Single-molecule Michaelis-Menten equations

This paper summarizes our present theoretical understanding of single-molecule kinetics associated with the Michaelis-Menten mechanism of enzymatic reactions. Single-molecule enzymatic turnover experiments typically measure the probability density f(t) of the stochastic waiting time t for individual turnovers. While f(t) can be reconciled with ensemble kinetics, it contains more information than the ensemble data; in particular, it provides crucial information on dynamic disorder, the apparent fluctuation of the catalytic rates due to the interconversion among the enzyme's conformers with different catalytic rate constants. In the presence of dynamic disorder, f(t) exhibits a highly stretched multiexponential decay at high substrate concentrations and a monoexponential decay at low substrate concentrations. We derive a single-molecule Michaelis-Menten equation for the reciprocal of the first moment of f(t), 1/, which shows a hyperbolic dependence on the substrate concentration [S], similar to the ensemble enzymatic velocity. We prove that this single-molecule Michaelis-Menten equation holds under many conditions, in particular when the intercoversion rates among different enzyme conformers are slower than the catalytic rate. However, unlike the conventional interpretation, the apparent catalytic rate constant and the apparent Michaelis constant in this single-molecule Michaelis-Menten equation are complicated functions of the catalytic rate constants of individual conformers. We also suggest that the randomness parameter r, defined as <(t - )2> / t2, can serve as an indicator for dynamic disorder in the catalytic step of the enzymatic reaction, as it becomes larger than unity at high substrate concentrations in the presence of dynamic disorder.     

Understanding dynamic disorder fluctuations in single-molecule enzymatic reactions

Single-molecule studies of enzymatic reactions reveal fluctuations in the reaction rate, which cannot be explained by classic Markovian dynamics. This dynamic disorder is attributed to slow transitions in enzyme conformations that take place over timescales longer than reaction cycle times. In this review we summarize current theoretical models for reaction kinetics in fluctuating, single enzyme systems. Also examined are some of the implications of single-molecule fluctuations for reaction rates in systems such as cells or biosensors that contain a moderate number of molecular copies. We conclude that the dynamic disorder in single-molecule enzyme systems is well-described by available models. However, more work is required to study the effect of single-molecule fluctuations on finite systems over limited periods of time.     

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.     

Millisecond dynamics in glutaredoxin during catalytic turnover is dependent on substrate binding and absent in the resting states

Conformational dynamics is important for enzyme function. Which motions of enzymes determine catalytic efficiency and whether the same motions are important for all enzymes, however, are not well understood. Here we address conformational dynamics in glutaredoxin during catalytic turnover with a combination of NMR magnetization transfer, R(2) relaxation dispersion, and ligand titration experiments. Glutaredoxins catalyze a glutathione exchange reaction, forming a stable glutathinoylated enzyme intermediate. The equilibrium between the reduced state and the glutathionylated state was biochemically tuned to exchange on the millisecond time scale. The conformational changes of the protein backbone during catalysis were followed by (15)N nuclear spin relaxation dispersion experiments. A conformational transition that is well described by a two-state process with an exchange rate corresponding to the glutathione exchange rate was observed for 23 residues. Binding of reduced glutathione resulted in competitive inhibition of the reduced enzyme having kinetics similar to that of the reaction. This observation couples the motions observed during catalysis directly to substrate binding. Backbone motions on the time scale of catalytic turnover were not observed for the enzyme in the resting states, implying that alternative conformers do not accumulate to significant concentrations. These results infer that the turnover rate in glutaredoxin is governed by formation of a productive enzyme-substrate encounter complex, and that catalysis proceeds by an induced fit mechanism rather than by conformer selection driven by intrinsic conformational dynamics.     

