MATHEMATICAL-ANALYSIS OF A PROTEOLYTIC POSITIVE-FEEDBACK LOOP - DEPENDENCE OF LAG TIME AND ENZYME YIELDS ON THE INITIAL CONDITIONS AND KINETIC-PARAMETERS

A model of a proteolytic positive-feedback loop, similar in general terms to feedback loops that occur in blood coagulation and other systems, has been examined by both explicit and numerical analysis. In this loop, modeled as a closed system, each enzyme (E1, E2) catalyzes the formation of the other from its respective zymogen (Z1, Z2), and both enzymes are subject to irreversible inhibition. The system shows three major characteristics. (1) No significant Z1 or Z2 activation occurs unless the combination of initial conditions and kinetic parameters is above a threshold level. This threshold occurs when the product of the enzyme generation rates equals the product of their inhibition rates. When the formation-rate product is less than the inhibition-rate product, there is no response: E1 and E2 generation is minimal and the lag time is effectively infinite. Conversely, when the generation-rate product exceeds the inhibition-rate product, explosive formation of both E1 and E2 is seen. For responses exceeding the threshold, the following obtain. (2) The lag time in E1 and E2 generation is a highly nonlinear function of the zymogen concentrations and the enzyme generation and inhibition rates. In contrast, there is a simple logarithmic relationship between the lag time and the initial trace concentration of the enzyme that is responsible for initiating the system; in this model, E1. (3) The extent of Z1 and Z2 activation is similarly a nonlinear function of the conditions and parameters but is independent of the initiating trace level of E1. Comparison of the predictions of explicit analysis with numerical simulation show that lag time and enzyme yield are acccurately described by the analytical functions, even to a large degree under conditions where the assumptions made in the mathematical analysis may not hold. Explicit expressions for the response threshold and the stimulus-response relationship hold great promise for our understanding of the role of positive-feedback loops in physiological systems.