Mathematical Analysis of Activation Thresholds in Enzyme-Catalyzed Positive Feedbacks: Application to the Feedbacks of Blood Coagulation (proteases / protease inhibitors)

Abstract

A hierarchy of enzyme-catalyzed positive feedback loops is examined by mathematical and numerical analysis. Four systems are described, from the simplest, in which an enzyme catalyzes its own formation from an inactive precursor, to the most complex, in which two feedback loops act in a cascade analogous to a section of the blood-coagulation system. In the latter we also examine the function of a long-range feedback, in which the final enzyme produced in the second loop activates the initial step in the first loop. When the enzymes generated are subject to inhibition or inactivation, all four systems exhibit threshold properties akin to other excitable systems like neuron firing. For those that are amenable to mathematical analysis, expressions are derived that relate the excitation threshold to the kinetics of enzyme generation and inhibition and the initial conditions. For the most complex system it was expedient to employ numerical simulation to demonstrate threshold behavior, and h..