Cooperativity and specificity in enzyme kinetics: a single-molecule time-based perspective

Monomeric Esterase: Insights into Cooperative Behavior, Hysteresis/Allokairy

Herein, we present a novel esterase enzyme, Ade1, isolated from a metagenomic library of Amazonian dark earths soils, demonstrating its broad substrate promiscuity by hydrolyzing ester bonds linked to aliphatic groups. The three-dimensional structure of the enzyme was solved in the presence and absence of substrate (tributyrin), revealing its classification within the α/β-hydrolase superfamily. Despite being a monomeric enzyme, enzymatic assays reveal a cooperative behavior with a sigmoidal profile (initial velocities vs substrate concentrations). Our investigation brings to light the allokairy/hysteresis behavior of Ade1, as evidenced by a transient burst profile during the hydrolysis of substrates such as p-nitrophenyl butyrate and p-nitrophenyl octanoate. Crystal structures of Ade1, coupled with molecular dynamics simulations, unveil the existence of multiple conformational structures within a single molecular state (E̅1). Notably, substrate binding induces a loop closure that traps the substrate in the catalytic site. Upon product release, the cap domain opens simultaneously with structural changes, transitioning the enzyme to a new molecular state (E̅2). This study advances our understanding of hysteresis/allokairy mechanisms, a temporal regulation that appears more pervasive than previously acknowledged and extends its presence to metabolic enzymes. These findings also hold potential implications for addressing human diseases associated with metabolic dysregulation.

On Kinetic Constraints That Catalysis Imposes on Elementary Processes

Catalysis, the acceleration of product formation by a substance that is left unchanged, typically results from multiple elementary processes, including diffusion of the reactants toward the catalyst, chemical steps, and release of the products. While efforts to design catalysts are often focused on accelerating the chemical reaction on the catalyst, catalysis is a global property of the catalytic cycle that involves all processes. These are controlled by both intrinsic parameters such as the composition and shape of the catalyst and extrinsic parameters such as the concentration of the chemical species at play. We examine here the conditions that catalysis imposes on the different steps of a reaction cycle and the respective role of intrinsic and extrinsic parameters of the system on the emergence of catalysis by using an approach based on first-passage times. We illustrate this approach for various decompositions of a catalytic cycle into elementary steps, including non-Markovian decompositions, which are useful when the presence and nature of intermediate states are a priori unknown. Our examples cover different types of reactions and clarify the constraints on elementary steps and the impact of species concentrations on catalysis.

Identifying the active sites in unequal iron-nitrogen single-atom catalysts

Single-atom catalysts (SACs) have become one of the most attractive frontier research fields in catalysis and energy conversion. However, due to the atomic heterogeneity of SACs and limitations of ensemble-averaged measurements, the essential active sites responsible for governing specific catalytic properties and mechanisms remain largely concealed. In this study, we develop a quantitative method of single-atom catalysis-fluorescence correlation spectroscopy (SAC-FCS), leveraging the atomic structure-dependent catalysis kinetics and single-turnover resolution of single-molecule fluorescence microscopy. This method enables us to investigate the oxidase-like single-molecule catalysis on unidentical iron-nitrogen (Fe-N) coordinated SACs, quantifying the active sites and their kinetic parameters. The findings reveal the significant differences of single sites from the average behaviors and corroborate the oxidase-like catalytic mechanism of the Fe-N active sites. We anticipate that the method will give essential insights into the rational design and application of SACs.

Rac-dependent feedforward autoactivation of NOX2 leads to oxidative burst

NADPH oxidase 2 (NOX2) produces the superoxide anion radical (O₂⁻), which has functions in both cell signaling and immune defense. NOX2 is a multimeric-protein complex consisting of several protein subunits including the GTPase Rac. NOX2 uniquely facilitates an oxidative burst, which is described by initially slow O₂⁻ production, which increases over time. The NOX2 oxidative burst is considered critical to immune defense because it enables expedited O₂⁻ production in response to infections. However, the mechanism of the initiation and progression of this oxidative burst and its implications for regulation of NOX2 have not been clarified. In this study, we show that the NOX2 oxidative burst is a result of autoactivation of NOX2 coupled with the redox function of Rac. NOX2 autoactivation begins when active Rac triggers NOX2 activation and the subsequent production of O₂⁻, which in turn activates redox-sensitive Rac. This activated Rac further activates NOX2, amplifying the feedforward cycle and resulting in a NOX2-mediated oxidative burst. Using mutagenesis-based kinetic and cell analyses, we show that enzymatic activation of Rac is exclusively responsible for production of the active Rac trigger that initiates NOX2 autoactivation, whereas redox-mediated Rac activation is the main driving force of NOX2 autoactivation and contributes to generation of ∼98% of the active NOX2 in cells. The results of this study provide insight into the regulation of NOX2 function, which could be used to develop therapeutics to control immune responses associated with dysregulated NOX2 oxidative bursts.

