H_∞ feedback-control theory in biochemical systems

In this paper we study the possible optimality of biochemical pathways in the H_∞ sense. We start by presenting simple linearized models of single enzymatic reaction systems, where we apply classical and modern tools of feedback-control theory. We then apply the results obtained by our analysis to a linearly unbranched enzyme pathway system, where we explore the effect of a negative feedback loop internally exerted on the system by a self-product of the pathway. We then probe the sensitivity of the enzymatic system to variations in certain variables and we deal with the problem of assessing the optimality of the static-output feedback control, in the H_∞ sense, inherent to the closed-loop system. In this point we demonstrate the applicability of our results via a theoretical example that provides an open-loop and closed-loop analysis of a four-block enzymatic system. We then apply the various tools we developed to the optimal analysis of the Threonine synthesis pathway which is regulated by three feedback loops. We demonstrate that this pathway is optimal in the H_∞ sense, in the face of considerable uncertainties in the various enzyme concentrations of the pathway.