Abstract With the high mutation rate in viruses, a mixture of closely related viral strains (called viral quasispecies) often co-infect an individual host. Reconstructing individual strains from viral quasispecies is a key step to characterizing the viral population, revealing strain-level genetic variability, and providing insights into biomedical and clinical studies. Reference-based approaches of reconstructing viral strains suffer from the lack of high-quality references due to high mutation rates and biased variant calling introduced by a selected reference. De novo methods require no references but face challenges due to errors in reads, the high similarity of quasispecies, and uneven abundance of strains. In this paper, we propose VStrains, a de novo approach for reconstructing strains from viral quasispecies. VStrains incorporates contigs, paired-end reads, and coverage information to iteratively extract the strain-specific paths from assembly graphs. We benchmark VStrains against multiple state-of-the-art de novo and reference-based approaches on both simulated and real datasets. Experimental results demonstrate that VStrains achieves the best overall performance on both simulated and real datasets under a comprehensive set of metrics such as genome fraction, duplication ratio, NGA50, error rate, etc. Availability: VStrains is freely available at https://github.com/ MetaGenTools/VStrains .

Mixed communities of organisms are found in many environments -- from the human gut to marine ecosystems -- and can have profound impact on human health and the environment. Metagenomics studies the genomic material of such communities through high-throughput sequencing that yields DNA subsequences for subsequent analysis. A fundamental problem in the standard workflow, called binning, is to discover clusters, of genomic subsequences, associated with the constituent organisms. Inherent noise in the subsequences, various biological constraints that need to be imposed on them and the skewed cluster size distribution exacerbate the difficulty of this unsupervised learning problem. In this paper, we present a new formulation using a graph where the nodes are subsequences and edges represent homophily information. In addition, we model biological constraints providing heterophilous signal about nodes that cannot be clustered together. We solve the binning problem by developing new algorithms for (i) graph representation learning that preserves both homophily relations and heterophily constraints (ii) constraint-based graph clustering method that addresses the problems of skewed cluster size distribution. Extensive experiments, on real and synthetic datasets, demonstrate that our approach, called RepBin, outperforms a wide variety of competing methods. Our constraint-based graph representation learning and clustering methods, that may be useful in other domains as well, advance the state-of-the-art in both metagenomics binning and graph representation learning.

Understanding genetic variation, e.g., through mutations, in organisms is crucial to unravel their effects on the environment and human health. A fundamental characterization can be obtained by solving the haplotype assembly problem, which yields the variation across multiple copies of chromosomes. Variations among fast evolving viruses that lead to different strains (called quasispecies) are also deciphered with similar approaches. In both these cases, high-throughput sequencing technologies that provide oversampled mixtures of large noisy fragments (reads) of genomes, are used to infer constituent components (haplotypes or quasispecies). The problem is harder for polyploid species where there are more than two copies of chromosomes. State-of-the-art neural approaches to solve this NP-hard problem do not adequately model relations among the reads that are important for deconvolving the input signal. We address this problem by developing a new method, called NeurHap, that combines graph representation learning with combinatorial optimization. Our experiments demonstrate the substantially better performance of NeurHap in real and synthetic datasets compared to competing approaches.

The development of DNA sequencing technologies makes it possible to obtain reads originated from both copies of a chromosome (two parental chromosomes, or haplotypes) of a single individual. Reconstruction of both haplotypes (i.e., haplotype phasing) plays a crucial role in genetic analysis and provides relationship information between genetic variation and disease susceptibility. With the emerging third-generation sequencing technologies, most existing approaches for haplotype phasing suffer from performance issues to handle long and error-prone reads. We develop a divide-and-conquer algorithm, DCHap, to phase haplotypes using third-generation reads. We benchmark DCHap against three state-of-the-art phasing tools on both PacBio SMRT data and ONT Nanopore data. The experimental results show that DCHap generates more accurate or comparable results (measured by the switch errors) while being scalable for higher coverage and longer reads. DCHap is a fast and accurate algorithm for haplotype phasing using third-generation sequencing data. As the third-generation sequencing platforms continue improving on their throughput and read lengths, accurate and scalable tools like DCHap are important to improve haplotype phasing from the advances of sequencing technologies. The source code is freely available at https://github.com/yanboANU/Haplotype-phasing.