Understanding the Optimal Cooperativity of Human Glucokinase: Kinetic Resonance in Nonequilibrium Conformational Fluctuations

The cooperativity of a monomeric enzyme arises from dynamic correlation instead of spatial correlation and is a consequence of nonequilibrium conformation fluctuations. We investigate the conformation-modulated kinetics of human glucokinase, a monomeric enzyme with important physiological functions, using a five-state kinetic model. We derive the non-Michealis-Menten (MM) correction term of the activity (i.e., turnover rate), predict its relationship to cooperativity, and reveal the violation of conformational detailed balance. Most importantly, we reproduce and explain the observed resonance effect in human glucokinase (i.e., maximal cooperativity when the conformational fluctuation rate is comparable to the catalytic rate). With the realistic parameters, our theoretical results are in quantitative agreement with the reported measurement by Miller and co-workers. The analysis can be extended to a general chemical network beyond the five-state model, suggesting the generality of kinetic cooperativity and resonance.

Protease circuits for processing biological information

Engineered biocircuits designed with biological components have the capacity to expand and augment living functions. Here we demonstrate that proteases can be integrated into digital or analog biocircuits to process biological information. We first construct peptide-caged liposomes that treat protease activity as two-valued (i.e., signal is 0 or 1) operations to construct the biological equivalent of Boolean logic gates, comparators and analog-to-digital converters. We use these modules to assemble a cell-free biocircuit that can combine with bacteria-containing blood, quantify bacteria burden, and then calculate and unlock a selective drug dose. By contrast, we treat protease activity as multi-valued (i.e., signal is between 0 and 1) by controlling the degree to which a pool of enzymes is shared between two target substrates. We perform operations on these analog values by manipulating substrate concentrations and combine these operations to solve the mathematical problem Learning Parity with Noise (LPN). These results show that protease activity can be used to process biological information by binary Boolean logic, or as multi-valued analog signals under conditions where substrate resources are shared.

Temperature, Dynamics, and Enzyme-Catalyzed Reaction Rates

We review the adaptations of enzyme activity to different temperatures. Psychrophilic (cold-adapted) enzymes show significantly different activation parameters (lower activation enthalpies and entropies) from their mesophilic counterparts. Furthermore, there is increasing evidence that the temperature dependence of many enzyme-catalyzed reactions is more complex than is widely believed. Many enzymes show curvature in plots of activity versus temperature that is not accounted for by denaturation or unfolding. This is explained by macromolecular rate theory: A negative activation heat capacity for the rate-limiting chemical step leads directly to predictions of temperature optima; both entropy and enthalpy are temperature dependent. Fluctuations in the transition state ensemble are reduced compared to the ground state. We show how investigations combining experiment with molecular simulation are revealing fundamental details of enzyme thermoadaptation that are relevant for understanding aspects of enzyme evolution. Simulations can calculate relevant thermodynamic properties (such as activation enthalpies, entropies, and heat capacities) and reveal the molecular mechanisms underlying experimentally observed behavior.

Nonequilibrium Thermodynamics in Cell Biology: Extending Equilibrium Formalism to Cover Living Systems

We discuss new developments in the nonequilibrium dynamics and thermodynamics of living systems, giving a few examples to demonstrate the importance of nonequilibrium thermodynamics for understanding biological dynamics and functions. We study single-molecule enzyme dynamics, in which the nonequilibrium thermodynamic and dynamic driving forces of chemical potential and flux are crucial for the emergence of non-Michaelis-Menten kinetics. We explore single-gene expression dynamics, in which nonequilibrium dissipation can suppress fluctuations. We investigate the cell cycle and identify the nutrition supply as the energy input that sustains the stability, speed, and coherence of cell cycle oscillation, from which the different vital phases of the cell cycle emerge. We examine neural decision-making processes and find the trade-offs among speed, accuracy, and thermodynamic costs that are important for neural function. Lastly, we consider the thermodynamic cost for specificity in cellular signaling and adaptation.

