The Gene-Duplication Problem: Near-Linear Time Algorithms for NNI-Based Local Searches

The gene-duplication problem is to infer a species supertree from a collection of gene trees that are confounded by complex histories of gene-duplication events. This problem is NP-complete and thus requires efficient and effective heuristics. Existing heuristics perform a stepwise search of the tree space, where each step is guided by an exact solution to an instance of a local search problem. A classical local search problem is the {tt NNI} search problem, which is based on the nearest neighbor interchange operation. In this work, we 1) provide a novel near-linear time algorithm for the {tt NNI} search problem, 2) introduce extensions that significantly enlarge the search space of the {tt NNI} search problem, and 3) present algorithms for these extended versions that are asymptotically just as efficient as our algorithm for the {tt NNI} search problem. The exceptional speedup achieved in the extended {tt NNI} search problems makes the gene-duplication problem more tractable for large-scale phylogenetic analyses. We verify the performance of our algorithms in a comparison study using sets of large randomly generated gene trees.     

Reconstructing Unrooted Phylogenetic Trees from Symbolic Ternary Metrics

Böcker and Dress (Adv Math 138:105–125, 1998) presented a 1-to-1 correspondence between symbolically dated rooted trees and symbolic ultrametrics. We consider the corresponding problem for unrooted trees. More precisely, given a tree T with leaf set X and a proper vertex coloring of its interior vertices, we can map every triple of three different leaves to the color of its median vertex. We characterize all ternary maps that can be obtained in this way in terms of 4- and 5-point conditions, and we show that the corresponding tree and its coloring can be reconstructed from a ternary map that satisfies those conditions. Further, we give an additional condition that characterizes whether the tree is binary, and we describe an algorithm that reconstructs general trees in a bottom-up fashion.     

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.     

Ancestral maximum likelihood of evolutionary trees is hard

Maximum likelihood (ML) (Neyman, 1971) is an increasingly popular optimality criterion for selecting evolutionary trees. Finding optimal ML trees appears to be a very hard computational task--in particular, algorithms and heuristics for ML take longer to run than algorithms and heuristics for maximum parsimony (MP). However, while MP has been known to be NP-complete for over 20 years, no such hardness result has been obtained so far for ML. In this work we make a first step in this direction by proving that ancestral maximum likelihood (AML) is NP-complete. The input to this problem is a set of aligned sequences of equal length and the goal is to find a tree and an assignment of ancestral sequences for all of that tree's internal vertices such that the likelihood of generating both the ancestral and contemporary sequences is maximized. Our NP-hardness proof follows that for MP given in (Day, Johnson and Sankoff, 1986) in that we use the same reduction from Vertex Cover; however, the proof of correctness for this reduction relative to AML is different and substantially more involved.     

Joint order batching and picker routing in single and multiple-cross-aisle warehouses using cluster-based tabu search algorithms

The organization of order picking operations is one of the most critical issues in warehouse management. In this paper, novel tabu search (TS) algorithms integrated with a novel clustering algorithm are proposed to solve the order batching and picker routing problems jointly for multiple-cross-aisle warehouse systems. A clustering algorithm that generates an initial solution for the TS algorithms is developed to provide fast and effective solutions to the order-batching problem. Unlike most common picker routing heuristics, we model the routing problem of pickers as a classical TSP and propose efficient Nearest Neighbor+Or-opt and Savings+2-Opt heuristics to meet the specific features for the problem. Various problem instances including the number of orders, weight of items, and picking coordinates are generated randomly, and detailed numerical experiments are carried out to evaluate the performances of the proposed methods. In conclusion, the TS algorithms come out to be the most efficient methods in terms of solution quality and computational efficiency.     

A fast and sensitive algorithm for aligning ESTs to the human genome

There is a pressing need to align the growing set of expressed sequence tags (ESTs) with the newly sequenced human genome. However, the problem is complicated by the exon/intron structure of eukaryotic genes misread nucleotides in ESTs, and the millions of repetitive sequences in genomic sequences. To solve this problem, algorithms that use dynamic programming have been proposed. In reality, however, these algorithms require an enormous amount of processing time. In an effort to improve the computational efficiency of these classical DP algorithms, we developed software that fully utilizes lookup-tables to detect the start- and endpoints of an EST within a given DNA sequence efficiently, and subsequently promptly identify exons and introns. In addition, the locations of all splice sites must be calculated correctly with high sensitivity and accuracy, while retaining high computational efficiency. This goal is hard to accomplish in practice, due to misread nucleotides in ESTs and repetitive sequences in the genome. Nevertheless, we present two heuristics that effectively settle this issue. Experimental results confirm that our technique improves the overall computation time by orders of magnitude compared with common tools, such as SIM4 and BLAT, and simultaneously attains high sensitivity and accuracy against a clean dataset of documented genes.     

