An efficient and accurate distance based algorithm to reconstruct tandem duplication trees

The problem of reconstructing the duplication tree of a set of tandemly repeated sequences which are supposed to have arisen through unequal recombination, was first introduced by Fitch (1977, Genetics, 86, 93-104), and has recently received a lot of attention. In this paper, we describe DTSCORE, a fast distance based algorithm to reconstruct tandem duplication trees, which is statistically consistent. As a cousin of the ADDTREE algorithm (Sattath and Tversky, 1977, Psychometrika, 42, 319-345), the raw DTSCORE has a time complexity in O(n(5)), where n is the number of observed repeated sequences. Through a series of algorithmic refinements, we improve its complexity to O(n(4)) in the worst case, but stress that the refined DTSCORE algorithm should perform faster with real data. We assess the topological accuracy of DTSCORE using simulated data sets, and compare it to existing reconstruction methods. The results clearly show that DTSCORE is more accurate than all the other methods we studied. Finally, we report the results of DTSCORE on a real dataset. Supplementary information: http://www.lirmm.fr/w3ifa/MAAS/     

Efficient substructure RMSD query algorithms.

Protein structure analysis is a very important research topic in the molecular biology of the post-genomic era. The root mean square deviation (RMSD) is the most frequently used measure for comparing two protein three-dimensional (3-D) structures. In this paper, we deal with two fundamental problems related to the RMSD. We first deal with a problem called the "range RMSD query" problem. Given an aligned pair of structures, the problem is to compute the RMSD between two aligned substructures of them without gaps. This problem has many applications in protein structure analysis. We propose a linear-time preprocessing algorithm that enables constant-time RMSD computation. Next, we consider a problem called the "substructure RMSD query" problem, which is a generalization of the above range RMSD query problem. It is a problem to compute the RMSD between any substructures of two unaligned structures without gaps. Based on the algorithm for the range RMSD problem, we propose an O(nm) preprocessing algorithm that enables constant-time RMSD computation, where n and m are the lengths of the given structures. Moreover, we propose O(nm log r/r)-time and O(nm/r)-space preprocessing algorithm that enables O(r) query, where r is an arbitrary integer such that 1 < or = r < or = min(n, m). We also show that our strategy also works for another measure called the unit-vector root mean square deviation (URMSD), which is a variant of the RMSD.     

Practicality and time complexity of a sparsified RNA folding algorithm

Commonly used RNA folding programs compute the minimum free energy structure of a sequence under the pseudoknot exclusion constraint. They are based on Zuker's algorithm which runs in time O(n(3)). Recently, it has been claimed that RNA folding can be achieved in average time O(n(2)) using a sparsification technique. A proof of quadratic time complexity was based on the assumption that computational RNA folding obeys the "polymer-zeta property". Several variants of sparse RNA folding algorithms were later developed. Here, we present our own version, which is readily applicable to existing RNA folding programs, as it is extremely simple and does not require any new data structure. We applied it to the widely used Vienna RNAfold program, to create sibRNAfold, the first public sparsified version of a standard RNA folding program. To gain a better understanding of the time complexity of sparsified RNA folding in general, we carried out a thorough run time analysis with synthetic random sequences, both in the context of energy minimization and base pairing maximization. Contrary to previous claims, the asymptotic time complexity of a sparsified RNA folding algorithm using standard energy parameters remains O(n(3)) under a wide variety of conditions. Consistent with our run-time analysis, we found that RNA folding does not obey the "polymer-zeta property" as claimed previously. Yet, a basic version of a sparsified RNA folding algorithm provides 15- to 50-fold speed gain. Surprisingly, the same sparsification technique has a different effect when applied to base pairing optimization. There, its asymptotic running time complexity appears to be either quadratic or cubic depending on the base composition. The code used in this work is available at: .     

Haplotypes histories as pathways of recombinations

MOTIVATION: The diversity of a haplotype, represented as a string of polymorphic sites along a DNA sequence, increases exponentially with the number of sites if recombinations are taking place. Reconstructing the history of recombinations compared with that of the polymorphic sites is thus extremely difficult. However, in the human genome, because of the relatively simple pattern of haplotype diversity dominated by a few ancestral haplotypes, the complexity of the recombinational network can be reduced, thus making its reconstruction feasible. We focus on the problem of inferring the recombination pathways starting with putative ancestral haplotypes and leading to new rare recombinant haplotypes. RESULTS: We describe classes of recombinations that represent the whole set of minimal recombination pathways leading to a new haplotype. We present an O(n(2)) algorithm that outputs such representative recombination pathways. We apply it to haplotypes of the 8 kb dystrophin gene segment dys44. AVAILABILITY: A software implementing the algorithm and some other extentions has been developed on a Java platform (JDK 1.3.1). It is freely available at http://www.iro.umontreal.ca/~mabrouk/     

An algorithm to enumerate sorting reversals for signed permutations.

