MATHEMATICAL-ANALYSIS OF ACTIVATION THRESHOLDS IN ENZYME-CATALYZED POSITIVE FEEDBACKS - APPLICATION TO THE FEEDBACKS OF BLOOD-COAGULATION

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 sequential feedb ack loops act in a cascade. In the latter we also examine the function of a long-range feedback, in which the final enzyme produced in the s econd loop activates the initial step in the first loop. When the enzy mes generated are subject to inhibition or inactivation, all four syst ems exhibit threshold properties akin to excitable systems like neuron firing. For those that are amenable to mathematical analysis, express ions are derived that relate the excitation threshold to the kinetics of enzyme generation and inhibition and the initial conditions. For th e most complex system, it was expedient to employ numerical simulation to demonstrate threshold behavior, and in this case long-range feedba ck was seen to have two distinct effects. At sufficiently high catalyt ic rates, this feedback is capable of exciting an otherwise subthresho ld system, At lower catalytic rates, where the long-range feedback doe s not significantly affect the threshold, it nonetheless has a major e ffect in potentiating the response above the threshold. In particular, oscillatory behavior observed in simulations of sequential feedback l oops is abolished when a long-range feedback is present.