  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.
