## Abstract

Enzymatic turnovers of single cholesterol oxidase molecules were observed in real time by monitoring the emission from the enzyme's fluorescent active site, flavin adenine dinucleotide (FAD). Statistical analyses of single-molecule trajectories revealed a significant and slow fluctuation in the rate of cholesterol oxidation by FAD. The static disorder and dynamic disorder of reaction rates, which are essentially indistinguishable in ensemble-averaged experiments, were determined separately by the real-time single-molecule approach. A molecular memory phenomenon, in which an enzymatic turnover was not independent of its previous turnovers because of a slow fluctuation of protein conformation, was evidenced by spontaneous spectral fluctuation of FAD.

Recent advances in fluorescence microscopy have allowed studies of single molecules in an ambient environment (1, 2). Single-molecule measurements can reveal the distribution of molecular properties in inhomogeneous systems (3–10). The distributions, which can be either static (3–7) or dynamical (8–10), cannot usually be determined by ensemble-averaged measurements. Moreover, stochastic trajectories of a single-molecule property can be recorded in real time, containing detailed dynamical information extractable through statistical analyses. Single-molecule trajectories of translational diffusion (11–13), rotational diffusion (14), spectral fluctuation (15), conformational motion (16), and photochemical changes (17, 18) have been demonstrated. Of particular interest is the real-time observation of chemical reactions of biomolecules. Enzymatic turnovers of a few motor protein systems have been monitored in real time (19–21). In the study reported here, we examined enzymatic turnovers of single flavoenzyme molecules by monitoring the fluorescence from their active sites. Statistical analyses of chemical dynamics at the single-molecule level revealed insights into enzymatic properties.

Flavoenzymes are ubiquitous and undergo redox reactions in a reversible manner (22). Cholesterol oxidase (COx) from*Brevibacterium* sp. is a 53-kD flavoenzyme that catalyzes the oxidation of cholesterol by oxygen (23) (Scheme 1). The active site of the enzyme (E) involves a flavin adenine dinucleotide (FAD), which is naturally fluorescent in its oxidized form but not in its reduced form. The FAD is first reduced by a cholesterol molecule to FADH_{2}, and is then oxidized by O_{2}, yielding H_{2}O_{2}. The crystal structure of COx (23) shows that the FAD is noncovalently and tightly bound to the center of the protein and is surrounded by a hydrophobic binding pocket for cholesterol, which is otherwise filled with 14 water molecules.

A fluorescence image of single COx molecules in their oxidized form (Fig. 1A) was taken with an inverted fluorescence microscope by raster-scanning the sample with a fixed He-Cd laser (442 nm, LiCONiX) focus and collecting the FAD emission (peaked at 520 nm) with high efficiency, similar to previous reports (3, 7, 15). The single molecules of COx were confined in agarose gel of 99% water (24) with no observable translational diffusion. However, the COx molecules were freely rotating within the gel, which was evidenced by a polarization modulation experiment, as previously described (25). As the in-plane polarization of the excitation light was modulated between two perpendicular directions at 20 Hz, no emission intensity modulation was seen, indicating that the molecule was freely tumbling at a much faster rate than the modulation frequency. Every COx molecule examined in the gel exhibited the fast rotational diffusion, and thus COx was not bound to the polymer matrix. Ensemble-averaged enzymatic assays for COx (Sigma) in agarose gels yielded turnover rates similar to those in aqueous solutions (26, 27). Unlike the enzyme molecules, small substrate molecules (cholesterol and oxygen) undergo essentially free translational diffusion within the gel.

