Proceedings of IEEE CDC 2003 Model Validation and Robust Stability Analysis of the Bacterial Heat Shock Response Using SOSTOOLS H. El-Samad † , S. Prajna ‡ , A. Papachristodoulou ‡ , M. Khammash † , J.C. Doyle ‡ † Mechanical and Environmental Engineering, UC Santa Barbara Santa Barbara, CA 93106 ‡ Control and Dynamical Systems, California Institute of Technology Pasadena, CA 91125 Abstract The complexity inherent in gene regulatory network models, as well as their nonlinear nature make them difficult to analyze or validate/invalidate using conven- tional tools. Combining ideas from robust control the- ory, real algebraic geometry, optimization and semi- definite programming, SOSTOOLS provides a promis- ing framework to answer these robustness and model validation questions algorithmically. We adopt these tools in the study of the heat shock response in bacte- ria. For this purpose, we use a reduced order model of the bacterial heat stress response. We study the robust stability properties of this system to parametric uncer- tainty, and address the model validation/invalidation problem by proving the necessity for the existence of certain feedback loops to reproduce the known time behavior of the system. 1 Introduction One of the predominant goals of systems biology is to uncover how all of the genetic information is organized in regulatory systems that control life, health, and dis- ease. The first step in this direction is to build accurate computer models that give reliable, both qualitative and quantitative, descriptions of the mechanisms un- der study. An intrinsic problem with this approach re- volves around model validation/invalidation. In biolog- ical modeling, model validation is usually carried out by comparing the model predictions to data. Implic- itly underlying this task is the assumption that models could be unambiguously compared with data, when in fact this comparison is even more computationally chal- lenging than modeling and analysis itself. Specifically, given a model with a large number of unknown pa- rameters, simulation plus local sensitivity analysis and search can sometimes produce parameter values that fit data or are locally maximally likely to fit. If this fails, however, there may be no short proof that the model is incompatible with the data, mirroring the NP versus coNP distinction. Furthermore, with sufficiently complex models, it is well-known that almost any finite amount of data can be fit. Furthermore, even if a model has been shown to agree well with the measured data, the ability to assess the robustness of such a model is still important for understanding the function of the overall system and the contribution of its component subsystems. Combining notions from dynamical sys- tems theory, real algebraic geometry and semi-definite programming, SOSTOOLS [9, 10] provides an ideal framework to handle these issues in an algorithmic way. The procedure is based on the construction of Lya- punov functions as certificates that guarantee stability for a system which may contain uncertain parameters, as well as barrier functions as invalidation certificates for proposed models. These barrier functions separate the evolution of the model from measured data. The positivity conditions in these methodologies, such as those in Lyapunov’s stability theorem, can be relaxed computationally to Sum of Squares conditions, as the former are in general NP-hard to test whereas the latter can be verified in polynomial time using Semi-Definite Programming. Several constraints on these conditions can be adjoined to the system using the general frame- work of Positivstellensatz, a central theorem in Real Algebraic Geometry. In this work, we approach the validation/invalidation and robustness analysis of a model of the bacterial heat stress response in the context of SOSTOOLS. The bac- terial heat shock response is a fairly complex, highly conserved regulatory network that is of crucial impor- tance in the survival of most organisms. This system possesses a hierarchy of feedforward and feedback loops that serve different functions, ranging from increasing robustness to parametric uncertainty, to achieving fast transients and rejecting intrinsic cellular noise. We re- port the results of a preliminary test pertaining to the validity of a reduced order model of the HS response, in addition to results relevant to its robust stability fea- tures to parametric uncertainty, all using SOSTOOLS. 1