When does the Michaelis-Menten equation hold for fluctuating enzymes?

Estimating and Assessing Differential Equation Models with Time-Course Data

Ordinary differential equation (ODE) models are widely used to describe chemical or biological processes. This Article considers the estimation and assessment of such models on the basis of time-course data. Due to experimental limitations, time-course data are often noisy, and some components of the system may not be observed. Furthermore, the computational demands of numerical integration have hindered the widespread adoption of time-course analysis using ODEs. To address these challenges, we explore the efficacy of the recently developed MAGI (MAnifold-constrained Gaussian process Inference) method for ODE inference. First, via a range of examples we show that MAGI is capable of inferring the parameters and system trajectories, including unobserved components, with appropriate uncertainty quantification. Second, we illustrate how MAGI can be used to assess and select different ODE models with time-course data based on MAGI's efficient computation of model predictions. Overall, we believe MAGI is a useful method for the analysis of time-course data in the context of ODE models, which bypasses the need for any numerical integration.

Single enzyme electroanalysis

Traditional studies of enzymatic activity rely on the combined kinetics of millions of enzyme molecules to produce a product, an experimental approach that may wash out heterogeneities that exist between individual enzymes. Evaluating these properties on an enzyme-by-enzyme basis represents an unambiguous means of elucidating heterogeneities; however, the quantification of enzymatic activity at the single-enzyme level is fundamentally limited by the maximum catalytic rate, k_cat, inherent to a given enzyme. For electrochemical methods measuring current, single enzymes must turn over greater than 10⁷ molecules per second to produce a measurable signal on the order of 10^-12 A. Enzymes with this capability are extremely rare in nature, with typical k_cat values for biologically relevant enzymes falling between 1 and 10 000 s^-1. Thus, clever amplification strategies are necessary to electrochemically detect the vast majority of enzymes. This review details the progress toward the electroanalytical detection and evaluation of single enzyme kinetics largely focused on the nanoimpact method, a chronoamperometric detection strategy that monitors the change in the current-time profile associated with stochastic collisions of freely diffusing entities (e.g., enzymes) onto a microelectrode or nanoelectrode surface. We discuss the experimental setups and methods developed in the last decade toward the quantification of single molecule enzymatic rates. Special emphasis is given to the limitations of measurement science in the observation of single enzyme activity and feasible methods of signal amplification with reasonable bandwidth.

Explicit Treatment of Non-Michaelis-Menten and Atypical Kinetics in Early Drug Discovery*

Biological systems are highly regulated. They are also highly resistant to sudden perturbations enabling them to maintain the dynamic equilibrium essential to sustain life. This robustness is conferred by regulatory mechanisms that influence the activity of enzymes/proteins within their cellular context to adapt to changing environmental conditions. However, the initial rules governing the study of enzyme kinetics were mostly tested and implemented for cytosolic enzyme systems that were easy to isolate and/or recombinantly express. Moreover, these enzymes lacked complex regulatory modalities. Now, with academic labs and pharmaceutical companies turning their attention to more-complex systems (for instance, multiprotein complexes, oligomeric assemblies, membrane proteins and post-translationally modified proteins), the initial axioms defined by Michaelis-Menten (MM) kinetics are rendered inadequate, and the development of a new kind of kinetic analysis to study these systems is required. This review strives to present an overview of enzyme kinetic mechanisms that are atypical and, oftentimes, do not conform to the classical MM kinetics. Further, it presents initial ideas on the design and analysis of experiments in early drug-discovery for such systems, to enable effective screening and characterisation of small-molecule inhibitors with desirable physiological outcomes.

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.

A Hybrid VOx Incorporated Hexacyanoferrate Nanostructured Hydrogel as a Multienzyme Mimetic via Cascade Reactions

