Safe and complete contig assembly via omnitigs

Contig assembly is the first stage that most assemblers solve when reconstructing a genome from a set of reads. Its output consists of contigs -- a set of strings that are promised to appear in any genome that could have generated the reads. From the introduction of contigs 20 years ago, assemblers have tried to obtain longer and longer contigs, but the following question was never solved: given a genome graph $G$ (e.g. a de Bruijn, or a string graph), what are all the strings that can be safely reported from $G$ as contigs? In this paper we finally answer this question, and also give a polynomial time algorithm to find them. Our experiments show that these strings, which we call omnitigs, are 66% to 82% longer on average than the popular unitigs, and 29% of dbSNP locations have more neighbors in omnitigs than in unitigs.Comment: Full version of the paper in the proceedings of RECOMB 201

Safe and Complete Contig Assembly Through Omnitigs

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Universal lossless source coding with the Burrows Wheeler transform

The Burrows Wheeler transform (1994) is a reversible sequence transformation used in a variety of practical lossless source-coding algorithms. In each, the BWT is followed by a lossless source code that attempts to exploit the natural ordering of the BWT coefficients. BWT-based compression schemes are widely touted as low-complexity algorithms giving lossless coding rates better than those of the Ziv-Lempel codes (commonly known as LZ'77 and LZ'78) and almost as good as those achieved by prediction by partial matching (PPM) algorithms. To date, the coding performance claims have been made primarily on the basis of experimental results. This work gives a theoretical evaluation of BWT-based coding. The main results of this theoretical evaluation include: (1) statistical characterizations of the BWT output on both finite strings and sequences of length n → ∞, (2) a variety of very simple new techniques for BWT-based lossless source coding, and (3) proofs of the universality and bounds on the rates of convergence of both new and existing BWT-based codes for finite-memory and stationary ergodic sources. The end result is a theoretical justification and validation of the experimentally derived conclusions: BWT-based lossless source codes achieve universal lossless coding performance that converges to the optimal coding performance more quickly than the rate of convergence observed in Ziv-Lempel style codes and, for some BWT-based codes, within a constant factor of the optimal rate of convergence for finite-memory source

Reconstructing the History of Syntenies Through Super-Reconciliation

Classical gene and species tree reconciliation, used to infer the history of gene gain and loss explaining the evolution of gene families, assumes an independent evolution for each family. While this assumption is reasonable for genes that are far apart in the genome, it is clearly not suited for genes grouped in syntenic blocks, which are more plausibly the result of a concerted evolution. Here, we introduce the Super-Reconciliation model, that extends the traditional Duplication-Loss model to the reconciliation of a set of trees, accounting for segmental duplications and losses. From a complexity point of view, we show that the associated decision problem is NP-hard. We then give an exact exponential-time algorithm for this problem, assess its time efficiency on simulated datasets, and give a proof of concept on the opioid receptor genes

When Can You Fold a Map?

We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds; the simplest one-layer simple fold rotates a portion of paper about a crease in the paper by +-180 degrees. We first consider the analogous questions in one dimension lower -- bending a segment into a flat object -- which lead to interesting problems on strings. We develop efficient algorithms for the recognition of simply foldable 1D crease patterns, and reconstruction of a sequence of simple folds. Indeed, we prove that a 1D crease pattern is flat-foldable by any means precisely if it is by a sequence of one-layer simple folds. Next we explore simple foldability in two dimensions, and find a surprising contrast: ``map'' folding and variants are polynomial, but slight generalizations are NP-complete. Specifically, we develop a linear-time algorithm for deciding foldability of an orthogonal crease pattern on a rectangular piece of paper, and prove that it is (weakly) NP-complete to decide foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper, (2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a square piece of paper, and (3) crease patterns without a mountain/valley assignment.Comment: 24 pages, 19 figures. Version 3 includes several improvements thanks to referees, including formal definitions of simple folds, more figures, table summarizing results, new open problems, and additional reference

Assembly complexity of prokaryotic genomes using short reads

<p>Abstract</p> <p>Background</p> <p>De Bruijn graphs are a theoretical framework underlying several modern genome assembly programs, especially those that deal with very short reads. We describe an application of de Bruijn graphs to analyze the global repeat structure of prokaryotic genomes.</p> <p>Results</p> <p>We provide the first survey of the repeat structure of a large number of genomes. The analysis gives an upper-bound on the performance of genome assemblers for <it>de novo </it>reconstruction of genomes across a wide range of read lengths. Further, we demonstrate that the majority of genes in prokaryotic genomes can be reconstructed uniquely using very short reads even if the genomes themselves cannot. The non-reconstructible genes are overwhelmingly related to mobile elements (transposons, IS elements, and prophages).</p> <p>Conclusions</p> <p>Our results improve upon previous studies on the feasibility of assembly with short reads and provide a comprehensive benchmark against which to compare the performance of the short-read assemblers currently being developed.</p

On the Approximability of Comparing Genomes with Duplicates

International audienceA central problem in comparative genomics consists in computing a (dis-)simi- larity measure between two genomes, e.g. in order to construct a phylogenetic tree. A large number of such measures has been proposed in the recent past: number of reversals, number of breakpoints, number of common or conserved intervals, SAD etc. In their initial definitions, all these measures suppose that genomes contain no duplicates. However, we now know that genes can be duplicated within the same genome. One possible approach to overcome this difficulty is to establish a one-to-one correspondence (i.e. a matching) between genes of both genomes, where the correspondence is chosen in order to optimize the studied measure. Then, after a gene relabeling according to this matching and a deletion of the unmatched signed genes, two genomes without duplicates are obtained and the measure can be computed. In this paper, we are interested in three measures (number of breakpoints, number of common intervals and number of conserved intervals) and three models of matching (exemplar model, maximum matching model and non maximum matching model). We prove that, for each model and each measure, computing a matching between two genomes that optimizes the measure is APX-Hard. We show that this result remains true even for two genomes G1 and G2 such that G1 contains no duplicates and no gene of G2 appears more than twice. Therefore, our results extend those of [5–7]. Finally, we propose a 4-approximation algorithm for a measure closely related to the number of breakpoints, the number of adjacencies, under the maximum matching model, in the case where genomes contain the same number of duplications of each gene