Single-molecule tracking of sub-millisecond domain motion in calmodulin

We used single-pair fluorescence resonance energy transfer (spFRET) to track distance changes between domains of fluorescently labeled calmodulin (CaM) on the sub-millisecond time scale. In most cases, CaM remained in the same conformational substate over time periods of up to 1 ms, showing that conformational interchange occurs on a longer time scale. However, in some instances, apparent transitions between conformational substates could be detected. The magnitude of sub-millisecond motion within the dominant conformational substate also revealed fluctuations in distance between domains that were dependent on pH and ionic strength.     

Freezing a single distal motion in dihydrofolate reductase

Constraining a single motion between distal residues separated by approximately 28 A in hybrid quantum/classical molecular dynamics simulations is found to increase the free energy barrier for hydride transfer in dihydrofolate reductase by approximately 3 kcal/mol. Our analysis indicates that a single distal constraint alters equilibrium motions throughout the enzyme on a wide range of time scales. This alteration of the conformational sampling of the entire system is sufficient to significantly increase the free energy barrier and decrease the rate of hydride transfer. Despite the changes in conformational sampling introduced by the constraint, the system assumes a similar transition state conformation with a donor-acceptor distance of approximately 2.72 A to enable the hydride transfer reaction. The modified thermal sampling leads to a substantial increase in the average donor-acceptor distance for the reactant state, however, thereby decreasing the probability of sampling the transition state conformations with the shorter distances required for hydride transfer. These simulations indicate that fast thermal fluctuations of the enzyme, substrate, and cofactor lead to conformational sampling of configurations that facilitate hydride transfer. The fast thermal motions are in equilibrium as the reaction progresses along the collective reaction coordinate, and the overall average equilibrium conformational changes occur on the slower time scale measured experimentally. Recent single molecule experiments suggest that at least some of these thermally averaged equilibrium conformational changes occur on the millisecond time scale of the hydride transfer reaction. Thus, introducing a constraint that modifies the conformational sampling of an enzyme could significantly impact its catalytic activity.     

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

Enzymes are dynamic entities: both their conformation and catalytic activity fluctuate over time. When such fluctuations are relatively fast, it is not surprising that the classical Michaelis-Menten (MM) relationship between the steady-state enzymatic velocity and the substrate concentration still holds. However, recent single-molecule experiments have shown that this is the case even for an enzyme whose catalytic activity fluctuates on the 10(-4)-10 s range. The purpose of this paper is to examine various scenarios in which slowly fluctuating enzymes would still obey the MM relationship. Specifically, we consider (1) the quasi-static condition (e.g., the conformational fluctuation of the enzyme-substrate complex is much slower than binding, catalysis, and the conformational fluctuations of the free enzyme), (2) the quasi-equilibrium condition (when the substrate dissociation is much faster than catalysis, irrespective of the time scales or amplitudes of conformational fluctuations), and (3) the conformational-equilibrium condition (when the dissociation and catalytic rates depend on the conformational coordinate in the same way). For each of these scenarios, the physical meaning of the apparent Michaelis constant and catalytic rate constant is provided. Finally, as an example, the theoretical analysis of a recent single-molecule enzyme assay is considered in light of the perspectives presented in this paper.     

Structural reorganization and preorganization in enzyme active sites: comparisons of experimental and theoretically ideal active site geometries in the multistep serine esterase reaction cycle

