Antagonistic regulation with a unique setpoint, integral and double integral action

Several biochemical species are in organisms controlled in a pairwise manner i.e., two different species (e.g., hormone, enzyme, transporter protein) work to control the concentration of a third chemical species. Such pairs are often antagonistic, meaning that one of the controller species acts to increase whereas the other controller species acts to decrease the amount of the controlled species. How antagonistic systems interact to achieve regulation and to avoid competing against each other is not fully understood. An issue is how two antagonistic hormones can agree upon one common setpoint. We present here a new type of antagonistic regulatory system that has a single unique setpoint inherently defined by the system. The regulatory system controls the concentration of a chemical species with both integral and double integral action, achieving tight control. We show by the use of an analytical stability analysis, using the principle of vanishing perturbations, that the setpoint is asymptotically stable. Finally the prospect of treating the presented system as a part of a larger family of antagonistic regulatory systems with unique setpoints, integral and double integral action is discussed.