Single-molecule Enzymology (1999)

Viewing a movie of an enzyme molecule made by molecular dynamics simulation, we see incredible details of molecular motions, be they changes of the conformation or actions during a chemical reaction. Molecular dynamics simulations have advanced our understanding of the dynamics of macromolecules in ways that would not be deducible from the static crystal structures (for review see Refs. 1Brooks III, C.L. Karplus M. Pettitt B.M. Proteins: A Theoretical Perspective of Dynamics, Structure, and Thermodynamics, Advances in Chemical Physics. LXXI. John Wiley & Sons, Inc., New York1998Google Scholar and 2McCammon J.A. Harvey S.C. Dynamics of Proteins and Nucleic Acids. Cambridge University Press, Cambridge, United Kingdom1987Crossref Google Scholar). Unfortunately, these "virtual movies" do not run long enough compared with the time scale of milliseconds to seconds in which most enzymatic reactions take place. In recent years, rapid advances in the patch clamp technique (for review see Ref. 3), atomic force microscopy (4Radmacher M. Fritz M. Hansma H.G. Hansma P.K. Science. 1994; 265: 1577-1579Crossref PubMed Scopus (338) Google Scholar, 5Rees W.A. Keller R.W. Vesenka J.P. Bustamante C. Science. 1993; 260: 1646-1649Crossref PubMed Scopus (204) Google Scholar), optical tweezers (6Ashkin A. Dziedzic J.M. Science. 1987; 235: 1517-1520Crossref PubMed Scopus (1749) Google Scholar, 7Perkins T.T. Quake S.R. Smith D.E. Chu S. Science. 1994; 264: 822-826Crossref PubMed Scopus (586) Google Scholar), and fluorescence microscopy (for review see Refs. 8Xie X.S. Trautman J.K. Annu. Rev. Phys. Chem. 1998; 59: 441-480Crossref Scopus (623) Google Scholar and 9Nie S. Zare R.N. Annu. Rev. Biophys. Biomol. Struct. 1997; 26: 567-596Crossref PubMed Scopus (417) Google Scholar) have permitted making single-molecule "movies" in situ at the millisecond to second time scale. Unlike molecular dynamics simulation, these techniques have low time resolutions, but their single-molecule sensitivities allow probing of slow conformational motions, which are otherwise masked in ensemble-averaged experiments. Moreover, chemical reactions can now be observed on a single-molecule basis. For example, enzymatic turnovers of a few motor proteins (10Funatsu T. Harada Y. Tokunaga M. Saito K. Yanagida T. Nature. 1995; 374: 555-559Crossref PubMed Scopus (898) Google Scholar, 11Vale R.D. Funatsu T. Pierce D.W. Romberg L. Harada Y. Yanagida T. Nature. 1996; 380: 451-453Crossref PubMed Scopus (614) Google Scholar, 12Ishijima A. Kojima H. Funatsu T. Tokunaga M. Higuchi H. Tanaka H. Yanagida T. Cell. 1998; 92: 161-171Abstract Full Text Full Text PDF PubMed Scopus (435) Google Scholar, 13Noji H. Yasuda R. Yoshida M. Kinosita K. Nature. 1997; 386: 299-302Crossref PubMed Scopus (1966) Google Scholar, 14Yasuda R. Moji H. Kinoshita K. Yoshida M. Cell. 1997; 93: 1117-1124Abstract Full Text Full Text PDF Scopus (713) Google Scholar), a nuclease (15Ha T. Ting A.Y. Liang J. Caldwell W.B. Deniz A.A. Chemla D.S. Schultz P.G. Weiss S. Proc. Natl. Acad. Sci. U. S. A. 1999; 96: 893-898Crossref PubMed Scopus (471) Google Scholar), and a flavoenzyme (16Lu H.P. Xun L. Xie X.S. Science. 1998; 282: 1877-1882Crossref PubMed Google Scholar) have been monitored optically in real time. Our knowledge of enzyme kinetics has come primarily from experiments conducted on large ensembles of enzyme molecules, in which concentration changes over time are measured. In a single-molecule experiment, the concentration of the molecule being studied becomes meaningless in discussing chemical kinetics. However, this does not negate the fundamental principles of chemical kinetics. As we shall show, chemical kinetics can be cast in terms of single-molecule probabilities. Thinking of chemical kinetics in terms of single molecules is not only pertinent to the ever increasing single-molecule studies but is also insightful and very often more informative. Such "single-molecule" thinking is also useful in understanding chemistry in living cells. In a living cell, the number of enzyme