Many enzymes catalyze reactions with multiple chemical steps, requiring the stabilization of multiple transition states during catalysis. Such enzymes must strike a balance between the conformational reorganization required to stabilize multiple transition states of a reaction and the confines of a preorganized active site in the polypeptide tertiary structure. Here we investigate the compromise between structural reorganization during the catalytic process and preorganization of the active site for a multistep enzyme-catalyzed reaction, the hydrolysis of esters by the Ser-His-Asp/Glu catalytic triad. Quantum mechanical transition states were used to generate ensembles of geometries that can catalyze each individual step in the mechanism. These geometries are compared to each other by superpositions of catalytic atoms to find "consensus" geometries that can catalyze all steps with minimal rearrangement. These consensus geometries are found to be excellent matches for the natural active site. Preorganization is therefore found to be the major defining characteristic of the active site, and reorganizational motions often proposed to promote catalysis have been minimized. The variability of enzyme active sites observed by X-ray crystallography was also investigated empirically. A catalog of geometrical parameters relating active site residues to each other and to bound inhibitors was collected from a set of crystal structures. The crystal-structure-derived values were then compared to the ranges found in quantum mechanically optimized structures along the entire reaction coordinate. The empirical ranges are found to encompass the theoretical ranges when thermal fluctuations are taken into account. Therefore, the active sites are preorganized to a geometry that can be objectively and quantitatively defined as minimizing conformational reorganization while maintaining optimal transition state stabilization for every step during catalysis. The results provide a useful guiding principle for de novo design of enzymes with multistep mechanisms.     

Role of protein dynamics in reaction rate enhancement by enzymes

An integrated view of protein structure, dynamics, and function is emerging, where proteins are considered as dynamically active assemblies and internal motions are closely linked to function such as enzyme catalysis. Further, the motion of solvent bound to external regions of protein impacts internal motions and, therefore, protein function. Recently, we discovered a network of protein vibrations in enzyme cyclophilin A, coupled to its catalytic activity of peptidyl-prolyl cis-trans isomerization. Detailed studies suggest that this network, extending from surface regions to active site, is a conserved part of enzyme structure and has a role in promoting catalysis. In this report, theoretical investigations of concerted conformational fluctuations occurring on microsecond and longer time scales within the discovered network are presented. Using a new technique, kinetic energy was added to protein vibrational modes corresponding to conformational fluctuations in the network. The results reveal that protein dynamics promotes catalysis by altering transition state barrier crossing behavior of reaction trajectories. An increase in transmission coefficient and number of productive trajectories with increasing amounts of kinetic energy in vibrational modes is observed. Variations in active site enzyme-substrate interactions near transition state are found to be correlated with barrier recrossings. Simulations also showed that energy transferred from first solvation shell to surface residues impacts catalysis through network fluctuations. The detailed characterization of network presented here indicates that protein dynamics plays a role in rate enhancement by enzymes. Therefore, coupled networks in enzymes have wide implications in understanding allostericity and cooperative effects, as well as protein engineering and drug design.     

A snapshot of enzyme catalysis using electrospray ionization mass spectrometry

Insights into the early molecular events involving protein-ligand/substrate interactions such as protein signaling and enzyme catalysis can be obtained by examining these processes on a very short, millisecond time scale. We have used time-resolved electrospray mass spectrometry to delineate the catalytic mechanism of a key enzyme in bacterial lipopolysaccharide biosynthesis, 3-deoxy-d-manno-2-octulosonate-8-phosphate synthase (KDO8PS). Direct real-time monitoring of the catalytic reaction under single enzyme turnover conditions reveals a novel hemiketal phosphate intermediate bound to the enzyme in a noncovalent complex that establishes the reaction pathway. This study illustrates the successful application of mass spectrometry to reveal transient biochemical processes and opens a new time domain that can provide detailed structural information of short-lived protein-ligand complexes.     

Measurement of the alpha-secondary kinetic isotope effect for the reaction catalyzed by mammalian protein farnesyltransferase

Protein farnesytransferase (FTase) catalyzes the transfer of a 15-carbon prenyl group from farnesyl diphosphate (FPP) to the cysteine residue of target proteins and is a member of the newest class of zinc metalloenzymes that catalyze sulfur alkylation. Common substrates of FTase include oncogenic Ras proteins, and therefore inhibitors are under development for the treatment of various cancers. An increased understanding of the salient features of the chemical transition state of FTase may aid in the design of potent inhibitors and enhance our understanding of the mechanism of this class of zinc enzymes. To investigate the transition state of FTase we have used transient kinetics to measure the alpha-secondary 3H kinetic isotope effect at the sensitive C1 position of FPP. The isotope effect for the FTase single turnover reaction using a peptide substrate that is farnesylated rapidly is near unity, indicating that a conformational change, rather than farnesylation, is the rate-limiting step. To look at the chemical step, the kinetic isotope effect was measured as 1.154 +/- 0.006 for a peptide that is farnesylated slowly, and these data suggest that FTase proceeds via a concerted mechanism with dissociative character.     

