Approches de programmation par contraintes pour l'analyse des propriétés structurelles des réseaux de Petri et application aux réseaux biochimiques
Résumé
Petri nets are a simple formalism for modelling concurrent computation. This formalism has been proposed as a promising tool to describe and analyse biochemical networks. In this thesis, we explore the structural properties of Petri nets as a mean to provide information about the biochemical system evolution and its dynamics, especially when kinetic data are missing, making simulations impossible. In particular, we consider the structural properties of siphons and traps. We show that these structures entail a family of particular stability properties which can be characterized by a fragment of CTL over infinite state structures. Mixed integer linear programs have been proposed and a state-of-the-art algorithm from the Petri net community has been described later to compute minimal sets of siphons and traps in Petri nets. We present a simple boolean model capturing these notions and compare SAT and CLP methods for enumerating the set of all minimal siphons and traps of a Petri net. Our methods are applied to a benchmark composed of the 403 models from the biomodels.net repository. We analyse why these programs perform so well on even very large biological models. We show that, in networks with bounded tree-width, the existence of a minimal siphon containing a given set of places can be decided in a linear time, and the Siphon-Trap property as well. Moreover, we consider two other Petri net structural properties: place and transition invariants. We present a simple method to extract minimal semi-positive invariants of a Petri net as a constraint satisfaction problem on finite domains using constraint programming with symmetry detection and breaking. This allows us to generalize well-known results about the steady-state analysis of symbolic Ordinary Differential Equations systems corresponding to biochemical reactions by taking into account the structure of the reaction network. The study of the underlying Petri net, initially introduced for metabolic flux analysis, provides classes of reaction systems for which the symbolic computation of steady states is possible.
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