Geometry and topology of parameter space: investigating measures of robustness in regulatory networks.

The concept of robustness of regulatory networks has been closely related to the nature of the interactions among genes, and the capability of pattern maintenance or reproducibility. Defining this robustness property is a challenging task, but mathematical models have often associated it to the volume of the space of admissible parameters. Not only the volume of the space but also its topology and geometry contain information on essential aspects of the network, including feasible pathways, switching between two parallel pathways or distinct/disconnected active regions of parameters. A method is presented here to characterize the space of admissible parameters, by writing it as a semi-algebraic set, and then theoretically analyzing its topology and geometry, as well as volume. This method provides a more objective and complete measure of the robustness of a developmental module. As a detailed case study, the segment polarity gene network is analyzed.