Abstract

The switch-like character of the dynamics of genetic regulatory networks has attracted much attention from mathematical biologists and researchers on hybrid systems alike. While powerful techniques for the analysis, verification, and control of hybrid systems have been developed, the specificities of the biological application domain pose a number of challenges. In particular, while most networks of biological interest are large and complex, quantitative information on the kinetic parameters and molecular concentrations are usually absent.

We introduce a method for the analysis of reachability properties of genetic regulatory networks that is based on a class of discontinuous piecewise-affine differential equations (PADE) well-adapted to the above constraints. More specifically, we introduce a partition of the phase space by hyperrectangular regions in each of which the derivatives of the concentration variables have a unique sign pattern. This partition forms the basis for the definition of a discrete abstraction transforming the continuous transition system associated with a PADE model into a discrete or qualitative transition system. The discrete transition system is a simulation of the continuous transition system, thus providing a conservative approximation of the qualitative dynamics of the network. Moreover, the discrete transition system can be easily computed in a symbolic manner from inequality constraints on the parameters.

The method has been implemented in a new prototype version of the computer tool Genetic Network Analyzer (GNA), which has been applied to the analysis of a regulatory system whose functioning is not well-understood by biologists, the nutritional stress response in the bacterium Escherichia coli.

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