneous pruning of subtrees (simplification of the tree to higher order clades) and saving of subtrees is possible. treegraph exports to the standard vector graphics formats Scalable Vector Graphics and PostScript.     

Protein fold comparison by the alignment of topological strings

Using the definitions of protein folds encoded in a text string, a dynamic programming algorithm was devised to compare these and identify their largest common substructure and calculate the distance (in terms of the number of edit operations) that this lay from each structure. This provided a metric on which the folds were clustered into a 'phylogenetic' tree. This construction differs from previous automatic structure clustering algorithms as it has explicit representation of the structures at 'ancestral' branching nodes, even when these have no corresponding known structure. The resulting tree was compared with that compiled by an 'expert' in the field and while there was broad agreement, differences were found that resulted from differing degrees of emphasis being placed on the types of operations that can be used to transform structures. Some concluding speculations on the relationship of such trees to the evolutionary history and folding of the proteins are advanced.     

LabKey Server: An open source platform for scientific data integration, analysis and collaboration

Background

This paper is devoted to distance measures for leaf-labelled trees on free leafset. A leaf-labelled tree is a data structure which is a special type of a tree where only leaves (terminal) nodes are labelled. This data structure is used in bioinformatics for modelling of evolution history of genes and species and also in linguistics for modelling of languages evolution history. Many domain specific problems occur and need to be solved with help of tree postprocessing techniques such as distance measures.

Results

Here we introduce the tree edit distance designed for leaf labelled trees on free leafset, which occurs to be a metric. It is presented together with tree edit consensus tree notion. We provide statistical evaluation of provided measure with respect to R-F, MAST and frequent subsplit based dissimilarity measures as the reference measures.

Conclusions

The tree edit distance was proven to be a metric and has the advantage of using different costs for contraction and pruning, therefore their properties can be tuned depending on the needs of the user. Two of the presented methods carry the most interesting properties. E(3,1) is very discriminative (having a wide range of values) and has a very regular distance distribution which is similar to a normal distribution in its shape and is good both for similar and non-similar trees. NFC(2,1) on the other hand is proportional or nearly proportional to the number of mutation operations used, irrespective of their type.     

Seeded tree alignment

The optimal transformation of one tree into another by means of elementary edit operations is an important algorithmic problem that has several interesting applications to computational biology. Here we introduce a constrained form of this problem in which a partial mapping of a set of nodes (the "seeds") in one tree to a corresponding set of nodes in the other tree is given, and present efficient algorithms for both ordered and unordered trees. Whereas ordered tree matching based on seeded nodes has applications in pattern matching of RNA structures, unordered tree matching based on seeded nodes has applications in co-speciation and phylogeny reconciliation. The latter involves the solution of the planar tanglegram layout problem, for which a polynomial-time algorithm is given here.     

Evolution of the chromosomal races of Mus musculus domesticus in the Rhaetian Alps: the roles of whole-arm reciprocal translocation and zonal raciation

The evolution of ten Robertsonian (Rb) races of the house mouse (Mus musculus domesticus) in the Rhaetian Alps of northern Italy and southern Switzerland is reconsidered. The mechanisms of centric fusion, zonal raciation and, for the first time, whole-arm reciprocal translocation (WART), are used in this non-mathematical approach to produce a phylogenetic tree (using chromosome fusions as characters) with the smallest number of steps. The shortest tree that we found (16 steps) is at least two to nine mutations shorter than previously published models. Three other trees (17 or 18 steps) are also considered, since they are geographically more sensible. In general, these four scenarios correspond more closely to the present distributions of the ten Rb races than previous trees. Our results suggest that zonal raciation and WARTs play an important role in the evolution of Rb races of the house mouse.     

Problems in using Robertsonian rearrangements in determining monophyly: examples from the genera Tatera and Gerbillurus

Chromosomal banding data on three species of Tatera from Kenya significantly alter the previous hypothesis of relationships between and within the genera Tatera and Gerbillurus based on cladistic analyses and the rule of parsimony (Qumsiyeh, 1986b). Of the many possible hypothetical relationships, the most parsimonious tree showed three homoplasies and allowed the genus Gerbillurus to be paraphyletic. The alternative trees, depicting larger number of homoplasies but with homoplasies restricted to fusion or fission events, were compatible with the morphological data in supporting the monophyly of Gerbillurus. To choose between the hypothesis based on the most parsimonious chromosomal tree and those supported by both morphology and slightly less parsimonious chromosomal trees, we performed an electrophoretic study on this group of gerbils. The conclusions are that the genus Gerbillurus is monophyletic and represents a branch that is closely related to the T. robusta group of Taterillini. The study documents that fissions and fusions must have occurred frequently and that in some cases the same fusions were acquired in two independent lineages in numbers exceeding those that are predicted by strict parsimony. The results raise questions about the validity of systematic conclusions based solely on fusion/fission data and utilizing the parsimony criterion.     

