Reconstruction of extended Petri nets from time-series data by using logical control functions.

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 328­346; Marwan et al. 2008 in Math Methods Oper Res 67:117­132) 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.

Reconstruction of extended Petri nets from time series data and its application to signal transduction and to gene regulatory networks.

BACKGROUND: Network 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 deliveries all possible alternative minimal networks in terms of simple place/transition Petri nets that are consistent with a given discrete time series data set. RESULTS: We 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. CONCLUSIONS: The 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.

Automatic reconstruction of molecular and genetic networks from discrete time series data.

We apply a mathematical algorithm which processes discrete time series data to generate a complete list of Petri net structures containing the minimal number of nodes required to reproduce the data set. The completeness of the list as guaranteed by a mathematical proof allows to define a minimal set of experiments required to discriminate between alternative network structures. This in principle allows to prove all possible minimal network structures by disproving all alternative candidate structures. The dynamic behaviour of the networks in terms of a switching rule for the transitions of the Petri net is part of the result. In addition to network reconstruction, the algorithm can be used to determine how many yet undetected components at least must be involved in a certain process. The algorithm also reveals all alternative structural modifications of a network that are required to generate a predefined behaviour.