Conformational-relaxation models of single-enzyme kinetics

Quantifying the flux as the driving force for nonequilibrium dynamics and thermodynamics in non-Michaelis-Menten enzyme kinetics

The driving force for active physical and biological systems is determined by both the underlying landscape and nonequilibrium curl flux. While landscape can be experimentally quantified from the histograms of the collected real-time trajectories of the observables, quantifying the experimental flux remains challenging. In this work, we studied the single-molecule enzyme dynamics of horseradish peroxidase with dihydrorhodamine 123 and hydrogen peroxide (H2O2) as substrates. Surprisingly, significant deviations in the kinetics from the conventional Michaelis-Menten reaction rate were observed. Instead of a linear relationship between the inverse of the enzyme kinetic rate and the inverse of substrate concentration, a nonlinear relationship between the two emerged. We identified nonequilibrium flux as the origin of such non-Michaelis-Menten enzyme rate behavior. Furthermore, we quantified the nonequilibrium flux from experimentally obtained fluorescence correlation spectroscopy data and showed this flux to led to the deviations from the Michaelis-Menten kinetics. We also identified and quantified the nonequilibrium thermodynamic driving forces as the chemical potential and entropy production for such non-Michaelis-Menten kinetics. Moreover, through isothermal titration calorimetry measurements, we identified and quantified the origin of both nonequilibrium dynamic and thermodynamic driving forces as the heat absorbed (energy input) into the enzyme reaction system. Furthermore, we showed that the nonequilibrium driving forces led to time irreversibility through the difference between the forward and backward directions in time and high-order correlations were associated with the deviations from Michaelis-Menten kinetics. This study provided a general framework for experimentally quantifying the dynamic and thermodynamic driving forces for nonequilibrium systems.

Minute-scale persistence of a GPCR conformation state triggered by non-cognate G protein interactions primes signaling

Despite the crowded nature of the cellular milieu, ligand-GPCR-G protein interactions are traditionally viewed as spatially and temporally isolated events. In contrast, recent reports suggest the spatial and temporal coupling of receptor-effector interactions, with the potential to diversify downstream responses. In this study, we combine protein engineering of GPCR-G protein interactions with affinity sequestration and photo-manipulation of the crucial Gα C terminus, to demonstrate the temporal coupling of cognate and non-cognate G protein interactions through priming of the GPCR conformation. We find that interactions of the Gαs and Gαq C termini with the β₂-adrenergic receptor (β₂-AR), targeted at the G-protein-binding site, enhance Gs activation and cyclic AMP levels. β₂-AR-Gα C termini interactions alter receptor conformation, which persists for ~90 s following Gα C terminus dissociation. Non-cognate G-protein expression levels impact cognate signaling in cells. Our study demonstrates temporal allostery in GPCRs, with implications for the modulation of downstream responses through the canonical G-protein-binding interface.

Modeling of Nanomachine/Micromachine Crowds: Interplay between the Internal State and Surroundings

The activity of biological cells is primarily based on chemical reactions and typically modeled as a reaction-diffusion system. Cells are, however, highly crowded with macromolecules, including a variety of molecular machines such as enzymes. The working cycles of these machines are often coupled with their internal motion (conformational changes). In the crowded environment of a cell, motion interference between neighboring molecules is not negligible, and this interference can affect the reaction dynamics through machine operation. To simulate such a situation, we propose a reaction-diffusion model consisting of particles whose shape depends on an internal state variable, for crowds of nano- to micromachines. The interference between nearby particles is naturally introduced through excluded volume repulsion. In the simulations, we observed segregation and flow-like patterns enhanced by crowding out of relevant molecules, as well as molecular synchronization waves and phase transitions. The presented model is simple and extensible for diverse molecular machinery and may serve as a framework to study the interplay between the mechanical stress/strain network and the chemical reaction network in the cell. Applications to more macroscopic systems, e.g., crowds of cells, are also discussed.

Single-molecule theory of enzymatic inhibition

The classical theory of enzymatic inhibition takes a deterministic, bulk based approach to quantitatively describe how inhibitors affect the progression of enzymatic reactions. Catalysis at the single-enzyme level is, however, inherently stochastic which could lead to strong deviations from classical predictions. To explore this, we take the single-enzyme perspective and rebuild the theory of enzymatic inhibition from the bottom up. We find that accounting for multi-conformational enzyme structure and intrinsic randomness should strongly change our view on the uncompetitive and mixed modes of inhibition. There, stochastic fluctuations at the single-enzyme level could make inhibitors act as activators; and we state—in terms of experimentally measurable quantities—a mathematical condition for the emergence of this surprising phenomenon. Our findings could explain why certain molecules that inhibit enzymatic activity when substrate concentrations are high, elicit a non-monotonic dose response when substrate concentrations are low.

