Structurally robust biological networks

In this paper, we propose a framework to analyze robust stability of equilibria in biological networks. We employ Lyapunov and invariant sets theory, focusing on the structure of ordinary differential equation models. Without resorting to extensive numerical simulations, often necessary to explore the behavior of a model in its parameter space, we provide rigorous proofs of robust stability of known bio-molecular networks. Our results are in line with existing literature.The impact of our results is twofold: on the one hand, we highlight that classical and simple control theory methods are extremely useful to characterize the behavior of biological networks analytically. On the other hand, we are able to demonstrate that some biological networks are robust thanks to their structure and some qualitative properties of the interactions, regardless of the specific values of their parameters.The complex biochemistry of living organisms very often outperforms electrical and mechanical devices in terms of adaptability and robustness. Mapping such intricate reaction networks to high level design principles is the goal of systems biology, and it requires an immense collaborative effort among different disciplines, such as physics, mathematics and engineering [1].The most classical example of robust molecular circuitry is probably given by bacterial chemotaxis [2,3]. The action of the flagellar motor of E. coli cells is driven by a cascade of signaling proteins, whose active or inactive state is determined by the presence of nutrient in the environment. Both analysis on a simplified ordinary differential equation (ODE) model [2] and experiments [3] showed how the flagellar motion of E. coli presents a robustly stable steady state: steps in the nutrient concentration only temporarily alter the motor equilibrium. Cells are therefore sensitive to nutrient gradients, but always return to their stable motion mode (such property is also referred to as adaptability). Such stable ste