With excess amounts of cholesterol (0.2 mM) and oxygen (saturated solution, 0.25 mM) in the gel, the single FAD emission exhibits on-off behavior (Fig. 1B). We attribute this phenomenon to chemical reactions of the enzyme molecule: Fluorescence turns on and off as the redox state of the flavin toggles between the oxidized (FAD) and reduced (FADH_{2}) states. Each on-off cycle corresponds to an enzymatic turnover. Several control experiments support this conclusion. (i) There is essentially no blinking of emission without cholesterol molecules. Extremely infrequent blinking was seen during long trajectories for only a few, but not all, COx molecules, which we attribute to impurity of substrate molecules or a low-quantum-yield photoinduced process (1, 4, 17). (ii) In the presence of cholesterol, the turnover rate is independent of excitation intensity; thus, the blinking is due only to the enzymatic reactions in the ground electronic state rather than photoinduced phenomena. (iii) The averaged turnover rates of the trajectories are the same as the ensemble-averaged turnover rates under similar conditions (26,27).

The length of the trajectories is limited by photobleaching through either photooxidation involving singlet oxygen or excited-state absorption (1). We observed a better photostability for the FAD chromophore in protein than for dye molecules, most likely due to the protection by the protein. Trajectories with more than 500 turnovers and 2 × 10^{7} emitted photons (detection efficiency 10%) were recorded; Fig. 1B shows only a portion of a trajectory. During the course of the long trajectories, the average turnover rates (the number of on-off cycles per unit time) did not decrease with time because substrate (cholesterol and oxygen) concentrations were orders of magnitude higher than those of the enzyme. The long trajectories permit detailed statistical analyses.

The most obvious feature of the turnover trajectory in Fig. 1B is its stochastic nature. On a single-molecule basis, the event of a chemical reaction takes place on the subpicosecond time scale and cannot be time resolved here. However, the time needed for diffusion, thermal activation, or both before such an event is usually much longer. The emission on-time and off-time correspond to the “waiting time” for the FAD reduction and oxidation reactions, respectively. The most straightforward analysis of the trajectories is the distribution of the on- and off-times. Similar trajectory analyses have been conducted in studies of single-ion channels with the patch-clamp technique (28). Figure 1C shows the on-time distribution, derived from a trajectory at a cholesterol concentration of 0.2 mM. A similar off-time distribution is also seen (29).

For a reversible chemical kinetic scheme(1)the distribution of on-time (or off-time) should be an exponential function, with a time constant 1/*k*
_{f}(or 1/*k*
_{b}) due to Poissonian statistics (28). In contrast, Fig. 1C shows a nonexponential distribution, which we attribute to a more complex kinetic scheme. In general, many two-substrate enzymes, such as COx, follow the “ping-pong” mechanism for the two-substrate binding processes, each obeying the Michaelis-Menten mechanism (30):(2)
(3)where S denotes the cholesterol substrate, and*k*
_{1} and *k*′_{1} denote pseudo-first-order rates proportional to the concentrations of S and O_{2}, respectively. An assumption is made that other kinetic steps not shown in Eqs. 2 and 3, such as diffusion of product molecules out of the enzyme, are not rate limiting.

For COx, if we assume *k*
_{–1} = 0, it follows from a kinetic analysis (31) that the probability distribution of on-times is the following:
(4)which is the convolution of two exponentials (*k*
_{1} and *k*
_{2}). A similar expression of off-time distribution,*p*
_{off}(*t*), can be deduced for the FADH_{2} oxidation reaction (Eq. 3).

Conventional enzymatic assays for two-substrate enzymes are often done by varying the concentration of one substrate while holding steady the concentrations of the others. The apparent Michaelis-Menten equations take a complex form (30). In order to study the half-reactions separately, the stop-flow technique is usually used. The in situ study of single-molecule turnover trajectories presented here allows the two half-reactions to be examined separately. We focused mostly on the FAD reduction half-reaction (Eq. 2).

A simulation of the *p*
_{on}(*t*) with Eq. 4is plotted in Fig. 1C with *k*
_{1} = 2.9 ± 0.3 s^{−1} and *k*
_{2} = 17 ± 4 s^{−1}. The nonexponential distribution arises because an intermediate (E-FAD•S) exists for the FAD reduction reaction (Eq. 2). When the concentration of the cholesterol substrate was increased from 0.2 mM to 2 mM, the on-time distribution (Fig. 1D) changed significantly whereas the off-time distribution remained unchanged. Simulation with Eq. 4 (Fig. 1D) yields*k*
_{1} = 33 ± 6 s^{−1} and*k*
_{2} = 17 ± 2 s^{−1}. Note that the pseudo-first-order rate constant *k*
_{1} is increased by a factor of 11 ± 3. The analyses of on-time distribution confirm the existence of the intermediate for the FAD reduction reaction, which is consistent with the Michaelis-Menten mechanism (Eq. 2). Similarly, the off-time distribution (29) is also consistent with the Michaelis-Menten mechanism (Eq. 3).