Plasmids are extra-chromosomal genetic materials with important markers that affect the function and behaviour of the microorganisms supporting their environmental adaptations. Hence the identification and recovery of such plasmid sequences from assemblies is a crucial task in metagenomics analysis. In the past, machine learning approaches have been developed to separate chromosomes and plasmids. However, there is always a compromise between precision and recall in the existing classification approaches. The similarity of compositions between chromosomes and their plasmids makes it difficult to separate plasmids and chromosomes with high accuracy. However, high confidence classifications are accurate with a significant compromise of recall, and vice versa. Hence, the requirement exists to have more sophisticated approaches to separate plasmids and chromosomes accurately while retaining an acceptable trade-off between precision and recall. We present GraphPlas, a novel approach for plasmid recovery using coverage, composition and assembly graph topology. We evaluated GraphPlas on simulated and real short read assemblies with varying compositions of plasmids and chromosomes. Our experiments show that GraphPlas is able to significantly improve accuracy in detecting plasmid and chromosomal contigs on top of popular state-of-the-art plasmid detection tools. The source code is freely available at: https://github.com/anuradhawick/GraphPlas.

Recent advances in RNA-seq technology have made identification of expressed genes affordable, and thus boosting repaid development of transcriptomic studies. Transcriptome assembly, reconstructing all expressed transcripts from RNA-seq reads, is an essential step to understand genes, proteins, and cell functions. Transcriptome assembly remains a challenging problem due to complications in splicing variants, expression levels, uneven coverage and sequencing errors. Here, we formulate the transcriptome assembly problem as path extraction on splicing graphs (or assembly graphs), and propose a novel algorithm MultiTrans for path extraction using mixed integer linear programming. MultiTrans is able to take into consideration coverage constraints on vertices and edges, the number of paths and the paired-end information simultaneously. We benchmarked MultiTrans against two state-of-the-art transcriptome assemblers, TransLiG and rnaSPAdes. Experimental results show that MultiTrans generates more accurate transcripts compared to TransLiG (using the same splicing graphs) and rnaSPAdes (using the same assembly graphs). MultiTrans is freely available at https://github.com/jzbio/MultiTrans.

Study of pairwise genetic interactions, such as mutually exclusive mutations, has led to understanding of underlying mechanisms in cancer. Investigation of various combinatorial motifs within networks of such interactions can lead to deeper insights into its mutational landscape and inform therapy development. One such motif called the Between-Pathway Model (BPM) represents redundant or compensatory pathways that can be therapeutically exploited. Finding such BPM motifs is challenging since most formulations require solving variants of the NP-complete maximum weight bipartite subgraph problem. In this paper we design an algorithm based on Integer Linear Programming (ILP) to solve this problem. In our experiments, our approach outperforms the best previous method to mine BPM motifs. Further, our ILP-based approach allows us to easily model additional application-specific constraints. We illustrate this advantage through a new application of BPM motifs that can potentially aid in finding combination therapies to combat cancer.