Dynamics and Location of the Allosteric Midazolam Site in Cytochrome P4503A4 in Lipid Nanodiscs

Promiscuous and allosteric drug interactions with cytochrome P450 3A4 (CYP3A4) are ubiquitous but incompletely understood at the molecular level. A classic allosteric CYP3A4 drug interaction includes the benzodiazepine midazolam (MDZ). MDZ exhibits homotropic and heterotropic allostery when metabolized to 1'-hydroxy and 4-hydroxy metabolites in varying ratios. The combination of hydrogen-deuterium exchange mass spectrometry (HDX-MS) and Gaussian accelerated molecular dynamics (GaMD) simulations of CYP3A4 in lipid nanodiscs and in a lipid bilayer, respectively, reveals MDZ-dependent changes in dynamics in a membrane environment. The F-, G-, and intervening helices, as well as the loop preceding the β1-sheets, display the largest observed changes in HDX. The GaMD suggests a potential allosteric binding site for MDZ in the F'- and G'-regions, which undergo significant increases in HDX at near-saturating MDZ concentrations. The HDX-MS and GaMD results confirm that changes in dynamics are most significant near the developing consensus allosteric site, and these changes are distinct from those observed previously with the nonallosteric inhibitor ketoconazole. The results suggest that the allosteric MDZ remains mobile in its binding site at the Phe-cluster. The results further suggest that this binding site remains dynamic or changes the depth of insertion in the membrane.

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.

Allosteric autoactivation of SOS and its kinetic mechanism

Son of Sevenless (SOS), one of guanine nucleotide exchange factors (GEFs), activates Ras. We discovered that the allosteric domain of SOS yields SOS to proceed a previously unrecognized autoactivation kinetics. Its essential feature is a time-dependent acceleration of SOS feedback activation with a reaction initiator or with the priming of active Ras. Thus, this mechanistic autoactivation feature explains the notion, previously only conjectured, of accelerative SOS activation followed by the priming of active Ras, an action produced by another GEF Ras guanyl nucleotide-releasing protein (RasGRP). Intriguingly, the kinetic transition from gradual RasGRP activation to accelerative SOS activation has been interpreted as an analog to digital conversion; however, from the perspective of autoactivation kinetics, it is a process of straightforward RasGRP-mediated SOS autoactivation. From the viewpoint of allosteric protein cooperativity, SOS autoactivation is a unique time-dependent cooperative SOS activation because it enables an active SOS to accelerate activation of other SOS as a function of time. This time-dependent SOS cooperativity does not belong to the classic steady-state protein cooperativity, which depends on ligand concentration. Although its hysteretic or sigmoid-like saturation curvature is a classic hallmark of steady-state protein cooperativity, its hyperbolic saturation figure typically represents protein noncooperativity. We also discovered that SOS autoactivation perturbs the previously predicted hysteresis of SOS activation in a steady state to produce a hyperbolic saturation curve. We interpret this as showing that SOS allostery elicits, through SOS autoactivation, cooperativity uniquely time-dependent but not ligand concentration dependent.

Alleviated Inhibition of Single Enzyme in Confined and Crowded Environment

Most proteins perform functions in intracellular milieu. The crowding, compartmentalized cytosol environment affects the protein structure, folding, conformational stability, substrate diffusion, and substrate-enzyme binding. Moreover, enzymes are available at single or very low copy numbers in a cell, and thus the conformation fluctuations of a single enzyme in a crowding environment could also greatly influence its kinetics. However, the crowding effect is poorly understood in the kinetical aspect of enzymatic reactions. In the present study, individual horseradish peroxidase (HRP) is encapsulated in a liposome containing crowding reagents as mimics of viscous cytosol. The confined crowding environment possesses a profound influence on both the catalytic activity and the product inhibition of enzymes. By analyzing the correlation between product generation and product inhibition, we find that the allosteric noncompetitive inhibition of HRP is alleviated in the crowded and confined milieu. Small-angle X-ray scattering experiments provide straightforward proofs of structural changes of enzymes in crowding environments, which are responsible for the reduced enzyme activity and increased enzyme-substrate affinity. We expect that this work may deepen the understanding of correlations between enzymatic conformations and activity performance in real cellular environments.

Extensions to Michaelis-Menten Kinetics for Single Parameters

Biochemical transformation kinetics is based on the formation of enzyme-substrate complexes. We developed a robust scheme based on unit productions of enzymes and reactants in cyclic events to comply with mass action law to form enzyme-substrate complexes. The developed formalism supports a successful application of Michaelis-Menten kinetics in all biochemical transformations of single parameters. It is an essential tool to overcome some challenging healthcare and environmental issues. In developing the formalism, we defined the substrate [S]= [Product]3/4 and rate of reaction based on rate and time perspectives. It allowed us to develop two quadratic equations. The first, represents a body entity that gave a useful relationship of enzyme E = 2S0.33, and the second nutrients/feed, each giving [Enzymes] and [Enzyme-substrate complexes], simulating rate of reaction, [substrate], and their differentials. By combining [Enzymes] and [Enzyme-substrate complexes] values, this quadratic equation derives a Michaelis-Menten hyperbolic function. Interestingly, we can derive the proportionate rate of reaction and [Enzymes] values of the quadratics resulting in another Michaelis-Menten hyperbolic. What is clear from these results is that between these two hyperbolic functions, in-competitive inhibitions exist, indicating metabolic activities and growth in terms of energy levels. We validated these biochemical transformations with examples applicable to day to day life.

