Statistical Analyses and Theoretical Models of Single-Molecule Enzymatic Dynamics

Abstract

Real-time observation of enzymatic turnovers of single molecules has revealed non-Markovian dynamical behavior. Although chemical kinetics (such as the Michaelis-Menten mechanism) are sufficient to describe the average behavior of an ensemble of molecules, statistical analysis of the single-molecule fluorescence time trace reveals fluctuations in the rate of the activation step. These fluctuations are attributed to slow fluctuations of protein conformations. In this paper, we discuss models of the dynamical disorder behavior and relate them to observables of single molecule experiments. Simulations based on a discrete multistate model and a diffusive model are compared with experiment data. The role of various correlation functions, including higher order time correlation functions, in the interpretation of the underlying dynamics is discussed. IntroductionRecent advances in single-molecule spectroscopy have spurred much research on the dynamical behavior of single molecules. [1][2][3][4] Chemical dynamics can now be probed on a single-molecule basis. For example, enzymatic turnovers of single cholesterol oxidase molecules have been observed in real time by monitoring the emission from the enzyme's fluorescent active site, flavin adenine dinucleotide (FAD). 5 Chemical kinetics (such as the Michaelis-Menten mechanism), can account for the ensembleaveraged behaviors; however, statistical analyses of the singlemolecule trajectories reveal fluctuations in the rate of the activation step of the reaction. This dynamic disorder behavior is beyond the scope of conventional chemical kinetics and originates from slow fluctuations of protein conformations. 5 Ensemble-averaged experiments would not distinguish this dynamic variation of reaction rates from static heterogeneity. In this work, we present theoretical models for statistical analyses of single-molecule enzymatic dynamics. In this section, we give an overview of recent theoretical work relevant to single-molecule analyses.Skinner and co-workers 6,7 have conducted statistical analyses of single-molecule spectral trajectories in order to understand a pioneering experiment on single-molecule spectral diffusion at cryogenic temperatures. 8 The underlying dynamics of that system is tunneling of two-level systems, which was described by a two-state jump model, 6-7 and analyzed with autocorrelation functions of the transition frequency. Silbey and co-workers 9 and Klafter and co-workers 10 have also pursued this stochastic approach to two-level systems.Prompted by the room-temperature observations of double exponential fluorescence decays of single dye molecules in DNA complexes, 11,12 Geva and Skinner recast a two-state jump model in terms of population distributions as a function of data collection times. 13 Similar to "motional narrowing" in molecular spectroscopy, 14 this approach allowed the extraction of the time scale of conformational transitions of the systems. The twostate jumping model is within the framework of a simple reversible kinetic scheme; a more sophi...

Transitions in genetic toggle switches driven by dynamic disorder in rate coefficients

“…8 The particular behavior outlined in that work shows that enzymatic rate constants can remain at a fixed value for an interval of time before jumping and resampling from a heavy tailed distribution. 8,9 Throughout this paper, we will model such a behavior by a stochastic process in which the time intervals between switching events are exponentially distributed with a parameter λ s −1 that we call the switching rate. The single molecule studies find that, interestingly, allosteric regulation can affect both the switching rate λ, and the shape of the probability distribution p(k) that the rate parameters draw from.…”

Markov Chain Monte Carlo in the Analysis of Single-Molecule Experimental Data

“…Furthermore, it has also been argued that the two-state model is too simplistic to reflect the nature because the energy barrier between the two states may fluctuate dynamically or there may be sub-states within each of the two states and these substates may communicate at different rates. With the current data resolution and existing inference methods, discerning different models and assessing their fit to the experimental data have remained difficult [18].…”

“…We note that formula (4) is applicable to any finite-state hidden Markov process model, such as the two-by-two model of [18].…”

“…It has been observed that for certain molecules (other than the DNA hairpin) the twostate model (1) is not accurate enough to describe the conformational details [18]. This phenomenon of "dynamic disorder" motivates models beyond the two-state.…”

“…This phenomenon of "dynamic disorder" motivates models beyond the two-state. The 2 2 model in [18] is such an attempt and can be analyzed by the same Bayesian data augmentation approach outlined in the previous two sections. The 2 2 model can be further generalized to a continuous diffusive model, which needs additional effort.…”