Single-molecule spectroscopy can monitor conformational changes of a macromolecule containing fluorophores whose photophysics is directly influenced by such changes, e.g., Forster resonance energy transfer (FRET) and fluorescence quenching. In addition, it can be used to study the influence of conformational changes on the kinetics of chemical reactions such as enzyme catalysis when one of the intermediates fluoresces. In both cases the experimental output is a photon trajectory which contains information about the nature and time scale of the underlying conformational changes. New kinds of experiments require new kinds of theories to analyze them. Classically, the emission of a photon is associated with a kinetic transition between two states and can be described by the rate equations of chemical kinetics. To analyze experiments, one must be able to describe the statistics of such transitions within the framework of a microscopic model of the dynamics. ? ? As described in last years report, we have developed the general formalism for obtaining the statistics of monitored transitions from a microscopic model when the dynamics is described by master or rate equations or their continuum analog, multidimensional reaction-diffusion equations (1). The focus was on the distribution of the number of transitions during a fixed observation time, the distribution of times between transitions, and the corresponding correlation functions. We showed how these quantities are related to each other and how they can be explicitly calculated in a straightforward way for both immobile and diffusing molecules. Our formalism reduces to renewal theory when the monitored transitions either go to or originate from a single state. The influence of dynamics slow compared with the time between monitored transitions is treated in a simple way, and the probability distributions are expressed in terms of Mandel-type formulas. ? ? This year we have applied the general theory to various systems studied experimentally by single-molecule spectroscopy. These include the analysis of (A) catalytic turnovers of enzymes (in collaboration with the X. Sunney Xie group, Harvard University), (B) ultrafast dynamics of unfolded proteins (in collaboration with the B. Schuler group, University of Zurich) and (C) FRET efficiency distributions in protein folding (in collaboration with the W. Eaton group, LCP/NIDDK). These three projects lead to joint publications ((2), (3), and (4)), the results of which are briefly described below.? ? (A) 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 0.0001-10 second range. We examined various scenarios in which slowly fluctuating enzymes would still obey the MM relationship (2). For this purpose, we have developed a theoretical formalism to describe the kinetics of a fluctuating enzyme that incorporates the condition of detailed balance for substrate binding and dissociation. Our focus is on the dependence of the steady-state rate of product formation on the substrate concentration, which can be influenced by the time scale and amplitude of enzyme fluctuations. It has been shown that although a fluctuating enzyme does not in general exhibit MM steady-state kinetics, it does so in a large region of parameter space, albeit with apparent Michaelis and catalytic rate constants that have different microscopic interpretations. Thus, deviations from MM behavior may occur rarely and/or be small and difficult to detect experimentally. Many enzymes for which the MM relation holds on the ensemble level exhibit dynamic fluctuations in protein conformation and catalytic activity with a broad range of time scales on the single-molecule level. These notions have implications to biochemistry and cell biology, especially for systems containing a low copy number of enzyme molecules. ? ? (B) The role of unfolded states in determining protein folding mechanisms is largely unknown. Their dynamics in the submicrosecond range have recently been probed by measuring the statistics of photon emission from single cold shock proteins from Thermotoga maritima (CspTm) (3).The protein was labeled terminally with a green fluorescent donor and a red fluorescent acceptor dye, and freely diffusing molecules were observed in confocal single-molecule experiments. During the transit of the protein through the observation volume, its donor chromophore is excited by the laser beam. Depending on the distance r to the acceptor, energy transfer results with a rate that determines the relative probabilities of photon emission from donor and acceptor. Correspondingly, distance dynamics within the protein can be measured by fluctuations in the transfer efficiency and thus in the fluorescence emission of the chromophores. ? ? To analyze the measured emission of photons in terms of protein dynamics, we describe the relative motion of the chain ends as a diffusive process on the potential of mean force that corresponds to the end-to-end distance distribution of the unfolded protein. By combining these diffusive chain dynamics with the distance-dependent stochastic photon emission from the coupled dye pair, the complete photon statistics of the system can be obtained. We calculate the intensity correlation functions of donor and acceptor emission numerically. By adjusting the measured and calculated intensity correlation function, the dynamics of an unfolded protein in terms of the diffusion coefficient and the corresponding reconfiguration time is determined (3). Our analysis shows that global reconfiguration of the chain occurs on a time scale of 50 nanosecons and slows down concomitant with chain collapse under folding conditions. These diffusive dynamics provide a missing link between the phenomenological chemical kinetics commonly used in protein folding and a physical description in terms of quantitative free energy surfaces.? ? (C) Single-molecule FRET is ideally suited to investigate the structure and dynamics of unfolded proteins because, unlike ensemble studies, it can resolve the folded and unfolded subpopulations in low denaturant. Quantitative analysis of single-molecule FRET measurements has been used to compare two-state proteins, protein L and cold-shock protein CspTm (4). The FRET efficiency distribution widths were analyzed to extract dynamical information on the unfolded protein chain on the millisecond time scale. Dynamics at times comparable to or longer than the interval between detected protons can increase the width of the FRET efficiency distribution beyond that expected from shot noise resulting from the discrete nature of photons. The folded protein peaks for protein L and CspTm have shot-noise limited widths, whereas those of the unfolded subpopulations of both proteins have widths in excess of the shot noise. Photophysical artifacts such as transient population of triplet states, acceptor blinking, or photobleaching would have to occur on a time scale comparable to or longer than the average interval between photons to affect the width. The additional widths must therefore arise from conformational microstates of the dye-labeled protein system and do not interconvert on time scales longer than 1 ms. It remains to be determined whether these heterogeneities in the unfolded state are intrinsic to the protein dynamics or result from proteindye interaction.
