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

Distribution of dwell times of a ribosome: effects of infidelity, kinetic proofreading and ribosome crowding

Ribosome is a molecular machine that polymerizes a protein where the sequence of the amino acid residues, the monomers of the protein, is dictated by the sequence of codons (triplets of nucleotides) on a messenger RNA (mRNA) that serves as the template. The ribosome is a molecular motor that utilizes the template mRNA strand also as the track. Thus, in each step the ribosome moves forward by one codon and, simultaneously, elongates the protein by one amino acid. We present a theoretical model that captures most of the main steps in the mechanochemical cycle of a ribosome. The stochastic movement of the ribosome consists of an alternating sequence of pause and translocation; the sum of the durations of a pause and the following translocation is the time of dwell of the ribosome at the corresponding codon. We derive the analytical expression for the distribution of the dwell times of a ribosome in our model. Wherever experimental data are available, our theoretical predictions are consistent with those results. We suggest appropriate experiments to test the new predictions of our model, particularly the effects of the quality control mechanism of the ribosome and that of their crowding on the mRNA track.

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

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

Protein dynamics and enzyme catalysis: insights from simulations

The role of protein dynamics in enzyme catalysis is one of the most active and controversial areas in enzymology today. Some researchers claim that protein dynamics are at the heart of enzyme catalytic efficiency, while others state that dynamics make no significant contribution to catalysis. What is the biochemist - or student - to make of the ferocious arguments in this area? Protein dynamics are complex and fascinating, as molecular dynamics simulations and experiments have shown. The essential question is: do these complex motions have functional significance? In particular, how do they affect or relate to chemical reactions within enzymes, and how are chemical and conformational changes coupled together? Biomolecular simulations can analyse enzyme reactions and dynamics in atomic detail, beyond that achievable in experiments: accurate atomistic modelling has an essential part to play in clarifying these issues. This article is part of a Special Issue entitled: Protein Dynamics: Experimental and Computational Approaches.

Monte Carlo simulations of single- and multistep enzyme-catalyzed reaction sequences: effects of diffusion, cell size, enzyme fluctuations, colocalization, and segregation

Following the discovery of slow fluctuations in the catalytic activity of an enzyme in single-molecule experiments, it has been shown that the classical Michaelis-Menten (MM) equation relating the average enzymatic velocity and the substrate concentration may hold even for slowly fluctuating enzymes. In many cases, the average velocity is that given by the MM equation with time-averaged values of the fluctuating rate constants and the effect of enzyme fluctuations is simply averaged out. The situation is quite different for a sequence of reactions. For colocalization of a pair of enzymes in a sequence to be effective in promoting reaction, the second must be active when the first is active or soon after. If the enzymes are slowly varying and only rarely active, the product of the first reaction may diffuse away before the second enzyme is active, and colocalization may have little value. Even for single-step reactions the interplay of reaction and diffusion with enzyme fluctuations leads to added complexities, but for multistep reactions the interplay of reaction and diffusion, cell size, compartmentalization, enzyme fluctuations, colocalization, and segregation is far more complex than for single-step reactions. In this paper, we report the use of stochastic simulations at the level of whole cells to explore, understand, and predict the behavior of single- and multistep enzyme-catalyzed reaction systems exhibiting some of these complexities. Results for single-step reactions confirm several earlier observations by others. The MM relationship, with altered constants, is found to hold for single-step reactions slowed by diffusion. For single-step reactions, the distribution of enzymes in a regular grid is slightly more effective than a random distribution. Fluctuations of enzyme activity, with average activity fixed, have no observed effects for simple single-step reactions slowed by diffusion. Two-step sequential reactions are seen to be slowed by segregation of the enzymes for each step, and results of the calculations suggest limits for cell size. Colocalization of enzymes for a two-step sequence is seen to promote reaction, and rates fall rapidly with increasing distance between enzymes. Low frequency fluctuations of the activities of colocalized enzymes, with average activities fixed, can greatly reduce reaction rates for sequential reactions.

