We consider the problem of assessing the sensitivity of uncertain biochemical systems in the presence of input perturbations (either constant or periodic) around a stable steady state. In particular, we propose approaches for the robust sensitivity analysis of systems with uncertain parameters assumed to take values in a hyper-rectangle. We highlight vertex results, which allow us to check whether a property is satisfied for all parameter choices in the hyper-rectangle by simply checking whether it is satisfied for all parameter choices at the vertices of the hyper-rectangle. We show that, for a vast class of systems, including (bio)chemical reaction networks with mass-action kinetics, the system Jacobian has a totally multiaffine structure (namely, all minors of the Jacobian matrix are multiaffine functions of the uncertain parameters), which can be exploited to obtain several vertex results. We consider different problems: robust non-singularity; robust stability of the steady-state; robust steady-state sensitivity analysis, in the case of constant perturbations; robust frequency-response sensitivity analysis, in the presence of periodic perturbations; and robust adaptation analysis. The developed theory is then applied to gain insight into some examples of uncertain biochemical systems, including the incoherent feed-forward loop, the coherent feed-forward loop, the Brusselator oscillator and the Goldbeter oscillator.

In systems biology, perfect adaptation (adaptation) denotes the property of a system reacting to a step input stimulus by completely (partially) restoring the pre-stimulus output value at steady state. We address the problem of predicting adaptation for uncertain dynamical systems. To this aim, we introduce a formal definition of adaptation tailored to the robust analysis of dynamical systems. Whilst the definition is more general and valid also for the step response analysis of nonlinear systems, in the linear case such a definition of adaptation reduces to the presence of a single real zero that dominates all poles. Based on this definition, we can assess robust adaptation by means of the robust real plot, which characterises the position of real zeros and poles for linear systems with parametric uncertainties.