Computing shortest paths is a fundamental operation in processing graph data. In many real-world applications, discovering shortest paths between two vertices empowers us to make full use of the underlying structure to understand how vertices are related in a graph, e.g. the strength of social ties between individuals in a social network. In this paper, we study the shortest-path-graph problem that aims to efficiently compute a shortest path graph containing exactly all shortest paths between any arbitrary pair of vertices on complex networks. Our goal is to design an exact solution that can scale to graphs with millions or billions of vertices and edges. To achieve high scalability, we propose a novel method, Query-by-Sketch (QbS), which efficiently leverages offline labelling (i.e., precomputed labels) to guide online searching through a fast sketching process that summarizes the important structural aspects of shortest paths in answering shortest-path-graph queries. We theoretically prove the correctness of this method and analyze its computational complexity. To empirically verify the efficiency of QbS, we conduct experiments on 12 real-world datasets, among which the largest dataset has 1.7 billion vertices and 7.8 billion edges. The experimental results show that QbS can answer shortest-path-graph queries in microseconds for million-scale graphs and less than half a second for billion-scale graphs.

Advancements in metagenomics sequencing allow the study of microbial communities directly from their environments. Metagenomics binning is a key step in the species characterisation of microbial communities. Next-generation sequencing reads are usually assembled into contigs for metagenomics binning mainly due to the limited information within short reads. Third-generation sequencing provides much longer reads that have lengths similar to the contigs assembled from short reads. However, existing contig-binning tools cannot be directly applied on long reads due to the absence of coverage information and the presence of high error rates. The few existing long-read binning tools either use only composition or use composition and coverage information separately. This may ignore bins that correspond to low-abundance species or erroneously split bins that correspond to species with non-uniform coverages. Here we present a reference-free binning approach, LRBinner, that combines composition and coverage information of complete long-read datasets. LRBinner also uses a distance-histogram-based clustering algorithm to extract clusters with varying sizes. The experimental results on both simulated and real datasets show that LRBinner achieves the best binning accuracy against the baselines. Moreover, we show that binning reads using LRBinner prior to assembly reduces computational resources for assembly while attaining satisfactory assembly qualities.

A bipartite network is a graph structure where nodes are from two distinct domains and only inter-domain interactions exist as edges. A large number of network embedding methods exist to learn vectorial node representations from general graphs with both homogeneous and heterogeneous node and edge types, including some that can specifically model the distinct properties of bipartite networks. However, these methods are inadequate to model multiplex bipartite networks (e.g., in e-commerce), that have multiple types of interactions (e.g., click, inquiry, and buy) and node attributes. Most real-world multiplex bipartite networks are also sparse and have imbalanced node distributions that are challenging to model. In this paper, we develop an unsupervised Dual HyperGraph Convolutional Network (DualHGCN) model that scalably transforms the multiplex bipartite network into two sets of homogeneous hypergraphs and uses spectral hypergraph convolutional operators, along with intra- and inter-message passing strategies to promote information exchange within and across domains, to learn effective node embeddings. We benchmark DualHGCN using four real-world datasets on link prediction and node classification tasks. Our extensive experiments demonstrate that DualHGCN significantly outperforms state-of-the-art methods, and is robust to varying sparsity levels and imbalanced node distributions.

SNP calling is a fundamental problem of genetic analysis and has many applications, such as gene-disease diagnosis, drug design, and ancestry inference. Prior approaches either require high-quality reference genome, or suffer from low recall/precision or high runtime. We develop a reference-free algorithm Kmer2SNP to call SNP directly from raw reads, an approach that models SNP calling into a maximum weight matching problem. We benchmark Kmer2SNP against reference-free methods including hybrid (assembly-based) and assembly-free methods on both simulated and real datasets. Experimental results show that Kmer2SNP achieves better SNP calling quality while being an order of magnitude faster than the state-of-the-art methods. Kmer2SNP shows the potential of calling SNPs only using k-mers from raw reads without assembly. The source code is freely available at https://github.com/yanboANU/Kmer2SNP.