Inspired by the cascade reactions occurring in micro-organelles of living systems, we have developed a hybrid hydrogel, a nanozyme that mimics three key enzymes including peroxidase, superoxide dismutase, and catalase. The organic/inorganic nanostructured hydrogel constituting VO_x incorporated hexacyanoferrate Berlin green analogue complex (VO_xBG) is prepared by a simple one-step hydrothermal process, and its composition, structure, and properties are thoroughly investigated. Polyvinylpyrrolidone, a low-cost and biocompatible polymer, was utilized as a scaffold to increase the surface area and dispersion of the highly active catalytic centers of the nanozyme. Compared to the widely used horseradish peroxidase in enzyme-linked immunosorbent assay, our VO_xBG analogue hydrogel displays an excellent affinity toward the chromogenic substrate that is used in these peroxidase-based assays. This higher affinity makes it a competent nanozyme for detection and oxidation of biomolecules, including glucose, in a cascade-like system which can be further used for hydrogel photolithography. The VO_xBG analogue hydrogel also holds a good ability for the rapid and efficient oxidative degradation of environmentally unfriendly recalcitrant substrates under light irradiation. Detailed mechanistic studies of this multifaceted material suggest that different complex catalytic processes and routes are involved in these photo-Fenton and Fenton reactions that are responsible for the generation as well as consumption of reactive oxygen species, which are effectively activated by a multienzyme mimetic of the VO_xBG analogue hydrogel.

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.

Effect of Substrate Number Fluctuations in Stochastic Enzyme Kinetics

Conventional studies on enzyme kinetics assume that the substrate concentration remains constant. However, for catalytic reactions taking place in intracellular compartments, the substrate molecules are fed in and out of the compartment and are catalyzed into product molecules within the compartment. In this work, we use a chemical master equation approach to study the stochastic kinetics of a single enzyme for different reaction pathways with one or more intermediate states. We obtain velocity expressions that deviate from the Michaelis-Menten expression. We study the coefficient of variation, which is a measure of the noise strength as a function of the mean substrate concentration for systems where there is influx or/and outflux of substrate molecules. Our results show that the noise strength decreases with the increase in the substrate concentration and finally remains the same when the substrate is present in abundance.

Relatively slow stochastic gene-state switching in the presence of positive feedback significantly broadens the region of bimodality through stabilizing the uninduced phenotypic state

Within an isogenic population, even in the same extracellular environment, individual cells can exhibit various phenotypic states. The exact role of stochastic gene-state switching regulating the transition among these phenotypic states in a single cell is not fully understood, especially in the presence of positive feedback. Recent high-precision single-cell measurements showed that, at least in bacteria, switching in gene states is slow relative to the typical rates of active transcription and translation. Hence using the lac operon as an archetype, in such a region of operon-state switching, we present a fluctuating-rate model for this classical gene regulation module, incorporating the more realistic operon-state switching mechanism that was recently elucidated. We found that the positive feedback mechanism induces bistability (referred to as deterministic bistability), and that the parameter range for its occurrence is significantly broadened by stochastic operon-state switching. We further show that in the absence of positive feedback, operon-state switching must be extremely slow to trigger bistability by itself. However, in the presence of positive feedback, which stabilizes the induced state, the relatively slow operon-state switching kinetics within the physiological region are sufficient to stabilize the uninduced state, together generating a broadened parameter region of bistability (referred to as stochastic bistability). We illustrate the opposite phenotype-transition rate dependence upon the operon-state switching rates in the two types of bistability, with the aid of a recently proposed rate formula for fluctuating-rate models. The rate formula also predicts a maximal transition rate in the intermediate region of operon-state switching, which is validated by numerical simulations in our model. Overall, our findings suggest a biological function of transcriptional “variations” among genetically identical cells, for the emergence of bistability and transition between phenotypic states.

Identifying the mechanism underlying the coexistence of multiple stable phenotypic states has been a challenging scientific problem for more than half a century, and an appropriate mathematical model at the single-cell level is also in high demand. Single-cell measurements conducted in the past ten years have shown that gene-state switching is slow relative to the typical rates of active transcription and translation; hence the recently proposed fluctuating-rate model is a good candidate for describing the single-cell dynamics. We use the classic gene regulation module of the lac operon as an archetype and build a specific fluctuating-rate model based on the recently identified operon-state switching mechanism. This model is analyzed to dissect the interplay between positive feedback and the stochastic switching of gene states in the emergence of bistability/multistablity and the transition between phenotypic states. We show that relatively slow operon-state switching stabilizes the uninduced state and that the positive feedback stabilizes the induced state. Thus, the parameter range for bistability is significantly broadened. In addition, recently proposed landscape theory and rate formula predict opposite phenotype-transition rate dependence on operon-state switching rates for the two types of bistability.

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.