molecules in a cellular component may not be large. Under this situation, the concentration in a small probe volume is no longer a constant but a fluctuating quantity, as molecules react or diffuse in and out of the probe volume. In fact, the reaction rate (and diffusion rate) can be extracted from the analyses of concentration fluctuations (17Magde D. Elson E. Webb W.W. Phys. Rev. Lett. 1972; 29: 705-708Crossref Scopus (1471) Google Scholar, 18Feher G. Weissman M. Proc. Natl. Acad. Sci. U. S. A. 1973; 70: 870-875Crossref PubMed Google Scholar). This approach is referred to as fluctuation correlation spectroscopy (for review see Ref. 19Maiti S. Haupts U. Webb W.W. Proc. Natl. Acad. Sci. U. S. A. 1997; 94: 11753-11757Crossref PubMed Scopus (298) Google Scholar) and has recently been conducted with single-molecule sensitivity (20Eigen M. Rigler R. Proc. Natl. Acad. Sci. U. S. A. 1994; 91: 5740-5747Crossref PubMed Scopus (907) Google Scholar). A typical fluctuation correlation spectroscopy trace, however, is averaged from a large number of molecules diffusing one or a few at a time in and out of a fixed probe volume. There are situations in which we need to focus on the behavior of a single molecule. For example, DNA exists as a "single molecule" inside a bacteria cell. The trajectory of a DNA-enzyme complex can be tracked. In another example, a single receptor protein at a particular spot in a membrane can be interrogated by optical or scanning probe microscopy. Studies in a similar line have been extensively carried out on ion channel proteins with the patch clamp technique (3). In this minireview, we utilize our recent work on a flavoenzyme (16Lu H.P. Xun L. Xie X.S. Science. 1998; 282: 1877-1882Crossref PubMed Google Scholar) to discuss the underlying principles of single-molecule kinetics and the information obtainable from single-molecule studies. What does one gain by doing single-molecule enzymatic studies? The stochastic events of individual molecules are not observable in conventional measurements, and the steady state concentrations of transient intermediates are usually too low to detect. The single-molecule experiments allow direct observations of individual steps or intermediates of biochemical reactions. The trajectories of motor proteins serve as good examples of the visualization of individual steps (10Funatsu T. Harada Y. Tokunaga M. Saito K. Yanagida T. Nature. 1995; 374: 555-559Crossref PubMed Scopus (898) Google Scholar, 11Vale R.D. Funatsu T. Pierce D.W. Romberg L. Harada Y. Yanagida T. Nature. 1996; 380: 451-453Crossref PubMed Scopus (614) Google Scholar, 12Ishijima A. Kojima H. Funatsu T. Tokunaga M. Higuchi H. Tanaka H. Yanagida T. Cell. 1998; 92: 161-171Abstract Full Text Full Text PDF PubMed Scopus (435) Google Scholar, 13Noji H. Yasuda R. Yoshida M. Kinosita K. Nature. 1997; 386: 299-302Crossref PubMed Scopus (1966) Google Scholar, 14Yasuda R. Moji H. Kinoshita K. Yoshida M. Cell. 1997; 93: 1117-1124Abstract Full Text Full Text PDF Scopus (713) Google Scholar). Single-molecule spectroscopy is capable of capturing reaction intermediates. Perhaps less obviously, single-molecule experiments allow determination of static and dynamic disorder. Seemingly identical copies of biomolecules often have non-identical properties. Static disorder is the stationary heterogeneity of a property within a large ensemble of molecules. Dynamic disorder (21Zwanzig R. Acc. Chem. Res. 1990; 23: 148-152Crossref Scopus (383) Google Scholar) is the time-dependent fluctuation of the property of an individual molecule. Distributions of molecular properties of an ensemble are usually broad because of both static and dynamic disorder. The distribution is difficult to determine by ensemble-averaged measurements. Furthermore, ensemble-averaged measurements cannot distinguish between static and dynamic disorders. A recent single-molecule enzymatic assay by Xue and Yeung (22Xue Q.F. Yeung E.S. Nature. 