On the Scalability of Dynamic Scheduling Scientific Applications with Adaptive Weighted Factoring

In heterogeneous environments, dynamic scheduling algorithms are a powerful tool towards performance improvement of scientific applications via load balancing. However, these scheduling techniques employ heuristics that require prior knowledge about workload via profiling resulting in higher overhead as problem sizes and number of processors increase. In addition, load imbalance may appear only at run-time, making profiling work tedious and sometimes even obsolete. Recently, the integration of dynamic loop scheduling algorithms into a number of scientific applications has been proven effective. This paper reports on performance improvements obtained by integrating the Adaptive Weighted Factoring, a recently proposed dynamic loop scheduling technique that addresses these concerns, into two scientific applications: computational field simulation on unstructured grids, and N-Body simulations. Reported experimental results confirm the benefits of using this methodology, and emphasize its high potential for future integration into other scientific applications that exhibit substantial performance degradation due to load imbalance.     

Genetic Algorithm Based Scheduling of Meta-Tasks with Stochastic Execution Times in Heterogeneous Computing Systemst1

In this study, we address the meta-task scheduling problem in heterogeneous computing (HC) systems, which is to find a task assignment that minimizes the schedule length of a meta-task composed of several independent tasks with no data dependencies. The fact that the meta-task scheduling problem in HC systems is NP-hard has motivated the development of many heuristic scheduling algorithms. These heuristic algorithms, however, neglect the stochastic nature of task execution times in an attempt to minimize a deterministic objective function, which is the maximum of the expected values of machine loads. Contrary to existing heuristics, we account for this stochastic nature by modeling task execution times as random variables. We, then, formulate a stochastic scheduling problem where the objective is to minimize the expected value of the maximum of machine loads. We prove that this new objective is underestimated by the deterministic objective function and that an optimal task assignment obtained with respect to the deterministic objective function could be inefficient in a real computing platform. In order to solve the stochastic scheduling problem posed, we develop a genetic algorithm based scheduling heuristic. Our extensive simulation studies show that the proposed genetic algorithm can produce better task assignments as compared to existing heuristics. Specifically, we observe a performance improvement on the relative cost heuristic (M.-Y. Wu and W. Shu, A high-performance mapping algorithm for heterogeneous computing systems, in: Int. Parallel and Distributed Processing Symposium, San Francisco, CA, April 2001) by up to 61%.     

An iterative method for faster sum-of-pairs multiple sequence alignment

MOTIVATION: Multiple sequence alignment is an important tool in computational biology. In order to solve the task of computing multiple alignments in affordable time, the most commonly used multiple alignment methods have to use heuristics. Nevertheless, the computation of optimal multiple alignments is important in its own right, and it provides a means of evaluating heuristic approaches or serves as a subprocedure of heuristic alignment methods. RESULTS: We present an algorithm that uses the divide-and-conquer alignment approach together with recent results on search space reduction to speed up the computation of multiple sequence alignments. The method is adaptive in that depending on the time one wants to spend on the alignment, a better, up to optimal alignment can be obtained. To speed up the computation in the optimal alignment step, we apply the alpha(*) algorithm which leads to a procedure provably more efficient than previous exact algorithms. We also describe our implementation of the algorithm and present results showing the effectiveness and limitations of the procedure.     

Efficient likelihood computations with nonreversible models of evolution

Recent advances in heuristics have made maximum likelihood phylogenetic tree estimation tractable for hundreds of sequences. Noticeably, these algorithms are currently limited to reversible models of evolution, in which Felsenstein's pulley principle applies. In this paper we show that by reorganizing the way likelihood is computed, one can efficiently compute the likelihood of a tree from any of its nodes with a nonreversible model of DNA sequence evolution, and hence benefit from cutting-edge heuristics. This computational trick can be used with reversible models of evolution without any extra cost. We then introduce nhPhyML, the adaptation of the nonhomogeneous nonstationary model of Galtier and Gouy (1998; Mol. Biol. Evol. 15:871-879) to the structure of PhyML, as well as an approximation of the model in which the set of equilibrium frequencies is limited. This new version shows good results both in terms of exploration of the space of tree topologies and ancestral G+C content estimation. We eventually apply it to rRNA sequences slowly evolving sites and conclude that the model and a wider taxonomic sampling still do not plead for a hyperthermophilic last universal common ancestor.     

