Quantitative interpretation of the randomness in single enzyme turnover times

Unraveling the dynamic slowdown in supercooled water: The role of dynamic disorder in jump motions

When a liquid is rapidly cooled below its melting point without inducing crystallization, its dynamics slow down significantly without noticeable structural changes. Elucidating the origin of this slowdown has been a long-standing challenge. Here, we report a theoretical investigation into the mechanism of the dynamic slowdown in supercooled water, a ubiquitous yet extraordinary substance characterized by various anomalous properties arising from local density fluctuations. Using molecular dynamics simulations, we found that the jump dynamics, which are elementary structural change processes, deviate from Poisson statistics with decreasing temperature. This deviation is attributed to slow variables competing with the jump motions, i.e., dynamic disorder. The present analysis of the dynamic disorder showed that the primary slow variable is the displacement of the fourth nearest oxygen atom of a jumping molecule, which occurs in an environment created by the fluctuations of molecules outside the first hydration shell. As the temperature decreases, the jump dynamics become slow and intermittent. These intermittent dynamics are attributed to the prolonged trapping of jumping molecules within extended and stable low-density domains. As the temperature continues to decrease, the number of slow variables increases due to the increased cooperative motions. Consequently, the jump dynamics proceed in a higher-dimensional space consisting of multiple slow variables, becoming slower and more intermittent. It is then conceivable that with further decreasing temperature, the slowing and intermittency of the jump dynamics intensify, eventually culminating in a glass transition.

A minimal kinetic model for the interpretation of complex catalysis in single enzyme molecules

Multi-exponential waiting-time distribution and randomness parameter greater than unity ascribe dynamic disorder in single-enzyme catalysis corroborated to the interplay of transforming conformers [English et al., Nat. Chem. Biol., 2006, 2, 87]. The associated multi-state model of enzymatic turnovers with statically heterogeneous catalytic rates misdescribes the non-linear uprising of the randomness parameter from unity in relation to the attributes of the fall-offs of the waiting-time distribution at different substrate concentrations. To resolve this crucial issue, we first employ a comprehensive stochastic reaction scenario and further rationalize and work out the minimal indispensable dynamic-disorder model that ensures the foregoing relationship upon comparison with the data. We elucidate that specific disregard for the transition rate coefficients in the multi-state model on account of the especially slow conformational transitions is the underlying reason for not achieving interrelation between the observables.

Dynamics of chemical reactions on single nanocatalysts with heterogeneous active sites

Modern chemical science and industries critically depend on the application of various catalytic methods. However, the underlying molecular mechanisms of these processes still remain not fully understood. Recent experimental advances that produced highly-efficient nanoparticle catalysts allowed researchers to obtain more quantitative descriptions, opening the way to clarify the microscopic picture of catalysis. Stimulated by these developments, we present a minimal theoretical model that investigates the effect of heterogeneity in catalytic processes at the single-particle level. Using a discrete-state stochastic framework that accounts for the most relevant chemical transitions, we explicitly evaluated the dynamics of chemical reactions on single heterogeneous nanocatalysts with different types of active sites. It is found that the degree of stochastic noise in nanoparticle catalytic systems depends on several factors that include the heterogeneity of catalytic efficiencies of active sites and distinctions between chemical mechanisms on different active sites. The proposed theoretical approach provides a single-molecule view of heterogeneous catalysis and also suggests possible quantitative routes to clarify some important molecular details of nanocatalysts.

Computational Aspects of Single-Molecule Kinetics for Coupled Catalytic Cycles: A Spectral Analysis

Catalysis from single active sites is analyzed using methods developed from single-molecule kinetics. Using a stochastic Markov-state description, the observable properties of general catalytic networks of reactions are expressed using an eigenvalue decomposition of the transition matrix for the Markov process. By the use of a sensitivity analysis, the necessary eigenvalues and eigenvectors are related to the energies of controlling barriers and wells located along the reaction routes. A generalization of the energetic span theory allows the eigenvalues to be computed from several activation energies corresponding to distinct barrier-well pairings. The formalism is demonstrated for model problems and for a physically realistic mechanism for an alkene hydrogenation reaction on a single-atom catalyst. The spectral analysis permits a hierarchy of timescales to be identified from the single-molecule signal, which correspond to specific relaxation modes in the network.