1995; 373: 681-683Crossref PubMed Scopus (322) Google Scholar) has revealed static disorder in enzymatic turnover rates of genetically identical and electrophoretically pure enzyme molecules. In a capillary tube containing a solution of highly diluted enzyme molecules (lactate dehydrogenase) and concentrated substrate molecules (lactate and NAD+), each enzyme molecule produced a discrete zone of thousands of NADH molecules after 1 h of incubation. The zones were then eluted by capillary electrophoresis and monitored by natural fluorescence of NADH. The enzyme molecules had a broad and asymmetrical distribution of activity, which was otherwise masked by ensemble-averaged measurements. The heterogeneity was found to be static at the hour time scale because the same enzyme molecule produces the same zone intensity after another incubation period. The microscopic origin of the static disorder observed is an interesting subject that deserves future research. Using a similar approach, Craiget al. (23Craig D.B. Arriaga E.A. Wong J.C.Y. Lu H. Dovichi N.J. J. Am. Chem. Soc. 1996; 118: 5245-5253Crossref Scopus (198) Google Scholar) have studied single alkaline phosphatase molecules and found an even broader, multipeak distribution of activities. The static disorder in this system was attributed to glycosylation and other post-translational modifications, which produce non-identical copies of enzyme in this system. Such experiments are capable of determining static disorder but not dynamic disorder. Dynamic disorder in enzymatic turnover rates has been observed by real time single-molecule experiments, as is discussed below. With the separation of static and dynamic disorders, relations between the dynamics and functions of enzyme molecules can be better interrogated with single-molecule real-time measurements. Consider the example of cholesterol oxidase, a 53-kDa flavoprotein that catalyzes the oxidation of cholesterol by oxygen (Fig.1). The active site of the enzyme (E) involves a FAD, which is naturally fluorescent in its oxidized form but not in its reduced form. The FAD is reduced by a cholesterol molecule to FADH2 and is then oxidized by molecular oxygen. As shown in Fig. 1, fluorescence turns on and off as the redox state of the FAD toggles between the oxidized and reduced states. Each on-off cycle corresponds to an enzymatic turnover. The turnover trajectory contains detailed information about the chemical dynamics. The single-molecule fluorescence measurements are carried out with an inverted fluorescence microscope, as described elsewhere (16Lu H.P. Xun L. Xie X.S. Science. 1998; 282: 1877-1882Crossref PubMed Google Scholar, 24Macklin J.J. Trautman J.K. Harris T.D. Brus L.E. Science. 1996; 272: 255-258Crossref Scopus (367) Google Scholar, 30Lu H.P. Xie X.S. Nature. 1997; 385: 143-146Crossref Scopus (339) Google Scholar). It is desirable to study immobilized molecules to avoid the complications of the diffusion process. The samples are thin films of agarose gel of 99% water. The single-enzyme molecules are confined in the gel with no noticeable translational diffusion. In contrast, small substrate molecules still diffuse freely. Though confined in the polymer matrix, the enzyme molecules freely rotate within the gel, which was evidenced by a polarization modulation experiment, as previously described (25Xie X.S. Dunn R.C. Science. 1994; 265: 361-364Crossref PubMed Scopus (560) Google Scholar, 26Ha M.T. Enderle Th. Chemla D.S. Selvin P.R. Weiss S. Phys. Rev. Lett. 1996; 77: 3979-3982Crossref PubMed Scopus (306) Google Scholar). This means that the enzyme molecules do not bind to the polymer matrix. Control experiments were done to ensure that the conventional enzymatic assays gave the same results in gel and in solution. Polyacrylamide gel, which has a smaller pore size, has also been used to confine proteins, such as green fluorescence proteins (GFP) 1The abbreviations used is: GFP, green fluorescence proteins. (27Dickson R.M. Norris D.J. Tzeng Y.L. Moerner W.E. Science. 