Metagenomic sequencing allows us to study structure, diversity and ecology in microbial communities without the necessity of obtaining pure cultures. In many metagenomics studies, the reads obtained from metagenomics sequencing are first assembled into longer contigs and these contigs are then binned into clusters of contigs where contigs in a cluster are expected to come from the same species. As different species may share common sequences in their genomes, one assembled contig may belong to multiple species. However, existing tools for contig binning only support non-overlapped binning, i.e., each contig is assigned to at most one bin (species). In this paper, we introduce GraphBin2 which refines the binning results obtained from existing tools and, more importantly, is able to assign contigs to multiple bins. GraphBin2 uses the connectivity and coverage information from assembly graphs to adjust existing binning results on contigs and to infer contigs shared by multiple species. Experimental results on both simulated and real datasets demonstrate that GraphBin2 not only improves binning results of existing tools but also supports to assign contigs to multiple bins.

MOTIVATION: The field of metagenomics has provided valuable insights into the structure, diversity and ecology within microbial communities. One key step in metagenomics analysis is to assemble reads into longer contigs which are then binned into groups of contigs that belong to different species present in the metagenomic sample. Binning of contigs plays an important role in metagenomics and most available binning algorithms bin contigs using genomic features such as oligonucleotide/k-mer composition and contig coverage. As metagenomic contigs are derived from the assembly process, they are output from the underlying assembly graph which contains valuable connectivity information between contigs that can be used for binning. RESULTS: We propose GraphBin, a new binning method that makes use of the assembly graph and applies a label propagation algorithm to refine the binning result of existing tools. We show that GraphBin can make use of the assembly graphs constructed from both the de Bruijn graph and the overlap-layout-consensus approach. Moreover, we demonstrate improved experimental results from GraphBin in terms of identifying mis-binned contigs and binning of contigs discarded by existing binning tools. To the best of our knowledge, this is the first time that the information from the assembly graph has been used in a tool for the binning of metagenomic contigs. AVAILABILITY AND IMPLEMENTATION: The source code of GraphBin is available at https://github.com/Vini2/GraphBin. CONTACT: [email protected] or [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: Metagenomics studies have provided key insights into the composition and structure of microbial communities found in different environments. Among the techniques used to analyse metagenomic data, binning is considered a crucial step to characterize the different species of micro-organisms present. The use of short-read data in most binning tools poses several limitations, such as insufficient species-specific signal, and the emergence of long-read sequencing technologies offers us opportunities to surmount them. However, most current metagenomic binning tools have been developed for short reads. The few tools that can process long reads either do not scale with increasing input size or require a database with reference genomes that are often unknown. In this article, we present MetaBCC-LR, a scalable reference-free binning method which clusters long reads directly based on their k-mer coverage histograms and oligonucleotide composition. RESULTS: We evaluate MetaBCC-LR on multiple simulated and real metagenomic long-read datasets with varying coverages and error rates. Our experiments demonstrate that MetaBCC-LR substantially outperforms state-of-the-art reference-free binning tools, achieving ∼13% improvement in F1-score and ∼30% improvement in ARI compared to the best previous tools. Moreover, we show that using MetaBCC-LR before long-read assembly helps to enhance the assembly quality while significantly reducing the assembly cost in terms of time and memory usage. The efficiency and accuracy of MetaBCC-LR pave the way for more effective long-read-based metagenomics analyses to support a wide range of applications. AVAILABILITY AND IMPLEMENTATION: The source code is freely available at: https://github.com/anuradhawick/MetaBCC-LR. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Answering exact shortest path distance queries is a fundamental task in graph theory. Despite a tremendous amount of research on the subject, there is still no satisfactory solution that can scale to billion-scale complex networks. Labelling-based methods are well-known for rendering fast response time to distance queries; however, existing works can only construct labelling on moderately large networks (million-scale) and cannot scale to large networks (billion-scale) due to their prohibitively large space requirements and very long preprocessing time. In this work, we present novel techniques to efficiently construct distance labelling and process exact shortest path distance queries for complex networks with billions of vertices and billions of edges. Our method is based on two ingredients: (i) a scalable labelling algorithm for constructing minimal distance labelling, and (ii) a querying framework that supports fast distance-bounded search on a sparsified graph. Thus, we first develop a novel labelling algorithm that can scale to graphs at the billion-scale. Then, we formalize a querying framework for exact distance queries, which combines our proposed highway cover distance labelling with distance-bounded searches to enable fast distance computation. To speed up the labelling construction process, we further propose a parallel labelling method that can construct labelling simultaneously for multiple landmarks. We evaluated the performance of the proposed methods on 12 real-world networks. The experiments show that the proposed methods can not only handle networks with billions of vertices, but also be up to 70 times faster in constructing labelling and save up to 90% of labelling space. In particular, our method can answer distance queries on a billion-scale network of around 8B edges in less than 1ms, on average.