Reaction-path statistical mechanics of enzymatic kinetics

We introduce a reaction-path statistical mechanics formalism based on the principle of large deviations to quantify the kinetics of single-molecule enzymatic reaction processes under the Michaelis-Menten mechanism, which exemplifies an out-of-equilibrium process in the living system. Our theoretical approach begins with the principle of equal a priori probabilities and defines the reaction path entropy to construct a new nonequilibrium ensemble as a collection of possible chemical reaction paths. As a result, we evaluate a variety of path-based partition functions and free energies by using the formalism of statistical mechanics. They allow us to calculate the timescales of a given enzymatic reaction, even in the absence of an explicit boundary condition that is necessary for the equilibrium ensemble. We also consider the large deviation theory under a closed-boundary condition of the fixed observation time to quantify the enzyme-substrate unbinding rates. The result demonstrates the presence of a phase-separation-like, bimodal behavior in unbinding events at a finite timescale, and the behavior vanishes as its rate function converges to a single phase in the long-time limit.

Understanding the Reaction Dynamics on Heterogeneous Catalysts Using a Simple Stochastic Approach

Recent experimental advances on investigating nanoparticle catalysts with multiple active sites provided a large amount of quantitative information on catalytic processes. These observations stimulated significant theoretical efforts, but the underlying molecular mechanisms are still not well-understood. We introduce a simple theoretical method to analyze the reaction dynamics on catalysts with multiple active sites based on a discrete-state stochastic description and obtain a comprehensive description of the dynamics of chemical reactions on such catalysts. We explicitly determine how the dynamics of catalyzed chemical reactions depend on the number of active sites, on the number of intermediate chemical transitions, and on the topology of underlying chemical reactions. It is argued that the theory provides quantitative bounds for realistic dynamic properties of catalytic processes that can be directly applied to analyze the experimental observations. In addition, this theoretical approach clarifies several important aspects of the molecular mechanisms of chemical reactions on catalysts.

A Revisit to Turnover Kinetics of Individual Escherichia coli β-Galactosidase Molecules

Single-molecule experiments on β-galactosidase from Escherichia coli that catalyzes the hydrolysis of resorufin-β-d-galactopyranoside revealed important observations like fluctuating catalytic rate, memory effects arising from temporal correlations between the enzymatic turnovers and nonexponential waiting time distributions. The root cause of the observed results is intrinsic fluctuations among the different conformers of the active species, during the course of the reaction, thereby imparting dynamic disorder in the system under investigation. Originally, a multistate stochastic kinetic theory was employed that, despite satisfying the measured waiting time distributions and the mean waiting times at different substrate concentrations, yields a constant estimate of the randomness parameter. Inevitably, this manifests a strong disagreement with the substrate-concentration-dependent time variations of the said distribution, which at the same time misinterprets the measured magnitudes of the randomness parameter at lower concentrations. Here, we suggest a dual approach to the single-enzyme reaction, independently, making important improvements over the parent study and the recently suggested two-state stochastic analyses followed by quantitative rationalization of the experimental data. In the first case, an off-pathway mechanism satisfied the Michaelis-Menten equation under the circumstance of prevailing disorder while tested against the single-molecule data. However, recovery of randomness data in the lower-concentration regime, albeit primarily marks a significant refinement, a qualitative agreement at the growing concentrations seems to be reasoned by an account of switching among the limited numbers of discrete conformers. Consequently, in the second case, we circumvented the conventional way of approaching the enzyme catalysis and mapped the dynamics of structural transitions of the biocatalyst with the temporal fluctuations of the spatial distance between the different locations along a coarse-grained polymer chain. Exploiting a general mechanism for dynamic disorder, a reaction-diffusion formalism yielded an analytical expression for the waiting time distribution of the enzymatic turnovers, from which the mean waiting time and the randomness parameter were readily determined. Application of our results to the findings of the experiment on single β-galactosidase shows a quantitative agreement in each case. This soundly validates the usefulness of accounting for a more rigorous microscopic description pertinent to the conformational multiplicity in rationalizing the real-time data over the routine state-based sketch of the reaction system.