Structural conditions on complex networks for the Michaelis-Menten input-output response

The Michaelis-Menten (MM) fundamental formula describes how the rate of enzyme catalysis depends on substrate concentration. The familiar hyperbolic relationship was derived by timescale separation for a network of three reactions. The same formula has subsequently been found to describe steady-state input-output responses in many biological contexts, including single-molecule enzyme kinetics, gene regulation, transcription, translation, and force generation. Previous attempts to explain its ubiquity have been limited to networks with regular structure or simplifying parametric assumptions. Here, we exploit the graph-based linear framework for timescale separation to derive general structural conditions under which the MM formula arises. The conditions require a partition of the graph into two parts, akin to a "coarse graining" into the original MM graph, and constraints on where and how the input variable occurs. Other features of the graph, including the numerical values of parameters, can remain arbitrary, thereby explaining the formula's ubiquity. For systems at thermodynamic equilibrium, we derive a necessary and sufficient condition. For systems away from thermodynamic equilibrium, especially those with irreversible reactions, distinct structural conditions arise and a general characterization remains open. Nevertheless, our results accommodate, in much greater generality, all examples known to us in the literature.

Generic Schemes for Single-Molecule Kinetics. 3: Self-Consistent Pathway Solutions for Nonrenewal Processes

We recently developed a pathway analysis framework (paper 1) for describing single-molecule kinetics for renewal (i.e., memoryless) processes based on the decomposition of a kinetic scheme into generic structures. In our approach, waiting time distribution functions corresponding to such structures are expressed in terms of self-consistent pathway solutions and concatenated to form measurable probability distribution functions (PDFs), affording a simple way to decompose and recombine a network. Here, we extend this framework to nonrenewal processes, which involve correlations between events, and employ it to formulate waiting time PDFs, including the first-passage time PDF, for a general kinetic network model. Our technique does not require the assumption of Poissonian kinetics, permitting a more general kinetic description than the usual rate approach, with minimal topological restrictiveness. To demonstrate the usefulness of this technique, we provide explicit calculations for our general model, which we adapt to two generic schemes for single-enzyme turnover with conformational interconversion. For each generic scheme, wherein the intermediate state(s) need not undergo Poissonian decay, the functional dependence of the mean first-passage time on the concentration of an external substrate is analyzed. When conformational detailed balance is satisfied, the enzyme turnover rate (related to the mean first-passage time) reduces to the celebrated Michaelis-Menten functional form, consistent with our previous work involving a similar scheme with all rate processes, thereby establishing further generality to this intriguing result. Our framework affords a general and intuitive approach for evaluating measurable waiting time PDFs and their moments, making it a potentially useful kinetic tool for a wide variety of single-molecule processes.

Single-Molecule Kinetics of an Enzyme in the Presence of Multiple Substrates

Herein, the catalytic activity of a single enzyme in the presence of multiple substrates is studied. Three different mechanisms of bisubstrate binding, namely, ordered sequential, random sequential and ping-pong nonsequential pathway, are broadly discussed. By means of the chemical master equation approach, exact expressions for the waiting-time distributions, the mean turnover time and the randomness parameter as a function of the substrate concentration, such that both concentrations are fixed, but one of them is changed quasi-statically are obtained. The randomness parameter is not equal to unity at intermediate to high substrate concentrations, which indicates the presence of multiple rate-limiting steps in the reaction pathway in all three modes of bisubstrate binding. This arises due to transitions between the free enzyme and the enzyme-substrate complexes that occur on comparable timescales. Such turnover statistics of the single enzyme can also distinguish between the different types of bisubstrate binding mechanisms.

Conformational Nonequilibrium Enzyme Kinetics: Generalized Michaelis-Menten Equation

In a conformational nonequilibrium steady state (cNESS), enzyme turnover is modulated by the underlying conformational dynamics. On the basis of a discrete kinetic network model, we use an integrated probability flux balance method to derive the cNESS turnover rate for a conformation-modulated enzymatic reaction. The traditional Michaelis-Menten (MM) rate equation is extended to a generalized form, which includes non-MM corrections induced by conformational population currents within combined cyclic kinetic loops. When conformational detailed balance is satisfied, the turnover rate reduces to the MM functional form, explaining its general validity. For the first time, a one-to-one correspondence is established between non-MM terms and combined cyclic loops with unbalanced conformational currents. Cooperativity resulting from nonequilibrium conformational dynamics can be achieved in enzymatic reactions, and we provide a novel, rigorous means of predicting and characterizing such behavior. Our generalized MM equation affords a systematic approach for exploring cNESS enzyme kinetics.