The rearrangement distance between single-chromosome genomes can be estimated as the minimum number of inversions required to transform the gene ordering observed in one into that observed in the other. This measure, known as "inversion distance," can be computed as the reversal distance between signed permutations. During the past decade, much progress has been made both on the problem of computing reversal distance and on the related problem of finding a minimum-length sequence of reversals, which is known as "sorting by reversals." For most problem instances, however, many minimum-length sequences of reversals exist, and in the absence of auxiliary information, no one is of greater value than the others. The problem of finding all minimum-length sequences of reversals is thus a natural generalization of sorting by reversals, yet it has received little attention. This problem reduces easily to the problem of finding all "sorting reversals" of one permutation with respect to another - that is, all reversals rho such that, if rho is applied to one permutation, then the reversal distance of that permutation from the other is decreased. In this paper, an efficient algorithm is derived to solve the problem of finding all sorting reversals, and experimental results are presented indicating that, while the new algorithm does not represent a significant improvement in asymptotic terms (it takes O(n(3)) time, for permutations of size n; the problem can now be solved by brute force in Theta(n(3)) time), it performs dramatically better in practice than the best known alternative. An implementation of the algorithm is available at www.cse.ucsc.edu/~acs.     

A fast algorithm to build a supertree with a set of gene trees

Important desired properties of an algorithm to construct a supertree (species tree) by reconciling input trees are its low complexity and applicability to large biological data. In its common statement the problem is proved to be NP-hard, i.e. to have an exponential complexity in practice. We propose a reformulation of the supertree building problem that allows a computationally effective solution. We introduce a biologically natural requirement that the supertree is sought for such that it does not contain clades incompatible with those existing in the input trees. The algorithm was tested with simulated and biological trees and was shown to possess an almost square complexity even if horizontal transfers are allowed. If HGTs are not assumed, the algorithm is mathematically correct and possesses the longest running time of n3 x[V0]3, where n is the number of input trees and [V0] is the total number of species. The authors are unaware of analogous solutions in published evidence. The corresponding inferring program, its usage examples and manual are freely available at http://lab6.iitp.ru/en/super3gl. The available program does not implement HGTs. The generalized case is described in the publication "A tree nearest in average to a set of trees" (Information Transmission Problems, 2011).     

Towards sub-quadratic time and space complexity solutions for the dated tree reconciliation problem

Background

Recent coevolutionary analysis has considered tree topology as a means to reduce the asymptotic complexity associated with inferring the complex coevolutionary interrelationships that arise between phylogenetic trees. Targeted algorithmic design for specific tree topologies has to date been highly successful, with one recent formulation providing a logarithmic space complexity reduction for the dated tree reconciliation problem.

Methods

In this work we build on this prior analysis providing a further asymptotic space reduction, by providing a new formulation for the dynamic programming table used by a number of popular coevolutionary analysis techniques. This model gives rise to a sub quadratic running time solution for the dated tree reconciliation problem for selected tree topologies, and is shown to be, in practice, the fastest method for solving the dated tree reconciliation problem for expected evolutionary trees. This result is achieved through the analysis of not only the topology of the trees considered for coevolutionary analysis, but also the underlying structure of the dynamic programming algorithms that are traditionally applied to such analysis.

Conclusion

The newly inferred theoretical complexity bounds introduced herein are then validated using a combination of synthetic and biological data sets, where the proposed model is shown to provide an \(O(\sqrt{n})\) space saving, while it is observed to run in half the time compared to the fastest known algorithm for solving the dated tree reconciliation problem. What is even more significant is that the algorithm derived herein is able to guarantee the optimality of its inferred solution, something that algorithms of comparable speed have to date been unable to achieve.

