Biochemical integral feedback control requires neutral stability and zeroth order reactions

Biochemical reaction networks perform a variety of signal processing functions, one of which is computing the integrals of biochemical signal values. This is often used in integral feedback control, where it enables a system output to respond to changing inputs and then return back to a predetermined setpoint value, for example, to maintain biochemical homeostasis. In order to analyze how these systems function, we used analytical theory and simulations to investigate several simple reaction networks that perform integral feedback control. By starting with specific models and then generalizing them, we found that these networks share key elements. First, the open-loop version of each system (with feedback removed) includes a species concentration, or a combination of species concentrations, that is neutrally stable, meaning that this value does not have a unique stable steady-state solution. Secondly, this neutral stability invariably arises from zeroth order production or degradation reactions. These results, which are easily derived but have not been described before for biochemical systems, offer a new perspective on biochemical integral feedback control. We present the mathematics with a relatively high level of detail in order to illustrate common approaches for reaction network analysis.