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Non-renewal statistics in the catalytic activity of enzyme molecules at mesoscopic concentrations

Recent fluorescence spectroscopy measurements of single-enzyme kinetics have shown that enzymatic turnovers form a renewal stochastic process in which the inverse of the mean waiting time between turnovers follows the Michaelis-Menten equation. Under typical physiological conditions, however, tens to thousands of enzymes react in catalyzing thousands to millions of substrates. We study enzyme kinetics at these physiologically relevant conditions through a master equation including stochasticity and molecular discreteness. From the exact solution of the master equation we find that the waiting times are neither independent nor are they identically distributed, implying that enzymatic turnovers form a non-renewal stochastic process. The inverse of the mean waiting time shows strong departures from the Michaelis-Menten equation. The waiting times between consecutive turnovers are anti-correlated, where short intervals are more likely to be followed by long intervals and vice versa. Correlations persist beyond consecutive turnovers indicating that multi-scale fluctuations govern enzyme kinetics.