Nonrenewal Statistics in the Catalytic Activity of Enzyme Molecules at Mesoscopic Concentrations

Abstract

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. We study enzyme kinetics at physiologically relevant mesoscopic concentrations using a master equation. From the exact solution of the master equation we find that the waiting times are neither independent nor identically distributed, implying that enzymatic turnovers form a nonrenewal stochastic process. The inverse of the mean waiting time shows strong departure from the Michaelis-Menten equation. The waiting times between consecutive turnovers are anticorrelated, where short intervals are more likely to be followed by long intervals and vice versa. Correlations persist beyond consecutive turnovers indicating that multiscale fluctuations govern enzyme kinetics.

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“…By constructing the master equation for these degenerate states, one can show that the waiting time distribution in minus-run/TOW is a multiexponential function [29]. This is similar to the recent discovery of non-Markovian behavior in enzyme kinetics characterized by multi-exponential waiting time distributions, when more than one enzyme is present in the system [30]. Because memory originates from degeneracy of states, it is natural to expect that µ will be non-zero in the more general K > 1, D > 1 cases also.…”

Comment on “Nonrenewal Statistics in the Catalytic Activity of Enzyme Molecules at Mesoscopic Concentrations”

“…However, in fact, the MM equation provides the correct steady-state rate for the model in Ref. [3] regardless of the number of enzymes. Using the relationship between the mean and characteristic function,…”

“…The authors of Ref. [3] further claimed that the mean catalytic rate of a system of enzymes at mesoscopic concentration does not obey the Michaelis-Menten (MM) equation, even if the MM equation is correct for a single enzyme as well as for a macroscopic number of enzymes [4][5][6]. However, in fact, the MM equation provides the correct steady-state rate for the model in Ref.…”

“…The authors of Ref. [3] obtained the result inconsistent with the MM equation because they investigated the enzyme system in a nonstationary state. For example, Eq.…”

“…(7) in Ref. [3] is the probability density of the time elapsed for the very first enzymatic turnover in N enzyme system under the synchronized initial condition ( Fig. 1(a)), under which all the enzymes in the system synchronously start catalytic reactions at time 0.…”