A linear constrained integral feedback for a class of reaction systems with absolute concentration robustness

In recent years, the concept of absolute concentration robust (ACR) biochemical reaction systems has been introduced and extensively studied in the reaction network literature. The biological relevance of ACR systems resides in the fact that the concentration of certain chemical species attains the same equilibrium level at any positive steady state, thus enabling a robust and predictive chemical response despite the initial conditions and the number of positive steady states. Sufficient structural conditions implying that a biochemical reaction system is ACR are shown in [17]. We prove that under the same conditions, there always exists a linear combination of the chemical species serving as a constrained integral feedback (CIF), with properties similar to that of a proper integral feedback provided that the species concentrations remain strictly positive. As a consequence, we show that the class of ACR systems studied in [17] are able to reject disturbances applied to the species concentrations over time. We then exploit these properties and demonstrate how such systems can be used as insulators due to their capacity for rejecting loading effects from downstream systems.