Mechanism of Action of ABC Importers: Conservation, Divergence, and Physiological Adaptations

The past decade has seen a remarkable surge in structural characterization of ATP binding cassette (ABC) transporters, which have spurred a more focused functional analysis of these elaborate molecular machines. As a result, it has become increasingly apparent that there is a substantial degree of mechanistic variation between ABC transporters that function as importers, which correlates with their physiological roles. Here, we summarize recent advances in ABC importers' structure-function studies and provide an explanation as to the origin of the different mechanisms of action.

High performance enzyme kinetics of turnover, activation and inhibition for translational drug discovery

INTRODUCTION

Enzymes are the macromolecular catalysts of many living processes and represent a sizable proportion of all druggable biological targets. Enzymology has been practiced just over a century during which much progress has been made in both the identification of new enzymes and the development of novel methodologies for enzyme kinetics. Areas covered: This review aims to address several key practical aspects in enzyme kinetics in reference to translational drug discovery research. The authors first define what constitutes a high performance enzyme kinetic assay. The authors then review the best practices for turnover, activation and inhibition kinetics to derive critical parameters guiding drug discovery. Notably, the authors recommend global progress curve analysis of dose/time dependence employing an integrated Michaelis-Menten equation and global curve fitting of dose/dose dependence. Expert opinion: The authors believe that in vivo enzyme and substrate abundance and their dynamics, binding modality, drug binding kinetics and enzyme's position in metabolic networks should be assessed to gauge the translational impact on drug efficacy and safety. Integrating these factors in a systems biology and systems pharmacology model should facilitate translational drug discovery.

Entropy production in mesoscopic stochastic thermodynamics: nonequilibrium kinetic cycles driven by chemical potentials, temperatures, and mechanical forces

Nonequilibrium thermodynamics (NET) investigates processes in systems out of global equilibrium. On a mesoscopic level, it provides a statistical dynamic description of various complex phenomena such as chemical reactions, ion transport, diffusion, thermochemical, thermomechanical and mechanochemical fluxes. In the present review, we introduce a mesoscopic stochastic formulation of NET by analyzing entropy production in several simple examples. The fundamental role of nonequilibrium steady-state cycle kinetics is emphasized. The statistical mechanics of Onsager's reciprocal relations in this context is elucidated. Chemomechanical, thermomechanical, and enzyme-catalyzed thermochemical energy transduction processes are discussed. It is argued that mesoscopic stochastic NET in phase space provides a rigorous mathematical basis of fundamental concepts needed for understanding complex processes in chemistry, physics and biology. This theory is also relevant for nanoscale technological advances.

The uncoupled ATPase activity of the ABC transporter BtuC2D2 leads to a hysteretic conformational change, conformational memory, and improved activity

ABC transporters comprise a large and ubiquitous family of proteins. From bacteria to man they translocate solutes at the expense of ATP hydrolysis. Unlike other enzymes that use ATP as an energy source, ABC transporters are notorious for having high levels of basal ATPase activity: they hydrolyze ATP also in the absence of their substrate. It is unknown what are the effects of such prolonged and constant activity on the stability and function of ABC transporters or any other enzyme. Here we report that prolonged ATP hydrolysis is beneficial to the ABC transporter BtuC2D2. Using ATPase assays, surface plasmon resonance interaction experiments, and transport assays we observe that the constantly active transporter remains stable and functional for much longer than the idle one. Remarkably, during extended activity the transporter undergoes a slow conformational change (hysteresis) and gradually attains a hyperactive state in which it is more active than it was to begin with. This phenomenon is different from stabilization of enzymes by ligand binding: the hyperactive state is only reached through ATP hydrolysis, and not ATP binding. BtuC2D2 displays a strong conformational memory for this excited state, and takes hours to return to its basal state after catalysis terminates.

Enzyme efficiency: An open reaction system perspective

A measure of enzyme efficiency is proposed for an open reaction network that, in suitable form, applies to closed systems as well. The idea originates from the description of classical enzyme kinetics in terms of cycles. We derive analytical expressions for the efficiency measure by treating the network not only deterministically but also stochastically. The latter accounts for any significant amount of noise that can be present in biological systems and hence reveals its impact on efficiency. Numerical verification of the results is also performed. It is found that the deterministic equation overestimates the efficiency, the more so for very small system sizes. Roles of various kinetics parameters and system sizes on the efficiency are thoroughly explored and compared with the standard definition k2/KM. Study of substrate fluctuation also indicates an interesting efficiency-accuracy balance.