FastJoin, an improved neighbor-joining algorithm

Reconstructing the evolutionary history of a set of species is an elementary problem in biology, and methods for solving this problem are evaluated based on two characteristics: accuracy and efficiency. Neighbor-joining reconstructs phylogenetic trees by iteratively picking a pair of nodes to merge as a new node until only one node remains; due to its good accuracy and speed, it has been embraced by the phylogeny research community. With the advent of large amounts of data, improved fast and precise methods for reconstructing evolutionary trees have become necessary. We improved the neighbor-joining algorithm by iteratively picking two pairs of nodes and merging as two new nodes, until only one node remains. We found that another pair of true neighbors could be chosen to merge as a new node besides the pair of true neighbors chosen by the criterion of the neighbor-joining method, in each iteration of the clustering procedure for the purely additive tree. These new neighbors will be selected by another iteration of the neighbor-joining method, so that they provide an improved neighbor-joining algorithm, by iteratively picking two pairs of nodes to merge as two new nodes until only one node remains, constructing the same phylogenetic tree as the neighbor-joining algorithm for the same input data. By combining the improved neighbor-joining algorithm with styles upper bound computation optimization of RapidNJ and external storage of ERapidNJ methods, a new method of reconstructing phylogenetic trees, FastJoin, was proposed. Experiments with sets of data showed that this new neighbor-joining algorithm yields a significant speed-up compared to classic neighbor-joining, showing empirically that FastJoin is superior to almost all other neighbor-joining implementations.     

A clique-based method for the edit distance between unordered trees and its application to analysis of glycan structures

Background

Measuring similarities between tree structured data is important for analysis of RNA secondary structures, phylogenetic trees, glycan structures, and vascular trees. The edit distance is one of the most widely used measures for comparison of tree structured data. However, it is known that computation of the edit distance for rooted unordered trees is NP-hard. Furthermore, there is almost no available software tool that can compute the exact edit distance for unordered trees.

Results

In this paper, we present a practical method for computing the edit distance between rooted unordered trees. In this method, the edit distance problem for unordered trees is transformed into the maximum clique problem and then efficient solvers for the maximum clique problem are applied. We applied the proposed method to similar structure search for glycan structures. The result suggests that our proposed method can efficiently compute the edit distance for moderate size unordered trees. It also suggests that the proposed method has the accuracy comparative to those by the edit distance for ordered trees and by an existing method for glycan search.

Conclusions

The proposed method is simple but useful for computation of the edit distance between unordered trees. The object code is available upon request.

     

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.     

Reconstructing evolution of sequences subject to recombination using parsimony

The parsimony principle states that a history of a set of sequences that minimizes the amount of evolution is a good approximation to the real evolutionary history of the sequences. This principle is applied to the reconstruction of the evolution of homologous sequences where recombinations or horizontal transfer can occur. First it is demonstrated that the appropriate structure to represent the evolution of sequences with recombinations is a family of trees each describing the evolution of a segment of the sequence. Two trees for neighboring segments will differ by exactly the transfer of a subtree within the whole tree. This leads to a metric between trees based on the smallest number of such operations needed to convert one tree into the other. An algorithm is presented that calculates this metric. This metric is used to formulate a dynamic programming algorithm that finds the most parsimonious history that fits a given set of sequences. The algorithm is potentially very practical, since many groups of sequences defy analysis by methods that ignore recombinations. These methods give ambiguous or contradictory results because the sequence history cannot be described by one phylogeny, but only a family of phylogenies that each describe the history of a segment of the sequences. The generalization of the algorithm to reconstruct gene conversions and the possibility for heuristic versions of the algorithm for larger data sets are discussed.     

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.     