New approaches to phylogenetic tree search and their application to large numbers of protein alignments

Phylogenetic tree estimation plays a critical role in a wide variety of molecular studies, including molecular systematics, phylogenetics, and comparative genomics. Finding the optimal tree relating a set of sequences using score-based (optimality criterion) methods, such as maximum likelihood and maximum parsimony, may require all possible trees to be considered, which is not feasible even for modest numbers of sequences. In practice, trees are estimated using heuristics that represent a trade-off between topological accuracy and speed. I present a series of novel algorithms suitable for score-based phylogenetic tree reconstruction that demonstrably improve the accuracy of tree estimates while maintaining high computational speeds. The heuristics function by allowing the efficient exploration of large numbers of trees through novel hill-climbing and resampling strategies. These heuristics, and other computational approximations, are implemented for maximum likelihood estimation of trees in the program Leaphy, and its performance is compared to other popular phylogenetic programs. Trees are estimated from 4059 different protein alignments using a selection of phylogenetic programs and the likelihoods of the tree estimates are compared. Trees estimated using Leaphy are found to have equal to or better likelihoods than trees estimated using other phylogenetic programs in 4004 (98.6%) families and provide a unique best tree that no other program found in 1102 (27.1%) families. The improvement is particularly marked for larger families (80 to 100 sequences), where Leaphy finds a unique best tree in 81.7% of families.     

Algorithmic approaches to clonal reconstruction in heterogeneous cell populations

Background: The reconstruction of clonal haplotypes and their evolutionary history in evolving populations is a common problem in both microbial evolutionary biology and cancer biology. The clonal theory of evolution provides a theoretical framework for modeling the evolution of clones.Results: In this paper, we review the theoretical framework and assumptions over which the clonal reconstruction problem is formulated. We formally define the problem and then discuss the complexity and solution space of the problem. Various methods have been proposed to find the phylogeny that best explains the observed data. We categorize these methods based on the type of input data that they use (space-resolved or time-resolved), and also based on their computational formulation as either combinatorial or probabilistic. It is crucial to understand the different types of input data because each provides essential but distinct information for drastically reducing the solution space of the clonal reconstruction problem. Complementary information provided by single cell sequencing or from whole genome sequencing of randomly isolated clones can also improve the accuracy of clonal reconstruction. We briefly review the existing algorithms and their relationships. Finally we summarize the tools that are developed for either directly solving the clonal reconstruction problem or a related computational problem.Conclusions: In this review, we discuss the various formulations of the problem of inferring the clonal evolutionary history from allele frequeny data, review existing algorithms and catergorize them according to their problem formulation and solution approaches. We note that most of the available clonal inference algorithms were developed for elucidating tumor evolution whereas clonal reconstruction for unicellular genomes are less addressed. We conclude the review by discussing more open problems such as the lack of benchmark datasets and comparison of performance between available tools.     

Uniform Sampling of Steady States in Metabolic Networks: Heterogeneous Scales and Rounding

The uniform sampling of convex polytopes is an interesting computational problem with many applications in inference from linear constraints, but the performances of sampling algorithms can be affected by ill-conditioning. This is the case of inferring the feasible steady states in models of metabolic networks, since they can show heterogeneous time scales. In this work we focus on rounding procedures based on building an ellipsoid that closely matches the sampling space, that can be used to define an efficient hit-and-run (HR) Markov Chain Monte Carlo. In this way the uniformity of the sampling of the convex space of interest is rigorously guaranteed, at odds with non markovian methods. We analyze and compare three rounding methods in order to sample the feasible steady states of metabolic networks of three models of growing size up to genomic scale. The first is based on principal component analysis (PCA), the second on linear programming (LP) and finally we employ the Lovazs ellipsoid method (LEM). Our results show that a rounding procedure dramatically improves the performances of the HR in these inference problems and suggest that a combination of LEM or LP with a subsequent PCA perform the best. We finally compare the distributions of the HR with that of two heuristics based on the Artificially Centered hit-and-run (ACHR), gpSampler and optGpSampler. They show a good agreement with the results of the HR for the small network, while on genome scale models present inconsistencies.     