Michaelis–Menten kinetics under non-isothermal conditions

We extend the celebrated Michaelis–Menten kinetics description of an enzymatic reaction taking into consideration the presence of a thermal driving force.

Statistics and Related Topics in Single-Molecule Biophysics

Since the universal acceptance of atoms and molecules as the fundamental constituents of matter in the early twentieth century, molecular physics, chemistry and molecular biology have all experienced major theoretical breakthroughs. To be able to actually "see" biological macromolecules, one at a time in action, one has to wait until the 1970s. Since then the field of single-molecule biophysics has witnessed extensive growth both in experiments and theory. A distinct feature of single-molecule biophysics is that the motions and interactions of molecules and the transformation of molecular species are necessarily described in the language of stochastic processes, whether one investigates equilibrium or nonequilibrium living behavior. For laboratory measurements following a biological process, if it is sampled over time on individual participating molecules, then the analysis of experimental data naturally calls for the inference of stochastic processes. The theoretical and experimental developments of single-molecule biophysics thus present interesting questions and unique opportunity for applied statisticians and probabilists. In this article, we review some important statistical developments in connection to single-molecule biophysics, emphasizing the application of stochastic-process theory and the statistical questions arising from modeling and analyzing experimental data.

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.

Exploration of the spontaneous fluctuating activity of single enzyme molecules

Single enzyme molecules display inevitable, stochastic fluctuations in their catalytic activity. In metabolism, for instance, the stochastic activity of individual enzymes is averaged out due to their high copy numbers per single cell. However, many processes inside cells rely on single enzyme activity, such as transcription, replication, translation, and histone modifications. Here we introduce the main theoretical concepts of stochastic single-enzyme activity starting from the Michaelis-Menten enzyme mechanism. Next, we discuss stochasticity of multi-substrate enzymes, of enzymes and receptors with multiple conformational states and finally, how fluctuations in receptor activity arise from fluctuations in signal concentration. This paper aims to introduce the exciting field of single-molecule enzyme kinetics and stochasticity to a wider audience of biochemists and systems biologists.

A COMPARISON BETWEEN THE INITIAL RATE EXPRESSIONS OBTAINED UNDER STRICT CONDITIONS AND THE RAPID EQUILIBRIUM ASSUMPTION USING, AS EXAMPLE, A FOUR SUBSTRATE ENZYME REACTION

The software WinStes, developed by our group, is used to derive the strict steady-state initial rate equation of the reaction mechanism of CTP:sn-glycerol-3-phosphate cytidylyltransferase [EC 2.7.7.39] from Bacillus subtilis. This enzyme catalyzes a reaction with two substrates and operates by a random ordered binding mechanism with two molecules of each substrate. The accuracy of the steady-state rate equation derived is checked by comparing the rate values it provides with those obtained from the simulated progress curves. To analyze the kinetics of this enzyme using the strict steady-state initial rate equation, several curves for different substrate concentrations and different rate constants are generated. A comparison of these curves with the curves obtained from the rapid equilibrium initial rate equation, with different substrate concentration values, serves to analyze how the strict steady-state rate equation values are closer to those of rapid equilibrium rate equations when rapid equilibrium conditions are fulfilled.

Allosteric activation of the phosphoinositide phosphatase Sac1 by anionic phospholipids

Sac family phosphoinositide phosphatases comprise an evolutionarily conserved family of enzymes in eukaryotes. Our recently determined crystal structure of the Sac phosphatase domain of yeast Sac1, the founding member of the Sac family proteins, revealed a unique conformation of the catalytic P-loop and a large positively charged groove at the catalytic site. We now report a unique mechanism for the regulation of its phosphatase activity. Sac1 is an allosteric enzyme that can be activated by its product phosphatidylinositol or anionic phospholipid phosphatidylserine. The activation of Sac1 may involve conformational changes of the catalytic P-loop induced by direct binding with the regulatory anionic phospholipids in the large cationic catalytic groove. These findings highlight the fact that lipid composition of the substrate membrane plays an important role in the control of Sac1 function.

