THRESHOLD, WAVE AND CELL-CELL AVALANCHE BEHAVIOR IN A CLASS OF SUBSTRATE-INHIBITION OSCILLATORS

We consider a simple realistic model of enzyme inhibition by a substrate and show that for a certain range of the parameters, the reaction mechanism has a unique positive stable steady state which exhibits a threshold phenomenon: that is a sufficiently large perturbation about the steady state results in a much larger perturbation in the concentrations of the reactants before the return to the steady state. The mechanism considered is qualitatively similar to the reaction between uric acid and oxygen catalysed by the enzyme uricase immobilized on an insoluble matrix. For the diffusive case we show that these excursions produce a stable finite amplitude wave in the concentration of one of the reactants. The wave travels without change of shape and the unique wave speed is found. Such a wave can diffuse through cell walls and so, because of the threshold behaviour of the reaction mechanism, can give rise to an avalanche effect. Such a reactiondiffusion scheme is a possible robust trigger mechanism for the transfer of biochemical information between cells. The mechanism of transmission of such a solitary front is quite different from that in Hodgkin-Huxley nerve action potentials. © 1979.