1996; 274: 966-998Crossref PubMed Scopus (343) Google Scholar). The turnover trajectories contain detailed dynamic information, which is extractable from statistical analyses. Good statistical analyses require long trajectories. The lengths of the trajectories are limited by photobleaching through photochemistry on the excited state (8Xie X.S. Trautman J.K. Annu. Rev. Phys. Chem. 1998; 59: 441-480Crossref Scopus (623) Google Scholar). We observed a better photostability for the FAD chromophore in protein than for dye molecules, most likely because of the protection by the protein. Trajectories with more than 500 turnovers and 2 × 106 detected photons (detection efficiency, 10%) have been recorded. Similar photostability has been seen for other natural fluorophores, such as those in GFP (27Dickson R.M. Norris D.J. Tzeng Y.L. Moerner W.E. Science. 1996; 274: 966-998Crossref PubMed Scopus (343) Google Scholar). In the case of GFP, emission of a single molecule blinks off because of photoinduced chemical reactions. We did not observe such photoinduced blinking of cholesterol oxidase. We have also done control experiments to make sure that repetitive excitation/de-excitation does not perturb the enzymatic reactions. Many two-substrate enzymes, such as cholesterol oxidase, follow the ping-pong mechanism for the two-substrate binding processes, obeying the Michaelis-Menten mechanism (28Palmer T. Understanding Enzymes. 4th Ed. Prentice Hall, New York1991Google Scholar). E­FAD+S⇄k−1k1E­FAD·S→k2E­FADH2+PEquation 1 ononoff E­FADH2+O2⇄k−1′k1′E­FADH2·O2→k2′E­FAD+H2O2Equation 2 offoffonThe most obvious feature of the turnover trajectory in Fig. 1 is its stochastic nature. The emission on-times and off-times correspond to the "waiting time" for the reductive and oxidative reactions, respectively. The simplest analysis of the trajectory is the distribution of the on- or off-times. We limit our discussion below to the on-time distributions although similar analyses can be done for off-times as well. First, take a simple case in which k 2 is rate-limiting. This situation can be created with a slowly reacting substrate (derivative of cholesterol) and a high concentration of the substrate. The FAD reduction reaction follows a simple kinetic scheme. E­FAD·S→k2E­FADH2+PEquation 3 onoffFor this scheme, the probability density of the on-time, τ, for a single-molecule turnover trajectory is an exponential function,p on(τ) = k 2exp(−k 2τ), with the average of the on-times being 1/k 2, the time constant of the exponential. The exponential function follows from the fact that Equation 3 is a Poisson process. One caveat is that this does not mean that zero on-time has the highest probability;p on(τ) is the probability density, whereas the probability for the on-time to be between τ and Δτ is given byp on(τ)Δτ, with the integrated area underp on(τ) being unity. A simple example of a Poisson process is the case of a telephone being turned on and off. The Poisson process corresponds to a situation in which all phone calls are independent. This results in an exponential probability density distribution of the lengths of the calls (on-times). In such an analogy, our real-time experiment determines the on-time distribution of a single phone, Yeung's experiment (22Xue Q.F. Yeung E.S. Nature. 1995; 373: 681-683Crossref PubMed Scopus (322) Google Scholar) determines the average on-time of a single phone, and an ensemble-averaged measurement gives the average on-time of all the phones in the world. Fig. 2 A shows the histogram of on-times derived from a trajectory with a cholesterol derivative of 2 mm concentration, which was fitted with a single exponential. The inset of Fig. 2 A shows the distribution of k 2 among 33 molecules examined, reflecting the large static disorder for the activation step. However, we did not observe static disorder for k 1,k′1, and k′2. The on-time histogram (or probability density) takes a more complex form when k 2 is not rate-limiting. Assuming thatk −1 = 0 in Equation 1, the probability density of the on-times is expected to be the convolution of the two exponential functions (time constants of 1/k 1and 1/k 2) with an exponential rise (the faster one of k 1 and k 2) and an exponential decay (the slower one of k 1 andk 2) (16Lu H.P. Xun L. Xie X.S. Science. 