Single molecule approaches demonstrated that enzymatic catalysis is stochastic which could lead to deviations from classical predictions. Here authors rebuild the theory of enzymatic inhibition to show that stochastic fluctuations on the single enzyme level could make inhibitors act as activators.

Adaptive Heat Engine

A major limitation of many heat engines is that their functioning demands on-line control and/or an external fitting between the environmental parameters (e.g., temperatures of thermal baths) and internal parameters of the engine. We study a model for an adaptive heat engine, where-due to feedback from the functional part-the engine's structure adapts to given thermal baths. Hence, no on-line control and no external fitting are needed. The engine can employ unknown resources; it can also adapt to results of its own functioning that make the bath temperatures closer. We determine resources of adaptation and relate them to the prior information available about the environment.

Shedding light on protein folding, structural and functional dynamics by single molecule studies

The advent of advanced single molecule measurements unveiled a great wealth of dynamic information revolutionizing our understanding of protein dynamics and behavior in ways unattainable by conventional bulk assays. Equipped with the ability to record distribution of behaviors rather than the mean property of a population, single molecule measurements offer observation and quantification of the abundance, lifetime and function of multiple protein states. They also permit the direct observation of the transient and rarely populated intermediates in the energy landscape that are typically averaged out in non-synchronized ensemble measurements. Single molecule studies have thus provided novel insights about how the dynamic sampling of the free energy landscape dictates all aspects of protein behavior; from its folding to function. Here we will survey some of the state of the art contributions in deciphering mechanisms that underlie protein folding, structural and functional dynamics by single molecule fluorescence microscopy techniques. We will discuss a few selected examples highlighting the power of the emerging techniques and finally discuss the future improvements and directions.

INSIGHTS IN ENZYME FUNCTIONAL DYNAMICS AND ACTIVITY REGULATION BY SINGLE MOLECULE STUDIES

The advent of advanced single molecule measurements heralded the arrival of a wealth of dynamic information revolutionizing our understanding of protein dynamics and behavior in ways not deducible by conventional bulk assays. They offered the direct observation and quantification of the abundance and life time of multiple states and transient intermediates in the energy landscape that are typically averaged out in non-synchronized ensemble measurements, thus providing unprecedented insights into complex biological processes. Here we survey the current state of the art in single-molecule fluorescence microscopy methodology for studying the mechanism of enzymatic activity and the insights on protein functional dynamics. We will initially discuss the strategies employed to date, their limitations and possible ways to overcome them, and finally how single enzyme kinetics can advance our understanding on mechanisms underlying function and regulation of proteins.

[Formula: see text]Special Issue Comment: This review focuses on functional dynamics of individual enzymes and is related to the review on ion channels by Lu,44 the reviews on mathematical treatment of Flomenbom45 and Sach et al.,46 and review on FRET by Ruedas-Rama et al.41

Variational Bayes analysis of a photon-based hidden Markov model for single-molecule FRET trajectories

Single-molecule fluorescence resonance energy transfer (smFRET) measurement is a powerful technique for investigating dynamics of biomolecules, for which various efforts have been made to overcome significant stochastic noise. Time stamp (TS) measurement has been employed experimentally to enrich information within the signals, while data analyses such as the hidden Markov model (HMM) have been successfully applied to recover the trajectories of molecular state transitions from time-binned photon counting signals or images. In this article, we introduce the HMM for TS-FRET signals, employing the variational Bayes (VB) inference to solve the model, and demonstrate the application of VB-HMM-TS-FRET to simulated TS-FRET data. The same analysis using VB-HMM is conducted for other models and the previously reported change point detection scheme. The performance is compared to other analysis methods or data types and we show that our VB-HMM-TS-FRET analysis can achieve the best performance and results in the highest time resolution. Finally, an smFRET experiment was conducted to observe spontaneous branch migration of Holliday-junction DNA. VB-HMM-TS-FRET was successfully applied to reconstruct the state transition trajectory with the number of states consistent with the nucleotide sequence. The results suggest that a single migration process frequently involves rearrangement of multiple basepairs.

Interpreting single turnover catalysis measurements with constrained mean dwell times

Observation of a chemical transformation at the single-molecule level yields a detailed view of kinetic pathways contributing to the averaged results obtained in a bulk measurement. Studies of a fluorogenic reaction catalyzed by gold nanoparticles have revealed heterogeneous reaction dynamics for these catalysts. Measurements on single nanoparticles yield binary trajectories with stochastic transitions between a dark state in which no product molecules are adsorbed and a fluorescent state in which one product molecule is present. The mean dwell time in either state gives information corresponding to a bulk measurement. Quantifying fluctuations from mean kinetics requires identifying properties of the fluorescence trajectory that are selective in emphasizing certain dynamic processes according to their time scales. We propose the use of constrained mean dwell times, defined as the mean dwell time in a state with the constraint that the immediately preceding dwell time in the other state is, for example, less than a variable time. Calculations of constrained mean dwell times for a kinetic model with dynamic disorder demonstrate that these quantities reveal correlations among dynamic fluctuations at different active sites on a multisite catalyst. Constrained mean dwell times are determined from measurements of single nanoparticle catalysis. The results indicate that dynamical fluctuations at different active sites are correlated, and that especially rapid reaction events produce particularly slowly desorbing product molecules.

