The role of membrane patch size and flow in regulating a proteolytic feedback threshold on a membrane: possible application in blood coagulation.

Positive feedback controls in proteolytic systems are characterized by thresholds which are regulated by the concentration of the initial stimulus and the kinetic parameters for enzyme generation and inhibition. Significant complexity is added when a positive feedback is localized on a membrane in contact with a flowing medium, such as seen in the early steps of blood coagulation. A partial differential equation model of an archetypal feedback loop is examined in which a proteolytic enzyme catalyzes its own formation from a zymogen on a membrane in contact with a flowing medium. As predicted from prior solution-phase and membrane-phase analyses, the threshold conditions for activation of the system are regulated by the kinetics of enzyme generation and inhibition and by the density of reactant-binding sites on the membrane; but the present analysis also establishes how the feedback threshold is controlled by the flow rate of the adjacent medium and the physical size of the membrane patch on which the feedback loop is localized. For given systems of particular kinetic properties, lower flow rates or larger active patches of membrane can result in the activation threshold being exceeded, whereas higher flow rates or smaller membrane patches can prevent initiation. In addition to numerical simulation, a simplified non-flowing model is analyzed to formulate an approximate mathematical statement of the dependence of the minimum activatable patch size on the kinetic and other parameters.