Identifying (un)controllable dynamical behavior in complex networks

We present a technique applicable in any dynamical framework to identify control-robust subsets of an interacting system. These robust subsystems, which we call stable modules, are characterized by constraints on the variables that make up the subsystem. They are robust in the sense that if the defining constraints are satisfied at a given time, they remain satisfied for all later times, regardless of what happens in the rest of the system, and can only be broken if the constrained variables are externally manipulated. We identify stable modules as graph structures in an expanded network, which represents causal links between variable constraints. A stable module represents a system “decision point”, or trap subspace. Using the expanded network, small stable modules can be composed sequentially to form larger stable modules that describe dynamics on the system level. Collections of large, mutually exclusive stable modules describe the system’s repertoire of long-term behaviors. We implement this technique in a broad class of dynamical systems and illustrate its practical utility via examples and algorithmic analysis of two published biological network models. In the segment polarity gene network of Drosophila melanogaster, we obtain a state-space visualization that reproduces by novel means the four possible cell fates and predicts the outcome of cell transplant experiments. In the T-cell signaling network, we identify six signaling elements that determine the high-signal response and show that control of an element connected to them cannot disrupt this response.Author summary: We show how to uncover the causal relationships between qualitative statements about the values of variables in ODE systems. We then show how these relationships can be used to identify subsystem behaviors that are robust to outside interventions. This informs potential system control strategies (e.g., in identifying drug targets). Typical analytical properties of biomolecular systems render them particularly amenable to our techniques. Furthermore, due to their often high dimension and large uncertainties, our results are particularly useful in biomolecular systems. We apply our methods to two quantitative biological models: the segment polarity gene network of Drosophila melanogaster and the T-cell signal transduction network.