Kinetic behavior of the general modifier mechanism of Botts and Morales with non-equilibrium binding

In this paper, we perform a complete analysis of the kinetic behavior of the general modifier mechanism of Botts and Morales in both equilibrium steady states and non-equilibrium steady states (NESS). Enlightened by the non-equilibrium theory of Markov chains, we introduce the net flux into discussion and acquire an expression of the rate of product formation in NESS, which has clear biophysical significance. Up till now, it is a general belief that being an activator or an inhibitor is an intrinsic property of the modifier. However, we reveal that this traditional point of view is based on the equilibrium assumption. A modifier may no longer be an overall activator or inhibitor when the reaction system is not in equilibrium. Based on the regulation of enzyme activity by the modifier concentration, we classify the kinetic behavior of the modifier into three categories, which are named hyperbolic behavior, bell-shaped behavior, and switching behavior, respectively. We show that the switching phenomenon, in which a modifier may convert between an activator and an inhibitor when the modifier concentration varies, occurs only in NESS. Effects of drugs on the Pgp ATPase activity, where drugs may convert from activators to inhibitors with the increase of the drug concentration, are taken as a typical example to demonstrate the occurrence of the switching phenomenon.

A perturbation analysis of rate theory of self-regulating genes and signaling networks

A thorough kinetic analysis of the rate theory for stochastic self-regulating gene networks is presented. The chemical master equation kinetic model in terms of a coupled birth-death process is deconstructed into several simpler kinetic modules. We formulate and improve upon the rate theory of self-regulating genes in terms of perturbation theory. We propose a simple five-state scheme as a faithful caricature that elucidates the full kinetics including the "resonance phenomenon" discovered by Walczak et al. [Proc. Natl. Acad. Sci. U.S.A. 102, 18926 (2005)]. The same analysis can be readily applied to other biochemical networks such as phosphorylation signaling with fluctuating kinase activity. Generalization of the present approach can be included in multiple time-scale numerical computations for large biochemical networks.

Dynamic coupling between the LID and NMP domain motions in the catalytic conversion of ATP and AMP to ADP by adenylate kinase

The catalytic conversion of adenosine triphosphate (ATP) and adenosine monophosphate (AMP) to adenosine diphosphate (ADP) by adenylate kinase (ADK) involves large amplitude, ligand induced domain motions, involving the opening and the closing of ATP binding domain (LID) and AMP binding domain (NMP) domains, during the repeated catalytic cycle. We discover and analyze an interesting dynamical coupling between the motion of the two domains during the opening, using large scale atomistic molecular dynamics trajectory analysis, covariance analysis, and multidimensional free energy calculations with explicit water. Initially, the LID domain must open by a certain amount before the NMP domain can begin to open. Dynamical correlation map shows interesting cross-peak between LID and NMP domain which suggests the presence of correlated motion between them. This is also reflected in our calculated two-dimensional free energy surface contour diagram which has an interesting elliptic shape, revealing a strong correlation between the opening of the LID domain and that of the NMP domain. Our free energy surface of the LID domain motion is rugged due to interaction with water and the signature of ruggedness is evident in the observed root mean square deviation variation and its fluctuation time correlation functions. We develop a correlated dynamical disorder-type theoretical model to explain the observed dynamic coupling between the motion of the two domains in ADK. Our model correctly reproduces several features of the cross-correlation observed in simulations.

Michaelis-Menten relations for complex enzymatic networks

Most biological processes are controlled by complex systems of enzymatic chemical reactions. Although the majority of enzymatic networks have very elaborate structures, there are many experimental observations indicating that some turnover rates still follow a simple Michaelis-Menten relation with a hyperbolic dependence on a substrate concentration. The original Michaelis-Menten mechanism has been derived as a steady-state approximation for a single-pathway enzymatic chain. The validity of this mechanism for many complex enzymatic systems is surprising. To determine general conditions when this relation might be observed in experiments, enzymatic networks consisting of coupled parallel pathways are investigated theoretically. It is found that the Michaelis-Menten equation is satisfied for specific relations between chemical rates, and it also corresponds to a situation with no fluxes between parallel pathways. Our results are illustrated for a simple model. The importance of the Michaelis-Menten relationship and derived criteria for single-molecule experimental studies of enzymatic processes are discussed.