Slow domain reconfiguration causes power-law kinetics in a two-state enzyme

Protein dynamics are typically captured well by rate equations that predict exponential decays for two-state reactions. Here, we describe a remarkable exception. The electron-transfer enzyme quiescin sulfhydryl oxidase (QSOX), a natural fusion of two functionally distinct domains, switches between open- and closed-domain arrangements with apparent power-law kinetics. Using single-molecule FRET experiments on time scales from nanoseconds to milliseconds, we show that the unusual open-close kinetics results from slow sampling of an ensemble of disordered domain orientations. While substrate accelerates the kinetics, thus suggesting a substrate-induced switch to an alternative free energy landscape of the enzyme, the power-law behavior is also preserved upon electron load. Our results show that the slow sampling of open conformers is caused by a variety of interdomain interactions that imply a rugged free energy landscape, thus providing a generic mechanism for dynamic disorder in multidomain enzymes.

Resolution of Submillisecond Kinetics of Multiple Reaction Pathways for Lactate Dehydrogenase

Enzymes are known to exhibit conformational flexibility. An important consequence of this flexibility is that the same enzyme reaction can occur via multiple reaction pathways on a reaction landscape. A model enzyme for the study of reaction landscapes is lactate dehydrogenase. We have previously used temperature-jump (T-jump) methods to demonstrate that the reaction landscape of lactate dehydrogenase branches at multiple points creating pathways with varied reactivity. A limitation of this previous work is that the T-jump method makes only small perturbations to equilibrium and may not report conclusively on all steps in a reaction. Therefore, interpreting T-jump results of lactate dehydrogenase kinetics has required extensive computational modeling work. Rapid mixing methods offer a complementary approach that can access large perturbations from equilibrium; however, traditional enzyme mixing methods like stopped-flow do not allow for the observation of fast protein dynamics. In this report, we apply a microfluidic rapid mixing device with a mixing time of <100 μs that allows us to study these fast dynamics and the catalytic redox step of the enzyme reaction. Additionally, we report UV absorbance and emission T-jump results with improved signal-to-noise ratio at fast times. The combination of mixing and T-jump results yields an unprecedented view of lactate dehydrogenase enzymology, confirming the timescale of substrate-induced conformational change and presence of multiple reaction pathways.

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.

On the generality of Michaelian kinetics

The reversible Michaelis-Menten equation is shown to follow from a very broad class of steady-state kinetic models involving enzymes that adopt a unique free (i.e., not complexed to substrate/product) state in solution. In the case of enzymes with multiple free states/conformations (e.g., fluctuating, hysteretic, or co-operative monomeric enzymes), Michaelian behavior is still assured if the relative steady-state populations of free enzyme states are independent of substrate and product concentration. Prior models for Michaelian behavior in multiple conformer enzymes are shown to be special cases of this single condition.

Dependence of the Enzymatic Velocity on the Substrate Dissociation Rate

Enzymes are biological catalysts that play a fundamental role in all living systems by supporting the selectivity and the speed for almost all cellular processes. While the general principles of enzyme functioning are known, the specific details of how they work at the microscopic level are not always available. Simple Michaelis-Menten kinetics assumes that the enzyme-substrate complex has only one conformation that decays as a single exponential. As a consequence, the enzymatic velocity decreases as the dissociation (off) rate constant of the complex increases. Recently, Reuveni et al. [ Proc. Natl. Acad. Sci. USA 2014 , 111 , 4391 - 4396 ] showed that it is possible for the enzymatic velocity to increase when the off rate becomes higher, if the enzyme-substrate complex has many conformations which dissociate with the same off rate constant. This was done using formal mathematical arguments, without specifying the nature of the dynamics of the enzyme-substrate complex. In order to provide a physical basis for this unexpected result, we derive an analytical expression for the enzymatic velocity assuming that the enzyme-substrate complex has multiple states and its conformational dynamics is described by rate equations with arbitrary rate constants. By applying our formalism to a complex with two conformations, we show that the unexpected off rate dependence of the velocity can be readily understood: If one of the conformations is unproductive, the system can escape from this "trap" by dissociating, thereby giving the enzyme another chance to form the productive enzyme-substrate complex. We also demonstrate that the nonmonotonic off rate dependence of the enzymatic velocity is possible not only when all off rate constants are identical, but even when they are different. We show that for typical experimentally determined rate constants, the nonmonotonic off rate dependence can occur for micromolar substrate concentrations. Finally, we discuss the relation of this work to the problem of optimizing the flux through singly occupied membrane channels and transporters.