     

Local sequence-structure motifs in RNA

Ribonuclic acid (RNA) enjoys increasing interest in molecular biology; despite this interest fundamental algorithms are lacking, e.g. for identifying local motifs. As proteins, RNA molecules have a distinctive structure. Therefore, in addition to sequence information, structure plays an important part in assessing the similarity of RNAs. Furthermore, common sequence-structure features in two or several RNA molecules are often only spatially local, where possibly large parts of the molecules are dissimilar. Consequently, we address the problem of comparing RNA molecules by computing an optimal local alignment with respect to sequence and structure information. While local alignment is superior to global alignment for identifying local similarities, no general local sequence-structure alignment algorithms are currently known. We suggest a new general definition of locality for sequence-structure alignments that is biologically motivated and efficiently tractable. To show the former, we discuss locality of RNA and prove that the defined locality means connectivity by atomic and non-atomic bonds. To show the latter, we present an efficient algorithm for the newly defined pairwise local sequence-structure alignment (lssa) problem for RNA. For molecules of lengthes n and m, the algorithm has worst-case time complexity of O(n2 x m2 x max(n,m)) and a space complexity of only O(n x m). An implementation of our algorithm is available at http://www.bio.inf.uni-jena.de. Its runtime is competitive with global sequence-structure alignment.     

Fast evaluation of internal loops in RNA secondary structure prediction.

MOTIVATION: Though not as abundant in known biological processes as proteins, RNA molecules serve as more than mere intermediaries between DNA and proteins. Research in the last 15 years demonstrates that RNA molecules serve in many roles, including catalysis. Furthermore, RNA secondary structure prediction based on free energy rules for stacking and loop formation remains one of the few major breakthroughs in the field of structure prediction, as minimum free energy structures and related quantities can be computed with full mathematical rigor. However, with the current energy parameters, the algorithms used hitherto suffer the disadvantage of either employing heuristics that risk (though highly unlikely) missing the optimal structure or becoming prohibitively time consuming for moderate to large sequences. RESULTS: We present a new method to evaluate internal loops utilizing currently used energy rules. This method reduces the time complexity of this part of the structure prediction from O(n4) to O(n3), thus reducing the overall complexity to O(n3). Even when the size of evaluated internal loops is bounded by k (a commonly used heuristic), the method presented has a competitive edge by reducing the time complexity of internal loop evaluation from O(k2n2) to O(kn2). The method also applies to the calculation of the equilibrium partition function. AVAILABILITY: Source code for an RNA secondary structure prediction program implementing this method is available at ftp://www.ibc.wustl.edu/pub/zuker/zuker .tar.Z     

Comparison of additive trees using circular orders.

It has been postulated that existing species have been linked in the past in a way that can be described using an additive tree structure. Any such tree structure reflecting species relationships is associated with a matrix of distances between the species considered which is called a distance matrix or a tree metric matrix. A circular order of elements of X corresponds to a circular (clockwise) scanning of the subset X of vertices of a tree drawn on a plane. This paper describes an optimal algorithm using circular orders to compare the topology of two trees given by their distance matrices. This algorithm allows us to compute the Robinson and Foulds topologic distance between two trees. It employs circular order tree reconstruction to compute an ordered bipartition table of the tree edges for both given distance matrices. These bipartition tables are then compared to determine the Robinson and Foulds topologic distance, known to be an important criterion of tree similarity. The described algorithm has optimal time complexity, requiring O(n(2)) time when performed on two n x n distance matrices. It can be generalized to get another optimal algorithm, which enables the strict consensus tree of k unrooted trees, given their distance matrices, to be constructed in O(kn(2)) time.     

Protein local structure alignment under the discrete Fréchet distance.

Protein structure alignment is a fundamental problem in computational and structural biology. While there has been lots of experimental/heuristic methods and empirical results, very few results are known regarding the algorithmic/complexity aspects of the problem, especially on protein local structure alignment. A well-known measure to characterize the similarity of two polygonal chains is the famous Fréchet distance, and with the application of protein-related research, a related discrete Fréchet distance has been used recently. In this paper, following the recent work of Jiang et al. we investigate the protein local structural alignment problem using bounded discrete Fréchet distance. Given m proteins (or protein backbones, which are 3D polygonal chains), each of length O(n), our main results are summarized as follows: * If the number of proteins, m, is not part of the input, then the problem is NP-complete; moreover, under bounded discrete Fréchet distance it is NP-hard to approximate the maximum size common local structure within a factor of n(1-epsilon). These results hold both when all the proteins are static and when translation/rotation are allowed. * If the number of proteins, m, is a constant, then there is a polynomial time solution for the problem.     