Enzyme dynamics point to stepwise conformational selection in catalysis

Recent data increasingly reveal that conformational dynamics are indispensable to enzyme function throughout the catalytic cycle, in substrate recruiting, chemical transformation, and product release. Conformational transitions may involve conformational selection and induced fit, which can be viewed as a special case in the catalytic network. NMR, X-ray crystallography, single-molecule FRET, and simulations clearly demonstrate that the free enzyme dynamics already encompass all the conformations necessary for substrate binding, preorganization, transition-state stabilization, and product release. Conformational selection and substate population shift at each step of the catalytic turnover can accommodate enzyme specificity and efficiency. Within such a framework, entropy can have a larger role in conformational dynamics than in direct energy transfer in dynamically promoted catalysis.

Dynamic disorder in quasi-equilibrium enzymatic systems

Conformations and catalytic rates of enzymes fluctuate over a wide range of timescales. Despite these fluctuations, there exist some limiting cases in which the enzymatic catalytic rate follows the macroscopic rate equation such as the Michaelis-Menten law. In this paper we investigate the applicability of macroscopic rate laws for fluctuating enzyme systems in which catalytic transitions are slower than ligand binding-dissociation reactions. In this quasi-equilibrium limit, for an arbitrary reaction scheme we show that the catalytic rate has the same dependence on ligand concentrations as obtained from mass-action kinetics even in the presence of slow conformational fluctuations. These results indicate that the timescale of conformational dynamics--no matter how slow--will not affect the enzymatic rate in quasi-equilibrium limit. Our numerical results for two enzyme-catalyzed reaction schemes involving multiple substrates and inhibitors further support our general theory.

Rate theories for biologists

Some of the rate theories that are most useful for modeling biological processes are reviewed. By delving into some of the details and subtleties in the development of the theories, the review will hopefully help the reader gain a more than superficial perspective. Examples are presented to illustrate how rate theories can be used to generate insight at the microscopic level into biomolecular behaviors. An attempt is made to clear up a number of misconceptions in the literature regarding popular rate theories, including the appearance of Planck's constant in the transition-state theory and the Smoluchowski result as an upper limit for protein-protein and protein-DNA association rate constants. Future work in combining the implementation of rate theories through computer simulations with experimental probes of rate processes, and in modeling effects of intracellular environments so that theories can be used for generating rate constants for systems biology studies is particularly exciting.

Macroscopic kinetic effect of cell-to-cell variation in biochemical reactions

Genetically identical cells under the same environmental conditions can show strong variations in protein copy numbers due to inherently stochastic events in individual cells. We here develop a theoretical framework to address how variations in enzyme abundance affect the collective kinetics of metabolic reactions observed within a population of cells. Kinetic parameters measured at the cell population level are shown to be systematically deviated from those of single cells, even within populations of homogeneous parameters. Because of these considerations, Michaelis-Menten kinetics can even be inappropriate to apply at the population level. Our findings elucidate a novel origin of discrepancy between in vivo and in vitro kinetics, and offer potential utility for analysis of single-cell metabolomic data.

Real-time DNA sequencing from single polymerase molecules

Pacific Biosciences has developed a method for real-time sequencing of single DNA molecules (Eid et al., 2009), with intrinsic sequencing rates of several bases per second and read lengths into the kilobase range. Conceptually, this sequencing approach is based on eavesdropping on the activity of DNA polymerase carrying out template-directed DNA polymerization. Performed in a highly parallel operational mode, sequential base additions catalyzed by each polymerase are detected with terminal phosphate-linked, fluorescence-labeled nucleotides. This chapter will first outline the principle of this single-molecule, real-time (SMRT) DNA sequencing method, followed by descriptions of its underlying components and typical sequencing run conditions. Two examples are provided which illustrate that, in addition to the DNA sequence, the dynamics of DNA polymerization from each enzyme molecules is directly accessible: the determination of base-specific kinetic parameters from single-molecule sequencing reads, and the characterization of DNA synthesis rate heterogeneities.