Cancer is known for its heterogeneity and is regarded as an evolutionary process driven by somatic mutations and clonal expansions. This evolutionary process can be modeled by a phylogenetic tree and phylogenetic analysis of multiple subclones of cancer cells can facilitate the study of the tumor variants progression. Copy-number aberration occurs frequently in many types of tumors in terms of segmental amplifications and deletions. In this paper, we developed a distance-based method for reconstructing phylogenies from copy-number profiles of cancer cells. We demonstrate the importance of distance correction from the edit (minimum) distance to the estimated actual number of events. Experimental results show that our approaches provide accurate and scalable results in estimating the actual number of evolutionary events between copy number profiles and in reconstructing phylogenies.

BACKGROUND: Pooling techniques, where multiple sub-samples are mixed in a single sample, are widely used to take full advantage of high-throughput DNA sequencing. Recently, Ranjard et al. (PLoS ONE 13:0195090, 2018) proposed a pooling strategy without the use of barcodes. Three sub-samples were mixed in different known proportions (i.e. 62.5%, 25% and 12.5%), and a method was developed to use these proportions to reconstruct the three haplotypes effectively. RESULTS: HaploJuice provides an alternative haplotype reconstruction algorithm for Ranjard et al.'s pooling strategy. HaploJuice significantly increases the accuracy by first identifying the empirical proportions of the three mixed sub-samples and then assembling the haplotypes using a dynamic programming approach. HaploJuice was evaluated against five different assembly algorithms, Hmmfreq (Ranjard et al., PLoS ONE 13:0195090, 2018), ShoRAH (Zagordi et al., BMC Bioinformatics 12:119, 2011), SAVAGE (Baaijens et al., Genome Res 27:835-848, 2017), PredictHaplo (Prabhakaran et al., IEEE/ACM Trans Comput Biol Bioinform 11:182-91, 2014) and QuRe (Prosperi and Salemi, Bioinformatics 28:132-3, 2012). Using simulated and real data sets, HaploJuice reconstructed the true sequences with the highest coverage and the lowest error rate. CONCLUSION: HaploJuice provides high accuracy in haplotype reconstruction, making Ranjard et al.'s pooling strategy more efficient, feasible, and applicable, with the benefit of reducing the sequencing cost.

BACKGROUND: The rapid accumulation of whole-genome data has renewed interest in the study of using gene-order data for phylogenetic analyses and ancestral reconstruction. Current software and web servers typically do not support duplication and loss events along with rearrangements. RESULTS: MLGO (Maximum Likelihood for Gene-Order Analysis) is a web tool for the reconstruction of phylogeny and/or ancestral genomes from gene-order data. MLGO is based on likelihood computation and shows advantages over existing methods in terms of accuracy, scalability and flexibility. CONCLUSIONS: To the best of our knowledge, it is the first web tool for analysis of large-scale genomic changes including not only rearrangements but also gene insertions, deletions and duplications. The web tool is available from http://www.geneorder.org/server.php .