Stochastic Kinetics of Nanocatalytic Systems

Catalytic reaction events occurring on the surface of a nanoparticle constitute a complex stochastic process. Although advances in modern single-molecule experiments enable direct measurements of individual catalytic turnover events occurring on a segment of a single nanoparticle, we do not yet know how to measure the number of catalytic sites in each segment or how the catalytic turnover counting statistics and the catalytic turnover time distribution are related to the microscopic dynamics of catalytic reactions. Here, we address these issues by presenting a stochastic kinetics for nanoparticle catalytic systems. We propose a new experimental measure of the number of catalytic sites in terms of the mean and variance of the catalytic event count. By considering three types of nanocatalytic systems, we investigate how the mean, the variance, and the distribution of the catalytic turnover time depend on the catalytic reaction dynamics, the heterogeneity of catalytic activity, and communication among catalytic sites. This work enables accurate quantitative analyses of single-molecule experiments for nanocatalytic systems and enzymes with multiple catalytic sites.

Theoretical Investigations of the Dynamics of Chemical Reactions on Nanocatalysts with Multiple Active Sites

Recent synthetic advances led to the development of new catalytic particles with well-defined atomic structures and multiple active sites, which are called nanocatalysts. Experimental studies of processes at nanocatalysts uncovered a variety of surprising effects, but the molecular mechanisms of these phenomena remain not well understood. We propose a theoretical method to investigate the dynamics of chemical reactions on catalytic particles with multiple active sites. It is based on a discrete-state stochastic description that allows us to explicitly evaluate dynamic properties of the system. It is found that for independently occurring chemical reactions, the mean turnover times are inversely proportional to the number of active sites, showing no stochastic effects. However, the molecular details of reactions and the number of active sites influence the higher moments of reaction times. Our theoretical method provides a way to quantify the molecular mechanisms of processes at nanocatalysts.

Frequency spectrum of chemical fluctuation: A probe of reaction mechanism and dynamics

Even in the steady-state, the number of biomolecules in living cells fluctuates dynamically, and the frequency spectrum of this chemical fluctuation carries valuable information about the dynamics of the reactions creating these biomolecules. Recent advances in single-cell techniques enable direct monitoring of the time-traces of the protein number in each cell; however, it is not yet clear how the stochastic dynamics of these time-traces is related to the reaction mechanism and dynamics. Here, we derive a rigorous relation between the frequency-spectrum of the product number fluctuation and the reaction mechanism and dynamics, starting from a generalized master equation. This relation enables us to analyze the time-traces of the protein number and extract information about dynamics of mRNA number and transcriptional regulation, which cannot be directly observed by current experimental techniques. We demonstrate our frequency spectrum analysis of protein number fluctuation, using the gene network model of luciferase expression under the control of the Bmal 1a promoter in mouse fibroblast cells. We also discuss how the dynamic heterogeneity of transcription and translation rates affects the frequency-spectra of the mRNA and protein number.

Anomalous Dynamics of in Vivo Cargo Delivery by Motor Protein Multiplexes

Vesicle transport conducted by motor protein multiplexes (MPMs), which is ubiquitous among eukaryotes, shows anomalous and stochastic dynamics qualitatively different from the dynamics of thermal motion and artificial active matter; the relationship between in vivo vesicle-delivery dynamics and the underlying physicochemical processes is not yet quantitatively understood. Addressing this issue, we perform accurate tracking of individual vesicles, containing upconverting nanoparticles, transported by kinesin-dynein-multiplexes along axonal microtubules. The mean-square-displacement of vesicles along the microtubule exhibits unusual dynamic phase transitions that are seemingly inconsistent with the scaling behavior of the mean-first-passage time over the travel length. These paradoxical results and the vesicle displacement distribution are quantitatively explained and predicted by a multimode MPM model, developed in the current work, where ATP-hydrolysis-coupled motion of MPM has both unidirectional and bidirectional modes.