Conformational Heterogeneity in the Michaelis Complex of Lactate Dehydrogenase: An Analysis of Vibrational Spectroscopy Using Markov and Hidden Markov Models

Lactate dehydrogenase (LDH) catalyzes the interconversion of pyruvate and lactate. Recent isotope-edited IR spectroscopy suggests that conformational heterogeneity exists within the Michaelis complex of LDH, and this heterogeneity affects the propensity toward the on-enzyme chemical step for each Michaelis substate. By combining molecular dynamics simulations with Markov and hidden Markov models, we obtained a detailed kinetic network of the substates of the Michaelis complex of LDH. The ensemble-average electric fields exerted onto the vibrational probe were calculated to provide a direct comparison with the vibrational spectroscopy. Structural features of the Michaelis substates were also analyzed on atomistic scales. Our work not only clearly demonstrates the conformational heterogeneity in the Michaelis complex of LDH and its coupling to the reactivities of the substates, but it also suggests a methodology to simultaneously resolve kinetics and structures on atomistic scales, which can be directly compared with the vibrational spectroscopy.

Protein Ensembles: How Does Nature Harness Thermodynamic Fluctuations for Life? The Diverse Functional Roles of Conformational Ensembles in the Cell

All soluble proteins populate conformational ensembles that together constitute the native state. Their fluctuations in water are intrinsic thermodynamic phenomena, and the distributions of the states on the energy landscape are determined by statistical thermodynamics; however, they are optimized to perform their biological functions. In this review we briefly describe advances in free energy landscape studies of protein conformational ensembles. Experimental (nuclear magnetic resonance, small-angle X-ray scattering, single-molecule spectroscopy, and cryo-electron microscopy) and computational (replica-exchange molecular dynamics, metadynamics, and Markov state models) approaches have made great progress in recent years. These address the challenging characterization of the highly flexible and heterogeneous protein ensembles. We focus on structural aspects of protein conformational distributions, from collective motions of single- and multi-domain proteins, intrinsically disordered proteins, to multiprotein complexes. Importantly, we highlight recent studies that illustrate functional adjustment of protein conformational ensembles in the crowded cellular environment. We center on the role of the ensemble in recognition of small- and macro-molecules (protein and RNA/DNA) and emphasize emerging concepts of protein dynamics in enzyme catalysis. Overall, protein ensembles link fundamental physicochemical principles and protein behavior and the cellular network and its regulation.

Variance-corrected Michaelis-Menten equation predicts transient rates of single-enzyme reactions and response times in bacterial gene-regulation

Many chemical reactions in biological cells occur at very low concentrations of constituent molecules. Thus, transcriptional gene-regulation is often controlled by poorly expressed transcription-factors, such as E.coli lac repressor with few tens of copies. Here we study the effects of inherent concentration fluctuations of substrate-molecules on the seminal Michaelis-Menten scheme of biochemical reactions. We present a universal correction to the Michaelis-Menten equation for the reaction-rates. The relevance and validity of this correction for enzymatic reactions and intracellular gene-regulation is demonstrated. Our analytical theory and simulation results confirm that the proposed variance-corrected Michaelis-Menten equation predicts the rate of reactions with remarkable accuracy even in the presence of large non-equilibrium concentration fluctuations. The major advantage of our approach is that it involves only the mean and variance of the substrate-molecule concentration. Our theory is therefore accessible to experiments and not specific to the exact source of the concentration fluctuations.

Roles of factorial noise in inducing bimodal gene expression

Some gene regulatory systems can exhibit bimodal distributions of mRNA or protein although the deterministic counterparts are monostable. This noise-induced bimodality is an interesting phenomenon and has important biological implications, but it is unclear how different sources of expression noise (each source creates so-called factorial noise that is defined as a component of the total noise) contribute separately to this stochastic bimodality. Here we consider a minimal model of gene regulation, which is monostable in the deterministic case. Although simple, this system contains factorial noise of two main kinds: promoter noise due to switching between gene states and transcriptional (or translational) noise due to synthesis and degradation of mRNA (or protein). To better trace the roles of factorial noise in inducing bimodality, we also analyze two limit models, continuous and adiabatic approximations, apart from the exact model. We show that in the case of slow gene switching, the continuous model where only promoter noise is considered can exhibit bimodality; in the case of fast switching, the adiabatic model where only transcriptional or translational noise is considered can also exhibit bimodality but the exact model cannot; and in other cases, both promoter noise and transcriptional or translational noise can cooperatively induce bimodality. Since slow gene switching and large protein copy numbers are characteristics of eukaryotic cells, whereas fast gene switching and small protein copy numbers are characteristics of prokaryotic cells, we infer that eukaryotic stochastic bimodality is induced mainly by promoter noise, whereas prokaryotic stochastic bimodality is induced primarily by transcriptional or translational noise.

