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 te rms to feedback loops that occur in blood coagulation and other system s, has been examined by both explicit and numerical analysis. In this loop, modeled as a closed system, each enzyme (E1, E2) catalyzes the f ormation of the other from its respective zymogen (Z1, Z2), and both e nzymes are subject to irreversible inhibition. The system shows three major characteristics. (1) No significant Z1 or Z2 activation occurs u nless 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 prod uct, there is no response: E1 and E2 generation is minimal and the lag time is effectively infinite. Conversely, when the generation-rate pr oduct exceeds the inhibition-rate product, explosive formation of both E1 and E2 is seen. For responses exceeding the threshold, the followi ng obtain. (2) The lag time in E1 and E2 generation is a highly nonlin ear function of the zymogen concentrations and the enzyme generation a nd inhibition rates. In contrast, there is a simple logarithmic relati onship 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 fu nction of the conditions and parameters but is independent of the init iating trace level of E1. Comparison of the predictions of explicit an alysis with numerical simulation show that lag time and enzyme yield a re 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 threshol d and the stimulus-response relationship hold great promise for our un derstanding of the role of positive-feedback loops in physiological sy stems.