1998; 282: 1877-1882Crossref PubMed Google Scholar). Fig. 2 B shows an on-time histogram for an enzyme molecule with 2 mm cholesterol, with the solid line being the convolution of two exponentials. The time lag in the histogram arises from the fact that there is an intermediate, E-FAD·S. There is noE-FADH2 generated until the intermediate emerges. In the telephone analogy, this corresponds to the caller having to wait for an operator to connect. If there is more than one fluorescent intermediate, the on-times are expected to have a narrower distribution. This is discussed in detail in Ref. 29Schnitzer M.J. Block S.M. Cold Spring Harbor Symp. Quant. Biol. 1995; 60: 793-802Crossref PubMed Scopus (110) Google Scholar. So far, the analyses of on-time distributions illustrate at the single-molecule level the validity of chemical kinetics, in particular, the Michaelis-Menten mechanism (Equation 1). What new information can we obtain from the trajectory analyses? Chemical kinetics holds for Markovian processes, implying that an enzyme molecule undergoing a turnover exhibits no memory of its preceding turnovers. Dynamic disorder is beyond the scope of chemical kinetics. Fig. 2 does not have a good enough signal-to-noise ratio to reveal multiexponential decay because of dynamic disorder. Furthermore, being a scrambled histogram, Fig. 2 is not sensitive to memory effects. What we need is a way to examine how a particular turnover turn is affected by its previous turnovers. We evaluated the conditional probability,p(X,Y), for a pair of on-times (X and Y) separated by a certain number of turnovers. Fig. 3, A andB, shows the two-dimensional histograms of a pair of on-times adjacent to each other and those separated by 10 turnovers, respectively. In the absence of dynamic disorder,p(X,Y) should be independent of the separation of turnovers. However, Fig. 3 A and Fig.3 B are clearly different. For the separation of 10 turnovers (Fig. 3 B), the loss of memory leads top(X,Y) =p(X)p(Y), wherep(X) and p(Y) are the same as in Fig. 3 A. For pairs of adjacent on-times (Fig.3 A), there is a diagonal feature, indicating that a short on-time tends to be followed by another short on-time, and a long on-time tends to be followed by another long on-time. This means that an enzymatic turnover is not independent of its previous turnovers. The memory effect arises from a slowly varying rate (k 2). Coming back to the telephone analogy, this corresponds to the average lengths of phone calls varying over the course of a day. Although the two-dimensional conditional probability plot provides a clear visual illustration, it needs to be constructed of a large number of turnovers of many molecules. A more practical and quantitative way to evaluate the dynamic disorder is the autocorrelation function of the on-times, r(m) = <Δτ(0)Δτ(m)>/<Δτ2>, wherem is an index number for the turnovers in a trajectory and Δτ(m) = τ(m) − <τ> and where the bracket denotes the average along the trajectory. The physical meaning of r(m) is as follows. In the absence of dynamic disorder, r(0) = 1 and r(m) = 0 (m > 0). In the presence of dynamic disorder,r(m) decays, with the initial (m= 1) amplitude reflecting the variance of k 2 and the decay time yielding the time scale of the k 2fluctuation. Fig. 3 C shows the r(m) derived from a single-molecule trajectory, with the decay constant being 1.6 ± 0.5 turnovers. We attribute the dynamic disorder behavior to a slow fluctuation of protein conformation, which was independently observed by spontaneous spectral fluctuation of FAD (16Lu H.P. Xun L. Xie X.S. Science. 1998; 282: 1877-1882Crossref PubMed Google Scholar, 30Lu H.P. Xie X.S. Nature. 1997; 385: 143-146Crossref Scopus (339) Google Scholar) on the same time scale ofk 2 fluctuation. Slow conformational fluctuations on a similar time scale have been observed on other systems with single-molecule experiments (4Radmacher M. Fritz M. Hansma H.G. Hansma P.K. Science. 