Shape, Size, and Robustness: Feasible Regions in the Parameter Space of Biochemical Networks

The concept of robustness of regulatory networks has received much attention in the last decade. One measure of robustness has been associated with the volume of the feasible region, namely, the region in the parameter space in which the system is functional. In this paper, we show that, in addition to volume, the geometry of this region has important consequences for the robustness and the fragility of a network. We develop an approximation within which we could algebraically specify the feasible region. We analyze the segment polarity gene network to illustrate our approach. The study of random walks in the parameter space and how they exit the feasible region provide us with a rich perspective on the different modes of failure of this network model. In particular, we found that, between two alternative ways of activating Wingless, one is more robust than the other. Our method provides a more complete measure of robustness to parameter variation. As a general modeling strategy, our approach is an interesting alternative to Boolean representation of biochemical networks.Author Summary: Developing models with a large number of parameters for describing the dynamics of a biochemical network is a common exercise today. The dependence of predictions of such a network model on the choice of parameters is important to understand for two reasons. For the purpose of fitting biological data and making predictions, we need to know which combinations of parameters are strongly constrained by observations and also which combinations seriously affect a particular prediction. In addition, we expect naturally evolved networks to be somewhat robust to parameter changes. If the functioning of the network requires fine-tuning in many parameters, then mutations causing changes in regulatory interactions could quickly make the network dysfunctional. For predictions involving gene products being ON or OFF, we found a method that facilitates the study parameter dependence. As an example, we analyzed several competing models of the segment polarity network in Drosophila. We explicitly describe the region in the parameter space where the wild-type expression pattern of key genes becomes feasible for each model. We also study how random walks in the parameter space exit from the feasible region of a network model, allowing us to compare the relative robustness of the alternative models.