Stochastic phenotype transition of a single cell in an intermediate region of gene state switching

Multiple phenotypic states often arise in a single cell with different gene-expression states that undergo transcription regulation with positive feedback. Recent experiments show that, at least in E.coli, the gene state switching can be neither extremely slow nor exceedingly rapid as many previous theoretical treatments assumed. Rather, it is in the intermediate region which is difficult to handle mathematically. Under this condition, from a full chemical-master-equation description we derive a model in which the protein copy number, for a given gene state, follows a deterministic mean-field description while the protein-synthesis rates fluctuate due to stochastic gene state switching. The simplified kinetics yields a nonequilibrium landscape function, which, similar to the energy function for equilibrium fluctuation, provides the leading orders of fluctuations around each phenotypic state, as well as the transition rates between the two phenotypic states. This rate formula is analogous to Kramers' theory for chemical reactions. The resulting behaviors are significantly different from the two limiting cases studied previously.

The dynamical nature of enzymatic catalysis

As is well-known, enzymes are proteins designed to accelerate specific life essential chemical reactions by many orders of magnitude. A folded protein is a highly dynamical entity, best described as a hierarchy or ensemble of interconverting conformations on all time scales from femtoseconds to minutes. We are just beginning to learn what role these dynamics play in the mechanism of chemical catalysis by enzymes due to extraordinary difficulties in characterizing the conformational space, that is, the energy landscape, of a folded protein. It seems clear now that their role is crucially important. Here we discuss approaches, based on vibrational spectroscopies of various sorts, that can reveal the energy landscape of an enzyme–substrate (Michaelis) complex and decipher which part of the typically very complicated landscape is relevant to catalysis. Vibrational spectroscopy is quite sensitive to small changes in bond order and bond length, with a resolution of 0.01 Å or less. It is this sensitivity that is crucial to its ability to discern bond reactivity.

Using isotope edited IR approaches, we have studied in detail the role of conformational heterogeneity and dynamics in the catalysis of hydride transfer by LDH (lactate dehydrogenase). Upon the binding of substrate, the LDH·substrate system undergoes a search through conformational space to find a range of reactive conformations over the microsecond to millisecond time scale. The ligand is shuttled to the active site via first forming a weakly bound enzyme·ligand complex, probably consisting of several heterogeneous structures. This complex undergoes numerous conformational changes spread throughout the protein that shuttle the enzyme·substrate complex to a range of conformations where the substrate is tightly bound. This ensemble of conformations all have a propensity toward chemistry, but some are much more facile for carrying out chemistry than others. The search for these tightly bound states is clearly directed by the forces that the protein can bring to bear, very much akin to the folding process to form native protein in the first place. In fact, the conformational subspace of reactive conformations of the Michaelis complex can be described as a “collapse” of reactive substates compared with that found in solution, toward a much smaller and much more reactive set.

These studies reveal how dynamic disorder in the protein structure can modulate the on-enzyme reactivity. It is very difficult to account for how the dynamical nature of the ground state of the Michaelis complex modulates function by transition state concepts since dynamical disorder is not a starting feature of the theory. We find that dynamical disorder may well play a larger or similar sized role in the measured Gibbs free energy of a reaction compared with the actual energy barrier involved in the chemical event. Our findings are broadly compatible with qualitative concepts of evolutionary adaptation of function such as adaptation to varying thermal environments. Our work suggests a methodology to determine the important dynamics of the Michaelis complex.

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.