BACKGROUND: In cell differentiation, a cell of a less specialized type becomes one of a more specialized type, even though all cells have the same genome. Transcription factors and epigenetic marks like histone modifications can play a significant role in the differentiation process. RESULTS: In this paper, we present a simple analysis of cell types and differentiation paths using phylogenetic inference based on ChIP-Seq histone modification data. We precisely defined the notion of cell-type trees and provided a procedure of building such trees. We propose new data representation techniques and distance measures for ChIP-Seq data and use these together with standard phylogenetic inference methods to build biologically meaningful cell-type trees that indicate how diverse types of cells are related. We demonstrate our approach on various kinds of histone modifications for various cell types, also using the datasets to explore various issues surrounding replicate data, variability between cells of the same type, and robustness. We use the results to get some interesting biological findings like important patterns of histone modification changes during cell differentiation process. CONCLUSIONS: We introduced and studied the novel problem of inferring cell type trees from histone modification data. The promising results we obtain point the way to a new approach to the study of cell differentiation. We also discuss how cell-type trees can be used to study the evolution of cell types.

We study the problem of sorting genomes under an evolutionary model that includes genomic rearrangements and segmental duplications. We propose an iterative algorithm to improve any initial evolutionary trajectory between two genomes in terms of parsimony. Our algorithm is based on a new graphical model, the trajectory graph, which models not only the final states of two genomes but also an existing evolutionary trajectory between them. We show that redundant rearrangements in the trajectory correspond to certain cycles in the trajectory graph, and prove that our algorithm converges to an optimal trajectory for any initial trajectory involving only rearrangements.

TIBA is a tool to reconstruct phylogenetic trees from rearrangement data that consist of ordered lists of synteny blocks (or genes), where each synteny block is shared with all of its homologues in the input genomes. The evolution of these synteny blocks, through rearrangement operations, is modelled by the uniform Double-Cut-and-Join model. Using a true distance estimate under this model and simple distance-based methods, TIBA reconstructs a phylogeny of the input genomes. Unlike any previous tool for inferring phylogenies from rearrangement data, TIBA uses novel methods of robustness estimation to provide support values for the edges in the inferred tree.

Computing the edit distance between two genomes under certain operations is a basic problem in the study of genome evolution. The double-cut-and-join (DCJ) model has formed the basis for most algorithmic research on rearrangements over the last few years. The edit distance under the DCJ model can be easily computed for genomes without duplicate genes. In this paper, we study the edit distance for genomes with duplicate genes under a model that includes DCJ operations, insertions and deletions. We prove that computing the edit distance is equivalent to finding the optimal cycle decomposition of the corresponding adjacency graph, and give an approximation algorithm with an approximation ratio of 1.5 + ∈ .

Comparing two or more phylogenetic trees is a fundamental task in computational biology. The simplest outcome of such a comparison is a pairwise measure of similarity, dissimilarity, or distance. A large number of such measures have been proposed, but so far all suffer from problems varying from computational cost to lack of robustness; many can be shown to behave unexpectedly under certain plausible inputs. For instance, the widely used Robinson-Foulds distance is poorly distributed and thus affords little discrimination, while also lacking robustness in the face of very small changes--reattaching a single leaf elsewhere in a tree of any size can instantly maximize the distance. In this paper, we introduce a new pairwise distance measure, based on matching, for phylogenetic trees. We prove that our measure induces a metric on the space of trees, show how to compute it in low polynomial time, verify through statistical testing that it is robust, and finally note that it does not exhibit unexpected behavior under the same inputs that cause problems with other measures. We also illustrate its usefulness in clustering trees, demonstrating significant improvements in the quality of hierarchical clustering as compared to the same collections of trees clustered using the Robinson-Foulds distance.

