Global stability analysis and robust design of multi-time-scale biological networks under parametric uncertainties

Biological networks are prone to internal parametric fluctuations and external noises. Robustness represents a crucial property of these networks, which militates the effects of internal fluctuations and external noises. In this paper biological networks are formulated as coupled nonlinear differential systems operating at different time-scales under vanishing perturbations. In contrast to previous work viewing biological parametric uncertain systems as perturbations to a known nominal linear system, the perturbed biological system is modeled as nonlinear perturbations to a known nonlinear idealized system and is represented by two time-scales (subsystems). In addition, conditions for the existence of a global uniform attractor of the perturbed biological system are presented. By using an appropriate Lyapunov function for the coupled system, a maximal upper bound for the fast time-scale associated with the fast state is derived. The proposed robust system design principles are potentially applicable to robust biosynthetic network design. Finally, two examples of two important biological networks, a neural network and a gene regulatory network, are presented to illustrate the applicability of the developed theoretical framework.