Stochastic ensembles, conformationally adaptive teamwork, and enzymatic detoxification

It has been appreciated for a long time that enzymes exist as conformational ensembles throughout multiple stages of the reactions they catalyze, but there is renewed interest in the functional implications. The energy landscape that results from conformationlly diverse poteins is a complex surface with an energetic topography in multiple dimensions, even at the transition state(s) leading to product formation, and this represents a new paradigm. At the same time there has been renewed interest in conformational ensembles, a new paradigm concerning enzyme function has emerged, wherein catalytic promiscuity has clear biological advantages in some cases. "Useful", or biologically functional, promiscuity or the related behavior of "multifunctionality" can be found in the immune system, enzymatic detoxification, signal transduction, and the evolution of new function from an existing pool of folded protein scaffolds. Experimental evidence supports the widely held assumption that conformational heterogeneity promotes functional promiscuity. The common link between these coevolving paradigms is the inherent structural plasticity and conformational dynamics of proteins that, on one hand, lead to complex but evolutionarily selected energy landscapes and, on the other hand, promote functional promiscuity. Here we consider a logical extension of the overlap between these two nascent paradigms: functionally promiscuous and multifunctional enzymes such as detoxification enzymes are expected to have an ensemble landscape with more states accessible on multiple time scales than substrate specific enzymes. Two attributes of detoxification enzymes become important in the context of conformational ensembles: these enzymes metabolize multiple substrates, often in substrate mixtures, and they can form multiple products from a single substrate. These properties, combined with complex conformational landscapes, lead to the possibility of interesting time-dependent, or emergent, properties. Here we demonstrate these properties with kinetic simulations of nonequilibrium steady state (NESS) behavior resulting from energy landscapes expected for detoxification enzymes. Analogous scenarios with other promiscuous enzymes may be worthy of consideration.

Metallic-Nanostructure-Enhanced Fluorescence of Single Flavin Cofactor and Single Flavoenzyme Molecules

The enzyme cofactors are intrinsically fluorescent and participate directly in the single molecule enzymology studies. Due to photobleaching, one cannot follow kinetics continuously by cofactor fluorescence for more than several minutes typically. Modification of spectral properties of fluorophores, such as the amplification of emission intensity, can be achieved through coupling with surface plasmons in close proximity to metallic nanostructures. This process, referred to as metal-enhanced fluorescence, offers promise for a range of applications, including bioassays, sensor technology, microarrays, and single-molecule studies. Here, we demonstrated up to a 100-fold increase in the emission of the single cofactors and flavoenzymes near silver nanostructures. Amplified fluorescence of different types of flavins and flavoenzymes has been interpreted by using time-resolved single molecule fluorescence data. The results show considerable promise for the studies of enzyme kinetics using the intrinsic fluorescence from the cofactors.

Cyclic conformational modification of an enzyme: serial engagement, energy relay, hysteretic enzyme, and Fischer's hypothesis

Reversible chemical modification of enzymes is one of the most important mechanisms in cellular signaling. We generalize this concept to include cyclic modification of enzyme conformations. The mechanism is fundamentally different from the ligand induced conformational change: It only requires a catalytic amount of ligand to activate an enzyme, but it does require an active chemical energy driving a "futile cycle" akin to the phosphorylation-dephosphorylation cycle. The mechanism covers several previously proposed models that include serial engagement for T-cell receptor activation, energy relay for proofreading in DNA replication and protein biosynthesis, the hysteretic enzyme, and Fischer's hypothesis on protein tyrosine phosphatase action. While for small proteins operating under a funnel-shaped energy landscape, multiple conformations with sufficiently long dwell times are not common, recent experiments suggest that for larger, multidomain proteins, cyclic conformational modification (CCM) is much more likely and evolution presumably finds a way to capitalize on this mode of regulation. CCM can be difficult to identify in cells; however, it is potentially an important, and yet overlooked, regulatory mechanism in cellular signal transduction. We suggest the serial engagement mechanism in T-cell activation to be a possible testing case for the CCM mechanism.

The chemical master equation approach to nonequilibrium steady-state of open biochemical systems: linear single-molecule enzyme kinetics and nonlinear biochemical reaction networks

We develop the stochastic, chemical master equation as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under a living environment. A living environment provides a continuous chemical energy input that sustains the reaction system in a nonequilibrium steady state with concentration fluctuations. We discuss the linear, unimolecular single-molecule enzyme kinetics, phosphorylation-dephosphorylation cycle (PdPC) with bistability, and network exhibiting oscillations. Emphasis is paid to the comparison between the stochastic dynamics and the prediction based on the traditional approach based on the Law of Mass Action. We introduce the difference between nonlinear bistability and stochastic bistability, the latter has no deterministic counterpart. For systems with nonlinear bistability, there are three different time scales: (a) individual biochemical reactions, (b) nonlinear network dynamics approaching to attractors, and (c) cellular evolution. For mesoscopic systems with size of a living cell, dynamics in (a) and (c) are stochastic while that with (b) is dominantly deterministic. Both (b) and (c) are emergent properties of a dynamic biochemical network; We suggest that the (c) is most relevant to major cellular biochemical processes such as epi-genetic regulation, apoptosis, and cancer immunoediting. The cellular evolution proceeds with transitions among the attractors of (b) in a "punctuated equilibrium" manner.