We evaluated the static disorder, that is, the static heterogeneity of rates among individual molecules. To evaluate the static disorder of*k*
_{2}, we made *k*
_{2} rate limiting. This condition is not quite achievable with a high concentration of cholesterol, but it is achievable with a high concentration of 5-pregene-3β-20α-diol substrate, a derivative of cholesterol (Fig. 2B, inset). The slower substrate results in a monotonic decay (*k*
_{2} = 3.9 ± 0.5 s^{−1}) in the on-time distribution for the COx molecule shown in Fig. 2A. A broad distribution of single exponential fits of *k*
_{2} for 33 COx molecules with the slow substrate (Fig. 2A, inset) is an apparently static disorder of*k*
_{2}, with *k*
_{2} being the time averages along the entire trajectories (10 to 20 min) of the molecules. Static disorder has been observed for genetically identical and electrophoretically pure enzyme molecules (5, 6) and has been postulated to arise from different conformers (5) or posttranslation modifications (6). Although the origin of the static disorder requires further investigation, we offer an alternative possibility for the COx system: proteolytic damages, such as oxidation, of the key residues of the enzyme (32).

Dynamic disorder (33), the fluctuation of reaction rate for a single molecule, is beyond the scope of conventional chemical kinetics. Chemical kinetics holds for Markovian processes (34), implying that an enzyme molecule undergoing a turnover exhibits no memory of its preceding turnovers. Dynamic disorder of proteins has been investigated in the context of single-molecule behavior with statistical theory (35) and molecular dynamics simulation (36). However, it is experimentally difficult, if not impossible, to distinguish static and dynamic disorder for chemical reactions in ensemble-averaged experiments. Statistical analyses of the turnover trajectories allow us to examine the validity of chemical kinetics at the single-molecule level and to distinguish static and dynamic disorders. To do so, we first chose the case of *k*
_{2} being the rate-limiting step (Fig. 2), so that the reaction scheme is reduced to Eq. 1, with forward and backward rates being *k*
_{f} =*k*
_{2} and *k*
_{b}, respectively.

Conventional kinetics experiments measure the relaxation of concentration of a large ensemble of molecules after a perturbation (such as fast mixing or a temperature jump). The fluctuation dissipation theorem applies to the reversible chemical reaction in Eq. 1 as follows (37):(5)where δ*c*(*t*) is the concentration change of E-FAD measured in ensemble-averaged relaxation experiments, and ζ(*t*) is a dynamical variable for a single enzyme molecule in dynamic equilibrium; ζ = 1 when the molecule is in the state of E-FAD, and ζ = 0 when the molecule is in the state of E-FADH_{2}. The bracket denotes averaging along a stationary trajectory, and Δζ(*t*) = ζ(*t*) − 〈ζ〉. Thus, Eq. 5 relates the macroscopic experimental observable to the autocorrelation function of ζ(*t*), a microscopic property that has, to date, only been obtainable from molecular dynamics simulation. Our single-molecule experiment allows the right side of Eq. 5 to be measured directly through the intensity autocorrelation function 〈Δ*I*(*t*)Δ*I*(0)〉. Figure 2B shows 〈Δ*I*(*t*)Δ*I*(0)〉 of the COx molecule in Fig. 2A.

