Convergence of the Number of Period Sets in Strings

Consider words of length n. The set of all periods of a word of length n is a subset of {0, 1, 2, . . ., n−1}. However, any subset of {0, 1, 2, . . ., n−1} is not necessarily a valid set of periods. In a seminal paper in 1981, Guibas and Odlyzko proposed to encode the set of periods of a word into an n long binary string, called an autocorrelation, where a one at position i denotes the period i. They considered the question of recognizing a valid period set, and also studied the number of valid period sets for strings of length n, denoted κn. They conjectured that ln(κn) asymptotically converges to a constant times ln2(n). Although improved lower bounds for ln(κn)/ln2(n) were proposed in 2001, the question of a tight upper bound has remained open since Guibas and Odlyzko’s paper. Here, we exhibit an upper bound for this fraction, which implies its convergence and closes this longstanding conjecture. Moreover, we extend our result to find similar bounds for the number of correlations: a generalization of autocorrelations which encodes the overlaps between two strings

Practical lower and upper bounds for the Shortest Linear Superstring

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DNA Slippage Occurs at Microsatellite Loci without Minimal Threshold Length in Humans: A Comparative Genomic Approach

The dynamics of microsatellite, or short tandem repeats (STRs), is well documented for long, polymorphic loci, but much less is known for shorter ones. For example, the issue of a minimum threshold length for DNA slippage remains contentious. Model-fitting methods have generally concluded that slippage only occurs over a threshold length of about eight nucleotides, in contradiction with some direct observations of tandem duplications at shorter repeated sites. Using a comparative analysis of the human and chimpanzee genomes, we examined the mutation patterns at microsatellite loci with lengths as short as one period plus one nucleotide. We found that the rates of tandem insertions and deletions at microsatellite loci strongly deviated from background rates in other parts of the human genome and followed an exponential increase with STR size. More importantly, we detected no lower threshold length for slippage. The rate of tandem duplications at unrepeated sites was higher than expected from random insertions, providing evidence for genome-wide action of indel slippage (an alternative mechanism generating tandem repeats). The rate of point mutations adjacent to STRs did not differ from that estimated elsewhere in the genome, except around dinucleotide loci. Our results suggest that the emergence of STR depends on DNA slippage, indel slippage, and point mutations. We also found that the dynamics of tandem insertions and deletions differed in both rates and size at which these mutations take place. We discuss these results in both evolutionary and mechanistic terms

Approximate Common Intervals in Multiple Genome Comparison

International audienceWe consider the problem of inferring approximate common intervals of multiple genomes. Genomes are modelled as sequences of homologous genes families identifiers, and approximate common intervals represent conserved regions possibly showing rearrangements, as well as repetitions, or insertions/deletions. This problem is already known, but existing approaches are not incremental and somehow limited to special cases. We adopt a simple, classical graph-based approach, where the vertices of the graph represent the exact common intervals of the sequences (\ie, regions containing the same gene set), and where edges link vertices that differ by less than $\delta$ elements (with $\delta$ being parameter). With this model, approximate gene clusters are maximal cliques of the graph: computing them can exploit known and well designed algorithms. For a proof of concept, we applied the method to several datasets of bacterial genomes and compared the two maximal cliques algorithms, a static and a dynamic one. While being quite flexible, this approach opens the way to a combinatorial characterization of genomic rearrangements in terms of graph substructures

Computing Phylo-k-mers

Phylogenetically informed k-mers, or phylo-k-mers for short, are k-mers that are predicted to appear within a given genomic region at predefined locations of a fixed phylogeny. Given a reference alignment for this genomic region and assuming a phylogenetic model of sequence evolution, we can compute a probability score for any given k-mer at any given tree node. The k-mers with sufficiently high probabilities can later be used to perform alignment-free phylogenetic classification of new sequences-a procedure recently proposed for the phylogenetic placement of metabarcoding reads and the detection of novel virus recombinants. While computing phylo-k-mers, we need to consider large numbers of k-mers at each tree node, which warrants the development of efficient enumeration algorithms. We consider a formal definition of the problem of phylo-k-mer computation: How to efficiently find all k-mers whose probability lies above a user-defined threshold for a given tree node? We describe and analyze algorithms for this problem, relying on branch-and-bound and divideand-conquer techniques. We exploit the redundancy of adjacent windows of the alignment and the structure of the probability matrix to save on computation. Besides computational complexity analyses, we provide an empirical evaluation of the relative performance of their implementations on real-world and simulated data. The divide-and-conquer algorithms, which to the best of our knowledge are novel, are found to be clear improvements over the branch-and-bound approach, especially when a large number of phylo-k-mers are found

CRAC: an integrated approach to the analysis of RNA-seq reads

International audienceA large number of RNA-sequencing studies set out to predict mutations, splice junctions or fusion RNAs. We propose a method, CRAC, that integrates genomic locations and local coverage to enable such predictions to be made directly from RNA-seq read analysis. A k-mer profiling approach detects candidate mutations, indels and splice or chimeric junctions in each single read. CRAC increases precision compared with existing tools, reaching 99:5% for splice junctions, without losing sensitivity. Importantly, CRAC predictions improve with read length. In cancer libraries, CRAC recovered 74% of validated fusion RNAs and predicted novel recurrent chimeric junctions. CRAC is available at http://crac.gforge.inria.fr

Accurate self-correction of errors in long reads using de Bruijn graphs

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LoRDEC : accurate and efficient long read error correction

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