Probing single-molecule enzyme active-site conformational state intermittent coherence

The relationship between protein conformational dynamics and enzymatic reactions has been a fundamental focus in modern enzymology. Using single-molecule fluorescence resonance energy transfer (FRET) with a combined statistical data analysis approach, we have identified the intermittently appearing coherence of the enzymatic conformational state from the recorded single-molecule intensity-time trajectories of enzyme 6-hydroxymethyl-7,8-dihydropterin pyrophosphokinase (HPPK) in catalytic reaction. The coherent conformational state dynamics suggests that the enzymatic catalysis involves a multistep conformational motion along the coordinates of substrate-enzyme complex formation and product releasing, presenting as an extreme dynamic behavior intrinsically related to the time bunching effect that we have reported previously. The coherence frequency, identified by statistical results of the correlation function analysis from single-molecule FRET trajectories, increases with the increasing substrate concentrations. The intermittent coherence in conformational state changes at the enzymatic reaction active site is likely to be common and exist in other conformation regulated enzymatic reactions. Our results of HPPK interaction with substrate support a multiple-conformational state model, being consistent with a complementary conformation selection and induced-fit enzymatic loop-gated conformational change mechanism in substrate-enzyme active complex formation.

Seeing the forest for the trees: fluorescence studies of single enzymes in the context of ensemble experiments

Enzymes are remarkable molecular machines that make many difficult biochemical reactions possible under mild biological conditions with incredible precision and efficiency. Our understanding of the working principles of enzymes, however, has not reached the level where one can readily deduce the mechanism and the catalytic rates from an enzyme's structure. Resolving the dynamics that relate the three-dimensional structure of an enzyme to its function has been identified as a key issue. While still challenging to implement, single-molecule techniques have emerged as one of the most useful methods for studying enzymes. We review enzymes studied using single-molecule fluorescent methods but placing them in the context of results from other complementary experimental work done on bulk samples. This review primarily covers three enzyme systems--flavoenzymes, dehydrofolate reductase, and adenylate kinase--with additional enzymes mentioned where appropriate. When the single-molecule experiments are discussed together with other methods aiming at the same scientific question, the weakness, strength, and unique contributions become clear.

Statistical properties of the dichotomous noise generated in biochemical processes

Dichotomous noise detected with the help of various single-molecule techniques convincingly reveals the actual occurrence of a multitude of conformational substates composing the native state of proteins. The nature of the stochastic dynamics of transitions between these substates is determined by the particular statistical properties of the noise observed. These involve nonexponential and possibly oscillatory time decay of the second order autocorrelation function, its relation to the third order autocorrelation function, and a relationship to dwell-time distribution densities and their correlations. Processes gated by specific conformational substates are distinguished from those with fluctuating barriers. This study throws light on the intriguing matter of the possibility of multiple stepping of the myosin motor along the actin filament per ATP molecule hydrolyzed.

Multiple-state reactions between the epidermal growth factor receptor and Grb2 as observed by using single-molecule analysis

Phosphorylation of the cytoplasmic tyrosine residues of the epidermal growth factor receptor (EGFR) upon binding of EGF induces recognition of various intracellular signaling molecules, including Grb2. Here, the reaction kinetics between EGFR and Grb2 was analyzed by visualizing single molecules of Grb2 conjugated to the fluorophore Cy3 (Cy3-Grb2). The plasma membrane fraction was purified from human epithelial carcinoma A431 cells after stimulation with EGF and attached to coverslips. Unitary events of association and dissociation of Cy3-Grb2 on the EGFR in the membrane fraction were observed at different concentrations of Grb2 (0.1-100 nM). The dissociation kinetics could be explained by using a multiple-exponential function with a major (>90%) dissociation rate of 8 s(-1) and a few minor components, suggesting the presence of multiple bound states. In contrast, the association kinetics could be described by a stretched exponential function, suggesting the presence of multiple reaction channels from many unbound substates. Transitions between the unbound substates were also suggested. Unexpectedly, the rate of association was not proportional to the Grb2 concentration: an increase in Cy3-Grb2 concentration by a factor of 10 induced an increase in the reaction frequency approximately by a factor of three. This effect can compensate for fluctuation of the signal transduction from EGFR to Grb2 caused by variations in the expression level of Grb2 in living cells.