Topac: alignment of gene regulatory networks using topology-aware coloring

We consider the problem of finding a subnetwork in a given biological network (i.e. target network) that is most similar to a given small query network. We aim to find the optimal solution (i.e. the subnetwork with the largest alignment score) with a provable confidence bound. There is no known polynomial time solution to this problem in the literature. Alon et al. has developed a state-of-the-art coloring method that reduces the cost of this problem. This method randomly colors the target network prior to alignment for many iterations until a user-supplied confidence is reached. Here we develop a novel coloring method, named k-hop coloring (k is a positive integer), that achieves a provable confidence value in a small number of iterations without sacrificing the optimality. Our method considers the color assignments already made in the neighborhood of each target network node while assigning a color to a node. This way, it preemptively avoids many color assignments that are guaranteed to fail to produce the optimal alignment. We also develop a filtering method that eliminates the nodes that cannot be aligned without reducing the alignment score after each coloring instance. We demonstrate both theoretically and experimentally that our coloring method outperforms that of Alon et al., which is also used by a number network alignment methods, including QPath and QNet, by a factor of three without reducing the confidence in the optimality of the result. Our experiments also suggest that the resulting alignment method is capable of identifying functionally enriched regions in the target network successfully.     

Maximum Parsimony on Phylogenetic networks

Background

Phylogenetic networks are generalizations of phylogenetic trees, that are used to model evolutionary events in various contexts. Several different methods and criteria have been introduced for reconstructing phylogenetic trees. Maximum Parsimony is a character-based approach that infers a phylogenetic tree by minimizing the total number of evolutionary steps required to explain a given set of data assigned on the leaves. Exact solutions for optimizing parsimony scores on phylogenetic trees have been introduced in the past.

Results

In this paper, we define the parsimony score on networks as the sum of the substitution costs along all the edges of the network; and show that certain well-known algorithms that calculate the optimum parsimony score on trees, such as Sankoff and Fitch algorithms extend naturally for networks, barring conflicting assignments at the reticulate vertices. We provide heuristics for finding the optimum parsimony scores on networks. Our algorithms can be applied for any cost matrix that may contain unequal substitution costs of transforming between different characters along different edges of the network. We analyzed this for experimental data on 10 leaves or fewer with at most 2 reticulations and found that for almost all networks, the bounds returned by the heuristics matched with the exhaustively determined optimum parsimony scores.

Conclusion

The parsimony score we define here does not directly reflect the cost of the best tree in the network that displays the evolution of the character. However, when searching for the most parsimonious network that describes a collection of characters, it becomes necessary to add additional cost considerations to prefer simpler structures, such as trees over networks. The parsimony score on a network that we describe here takes into account the substitution costs along the additional edges incident on each reticulate vertex, in addition to the substitution costs along the other edges which are common to all the branching patterns introduced by the reticulate vertices. Thus the score contains an in-built cost for the number of reticulate vertices in the network, and would provide a criterion that is comparable among all networks. Although the problem of finding the parsimony score on the network is believed to be computationally hard to solve, heuristics such as the ones described here would be beneficial in our efforts to find a most parsimonious network.     

ESPRIT-Tree: hierarchical clustering analysis of millions of 16S rRNA pyrosequences in quasilinear computational time

Taxonomy-independent analysis plays an essential role in microbial community analysis. Hierarchical clustering is one of the most widely employed approaches to finding operational taxonomic units, the basis for many downstream analyses. Most existing algorithms have quadratic space and computational complexities, and thus can be used only for small or medium-scale problems. We propose a new online learning-based algorithm that simultaneously addresses the space and computational issues of prior work. The basic idea is to partition a sequence space into a set of subspaces using a partition tree constructed using a pseudometric, then recursively refine a clustering structure in these subspaces. The technique relies on new methods for fast closest-pair searching and efficient dynamic insertion and deletion of tree nodes. To avoid exhaustive computation of pairwise distances between clusters, we represent each cluster of sequences as a probabilistic sequence, and define a set of operations to align these probabilistic sequences and compute genetic distances between them. We present analyses of space and computational complexity, and demonstrate the effectiveness of our new algorithm using a human gut microbiota data set with over one million sequences. The new algorithm exhibits a quasilinear time and space complexity comparable to greedy heuristic clustering algorithms, while achieving a similar accuracy to the standard hierarchical clustering algorithm.     