According to chemical kinetics (35), δ*c*(*t*)/δ*c*(0) = exp[−(*k*
_{b} +*k*
_{f})*t*], that is, a single exponential decay is expected for 〈Δ*I*(*t*)Δ*I*(0)〉, with a decay rate being the sum of the forward and backward rates. In fact, fluorescence intensity autocorrelation functions have been used to extract kinetics constants of chemical reactions with fluorescence correlation spectroscopy for a small number of molecules (38), and for one molecule at a time but averaging many molecules (39). The measured autocorrelation function (Fig. 2B, solid line) for a long trajectory of a single immobilized molecule was not a single exponential decay (Fig. 2B, dashed line). The multiexponential decay of the autocorrelation function can arise from the dynamic disorder of *k*
_{2} or*k*
_{f}, or both, or from a more complex kinetic scheme (Eq. 3) than the backward reaction in Eq. 1.

The *p*
_{on}(*t*) in Fig. 2A has the advantage of only reflecting the forward reaction, but it does not have a good enough signal-to-noise ratio to reveal a multiexponential decay due to the dynamic disorder of *k*
_{2}. Being a scrambled histogram, Fig. 2A is not sensitive to memory effects. We present a statistical analysis that provides definitive evidence for dynamic disorder in *k*
_{2}. We evaluated the conditional probability distribution,*p*(*x*,*y*), for pairs of on-times (*x* and *y*) separated by a certain number of turnovers. Two-dimensional (2D) histograms of a pair of on-times adjacent to each other (Fig. 3A) and separated by 10 turnovers (Fig. 3B) are shown for the sum of 33 COx molecules, thus reflecting the ensemble average of a single-molecule property. In the absence of dynamic disorder, the two on-times should be independent of each other, that is, *p*(*x,y*) =*p*(*x*)*p*(*y*).

In contrast, Figs. 3A and 3B are clearly different, indicating a memory effect. For adjacent pairs of on-times (Fig. 3A), 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. For the separation of 10 turnovers (Fig. 3B), the distribution becomes independent. The memory effect arises from a slowly varying rate (*k*
_{2} being rate limiting). The rate equation for the forward reaction in Eq. 1 is written as(6)where *k*
_{2}(*t*) is a stochastic variable with a mean, 〈*k*
_{2}〉. If the fluctuation of *k*
_{2} is very fast,*k*
_{2}(*t*) can be replaced by 〈*k*
_{2}〉. Dynamic disorder arises when the time scale of the *k*
_{2} fluctuation is comparable to or slower than 1/*k*
_{2}. The fluctuation of*k*
_{2}(*t*) can be characterized by the*k*
_{2} variance and correlation time, that is, the memory time.

We quantitatively analyzed the visual difference between Figs. 3A and3B by a covariance parameter (40, 41) defined as
(7)where *i* is an index number for a total of *n* + *m* turnovers in a trajectory;*t _{i}
* is the experimentally determined on-time; and

*m*is the separation between the pairs of on-times. Intuitively,

*r*= 1 for a diagonal line,

*r*= −1 for an off-diagonal line in the 2D distribution, and

*r*= 0 for independent

*x*and

*y*. In fact,

*r*(

*m*) is the autocorrelation function of the on-times (the last equality with the bracket denoting the trajectory average). In the absence of dynamic disorder,

*r*(0) = 1 and

*r*(

*m*) = 0 (

*m*> 0). In the presence of dynamic disorder,

*r*(

*m*) is a decay with the initial (

*m*= 1) amplitude reflecting the variance of

*k*

_{2}and the decay time yielding the correlation time of

*k*

_{2}.

The *r*(*m*) for a single enzyme molecule with 2 mM 5-pregene-3β-20α-diol (Fig. 4A) yields the correlation time of *k*
_{2}, τ = 1.0 ± 0.3 s (1.6 turnovers in this trajectory). The fluctuation of *k*
_{2} was otherwise masked in the ensemble-averaged results, as well as in the scrambled 1D histogram [*p*
_{on}(*t*) in Fig. 2A]. We found every enzyme molecule in the system exhibited similar*r*(*m*). The decay of *r*(*m*) remained unchanged as the excitation intensity was doubled, so the phenomenon is not photoinduced. The high sensitivity of*r*(*m*) to the memory effect allowed the observation of dynamic disorder even when cholesterol was the substrate (*k*
_{2} was not quite rate limiting). Figure 4B shows the decay of *r*(*m*) for the molecule in Fig. 1D with 2 mM cholesterol. In contrast, the *r*(*m*) for the molecule in Fig. 1C with 0.2 mM cholesterol remains zero (Fig. 4C) as *k*
_{1} becomes rate limiting. There is no dynamic disorder in *k*
_{1}. For the FADH_{2} oxidation half-reaction (Eq. 3), we found that the autocorrelation function for the off-times is zero for*m* > 0. In addition, there is no correlation between adjacent on- and off-times. Therefore, the FADH_{2} oxidation reaction (*k*′_{1} and *k*′_{2}) involving oxygen does not exhibit dynamic disorder.

