Chapter 2 Structural Analysis of Biological Networks Franco Blanchini and Elisa Franco Abstract We introduce the idea of structural analysis of biological network models. In general, mathematical representations of molecular systems are affected by para- metric uncertainty: experimental validation of models is always affected by errors and intrinsic variability of biological samples. Using uncertain models for predic- tions is a delicate task. However, given a plausible representation of a system, it is often possible to reach general analytical conclusions on the system’s admissible dynamic behaviors, regardless of specific parameter values: in other words, we say that certain behaviors are structural for a given model. Here we describe a parameter- free, qualitative modeling framework and we focus on several case studies, showing how many paradigmatic behaviors such as multistationarity or oscillations can have a structural nature. We highlight that classical control theory methods are extremely helpful in investigating structural properties. Keywords Biological network · Control theory · Structural analysis · Structural property · Enzymatic networks · Jacobian · Eigenvalue · Chemical reaction network · Robustness · Set invariance · Mitogen activated protein kinase (MAPK) 2.1 Introduction Structural analysis of a dynamical system aims at revealing behavioral patterns that occur regardless of the adopted parameters, or, at least, for wide parameters ranges. Due to their parametric variability, biological models are often subject to structural F. Blanchini Dipartimento di Matematica ed Informatica, Università degli Studi di Udine, Via delle Scienze 206, 33100 Udine, Italy e-mail: blanchini@uniud.it E. Franco (B ) Department of Mechanical Engineering, University of California at Riverside, 900 University Avenue, Riverside, CA 92521, USA e-mail: efranco@engr.ucr.edu V. V. Kulkarni et al. (eds.), A Systems Theoretic Approach to Systems and Synthetic 47 Biology I: Models and System Characterizations, DOI: 10.1007/978-94-017-9041-3_2, © Springer Science+Business Media Dordrecht 2014