Recent studies in single-molecule enzyme kinetics reveal that the turnover statistics of a single enzyme is governed by the waiting time distribution that decays as mono-exponential at low substrate concentration and multi-exponential at high substrate concentration. The multi-exponentiality arises due to protein conformational fluctuations, which act on the time scale longer than or comparable to the catalytic reaction step, thereby inducing temporal fluctuations in the catalytic rate resulting in dynamic disorder. In this work, we study the turnover statistics of a single enzyme in the presence of inhibitors to show that the multi-exponentiality in the waiting time distribution can arise even when protein conformational fluctuations do not influence the catalytic rate. From the Michaelis-Menten mechanism of inhibited enzymes, we derive exact expressions for the waiting time distribution for competitive, uncompetitive, and mixed inhibitions to quantitatively show that the presence of inhibitors can induce dynamic disorder in all three modes of inhibitions resulting in temporal fluctuations in the reaction rate. In the presence of inhibitors, dynamic disorder arises due to transitions between active and inhibited states of enzymes, which occur on time scale longer than or comparable to the catalytic step. In this limit, the randomness parameter (dimensionless variance) is greater than unity indicating the presence of dynamic disorder in all three modes of inhibitions. In the opposite limit, when the time scale of the catalytic step is longer than the time scale of transitions between active and inhibited enzymatic states, the randomness parameter is unity, implying no dynamic disorder in the reaction pathway.

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.

The mean first passage time of cyclization of a semiflexible polymer with reactive ends is calculated using the diffusion-reaction formalism of Wilemski and Fixman ͓J. Chem. Phys. 60, 866 ͑1974͔͒. The approach is based on a Smoluchowski-type equation for the time evolution, in the presence of a sink, of a many-body probability distribution function. In the present calculations, which are an extension of work carried out by Pastor et al. ͓J. Chem. Phys. 105, 3878 ͑1996͔͒ on completely flexible Gaussian chains, the polymer is modeled as a continuous curve with a nonzero energy of bending. Inextensibility is enforced on average through chain-end contributions that suppress the excess fluctuations that lead to departures from the Kratky-Porod result for the mean-square end-to-end distance. The sink term in the generalized diffusion equation that describes the dynamics of the chain is modeled as a modified step function along the lines suggested by Pastor et al. Detailed calculations of as a function of the chain length N, the reaction distance a, and the stiffness parameter z are presented. Among other results, is found to be a power law in N, with a z-dependent scaling exponent that ranges between about 2.2-2.4. a͒

A recent experiment has probed the electron transfer kinetics in the early stage of photosynthesis in Rhodobacter sphaeroides for the reaction center of wild type and different mutants [Science 316, 747 (2007)]. By monitoring the changes in the transient absorption of the donor-acceptor pair at 280 and 930 nm, both of which show non-exponential temporal decay, the experiment has provided a strong evidence that the initial electron transfer kinetics is modulated by the dynamics of protein backbone. In this work, we present a model where the electron transfer kinetics of the donor-acceptor pair is described along the reaction coordinate associated with the distance fluctuations in a protein backbone. The stochastic evolution of the reaction coordinate is described in terms of a non-Markovian generalized Langevin equation with a memory kernel and Gaussian colored noise, both of which are completely described in terms of the microscopics of the protein normal modes. This model provides excellent fits to the transient absorption signals at 280 and 930 nm associated with protein distance fluctuations and protein dynamics modulated electron transfer reaction, respectively. In contrast to previous models, the present work explains the microscopic origins of the non-exponential decay of the transient absorption curve at 280 nm in terms of multiple time scales of relaxation of the protein normal modes. Dynamic disorder in the reaction pathway due to protein conformational fluctuations which occur on time scales slower than or comparable to the electron transfer kinetics explains the microscopic origin of the non-exponential nature of the transient absorption decay at 930 nm. The theoretical estimates for the relative driving force for five different mutants are in close agreement with the experimental estimates obtained using electrochemical measurements.

Recent fluorescence spectroscopy measurements of the turnover time distribution of single-enzyme turnover kinetics of β-galactosidase provide evidence of Michaelis-Menten kinetics at low substrate concentration. However, at high substrate concentrations, the dimensionless variance of the turnover time distribution shows systematic deviations from the Michaelis-Menten prediction. This difference is attributed to conformational fluctuations in both the enzyme and the enzyme-substrate complex and to the possibility of both parallel- and off-pathway kinetics. Here, we use the chemical master equation to model the kinetics of a single fluctuating enzyme that can yield a product through either parallel- or off-pathway mechanisms. An exact expression is obtained for the turnover time distribution from which the mean turnover time and randomness parameters are calculated. The parallel- and off-pathway mechanisms yield strikingly different dependences of the mean turnover time and the randomness parameter on the substrate concentration. In the parallel mechanism, the distinct contributions of enzyme and enzyme-substrate fluctuations are clearly discerned from the variation of the randomness parameter with substrate concentration. From these general results, we conclude that an off-pathway mechanism, with substantial enzyme-substrate fluctuations, is needed to rationalize the experimental findings of single-enzyme turnover kinetics of β-galactosidase.