A recursive method for solving haplotype frequencies with application to genetics

Multiple loci analysis has become popular with the advanced developments in biological experiments. A lot of studies have been focused on the biological and the statistical properties of such multiple loci analysis. In this paper, we study one of the important computational problems: solving the probabilities of haplotype classes from a large linear system Ax = b derived from the recombination events in multiple loci analysis. Since the size of the recombination matrix A increases exponentially with respect to the number of loci, fast solvers are required to deal with a large number of loci in the analysis. By exploiting the nice structure of the matrix A, we develop an efficient recursive algorithm for solving such structured linear systems. In particular, the complexity of the proposed algorithm for the n loci problem is of O(n2(n)) operations and the memory requirement is of O(2(n)) locations for the 2(n)-by-2(n) matrix A. Numerical examples are given to demonstrate the effectiveness of our efficient solver. Finally, we apply our proposed method to analyze the haplotype classes for a set of single nucleotides polymorphisms (SNPs) from Hapmap data.     

LS Bound based gene selection for DNA microarray data

MOTIVATION: One problem with discriminant analysis of DNA microarray data is that each sample is represented by quite a large number of genes, and many of them are irrelevant, insignificant or redundant to the discriminant problem at hand. Methods for selecting important genes are, therefore, of much significance in microarray data analysis. In the present study, a new criterion, called LS Bound measure, is proposed to address the gene selection problem. The LS Bound measure is derived from leave-one-out procedure of LS-SVMs (least squares support vector machines), and as the upper bound for leave-one-out classification results it reflects to some extent the generalization performance of gene subsets. RESULTS: We applied this LS Bound measure for gene selection on two benchmark microarray datasets: colon cancer and leukemia. We also compared the LS Bound measure with other evaluation criteria, including the well-known Fisher's ratio and Mahalanobis class separability measure, and other published gene selection algorithms, including Weighting factor and SVM Recursive Feature Elimination. The strength of the LS Bound measure is that it provides gene subsets leading to more accurate classification results than the filter method while its computational complexity is at the level of the filter method. AVAILABILITY: A companion website can be accessed at http://www.ntu.edu.sg/home5/pg02776030/lsbound/. The website contains: (1) the source code of the gene selection algorithm; (2) the complete set of tables and figures regarding the experimental study; (3) proof of the inequality (9). CONTACT: ekzmao@ntu.edu.sg.     

PartTree: an algorithm to build an approximate tree from a large number of unaligned sequences

MOTIVATION: To construct a multiple sequence alignment (MSA) of a large number (> approximately 10,000) of sequences, the calculation of a guide tree with a complexity of O(N2) to O(N3), where N is the number of sequences, is the most time-consuming process. RESULTS: To overcome this limitation, we have developed an approximate algorithm, PartTree, to construct a guide tree with an average time complexity of O(N log N). The new MSA method with the PartTree algorithm can align approximately 60,000 sequences in several minutes on a standard desktop computer. The loss of accuracy in MSA caused by this approximation was estimated to be several percent in benchmark tests using Pfam. AVAILABILITY: The present algorithm has been implemented in the MAFFT sequence alignment package (http://align.bmr.kyushu-u.ac.jp/mafft/software/). SUPPLEMENTARY INFORMATION: Supplementary information is available at Bioinformatics online.     

An approximate matching algorithm for finding (sub-)optimal sequences in S-attributed grammars

MOTIVATION: S-attributed grammars (a generalization of classical Context-Free grammars) provide a versatile formalism for sequence analysis which allows to express long range constraints: the RNA folding problem is a typical example of application. Efficient algorithms have been developed to solve problems expressed with these tools, which generally compute the optimal attribute of the sequence w.r.t. the grammar. However, it is often more meaningful and/or interesting from the biological point of view to consider almost optimal attributes as well as approximate sequences; we thus need more flexible and powerful algorithms able to perform these generalized analyses. RESULTS: In this paper we present a basic algorithm which, given a grammar G and a sequence omega, computes the optimal attribute for all (approximate) strings omega(') in L(G) such that d(omega, omega(')) < or = M, and whose complexity is O(n(r + 1)) in time and O(n(2)) in space (r is the maximal length of the right-hand side of any production of G). We will also give some extensions and possible improvements of this algorithm.     

Computing phylogenetic diversity for split systems

In conservation biology it is a central problem to measure, predict, and preserve biodiversity as species face extinction. In 1992 Faith proposed measuring the diversity of a collection of species in terms of their relationships on a phylogenetic tree, and to use this information to identify collections of species with high diversity. Here we are interested in some variants of the resulting optimization problem that arise when considering species whose evolution is better represented by a network rather than a tree. More specifically, we consider the problem of computing phylogenetic diversity relative to a split system on a collection of species of size n. We show that for general split systems this problem is NP-hard. In addition we provide some efficient algorithms for some special classes of split systems, in particular presenting an optimal O(n) time algorithm for phylogenetic trees and an O(n log n + nk) time algorithm for choosing an optimal subset of size k relative to a circular split system.     