BACKGROUND: The rapidly increasing availability of whole-genome sequences has enabled the study of whole-genome evolution. Evolutionary mechanisms based on genome rearrangements have attracted much attention and given rise to many models; somewhat independently, the mechanisms of gene duplication and loss have seen much work. However, the two are not independent and thus require a unified treatment, which remains missing to date. Moreover, existing rearrangement models do not fit the dichotomy between most prokaryotic genomes (one circular chromosome) and most eukaryotic genomes (multiple linear chromosomes). RESULTS: To handle rearrangements, gene duplications and losses, we propose a new evolutionary model and the corresponding method for estimating true evolutionary distance. Our model, inspired from the DCJ model, is simple and the first to respect the prokaryotic/eukaryotic structural dichotomy. Experimental results on a wide variety of genome structures demonstrate the very high accuracy and robustness of our distance estimator. CONCLUSION: We give the first robust, statistically based, estimate of genomic pairwise distances based on rearrangements, duplications and losses, under a model that respects the structural dichotomy between prokaryotic and eukaryotic genomes. Accurate and robust estimates in true evolutionary distances should translate into much better phylogenetic reconstructions as well as more accurate genomic alignments, while our new model of genome rearrangements provides another refinement in simplicity and verisimilitude.

BACKGROUND: The study of genome rearrangements has become a mainstay of phylogenetics and comparative genomics. Fundamental in such a study is the median problem: given three genomes find a fourth that minimizes the sum of the evolutionary distances between itself and the given three. Many exact algorithms and heuristics have been developed for the inversion median problem, of which the best known is MGR. RESULTS: We present a unifying framework for median heuristics, which enables us to clarify existing strategies and to place them in a partial ordering. Analysis of this framework leads to a new insight: the best strategies continue to refer to the input data rather than reducing the problem to smaller instances. Using this insight, we develop a new heuristic for inversion medians that uses input data to the end of its computation and leverages our previous work with DCJ medians. Finally, we present the results of extensive experimentation showing that our new heuristic outperforms all others in accuracy and, especially, in running time: the heuristic typically returns solutions within 1% of optimal and runs in seconds to minutes even on genomes with 25'000 genes--in contrast, MGR can take days on instances of 200 genes and cannot be used beyond 1'000 genes. CONCLUSION: Finding good rearrangement medians, in particular inversion medians, had long been regarded as the computational bottleneck in whole-genome studies. Our new heuristic for inversion medians, ASM, which dominates all others in our framework, puts that issue to rest by providing near-optimal solutions within seconds to minutes on even the largest genomes.

MOTIVATION: Modern techniques can yield the ordering and strandedness of genes on each chromosome of a genome; such data already exists for hundreds of organisms. The evolutionary mechanisms through which the set of the genes of an organism is altered and reordered are of great interest to systematists, evolutionary biologists, comparative genomicists and biomedical researchers. Perhaps the most basic concept in this area is that of evolutionary distance between two genomes: under a given model of genomic evolution, how many events most likely took place to account for the difference between the two genomes? RESULTS: We present a method to estimate the true evolutionary distance between two genomes under the 'double-cut-and-join' (DCJ) model of genome rearrangement, a model under which a single multichromosomal operation accounts for all genomic rearrangement events: inversion, transposition, translocation, block interchange and chromosomal fusion and fission. Our method relies on a simple structural characterization of a genome pair and is both analytically and computationally tractable. We provide analytical results to describe the asymptotic behavior of genomes under the DCJ model, as well as experimental results on a wide variety of genome structures to exemplify the very high accuracy (and low variance) of our estimator. Our results provide a tool for accurate phylogenetic reconstruction from multichromosomal gene rearrangement data as well as a theoretical basis for refinements of the DCJ model to account for biological constraints. AVAILABILITY: All of our software is available in source form under GPL at http://lcbb.epfl.ch.

One of the main goals in the analysis of microarray data is to identify groups of genes and groups of experimental conditions (including environments, individuals, and tissues) that exhibit similar expression patterns. This is the so-called biclustering problem. In this paper, we consider two variations of the biclustering problem: the consensus submatrix problem and the bottleneck submatrix problem. The input of the problems contains an $m\times n$ matrix A and integers l and k. The consensus submatrix problem is to find an $l\times k$ submatrix with $l