1994; 265: 1577-1579Crossref PubMed Scopus (338) Google Scholar, 31Dapprich J. Mets U. Simm W. Eigen M. Rigler R. Exp. Tech. Phys. 1995; 41: 259-264Google Scholar, 32Wennmalm S. Edman L. Rigler R. Proc. Natl. Acad. Sci. U. S. A. 1997; 94: 10641-10646Crossref PubMed Scopus (180) Google Scholar, 33Jia Y. Sytnik A. Li L. Vladimarov S. Coopeman B.S. Hochstrasser R.M. Proc. Natl. Acad. Sci. U. S. A. 1997; 94: 7932-7936Crossref PubMed Scopus (105) Google Scholar, 34Geva E. Skinner J.L. Chem. Phys. Lett. 1998; 288: 225-229Crossref Scopus (82) Google Scholar). The simplest model we proposed involved (at least) two slowly converting conformational states (E and E′). E⇄kE′kEE′ E­FAD+S→k1E­FAD·S→k21E­FADH2+P↕↕↕E′­FAD+S→k1E′­FAD·S→k22E′­FADH2+P SCHEME1The dynamic disorder of k 2 for the Michaelis-Menten mechanism (Equation 2) can be accounted for by the more complicated kinetics scheme (Scheme 1) with time-independent ratesk 21 and k 22. A simulation of r(m) based on this kinetic scheme, assuming that kE/kE′ = 1 andk = the H. and S. in In although the Michaelis-Menten mechanism provides a good for the averaged of many molecules and for the averaged of many turnovers of a single it does not an of the real-time behavior of a single molecule. a single molecule the rate for the activation is and this is a not a small The of conformational dynamics on protein functions has been a subject of studies Nature. PubMed Scopus Google Scholar, Struct. Biol. 1996; PubMed Scopus Google Scholar, E. Karplus M. Chem. Phys. Lett. 1997; Scopus Google Scholar, J.A. H. J. and in Proteins and Nucleic Acids. University Press, Scholar, Acc. Chem. Res. Scopus Google Scholar, J. Science. PubMed Scopus Google Scholar, W.A. J. Chem. Phys. 1996; Scopus Google Scholar, H. R.D. Annu. Rev. Biophys. Biophys. Chem. PubMed Scopus Google Scholar). Our observations have revealed the slow conformational that the enzymatic However, we still need to the microscopic origin of the slow conformational of cholesterol oxidase, which has a large activation between the two conformational states. Although the model to for the fluctuation of k 2 and the spectral mean in our initial there can be more than two conformational or even a distribution of conformational each with a k Our observations that conformational take among states in the H. P.G. Science. PubMed Scopus Google Scholar). based on the model and the J.J. J. Chem. Phys. Scopus (343) Google Scholar, J. Phys. Rev. Lett. 1995; PubMed Scopus Google Scholar) model a experiments are to probe the distribution of conformational the distribution of and their The of the is of particular Similar slow conformational changes have been from other enzyme which were to be with enzymatic PubMed Scopus Google Scholar, C. Annu. Rev. PubMed Scopus Google Scholar, J. J. J. J. PubMed Scopus Google Scholar). an ensemble-averaged enzymatic measurement of cholesterol has revealed a of the enzymatic rate on the substrate concentration T. T. 1996; 60: Google Scholar). Our simulation that the proposed mechanism in 1 for this The of this on the behavior of the enzyme is in 1 is the of conformational changes by substrate binding the redox reactions, which can to new conformational states or the between the conformational states PubMed Scopus Google Scholar, C. Annu. Rev. PubMed Scopus Google Scholar, J. J. J. J. PubMed Scopus Google Scholar). changes with the have been D.E. 1998; Scopus Google Scholar). this our single-molecule experiments have not detected the conformational A study of between spectral and turnover trajectories is way to this Thinking of chemical kinetics in terms of single molecules is a as new of microscopy allow of the dynamics of individual molecules. We have shown that statistical analyses of turnover trajectories of single-enzyme molecules can reveal detailed information in ensemble-averaged measurements. The analyses described be useful for other enzymatic New enzymatic assays be reactions, such as K. Science. 1998; PubMed Scopus Google Scholar) and can be observed Single-molecule real-time allow of molecular and cellular dynamics at a level of We Xun and for and for of the