Enzymatic turnover of macromolecules generates long-lasting protein-water-coupled motions beyond reaction steady state

The main focus of enzymology is on the enzyme rates, substrate structures, and reactivity, whereas the role of solvent dynamics in mediating the biological reaction is often left aside owing to its complex molecular behavior. We used integrated X-ray- and terahertz- based time-resolved spectroscopic tools to study protein-water dynamics during proteolysis of collagen-like substrates by a matrix metalloproteinase. We show equilibration of structural kinetic transitions in the millisecond timescale during degradation of the two model substrates collagen and gelatin, which have different supersecondary structure and flexibility. Unexpectedly, the detected changes in collective enzyme-substrate-water-coupled motions persisted well beyond steady state for both substrates while displaying substrate-specific behaviors. Molecular dynamics simulations further showed that a hydration funnel (i.e., a gradient in retardation of hydrogen bond (HB) dynamics toward the active site) is substrate-dependent, exhibiting a steeper gradient for the more complex enzyme-collagen system. The long-lasting changes in protein-water dynamics reflect a collection of local energetic equilibrium states specifically formed during substrate conversion. Thus, the observed long-lasting water dynamics contribute to the net enzyme reactivity, impacting substrate binding, positional catalysis, and product release.

Enzyme molecules in solitary confinement

Large arrays of homogeneous microwells each defining a femtoliter volume are a versatile platform for monitoring the substrate turnover of many individual enzyme molecules in parallel. The high degree of parallelization enables the analysis of a statistically representative enzyme population. Enclosing individual enzyme molecules in microwells does not require any surface immobilization step and enables the kinetic investigation of enzymes free in solution. This review describes various microwell array formats and explores their applications for the detection and investigation of single enzyme molecules. The development of new fabrication techniques and sensitive detection methods drives the field of single molecule enzymology. Here, we introduce recent progress in single enzyme molecule analysis in microwell arrays and discuss the challenges and opportunities.

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.

Potential-dependent single molecule blinking dynamics for flavin adenine dinucleotide covalently immobilized in zero-mode waveguide array of working electrodes

Single molecules exhibit a set of behaviors that are characteristic and distinct from larger ensembles. Blinking is one such behavior that involves episodic transitions between luminescent and dark states. In addition to the common blinking mechanisms, flavin adenine dinucleotide (FAD), a cofactor in many common redox enzymes, exhibits blinking by cycling between a highly fluorescent oxidized state and a dark reduced state. In contrast to its behavior in flavoenzymes, where the transitions are coupled to chemical redox events, here we study single FAD molecules that are chemically immobilized to the Au region of a zero-mode waveguide (ZMW) array through a pyrroloquinoline quinone (PQQ) linker. In this structure, the Au functions both to confine the optical field in the ZMW and as the working electrode in a potentiostatically controlled 3-elecrode system, thus allowing potential-dependent blinking to be studied in single FAD molecules. The subset of ZMW nanopores housing a single molecule were identified statistically, and these were subjected to detailed study. Using equilibrium potential, E(eq), values determined from macroscopic planar Au electrodes, single molecule blinking behavior was characterized at potentials E < E(eq), E - E(eq), and E > E(eq). The probability of observing a reduced (oxidized) state is observed to increase (decrease) as the potential is scanned cathodic of E(eq). This is understood to reflect the potential-dependent probability of electron transfer for single FAD molecules. Furthermore, the observed transition rate reaches a maximum near E(eq) and decreases to either anodic or cathodic values, as expected, since the rate is dependent on having significant probabilities for both redox states, a condition that is obtained only near E(eq).

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.

Nonequilibrium dynamics of helix reorganization observed by transient 2D IR spectroscopy

The relaxation of helical structures very close to equilibrium is observed via transient 2D IR spectroscopy. An initial distribution of synthetically distorted helices having an unnatural bridge linking the 10th and 12th residues of an alanine-rich α-helix is released to evolve into the equilibrium distribution of α-helix conformations. The bridge constrains the structure to be slightly displaced from the full α-helix equilibrium near these residues, yet the peptide is not unfolded completely. The release is accomplished by a subpicosecond pulse of UV irradiation. The resulting 2D IR signals are used to obtain snapshots of the ∼100-ps helical conformational reorganization of the distorted dihedral angle and distance between amide units at chemical bond length-scale resolution. The decay rates of the angle between the dipoles, dihedral angles, and distance autocorrelations obtained from molecular dynamics simulations support the experiments, providing evidence that the final helix collapse conforms to linear response theory.

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.