We attribute the fluctuation of *k*
_{2} to conformational fluctuation of the protein, which can be probed by the emission spectra of the FAD active site. Figure 5A shows a trajectory of the spectral mean of the emission spectra of a single enzyme molecule in the absence of cholesterol substrate molecules. The slow fluctuation of the emission spectrum (Fig. 5B) reflects conformational changes around the FAD, a phenomenon that is otherwise hidden in ensemble-averaged measurements. Similar room-temperature spectral fluctuation of single dye molecules in polymer has been studied in detail, providing information regarding the energy landscape (15). The autocorrelation function of the spectral mean trajectory is shown inFig. 5C. The decay curve was independent of excitation rate, indicating a spontaneous (rather than photoinduced) conformational fluctuation. Interestingly, the decay constant is 1.3 ± 0.3 s, which is on the same time scale as the correlation time of*k*
_{2}, τ. This provides strong evidence that conformational fluctuation results in variation of the enzymatic rate*k*
_{2}.

The simplest model that can account for the fluctuation of*k*
_{2} involves two conformational states of the enzyme, E and E′, which have different rates (*k*
_{21} and *k*
_{22}) for the activation step and interchange with a rate 1/τ =*k*
_{E} + *k*
_{E′} slower than 〈*k*
_{2}〉:
(8)A simulation of *r*(*m*) based on this kinetic scheme assuming*k*
_{E}/*k*
_{E′} = 1 and*k*
_{21}/*k*
_{22} = 5 matches the experimental curve (Fig. 4A), indicating a substantial difference in*k*
_{21} and *k*
_{22}(42). We note that the dynamic disorder of*k*
_{2} for the Michaelis-Menten mechanism (Eq. 2), a non-Markovian behavior, can be accounted for by the more complicated kinetics scheme (Eq. 8) with the constant rates*k*
_{21} and *k*
_{22}.

Similar molecular memory phenomena deviating from the Michaelis-Menten mechanism have been inferred previously by ensemble-averaged experiments for other monomeric enzyme systems (43). A consequence of the memory effect is a sigmoidal dependence of the enzymatic reaction velocity on the substrate concentration. It was postulated that such “kinetic cooperativity” of the monomeric enzymes facilitates physiological regulation. Interestingly, a sigmoidal dependence on cholesterol concentration was reported for COx (27), which could be related to our finding of the memory effect.

Although the two-state model suffices to account for the fluctuation of *k*
_{2} and the spectral mean, there can be more than two conformational states, or even a broad distribution of conformational states (44) with distinctly different *k*
_{2}. For the two-state model, a simple kinetic scheme for the interconverting conformers can account for the dynamics (45). For the continuous model, a diffusive motion along a conformational coordinate perpendicular to the reaction coordinate induces a fluctuating barrier height (46). On the basis of this model, a simulation of the stochastic trajectory and its*r*(*m*) yields the *k*
_{2}variance, 〈*k*
_{2}
^{2}〉 = 12.5 s^{−2} (42), corresponding to a significant amplitude of fluctuation (standard deviation being 70% of 〈*k*
_{2}〉) (47). Detailed microscopic pictures of conformational dynamics and its influence on enzymatic reactions are expected to emerge from single molecule studies.

↵* To whom correspondence should be addressed. E-mail: xsxie{at}pnl.gov