Dynamic disorder-driven substrate inhibition and bistability in a simple enzymatic reaction

Conformations and catalytic rates of enzymes (biological catalysts) fluctuate over a wide range of time scales. Recent experimental and theoretical investigations demonstrated case studies where the enzymatic catalysis rate follows the Michaelis-Menten (MM) rate law despite molecular fluctuations. In this paper, we investigate deviations from MM law and their effects on the dynamical behavior of the enzymatic network. We consider a simple kinetic scheme for a single substrate enzymatic reaction in which the product release step is treated explicitly. We examine how conformational fluctuations affect the underlying rate law in the quasi-static limit when conformational dynamics is very slow in one of the states. Our numerical results and analytically solvable model indicate that slow conformational fluctuations of the enzyme-substrate complex lead to non-MM behavior, substrate inhibition, and possible bistability of the reaction network.

Stochastic kinetics of ribosomes: single motor properties and collective behavior

Syntheses of protein molecules in a cell are carried out by ribosomes. A ribosome can be regarded as a molecular motor which utilizes the input chemical energy to move on a messenger RNA (mRNA) track that also serves as a template for the polymerization of the corresponding protein. The forward movement, however, is characterized by an alternating sequence of translocation and pause. Using a quantitative model, which captures the mechanochemical cycle of an individual ribosome, we derive an exact analytical expression for the distribution of its dwell times at the successive positions on the mRNA track. Inverse of the average dwell time satisfies a "Michaelis-Menten-type" equation and is consistent with the general formula for the average velocity of a molecular motor with an unbranched mechanochemical cycle. Extending this formula appropriately, we also derive the exact force-velocity relation for a ribosome. Often many ribosomes simultaneously move on the same mRNA track, while each synthesizes a copy of the same protein. We extend the model of a single ribosome by incorporating steric exclusion of different individuals on the same track. We draw the phase diagram of this model of ribosome traffic in three-dimensional spaces spanned by experimentally controllable parameters. We suggest new experimental tests of our theoretical predictions.

Fine tuning gene expression through short DNA-protein binding cycles

Certain transcription factors have recently been shown to interact with DNA in living cells, through very short binding cycles, contrasting with the data previously obtained in vitro, and with the view of a stepwise building of transcription initiation complexes. These short cycles are triggered by active dissociation mechanisms, suggesting that they ensure important biological functions. Various interpretations of these observations have been proposed, including a mechanism allowing the cell to switch off gene expression after removal of the inducer, or increasing the availability of free transcription factors. The interpretation examined here is that the brevity of the transcription factor turnovers favors the determinism of gene expression. For the genes with open chromatin and subject to this mode of interaction, the differential dynamics between promoter occupancy and the following processes mediating protein accumulation, can be essential for the dosage of gene expression. Biological activities and quantitative conditions allowing to increase the frequency of DNA-protein binding cycles are proposed. The unexpected dynamics of certain DNA-protein interactions can provide a concrete example of the notion of apparent gradation of single-site occupancy, which is a general solution allowing to extend the mass action determinism to low copy number molecules.

Stochastic bifurcation, slow fluctuations, and bistability as an origin of biochemical complexity

We present a simple, unifying theory for stochastic biochemical systems with multiple time-scale dynamics that exhibit noise-induced bistability in an open-chemical environment, while the corresponding macroscopic reaction is unistable. Nonlinear stochastic biochemical systems like these are fundamentally different from classical systems in equilibrium or near-equilibrium steady state whose fluctuations are unimodal following Einstein-Onsager-Lax-Keizer theory. We show that noise-induced bistability in general arises from slow fluctuations, and a pitchfork bifurcation occurs as the rate of fluctuations decreases. Since an equilibrium distribution, due to detailed balance, has to be independent of changes in time-scale, the bifurcation is necessarily a driven phenomenon. As examples, we analyze three biochemical networks of currently interest: self-regulating gene, stochastic binary decision, and phosphorylation-dephosphorylation cycle with fluctuating kinase. The implications of bistability to biochemical complexity are discussed.

The effects of anomalous diffusion on power-law blinking statistics of CdSe nanorods

In this study of fluorescence blinking statistics for nanorods, we present a diffusion-controlled reaction model that leads to a more general formula: t(-m) exp[-(Gammat)(n)]. This formula describes a short-time power law with a crossover to a stretched exponential decay at later times. Based on quantum Brownian motion for a coupled central harmonic oscillator coupled to heat bath oscillators, we show that the position distribution follows anomalous diffusion with time-dependent diffusion coefficient and drift coefficient. The first and the second moments of the energy fluctuations are shown to be related to the exponent m and n for the blinking statistics.