Molecular synchronization waves in arrays of allosterically regulated enzymes

Spatiotemporal pattern formation in a product-activated enzymic reaction at high enzyme concentrations is investigated. Stochastic simulations show that catalytic turnover cycles of individual enzymes can become coherent and that complex wave patterns of molecular synchronization can develop. The analysis based on the mean-field approximation indicates that the observed patterns result from the presence of Hopf and wave bifurcations in the considered system.

Nonlinear relaxation dynamics in elastic networks and design principles of molecular machines

Analyzing nonlinear conformational relaxation dynamics in elastic networks corresponding to two classical motor proteins, we find that they respond by well defined internal mechanical motions to various initial deformations and that these motions are robust against external perturbations. We show that this behavior is not characteristic for random elastic networks. However, special network architectures with such properties can be designed by evolutionary optimization methods. Using them, an example of an artificial elastic network, operating as a cyclic machine powered by ligand binding, is constructed.

Simulated data sets for single molecule kinetics: some limitations and complications of data analysis

When the fluorescence intensity of a chromophore attached to or bound in an enzyme relates to a specific reactive step in the enzymatic reaction, a single molecule fluorescence study of the process reveals a time sequence in the fluorescence emission that can be analyzed to derive kinetic and mechanistic information. Reports of various experimental results and corresponding theoretical studies have provided a basis for interpreting these data and understanding the methodology. We have found it useful to parallel experiments with Monte Carlo simulations of potential models hypothesized to describe the reaction kinetics. The simulations can be adapted to include experimental limitations, such as limited data sets, and complexities such as dynamic disorder, where reaction rates appear to change over time. By using models that are known a priori, the simulations reveal some of the challenges of interpreting finite single-molecule data sets by employing various statistical signatures that have been identified.

Functional conformational motions in the turnover cycle of cholesterol oxidase

Reexamining experimental data of single-molecule fluorescence correlation spectroscopy for cholesterol oxidase, we find that the existing Michaelis-Menten models with dynamical disorder cannot explain strong correlations between subsequent turnover cycles revealed in the diagonal feature in the joint statistical distribution of adjacent "on" times of this enzyme. We suggest that functional conformational motions representing ordered sequences of transitions between a set of conformational substates are involved, along with equilibrium conformational fluctuations in the turnover cycle of cholesterol oxidase. A two-channel model of single-enzyme dynamics, including a slow functional conformational motion in one of the channels, is proposed that allows us to reproduce such strong correlations.

Altering the interaction between sigma70 and RNA polymerase generates complexes with distinct transcription-elongation properties

We compare the elongation behavior of native Escherichia coli RNA polymerase holoenzyme assembled in vivo, holoenzyme reconstituted from sigma70 and RNA polymerase in vitro, and holoenzyme with a specific alteration in the interface between sigma70 and RNA polymerase. Elongating RNA polymerase from each holoenzyme has distinguishable properties, some of which cannot be explained by differential retention or rebinding of sigma70 during elongation, or by differential presence of elongation factors. We suggest that interactions between RNA polymerase and sigma70 may influence the ensemble of conformational states adopted by RNA polymerase during initiation. These states, in turn, may affect the conformational states adopted by the elongating enzyme, thereby physically and functionally imprinting RNA polymerase.

Extraction of Kinetic Information from Single-Molecule Experiments

We discuss two possible approaches for extracting kinetic information from single-molecule experiments. The first approach is based on computing correlation functions from measured fluorescence signals, and the second on studying the statistics of on and off times of the same fluorescence signal. We show that in both cases it is possible to extract kinetic information about the nature of intramolecular fluctuations of the single molecule. We show that for single-molecule kinetics the intramolecular fluctuations produce stochastic memory effects which lead to new dynamic features that do not exist in traditional chemical kinetics. In particular, we investigate a new type of chemical oscillations in correlation functions observed experimentally by Edman and Rigler (Proc. Natl. Acad. Sci. USA 2000, 97, 8266).

Zero-mode waveguides for single-molecule analysis at high concentrations

Optical approaches for observing the dynamics of single molecules have required pico- to nanomolar concentrations of fluorophore in order to isolate individual molecules. However, many biologically relevant processes occur at micromolar ligand concentrations, necessitating a reduction in the conventional observation volume by three orders of magnitude. We show that arrays of zero-mode waveguides consisting of subwavelength holes in a metal film provide a simple and highly parallel means for studying single-molecule dynamics at micromolar concentrations with microsecond temporal resolution. We present observations of DNA polymerase activity as an example of the effectiveness of zero-mode waveguides for performing single-molecule experiments at high concentrations.