Searching for Most Parsimonious Trees with Simulated Evolutionary Optimization

This study describes novel algorithms for searching for most parsimonious trees. These algorithms are implemented as a parsimony computer program, PARSIGAL, which performs well even with difficult data sets. For high level search, PARSIGAL uses an evolutionary optimization algorithm, which feeds good tree candidates to a branch-swapping local search procedure. This study also describes an extremely fast method of recomputing state sets for binary characters (additive or nonadditive characters with two states), based on packing 32 characters into a single memory word and recomputing the tree simultaneously for all 32 characters using fast bitwise logical operations. The operational principles of PARSIGAL are quite different from those previously published for other parsimony computer programs. Hence it is conceivable that PARSIGAL may be able to locate islands of trees that are different from those that are easily located with existing parsimony computer programs.     

Identification of phylogenetic trees of minimal length.

The problem of determining an optimal phylogenetic tree from a set of data is an example of the Steiner problem in graphs. There is no efficient algorithm for solving this problem with reasonably large data sets. In the present paper an approach is described that proves in some cases that a given tree is optimal without testing all possible trees. The method first uses a previously described heuristic algorithm to find a tree of relatively small total length. The second part of the method independently analyses subsets of sites to determine a lower bound on the length of any tree. We simultaneously attempt to reduce the total length of the tree and increase the lower bound. When these are equal it is not possible to make a shorter tree with a given data set and given criterion. An example is given where the only two possible minimal trees are found for twelve different mammalian cytochrome c sequences. The criterion of finding the smallest number of minimum base changes was used. However, there is no general method of guaranteeing that a solution will be found in all cases and in particular better methods of improving the estimate of the lower bound need to be developed.     

A stepwise algorithm for finding minimum evolution trees

A stepwise algorithm for reconstructing minimum evolution (ME) trees from evolutionary distance data is proposed. In each step, a taxon that potentially has a neighbor (another taxon connected to it with a single interior node) is first chosen and then its true neighbor searched iteratively. For m taxa, at most (m-1)!/2 trees are examined and the tree with the minimum sum of branch lengths (S) is chosen as the final tree. This algorithm provides simple strategies for restricting the tree space searched and allows us to implement efficient ways of dynamically computing the ordinary least squares estimates of S for the topologies examined. Using computer simulation, we found that the efficiency of the ME method in recovering the correct tree is similar to that of the neighbor-joining method (Saitou and Nei 1987). A more exhaustive search is unlikely to improve the efficiency of the ME method in finding the correct tree because the correct tree is almost always included in the tree space searched with this stepwise algorithm. The new algorithm finds trees for which S values may not be significantly different from that of the ME tree if the correct tree contains very small interior branches or if the pairwise distance estimates have large sampling errors. These topologies form a set of plausible alternatives to the ME tree and can be compared with each other using statistical tests based on the minimum evolution principle. The new algorithm makes it possible to use the ME method for large data sets.      

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.     

Influence of Fruit Availability on the Fission–Fusion Dynamics of Spider Monkeys (<Emphasis Type="Italic">Ateles geoffroyi</Emphasis>)

Socioecological theory proposes that the flexibility in grouping patterns afforded by fission–fusion dynamics allows animals to cope with spatiotemporal variability in food abundance. We investigate the influence of fruit tree abundance and foraging environment heterogeneity on fission–fusion dynamics in a group of spider monkeys (Ateles geoffroyi) in the Yucatan peninsula, Mexico. We collected 1300 h of behavioral data and 23 samples of biweekly ecological data from August 2009 to July 2010. We measured fission–fusion dynamics through the temporal variation in the size and composition of subgroups, the spatial dispersion within and between subgroups, and the frequency of fissions and fusions. We measured habitat-wide food abundance of preferred species, including two that differ greatly in their relative abundance: Brosimum alicastrum (a hyperabundant resource) and Ficus spp. (a not so abundant resource but often represented by large trees). We evaluated the foraging environment heterogeneity through the variance in the number of trees with fruit between species. Our results show that, although habitat-wide food abundance is important, the availability of key resources strongly influences the spider monkeys’ fission–fusion dynamics. When there was a high abundance of fruit of Brosimum, subgroups tended to be more stable, smaller, and mixed sex, and their members remained close. In contrast, when Brosimum trees with fruit were scarce, females often formed large, more fluid and dispersed subgroups. Foraging environment heterogeneity had a positive effect on within-subgroup spatial dispersion and rates of fission and fusion. The complex relationships we have uncovered suggest that the flexibility afforded by fission–fusion dynamics is an adaptation to highly variable foraging environments.