Single-enzyme analysis in a droplet-based micro- and nanofluidic system

The kinetic activity of individual enzyme molecules was determined in aqueous droplets generated in a nano- and microfluidic device. To avoid high background noise, the enzyme and substrate solution was confined into femtoliter carriers, achieving high product concentrations from single-molecule encapsulation. The tiny droplets (φ ~ 2.5-3 μm) generated from this fluidic system were highly monodisperse, beneficial for an analysis of single enzyme activity. The method presented here allows to follow large numbers of individual droplets over time. The instrumental requirements are furthermore modest, since the small droplet size allows to use of standard microscope and standard Pyrex glass chips as well as the use of relatively high enzyme concentrations (nM range) for single molecule encapsulation.

Bistability in the chemical master equation for dual phosphorylation cycles

Dual phospho/dephosphorylation cycles, as well as covalent enzymatic-catalyzed modifications of substrates are widely diffused within cellular systems and are crucial for the control of complex responses such as learning, memory, and cellular fate determination. Despite the large body of deterministic studies and the increasing work aimed at elucidating the effect of noise in such systems, some aspects remain unclear. Here we study the stationary distribution provided by the two-dimensional chemical master equation for a well-known model of a two step phospho/dephosphorylation cycle using the quasi-steady state approximation of enzymatic kinetics. Our aim is to analyze the role of fluctuations and the molecules distribution properties in the transition to a bistable regime. When detailed balance conditions are satisfied it is possible to compute equilibrium distributions in a closed and explicit form. When detailed balance is not satisfied, the stationary non-equilibrium state is strongly influenced by the chemical fluxes. In the last case, we show how the external field derived from the generation and recombination transition rates, can be decomposed by the Helmholtz theorem, into a conservative and a rotational (irreversible) part. Moreover, this decomposition allows to compute the stationary distribution via a perturbative approach. For a finite number of molecules there exists diffusion dynamics in a macroscopic region of the state space where a relevant transition rate between the two critical points is observed. Further, the stationary distribution function can be approximated by the solution of a Fokker-Planck equation. We illustrate the theoretical results using several numerical simulations.

Correlation Analysis of Enzymatic Reaction of a Single Protein Molecule

New advances in nano sciences open the door for scientists to study biological processes on a microscopic molecule-by-molecule basis. Recent single-molecule biophysical experiments on enzyme systems, in particular, reveal that enzyme molecules behave fundamentally differently from what classical model predicts. A stochastic network model was previously proposed to explain the experimental discovery. This paper conducts detailed theoretical and data analyses of the stochastic network model, focusing on the correlation structure of the successive reaction times of a single enzyme molecule. We investigate the correlation of experimental fluorescence intensity and the correlation of enzymatic reaction times, and examine the role of substrate concentration in enzymatic reactions. Our study shows that the stochastic network model is capable of explaining the experimental data in depth.

Physics of bacterial morphogenesis

Bacterial cells utilize three-dimensional (3D) protein assemblies to perform important cellular functions such as growth, division, chemoreception, and motility. These assemblies are composed of mechanoproteins that can mechanically deform and exert force. Sometimes, small-nucleotide hydrolysis is coupled to mechanical deformations. In this review, we describe the general principle for an understanding of the coupling of mechanics with chemistry in mechanochemical systems. We apply this principle to understand bacterial cell shape and morphogenesis and how mechanical forces can influence peptidoglycan cell wall growth. We review a model that can potentially reconcile the growth dynamics of the cell wall with the role of cytoskeletal proteins such as MreB and crescentin. We also review the application of mechanochemical principles to understand the assembly and constriction of the FtsZ ring. A number of potential mechanisms are proposed, and important questions are discussed.

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.

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.

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.

Michaelis-Menten equation and detailed balance in enzymatic networks

Many enzymatic reactions in biochemistry are far more complex than the celebrated Michaelis-Menten scheme, but the observed turnover rate often obeys the hyperbolic dependence on the substrate concentration, a relation established almost a century ago for the simple Michaelis-Menten mechanism. To resolve the longstanding puzzle, we apply the flux balance method to predict the functional form of the substrate dependence in the mean turnover time of complex enzymatic reactions and identify detailed balance (i.e., the lack of unbalanced conformational current) as a sufficient condition for the Michaelis-Menten equation to describe the substrate concentration dependence of the turnover rate in an enzymatic network. This prediction can be verified in single-molecule event-averaged measurements using the recently proposed signatures of detailed balance violations. The finding helps analyze recent single-molecule studies of enzymatic networks and can be applied to other external variables, such as force-dependence and voltage-dependence.