Section: Research Program

Séquences

Participants : Mireille Régnier, Philippe Chassignet, Yann Ponty, Jean-Marc Steyaert, Alice Héliou, Antoine Soulé.

String searching and pattern matching is a classical area in computer science, enhanced by potential applications to genomic sequences. In Cpm/Spire community, a focus is given to general string algorithms and associated data structures with their theoretical complexity. Our group specialized in a formalization based on languages, weighted by a probabilistic model. Team members have a common expertise in enumeration and random generation of combinatorial sequences or structures, that are admissible according to some given constraints. A special attention is paid to the actual computability of formula or the efficiency of structures design, possibly to be reused in external software.

As a whole, motif detection in genomic sequences is a hot subject in computational biology that allows to address some key questions such as chromosome dynamics or annotation. Among specific motifs involved in molecular interactions, one may cite protein-DNA (cis-regulation), protein-protein (docking), RNA-RNA (miRNA, frameshift, circularisation). This area is being renewed by high throughput data and assembly issues. New constraints, such as energy conditions, or sequencing errors and amplification bias that are technology dependent, must be introduced in the models. A collaboration has beenestablished with Lob , at Ecole Polytechnique, who bought a sequencing machine, through the co-advised thesis of Alice Héliou. An other aim is to combine statistical sampling with a fragment based approach for decomposing structures, such as the cycle decomposition used within F. Major's group  [34]. In general, in the future, our methods for sampling and sequence data analysis should be extended to take into account such constraints, that are continuously evolving.

Combinatorial Algorithms and motifs

Participants : Mireille Régnier, Philippe Chassignet, Alice Héliou.

Besides applications [41] of analytic combinatorics to computational biology problems, the team addressed general combinatorial problems on words and fundamental issues on languages and data structures. Motif detection combines an algorithmic search of potential sites and a significance assessment. To assess the significance of an observation usually requires the evaluation of a quantitative criterion such as the P-value. In the recent years, a general scheme of derivation of analytic formula for the P-value under different constraints ($k$-occurrence, first occurrence, overrepresentation in large sequences,...) has been provided. It relies on a representation of continuous sequences of overlapping words, currently named clumps or clusters in a graph [35]. Recursive equations to compute $p$-values may be reduced to a traversal of that graph, leading to a linear algorithm. This improves over the space and time complexity of the generating function approach or previous probabilistic weighted automata.

In [45], it is claimed that half of the genome consists of different types of repeats. One may cite microsatellites, DNA transposons, transposons, long terminal repeats (Ltr ), long interspersed nuclear elements (Line ), ribosomal DNA, short interspersed nuclear elements (Sine ). Therefore, knowledge about the length of repeats is a key issue in several genomic problems, notably assembly or re-sequencing. Preliminary theoretical results are given in [29], and, recently, heuristics have been proposed and implemented [26], [40], [23]. A dual problem is the length of minimal absent words. Minimal absent words are words that do not occur but whose proper factors all occur in the sequence. Their computation is extremly related to finding maximal repeats (repeat that can not be extended on the right nor on the left). The comparison of the sets of minimal absent words provides a fast alternative for measuring approximation in sequence comparison [22], [24].

Recently, it was shown that considering the words which occur in one sequence but do no in another can be used to detect biologically significant events [44]. We have studied the computation of minimal absent words and we have provided new linear implementations [20]. We are now working on a dynamic approach to compute minimal absent words for a sliding window. For a sequence of size n, we expect a complexity of $O\left(n\right)$ time and space, independent of the size of the window. This approach could be use to align a sequence on a larger sequence using minimal absent words for comparison.

Random generation

Participants : Yann Ponty, Juraj Michalik, Christelle Rovetta.

Analytical methods may fail when both sequential and structural constraints of sequences are to be modelled or, more generally, when molecular structures such as RNA structures have to be handled. The random generation of combinatorial objects is a natural, alternative, framework to assess the significance of observed phenomena. General and efficient techniques have been developed over the last decades to draw objects uniformly at random from an abstract specification. However, in the context of biological sequences and structures, the uniformity assumption becomes unrealistic, and one has to consider non-uniform distributions in order to derive relevant estimates. Typically, context-free grammars can handle certain kinds of long-range interactions such as base pairings in secondary RNA structures.

In 2005, a new paradigm appeared in the ab initio secondary structure prediction [27]: instead of formulating the problem as a classic optimization, this new approach uses statistical sampling within the space of solutions. Besides giving better, more robust, results, it allows for a fruitful adaptation of tools and algorithms derived in a purely combinatorial setting.

We also introduced algorithms and data structures for a non-redundant generation of combinatorial objects. In situtations where the search space of a problem can be unambigously explored using dynamic programming, such algorithms generate objects within a postulated distribution, conditioned to avoid previously generated objects. This method can be used to probe objects having lower probabilities, a desirable property in the context of RNA kinetics studies, or could lead to better estimators in context where the exact emission probability of each object can be computed.