Dynamic distance disorder in proteins is caused by trapping

Dynamic disorder in proteins, as demonstrated by variations in single-molecule electron transfer rates, is investigated by molecular dynamics simulations. The potential of mean force for the fluctuating donor-acceptor distance is calculated for the NAD(P)H:flavin oxidoreductase (Fre) complex with flavin adenine dinucleotide (FAD) and is found to be in agreement with that estimated from electron transfer experiments. The calculated autocorrelation function of the distance fluctuations has a simple exponential behavior at low temperatures and stretched exponential behavior at higher temperatures on femtosecond to nanosecond time scales. This indicates that the calculated dynamic disorder arises from a wide range of trapping times in potential wells on the protein energy landscape and suggests a corresponding origin for the stretched exponential behavior observed experimentally on longer time scales.     

Molecular dynamics simulation of cellobiose in water

The conformational behavior of cellobiose was studied by molecular dynamics simulation in a periodic box of waters. Several different initial conformations were used and the results compared with equivalent vacuum simulations. The average positions and rms fluctuations within single torsional conformations of cellobiose were affected only slightly by the solvent. However, water damped local torsional librations and transitions. The conformational energies of the solute and their fluctuations were also sensitive to the presence of solvent. Intramolecular hydrogen bonding was weakened relative to that observed in vacuo due to competition with solvating waters. All cellobiose hydroxyl groups participated in intermolecular hydrogen bonds with water, with approximately eight hydrogen bonds formed per glucose ring. The hydrogen bonding was predominantly between water hydrogens and solute hydroxyl oxygens. Intermolecular hydrogen bonding to ring and bridge oxygens was seldom present. The diffusion coefficients of both water and solute agree closely with experimental values. Water interchanged rapidly between the solvating first shell and the bulk on the picosecond time scale. © 1993 John Wiley & Sons, Inc.     

Understanding the molecular mechanism of enzyme dynamics of ribonuclease A through protonation/deprotonation of HIS48

Molecular dynamics simulation is carried out to investigate the enzyme dynamics of RNase A with the HIS48 in three different states (HIP48 (protonated), HID48 (deprotonated), and H48A mutant). Insights derived from the current theoretical study, combined with the available experimental observations, enabled us to provide a microscopic picture for the efficient enzyme dynamics. Specifically, in the "closed" state or HIP48, the N-terminal hinge loop is intact and the enzyme remains in a relatively stable conformation which is preferred for catalytic reaction. Deprotonation of HIS48 induces the denaturing of this hinge-loop into a 3(10)-helix, causing it to break the original interaction network around the loop-1 and drive the partial unfolding of the N-terminal. The enhanced dynamic motion of the N-terminal helix facilitates the release of the catalytic product (the rate limiting step) and speeds up the overall catalytic process. The current study established that HIS49 acts as a modulator for the transformation of conformational states through the perturbing of hydrogen bond networks across loop-1, the N-terminal helix, and other residues nearby. Our study suggests that HIS48 may also serve to transport loop-1's kinetic energy to the reaction center.     

