Precision and Sensitivity in Detailed-Balance Reaction Networks

We study two specific measures of quality of chemical reaction networks, Precision and Sensitivity. The two measures arise in the study of sensory adaptation, in which the reaction network is viewed as an input-output system. Given a step change in input, Sensitivity is a measure of the magnitude of the response, while Precision is a measure of the degree to which the system returns to its original output for large time. High values of both are necessary for high-quality adaptation. We focus on reaction networks without dissipation, which we interpret as detailed-balance, mass-action networks. We give various upper and lower bounds on the optimal values of Sensitivity and Precision, characterized in terms of the stoichiometry, by using a combination of ideas from matroid theory and differential-equation theory. Among other results, we show that this class of nondissipative systems contains networks with arbitrarily high values of both Sensitivity and Precision. This good performance does come at a cost, however; since certain ratios of concentrations need to be large, the network has to contain many species and reactions, or the network should show strongly different time scales.