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BackgroundNetwork inference methods reconstruct mathematical models of molecular or genetic networks directly from experimental data sets. We have previously reported a mathematical method which is exclusively data-driven, does not involve any heuristic decisions within the reconstruction process, and deliveres all possible alternative minimal networks in terms of simple place/transition Petri nets that are consistent with a given discrete time series data set.ResultsWe fundamentally extended the previously published algorithm to consider catalysis and inhibition of the reactions that occur in the underlying network. The results of the reconstruction algorithm are encoded in the form of an extended Petri net involving control arcs. This allows the consideration of processes involving mass flow and/or regulatory interactions. As a non-trivial test case, the phosphate regulatory network of enterobacteria was reconstructed using in silico-generated time-series data sets on wild-type and in silico mutants.ConclusionsThe new exact algorithm reconstructs extended Petri nets from time series data sets by finding all alternative minimal networks that are consistent with the data. It suggested alternative molecular mechanisms for certain reactions in the network. The algorithm is useful to combine data from wild-type and mutant cells and may potentially integrate physiological, biochemical, pharmacological, and genetic data in the form of a single model.

The aim of this work is to extend a previously presented algorithm (Durzinsky et al. 2008b in Computational methods in systems biology, LNCS, vol 5307. Springer, Heidelberg, pp 328–346; Marwan et al. 2008 in Math Methods Oper Res 67:117–132) for the reconstruction of standard place/transition Petri nets from time-series of experimental data sets. This previously reported method finds provably all networks capable to reproduce the experimental observations. In this paper we enhance this approach to generate extended Petri nets involving mechanisms formally corresponding to catalytic or inhibitory dependencies that mediate the involved reactions. The new algorithm delivers the set of all extended Petri nets being consistent with the time-series data used for reconstruction. It is illustrated using the phosphate regulatory network of enterobacteria as a case study.

Webs and antiwebs are natural generalizations of odd holes and odd antiholes with circular symmetry of their maximum cliques and stable sets. Webs and antiwebs turned out to play a crucial role for describing the stable set polytopes for larger graph classes. In this short note we obtain, with the help of a result of Shepherd (1995), a complete description of the stable set polytopes for antiwebs showing that antiwebs are rank-perfect.The stable set polytope STAB(G) of a graph G = (V , E) is defined as the convex hull of the incidence vectors of all stable sets of G (a set V ⊆ V is stable if the nodes in V are mutually nonadjacent). A canonical relaxation of STAB(G) is the fractional stable set polytope QSTAB(G) given by all "trivial" facets, the nonnegativity constraintsfor all nodes i of G and by the clique constraintsfor all cliques Q ⊆ G (a set V ⊆ V is a clique if the nodes in V are mutually adjacent). A clique and a stable set have at most one node in common. Therefore, QSTAB(G) contains all incidence vectors of stable sets of G and STAB(G) ⊆ QSTAB(G) holds for all graphs G. The two polytopes coincide precisely for perfect graphs (Chvátal 1975;Padberg 1974).

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