Threshold excitations, relaxation oscillations, and effect of noise in an enzyme reaction

We study a deprotonation reaction by an enzyme with activity dependent on pH. The rate and transport equations are simplified with a number of assumptions, are analyzed according to the presence of different time scales, and are solved numerically to show relaxation oscillation and threshold excitation, for different choices of parameters. The imposition of fluctuations (noise) on the deterministic equations for threshold excitation conditions leads to random occurrence of an excitation and return to steady state at low noise level and to large, random variations in concentrations at high noise level. At intermediate noise levels (of the order of the threshold excitation), however, we find quasi-periodic concentration oscillations. Thus, critical values of external constraints necessary for oscillations are altered by the presence of noise.