The Chemical Fluctuation Theorem governing gene expression

Gene expression is a complex stochastic process composed of numerous enzymatic reactions with rates coupled to hidden cell-state variables. Despite advances in single-cell technologies, the lack of a theory accurately describing the gene expression process has restricted a robust, quantitative understanding of gene expression variability among cells. Here we present the Chemical Fluctuation Theorem (CFT), providing an accurate relationship between the environment-coupled chemical dynamics of gene expression and gene expression variability. Combined with a general, accurate model of environment-coupled transcription processes, the CFT provides a unified explanation of mRNA variability for various experimental systems. From this analysis, we construct a quantitative model of transcription dynamics enabling analytic predictions for the dependence of mRNA noise on the mRNA lifetime distribution, confirmed against stochastic simulation. This work suggests promising new directions for quantitative investigation into cellular control over biological functions by making complex dynamics of intracellular reactions accessible to rigorous mathematical deductions.

A unified framework to understand gene expression noise is still lacking. Here the authors derive a universal theorem relating the biological noise with dynamics of birth and death processes and present a model of transcription dynamics, allowing analytical prediction of the dependence of mRNA noise on mRNA lifetime variability.

Nonclassical Kinetics of Clonal yet Heterogeneous Enzymes

Enzyme-to-enzyme variation in the catalytic rate is ubiquitous among single enzymes created from the same genetic information, which persists over the lifetimes of living cells. Despite advances in single-enzyme technologies, the lack of an enzyme reaction model accounting for the heterogeneous activity of single enzymes has hindered a quantitative understanding of the nonclassical stochastic outcome of single enzyme systems. Here we present a new statistical kinetics and exactly solvable models for clonal yet heterogeneous enzymes with possibly nonergodic state dynamics and state-dependent reactivity, which enable a quantitative understanding of modern single-enzyme experimental results for the mean and fluctuation in the number of product molecules created by single enzymes. We also propose a new experimental measure of the heterogeneity and nonergodicity for a system of enzymes.

Optimal Stochastic Restart Renders Fluctuations in First Passage Times Universal

Stochastic restart may drastically reduce the expected run time of a computer algorithm, expedite the completion of a complex search process, or increase the turnover rate of an enzymatic reaction. These diverse first-passage-time (FPT) processes seem to have very little in common but it is actually quite the other way around. Here we show that the relative standard deviation associated with the FPT of an optimally restarted process, i.e., one that is restarted at a constant (nonzero) rate which brings the mean FPT to a minimum, is always unity. We interpret, further generalize, and discuss this finding and the implications arising from it.

Michaelis-Menten reaction scheme as a unified approach towards the optimal restart problem

We study the effect of restart, and retry, on the mean completion time of a generic process. The need to do so arises in various branches of the sciences and we show that it can naturally be addressed by taking advantage of the classical reaction scheme of Michaelis and Menten. Stopping a process in its midst-only to start it all over again-may prolong, leave unchanged, or even shorten the time taken for its completion. Here we are interested in the optimal restart problem, i.e., in finding a restart rate which brings the mean completion time of a process to a minimum. We derive the governing equation for this problem and show that it is exactly solvable in cases of particular interest. We then continue to discover regimes at which solutions to the problem take on universal, details independent forms which further give rise to optimal scaling laws. The formalism we develop, and the results obtained, can be utilized when optimizing stochastic search processes and randomized computer algorithms. An immediate connection with kinetic proofreading is also noted and discussed.

