Dynamical Systems and Control Theory Inspired by Molecular Biology Fa9550-08-1-0053

This project aims to develop new concepts, theory, and algorithms for control and signal processing using ideas inspired by molecular systems biology. Cell biology provides a wide repertoire of systems that are strongly fault-tolerant, nonlinear, feedback-rich, and truly hybrid, while making effective use of highly heterogeneous sensing and actuation channels. Advances in genomic/proteomics and molecular systems biology research are continually adding detailed knowledge of such systems’ architecture and operation, thus offering, in principle, a powerful source of inspiration for innovative solutions to problems of control and communication, sensor and actuator design, and systems integration. Often, problems in systems biology may superficially resemble standard problems in control theory but, on closer inspection, they differ in fundamental ways. These differences are challenging and worth exploring, and lead to and interesting and highly rewarding new directions of research. Summary of Findings: Systems that are “close” to monotone.: Monotone systems (which arose from the study of gene regulation and other biological systems) have proved a powerful tool for analyzing many nonlinear systems. However, monotonicity is a restrictive property vis a vis general biological as well as engineering systems. During the past few years, the PI and his collaborators and students have developed an approach (monotone I/O theory) that allows one to exploit the monotonicity of components even when the overall system is not monotone. In this context, a main line of work in this project is that of understanding the dynamics of systems that are not necessarily monotone but which, in some manner, are “close” to being so. In monotone systems, every net feedback loop is positive. On the other hand, negative feedback loops are important features of many systems, since they are required for adaptation and precision. The paper [19] used geometric singular perturbation theory to show that, provided that these negative loops act at a comparatively fast time scale, the main dynamical property of (strongly) monotone systems, convergence to steady states, is still valid. An application was worked out to a double-phosphorylation “futile cycle” motif which plays a central role in eukaryotic cell signaling. This work was complemented by [18], which studied the number of positive steady states in biomolecular reactions consisting of activation/deactivation futile cycles, such as those arising from phosphorylations and dephosphorylations at each level of a MAPK cascade. It was shown that for some parameter ranges, there are at least n+1 (if n is even) or n (if n is odd) steady states, there never are more than 2n − 1 steady states, that for parameters near the standard MichaelisMenten quasi-steady state conditions, there are at most n + 1 steady states and that for parameters far from the standard Michaelis-Menten quasi-steady state conditions, there is at most one steady state.