EMILiO: a fast algorithm for genome-scale strain design

Systems-level design of cell metabolism is becoming increasingly important for renewable production of fuels, chemicals, and drugs. Computational models are improving in the accuracy and scope of predictions, but are also growing in complexity. Consequently, efficient and scalable algorithms are increasingly important for strain design. Previous algorithms helped to consolidate the utility of computational modeling in this field. To meet intensifying demands for high-performance strains, both the number and variety of genetic manipulations involved in strain construction are increasing. Existing algorithms have experienced combinatorial increases in computational complexity when applied toward the design of such complex strains. Here, we present EMILiO, a new algorithm that increases the scope of strain design to include reactions with individually optimized fluxes. Unlike existing approaches that would experience an explosion in complexity to solve this problem, we efficiently generated numerous alternate strain designs producing succinate, l-glutamate and l-serine. This was enabled by successive linear programming, a technique new to the area of computational strain design.     

Combining many multiple alignments in one improved alignment

MOTIVATION: The fact that the multiple sequence alignment problem is of high complexity has led to many different heuristic algorithms attempting to find a solution in what would be considered a reasonable amount of computation time and space. Very few of these heuristics produce results that are guaranteed always to lie within a certain distance of an optimal solution (given a measure of quality, e.g. parsimony). Most practical heuristics cannot guarantee this, but nevertheless perform well for certain cases. An alignment, obtained with one of these heuristics and with a bad overall score, is not unusable though, it might contain important information on how substrings should be aligned. This paper presents a method that extracts qualitatively good sub-alignments from a set of multiple alignments and combines these into a new, often improved alignment. The algorithm is implemented as a variant of the traditional dynamic programming technique. RESULTS: An implementation of ComAlign (the algorithm that combines multiple alignments) has been run on several sets of artificially generated sequences and a set of 5S RNA sequences. To assess the quality of the alignments obtained, the results have been compared with the output of MSA 2.1 (Gupta et al., Proceedings of the Sixth Annual Symposium on Combinatorial Pattern Matching, 1995; Kececioglu et al., http://www.techfak.uni-bielefeld. de/bcd/Lectures/kececioglu.html, 1995). In all cases, ComAlign was able to produce a solution with a score comparable to the solution obtained by MSA. The results also show that ComAlign actually does combine parts from different alignments and not just select the best of them. AVAILABILITY: The C source code (a Smalltalk version is being worked on) of ComAlign and the other programs that have been implemented in this context are free and available on WWW (http://www.daimi.au.dk/ ?caprani). CONTACT: klaus@bucka-lassen.dk; jotun@pop.bio.au.dk;ocaprani@daimi.au.dk     

Discovering topological motifs using a compact notation.

Discovering topological motifs or common topologies in one or more graphs is an important as well as an interesting problem. It had been classically viewed as the subgraph isomorphism problem. This problem and its various flavors are known to be NP-Complete. However, this does not minimize the importance of solving this problem accurately in application areas such as bioinformatics or even larger network studies. The explosion in the size of the output is usually caused by isomorphisms in the motif or graph: we present a method to handle this without sacrificing the correct answers. In this paper, we apply the natural notion of maximality, used extensively in strings, to graphs and present a simple three-step approach to solving this problem completely and exactly (without resorting to heuristics). We handle the natural combinatorial explosion due to isomorphisms inherent in the problem (which could result in output size being exponential in the input size) by the use of "compact location lists." In other words, instead of enumerating k elements out of n, we use the ((n)(k)) form in an implicit manner (k immediate neighbors of a vertex out of n possible immediate neighbors). This drastically reduces the size of the output without any loss of information. The algorithm we present is linear in terms of the size of the output encoded as compact lists.     

Fast, optimal alignment of three sequences using linear gap costs

Alignment algorithms can be used to infer a relationship between sequences when the true relationship is unknown. Simple alignment algorithms use a cost function that gives a fixed cost to each possible point mutation-mismatch, deletion, insertion. These algorithms tend to find optimal alignments that have many small gaps. It is more biologically plausible to have fewer longer gaps rather than many small gaps in an alignment. To address this issue, linear gap cost algorithms are in common use for aligning biological sequence data. More reliable inferences are obtained by aligning more than two sequences at a time. The obvious dynamic programming algorithm for optimally aligning k sequences of length n runs in O(n(k)) time. This is impractical if k>/=3 and n is of any reasonable length. Thus, for this problem there are many heuristics for aligning k sequences, however, they are not guaranteed to find an optimal alignment. In this paper, we present a new algorithm guaranteed to find the optimal alignment for three sequences using linear gap costs. This gives the same results as the dynamic programming algorithm for three sequences, but typically does so much more quickly. It is particularly fast when the (three-way) edit distance is small. Our algorithm uses a speed-up technique based on Ukkonen's greedy algorithm (Ukkonen, 1983) which he presented for two sequences and simple costs.