An optimal algorithm for perfect phylogeny haplotyping.

Inferring haplotype data from genotype data is a crucial step in linking SNPs to human diseases. Given n genotypes over m SNP sites, the haplotype inference (HI) problem deals with finding a set of haplotypes so that each given genotype can be formed by a combining a pair of haplotypes from the set. The perfect phylogeny haplotyping (PPH) problem is one of the many computational approaches to the HI problem. Though it was conjectured that the complexity of the PPH problem was O(nm), the complexity of all the solutions presented until recently was O(nm (2)). In this paper, we make complete use of the column-ordering that was presented earlier and show that there must be some interdependencies among the pairwise relationships between SNP sites in order for the given genotypes to allow a perfect phylogeny. Based on these interdependencies, we introduce the FlexTree (flexible tree) data structure that represents all the pairwise relationships in O(m) space. The FlexTree data structure provides a compact representation of all the perfect phylogenies for the given set of genotypes. We also introduce an ordering of the genotypes that allows the genotypes to be added to the FlexTree sequentially. The column ordering, the FlexTree data structure, and the row ordering we introduce make the O(nm) OPPH algorithm possible. We present some results on simulated data which demonstrate that the OPPH algorithm performs quiet impressively when compared to the previous algorithms. The OPPH algorithm is one of the first O(nm) algorithms presented for the PPH problem.     

Computing fragmentation trees from metabolite multiple mass spectrometry data

Since metabolites cannot be predicted from the genome sequence, high-throughput de novo identification of small molecules is highly sought. Mass spectrometry (MS) in combination with a fragmentation technique is commonly used for this task. Unfortunately, automated analysis of such data is in its infancy. Recently, fragmentation trees have been proposed as an analysis tool for such data. Additional fragmentation steps (MS(n)) reveal more information about the molecule. We propose to use MS(n) data for the computation of fragmentation trees, and present the Colorful Subtree Closure problem to formalize this task: There, we search for a colorful subtree inside a vertex-colored graph, such that the weight of the transitive closure of the subtree is maximal. We give several negative results regarding the tractability and approximability of this and related problems. We then present an exact dynamic programming algorithm, which is parameterized by the number of colors in the graph and is swift in practice. Evaluation of our method on a dataset of 45 reference compounds showed that the quality of constructed fragmentation trees is improved by using MS(n) instead of MS2 measurements.     

Fast molecular shape matching using contact maps.

In this paper, we study the problem of computing the similarity of two protein structures by measuring their contact-map overlap. Contact-map overlap abstracts the problem of computing the similarity of two polygonal chains as a graph-theoretic problem. In R3, we present the first polynomial time algorithm with any guarantee on the approximation ratio for the 3-dimensional problem. More precisely, we give an algorithm for the contact-map overlap problem with an approximation ratio of sigma where sigma = min{sigma(P1), sigma(P2)} 0, is hard.     

The threshold bootstrap clustering: a new approach to find families or transmission clusters within molecular quasispecies

Background

Phylogenetic methods produce hierarchies of molecular species, inferring knowledge about taxonomy and evolution. However, there is not yet a consensus methodology that provides a crisp partition of taxa, desirable when considering the problem of intra/inter-patient quasispecies classification or infection transmission event identification. We introduce the threshold bootstrap clustering (TBC), a new methodology for partitioning molecular sequences, that does not require a phylogenetic tree estimation.

Methodology/Principal Findings

The TBC is an incremental partition algorithm, inspired by the stochastic Chinese restaurant process, and takes advantage of resampling techniques and models of sequence evolution. TBC uses as input a multiple alignment of molecular sequences and its output is a crisp partition of the taxa into an automatically determined number of clusters. By varying initial conditions, the algorithm can produce different partitions. We describe a procedure that selects a prime partition among a set of candidate ones and calculates a measure of cluster reliability. TBC was successfully tested for the identification of type-1 human immunodeficiency and hepatitis C virus subtypes, and compared with previously established methodologies. It was also evaluated in the problem of HIV-1 intra-patient quasispecies clustering, and for transmission cluster identification, using a set of sequences from patients with known transmission event histories.

Conclusion

TBC has been shown to be effective for the subtyping of HIV and HCV, and for identifying intra-patient quasispecies. To some extent, the algorithm was able also to infer clusters corresponding to events of infection transmission. The computational complexity of TBC is quadratic in the number of taxa, lower than other established methods; in addition, TBC has been enhanced with a measure of cluster reliability. The TBC can be useful to characterise molecular quasipecies in a broad context.