Quantifying multiscale noise sources in single-molecule time series

When analyzing single-molecule data, a low-dimensional set of system observables typically serves as the observational data. We calibrate stochastic dynamical models from time series that record such observables. Numerical techniques for quantifying noise from multiple time scales in a single trajectory, including experimental instrument and inherent thermal noise, are demonstrated. The techniques are applied to study time series coming from both simulations and experiments associated with the nonequilibrium mechanical unfolding of titin's I27 domain. The estimated models can be used for several purposes, (1) detect dynamical signatures of "rare events" by analyzing the effective diffusion and force as a function of the monitored observable, (2) quantify the influence that conformational degrees of freedom, which are typically difficult to directly monitor experimentally, have on the dynamics of the monitored observable, (3) quantitatively compare the inherent thermal noise to other noise sources, for example, instrument noise, variation induced by conformational heterogeneity, and so forth, (4) simulate random quantities associated with repeated experiments, and (5) apply pathwise, that is, trajectory-wise, hypothesis tests to assess the goodness-of-fit of the models and even detect conformational transitions in noisy signals. These items are all illustrated with several examples.

Extracting Kinetic and Stationary Distribution Information from Short MD Trajectories via a Collection of Surrogate Diffusion Models

Low-dimensional stochastic models can summarize dynamical information and make long time predictions associated with observables of complex atomistic systems. Maximum likelihood based techniques for estimating low-dimensional surrogate diffusion models from relatively short time series are presented. It is found that a heterogeneous population of slowly evolving conformational degrees of freedom modulates the dynamics. This underlying heterogeneity results in a collection of estimated low-dimensional diffusion models. Numerical techniques for exploiting this finding to approximate skewed histograms associated with the simulation are presented. In addition, statistical tests are also used to assess the validity of the models and determine physically relevant sampling information, e.g. the maximum sampling frequency at which one can discretely sample from an atomistic time series and have a surrogate diffusion model pass goodness-of-fit tests. The information extracted from such analyses can possibly be used to assist umbrella sampling computations as well as help in approximating effective diffusion coefficients. The techniques are demonstrated on simulations of Adenylate Kinase.

Decay times in turnover statistics of single enzymes

The first passage times for enzymatic turnovers in nonequilibrium steady state display a statistical symmetry property related to nonequilibrium fluctuation theorems, which makes it possible to extract the chemical driving force from single molecule trajectories in nonequilibrium steady state. Below, we show that the number of decay constants needed to describe the first passage time distribution of this system is not equal to the number of states in the first passage problem, as one would generally expect. Instead, the structure of the kinetic mechanism makes half of the decay times vanish identically from the turnover time distribution. The terms that cancel out correspond to the eigenvalues of a certain submatrix of the master equation matrix for the first exit time problem. We discuss how these results make modeling and data analysis easier for such systems, and how the turnovers can be measured.

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

An alternative theoretical approach to enzyme kinetics that is particularly applicable to single-molecule enzymology is presented. The theory, originated by Van Slyke and Cullen in 1914, develops enzyme kinetics from a "time perspective" rather than the traditional "rate perspective" and emphasizes the nonequilibrium steady-state nature of enzymatic reactions and the significance of small copy numbers of enzyme molecules in living cells. Sigmoidal cooperative substrate binding to slowly fluctuating, monomeric enzymes is shown to arise from association pathways with very small probability but extremely long passage time, which would be disregarded in the traditional rate perspective: A single enzyme stochastically takes alternative pathways in serial order rather than different pathways in parallel. The theory unifies dynamic cooperativity and Hopfield-Ninio's kinetic proofreading mechanism for specificity amplification.

Focus on function: single molecule RNA enzymology

The ability of RNA to catalyze chemical reactions was first demonstrated 25 years ago with the discovery that group I introns and RNase P function as RNA enzymes (ribozymes). Several additional ribozymes were subsequently identified, most notably the ribosome, followed by intense mechanistic studies. More recently, the introduction of single molecule tools has dissected the kinetic steps of several ribozymes in unprecedented detail and has revealed surprising heterogeneity not evident from ensemble approaches. Still, many fundamental questions of how RNA enzymes work at the molecular level remain unanswered. This review surveys the current status of our understanding of RNA catalysis at the single molecule level and discusses the existing challenges and opportunities in developing suitable assays.