Role of substrate unbinding in Michaelis-Menten enzymatic reactions

The Michaelis-Menten equation provides a hundred-year-old prediction by which any increase in the rate of substrate unbinding will decrease the rate of enzymatic turnover. Surprisingly, this prediction was never tested experimentally nor was it scrutinized using modern theoretical tools. Here we show that unbinding may also speed up enzymatic turnover--turning a spotlight to the fact that its actual role in enzymatic catalysis remains to be determined experimentally. Analytically constructing the unbinding phase space, we identify four distinct categories of unbinding: inhibitory, excitatory, superexcitatory, and restorative. A transition in which the effect of unbinding changes from inhibitory to excitatory as substrate concentrations increase, and an overlooked tradeoff between the speed and efficiency of enzymatic reactions, are naturally unveiled as a result. The theory presented herein motivates, and allows the interpretation of, groundbreaking experiments in which existing single-molecule manipulation techniques will be adapted for the purpose of measuring enzymatic turnover under a controlled variation of unbinding rates. As we hereby show, these experiments will not only shed first light on the role of unbinding but will also allow one to determine the time distribution required for the completion of the catalytic step in isolation from the rest of the enzymatic turnover cycle.

Extracting signal from noise: kinetic mechanisms from a Michaelis-Menten-like expression for enzymatic fluctuations

Enzyme-catalyzed reactions are naturally stochastic, and precision measurements of these fluctuations, made possible by single-molecule methods, promise to provide fundamentally new constraints on the possible mechanisms underlying these reactions. We review some aspects of statistical kinetics: a new field with the goal of extracting mechanistic information from statistical measures of fluctuations in chemical reactions. We focus on a widespread and important statistical measure known as the randomness parameter. This parameter is remarkably simple in that it is the squared coefficient of variation of the cycle completion times, although it places significant limits on the minimal complexity of possible enzymatic mechanisms. Recently, a general expression has been introduced for the substrate dependence of the randomness parameter that is for rate fluctuations what the Michaelis-Menten expression is for the mean rate of product generation. We discuss the information provided by the new kinetic parameters introduced by this expression and demonstrate that this expression can simplify the vast majority of published models.

Imitating chemical motors with optimal information motors

To induce transport, detailed balance must be broken. A common mechanism is to bias the dynamics with a thermodynamic fuel, such as chemical energy. An intriguing, alternative strategy is for a Maxwell demon to effect the bias using feedback. We demonstrate that these two different mechanisms lead to distinct thermodynamics by contrasting a chemical motor and information motor with identical dynamics. To clarify this difference, we study both models within one unified framework, highlighting the role of the interaction between the demon and the motor. This analysis elucidates the manner in which information is incorporated into a physical system.

Reexamining Michaelis-Menten enzyme kinetics for xanthine oxidase

Abbreviated expressions for enzyme kinetic expressions, such as the Michaelis-Menten (M-M) equations, are based on the premise that enzyme concentrations are low compared with those of the substrate and product. When one does progress experiments, where the solute is consumed during conversion to form a series of products, the idealized conditions are violated. Here, we analyzed data of xanthine oxidase in vitro from Escribano et al. (Biochem J 254: 829, 1988) on two conversions in series, hypoxanthine to xanthine to uric acid. Analyses were done using four models: standard irreversible M-M reactions (model 1), Escribano et al.'s M-M forward reaction expressions with product inhibition (model 2), fully reversible M-M equations (model 3), and standard differential equations allowing forward and backward reactions with mass balance accounting for binding (model 4). The results showed that the need for invoking product inhibition vanishes with more complete analyses. The reactions were not quite irreversible, so the backward reaction had a small effect. Even though the enzyme concentration was only 1-2% of the initial substrate concentrations, accounting for the fraction of solutes bound to the enzyme did influence the parameter estimates, but in this case, the M-M model overestimated Michaelis constant values by only about one-third. This article also presents the research and models in a reproducible and publicly available form.