Vertex results for the robust analysis of uncertain biochemical systems

Abstract

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.

Robust Microphase Separation Through Chemical Reaction Networks

“…Proof: If − is initially positive, then for all , p(s, κ, ) defined in (10) is initially positive. Lemma 2 ensures that ρ( , κ) < 0 in a right neighborhood of zero, for all .…”

“…Here we consider the problem of inducing MS via chemical reactions through a control-theoretic approach. Given a CRN structure, whose rate parameters are unknown but bounded in a known range, we rely on parametric robustness approaches [5] and vertex results [9], [10], [17] to provide conditions ensuring that MS is: impossible; possible for some parameter values; or robustly guaranteed for all parameter values in the range.…”

“…Proof: For any value of κ, the maximum and minimum in ( 6) and ( 7) are achieved on the vertices D of the hyper-rectangle D, because the determinants are multilinear functions of the uncertain parameters i and any multilinear function defined on a hyper-rectangle achieves both its minimum and its maximum value on a vertex [5], [9], [10].…”