Two-dimensional reaction free energy surfaces of catalytic reaction: effects of protein conformational dynamics on enzyme catalysis

We introduce a two-dimensional (2D) multisurface reaction free energy description of the catalytic cycle that explicitly connects the recently observed multi-time-scale conformational dynamics as well as dispersed enzymatic kinetics to the classical Michaelis-Menten equation. A slow conformational motion on a collective enzyme coordinate Q facilitates the catalytic reaction along the intrinsic reaction coordinate X, providing a dynamic realization of Pauling's well-known idea of transition-state stabilization. The catalytic cycle is modeled as transitions between multiple displaced harmonic wells in the XQ space representing different states of the cycle, which is constructed according to the free energy driving force of the cycle. Subsequent to substrate association with the enzyme, the enzyme-substrate complex under strain exhibits a nonequilibrium relaxation toward a new conformation that lowers the activation energy of the reaction, as first proposed by Haldane. The chemical reaction in X is thus enslaved to the down hill slow motion on the Q surface. One consequence of the present theory is that, in spite of the existence of dispersive kinetics, the Michaelis-Menten expression of the catalysis rate remains valid under certain conditions, as observed in recent single-molecule experiments. This dynamic theory builds the relationship between the protein conformational dynamics and the enzymatic reaction kinetics and offers a unified description of enzyme fluctuation-assisted catalysis.

Optimal signal processing in small stochastic biochemical networks

We quantify the influence of the topology of a transcriptional regulatory network on its ability to process environmental signals. By posing the problem in terms of information theory, we do this without specifying the function performed by the network. Specifically, we study the maximum mutual information between the input (chemical) signal and the output (genetic) response attainable by the network in the context of an analytic model of particle number fluctuations. We perform this analysis for all biochemical circuits, including various feedback loops, that can be built out of 3 chemical species, each under the control of one regulator. We find that a generic network, constrained to low molecule numbers and reasonable response times, can transduce more information than a simple binary switch and, in fact, manages to achieve close to the optimal information transmission fidelity. These high-information solutions are robust to tenfold changes in most of the networks' biochemical parameters; moreover they are easier to achieve in networks containing cycles with an odd number of negative regulators (overall negative feedback) due to their decreased molecular noise (a result which we derive analytically). Finally, we demonstrate that a single circuit can support multiple high-information solutions. These findings suggest a potential resolution of the "cross-talk" phenomenon as well as the previously unexplained observation that transcription factors that undergo proteolysis are more likely to be auto-repressive.

Dynamic disorder in single-molecule Michaelis-Menten kinetics: The reaction-diffusion formalism in the Wilemski-Fixman approximation

Single-molecule equations for the Michaelis-Menten [Biochem. Z. 49, 333 (1913)] mechanism of enzyme action are analyzed within the Wilemski-Fixman [J. Chem. Phys. 58, 4009 (1973); 60, 866 (1974)] approximation after the effects of dynamic disorder--modeled by the anomalous diffusion of a particle in a harmonic well--are incorporated into the catalytic step of the reaction. The solution of the Michaelis-Menten equations is used to calculate the distribution of waiting times between successive catalytic turnovers in the enzyme beta-galactosidase. The calculated distribution is found to agree qualitatively with experimental results on this enzyme obtained at four different substrate concentrations. The calculations are also consistent with measurements of correlations in the fluctuations of the fluorescent light emitted during the course of catalysis, and with measurements of the concentration dependence of the randomness parameter.

Manipulating single enzymes by an external harmonic force

We study a Michaelis-Menten reaction for a single two-state enzyme molecule, whose transition rates between the two conformations are modulated by an harmonically oscillating external force. In particular, we obtain a range of optimal driving frequencies for changing the conformation of the enzyme, thereby controlling the enzymatic activity (i.e., product formation). This analysis demonstrates that it is, in principle, possible to obtain information about particular rates within the kinetic scheme.