Nonrenewal 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. 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.

Figure 1

A trajectory of Eq. (1) for $N=2$ enzymes. The $p$th product is generated at time $T_p$. The waiting time between the $p$th and ($p+1$)th product is ${\tau }_p=T_{p+1}-T_p$.

Figure 2

Waiting time distributions for $N=1000$ and $N=1$ enzymes. The waitings times are identically distributed for a single enzyme (inset), but vary with the turnover number $p$ for multiple enzymes. Solid lines are analytical results obtained from Eq. (7) while the symbols are simulation data.

Figure 3

First moment of $T_1$ plotted in Lineweaver-Burk fashion against the inverse rate constant $k_a$. Solid lines are first moments of Eq. (7) while symbols are simulation data. With more than one enzyme, the MM equation (circles) is obeyed only in the limit of infinite substrate concentration or equivalently for $1/k_a\to 0$.

Figure 4

The correlation coefficient of ${\tau }_p$ and ${\tau }_{p+q}$ as a function of lag $q$ for $N=10$ (circles), $N=50$ (triangles) and $N=100$ (squares). The inset shows the joint distribution $w({\tau }_p,{\tau }_{p+1})$ for $N=10$.