Mass-dependent bond vibrational dynamics influence catalysis by HIV-1 protease

Protein motions that occur on the microsecond to millisecond time scale have been linked to enzymatic rates observed for catalytic turnovers, but not to transition-state barrier crossing. It has been hypothesized that enzyme motions on the femtosecond time scale of bond vibrations play a role in transition state formation. Here, we perturb femtosecond motion by substituting all nonexchangeable carbon, nitrogen, and hydrogen atoms with (13)C, (15)N, and (2)H and observe the catalytic effects in HIV-1 protease. According to the Born-Oppenheimer approximation, isotopic substitution alters vibrational frequency with unchanged electrostatic properties. With the use of a fluorescent peptide to report on multiple steps in the reaction, we observe significantly reduced rates in the heavy enzyme relative to the light enzyme. A possible interpretation of our results is that there exists a dynamic link between mass-dependent bond vibrations of the enzyme and events in the reaction coordinate.     

Rate processes with non-Markovian dynamical disorder

Rate processes with dynamical disorder are investigated within a simple framework provided by unidirectional electron transfer (ET) with fluctuating transfer rate. The rate fluctuations are assumed to be described by a non-Markovian stochastic jump process which reflects conformational dynamics of an electron transferring donor-acceptor molecular complex. A tractable analytical expression is obtained for the relaxation of the donor population (in the Laplace-transformed time domain) averaged over the stationary conformational fluctuations. The corresponding mean transfer time is also obtained in an analytical form. The case of two-state fluctuations is studied in detail for a model incorporating substate diffusion within one of the conformations. It is shown that an increase of the conformational diffusion time results in a gradual transition from the regime of fast modulation characterized by the averaged ET rate to the regime of quasistatic disorder. This transition occurs at the conformational mean residence time intervals fixed much less than the inverse of the corresponding ET rates. An explanation of this paradoxical effect is provided. Moreover, its presence is also manifested for the simplest, exactly solvable non-Markovian model with a biexponential distribution of the residence times in one of the conformations. The nontrivial conditions for this phenomenon to occur are found.     

The hydrogen-bonding network in heme oxygenase also functions as a modulator of enzyme dynamics: chaotic motions upon disrupting the H-bond network in heme oxygenase from Pseudomonas aeruginosa

Relaxation compensated Carr-Purcell-Meiboom-Gill (rc-CPMG) NMR experiments have been used to investigate micros-ms motions in heme oxygenase from Pseudomonas aeruginosa (pa-HO) in its ferric state, inhibited by CN- (pa-HO-CN) and N3- (pa-HO-N3), and in its ferrous state, inhibited by CO (pa-HO-CO). Comparative analysis of the data from the three forms indicates that the nature of the coordinated distal ligand affects the micros-ms conformational freedom of the polypeptide in regions of the enzyme far removed from the heme iron and distal ligand. Interpretation of the dynamical information in the context of the crystal structure of resting state pa-HO shows that residues involved in the network of structural hydrogen-bonded waters characteristic of HOs undergo micros-ms motions in pa-HO-CN, which was studied as a model of the highly paramagnetic S = 5/2 resting state form. In comparison, similar motions are suppressed in the pa-HO-CO and pa-HO-N3 complexes, which were studied as mimics of the obligatory oxyferrous and ferric hydroperoxide intermediates, respectively, in the catalytic cycle of heme degradation. These findings suggest that in addition to proton delivery to the nascent Fe(III)-OO(-) intermediate during catalysis, the hydrogen-bonding network serves two additional roles: (i) propagate the electronic state (reactive state) in each of the distinct steps of the catalytic cycle to key but remote sections of the polypeptide via small rearrangements in the network of hydrogen bonds and (ii) modulate the conformational freedom of the enzyme, thus allowing it to adapt to the demanding changes in axial coordination state and substrate transformations that take place during the catalytic cycle. This idea was probed by disrupting the hydrogen-bonding network in pa-HO by replacing R80 with L. NMR spectroscopic studies conducted with R80L-pa-HO-N3 and R80L-pa-HO-CO revealed that the mutant exhibits nearly global conformational disorder, which is absent in the equivalent complexes of the wild type enzyme. The "chaotic" disorder in the R80L mutant is likely related to its significantly lower efficiency to hydroxylate heme in the presence of H2O2, relative to the wild type enzyme.     

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

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