Single-molecule Forster resonance energy transfer (FRET) between fluorescent donor and acceptor labels attached to a protein or nucleic acid is widely used to probe intramolecular distances and study the structure, dynamics and function of macromolecules. In these experiments, a molecule is illuminated by a laser, and the donor fluorophore is excited. The donor can emit a photon or transfer the excitation to an acceptor which then can emit a photon of a different color. The rate of transfer depends on the interdye distance and this is why there is information about conformational dynamics. The output of these experiments is a sequence of photons with recorded colors and arrival times. When a single molecule is excited by a pulsed laser, it is also possible to detect the time interval between the laser pulse and the photon. This so-called delay time is related to the fluorescence lifetime of the donor fluorophore. The distances between fluorescence labels attached to a molecule fluctuate due to conformational dynamics on a wide range of time scales. Extracting information about the dynamics is particularly challenging when the fluctuations are as fast as the time between photons. During the last year we have been working on extending our previous work to three color FRET in which three dyes are attached to the protein of interest (one donor and two acceptors). These experiments contain more information (three instead of one distances) than the usual two color FRET, but the theory required to analyze them is also more challenging. The theory and analysis of three-color FRET have been applied to probe conformational dynamics of a fast-folding protein, 3D, in collaboration with Dr. Hoi Sung Chung and coworkers, LCP (reference 1). Since the folded and unfolded states cannot be distinguished in binned fluorescence trajectories, we used a maximum likelihood method that analyzes photons without binning. This method has been previously developed for two-color FRET, and we extended it to three-color photon sequences. The extracted kinetic parameters agree very well with the previously measured parameters for the same protein with two-color FRET. From the extracted fractions of acceptor photon counts, the FRET efficiencies for all three dye pairs were calculated after various corrections. We found that the FRET efficiencies obtained in three-color and two-color measurements can be different because fluctuations of all three interdye distances contribute to the FRET efficiency measured in three-color FRET. We show that this difference can be accounted for by using the Gaussian chain model for the unfolded state with the parameters obtained from the analysis of two-color segments. In addition to photon counts, the delay times of photons from the laser pulse were analyzed and fluorescence lifetimes were determined using the maximum likelihood analysis. The correlation between FRET efficiencies and lifetimes of the donor, acceptor 1, and acceptor 2 was visualized in two-dimensional FRET efficiencylifetime histograms. These histograms demonstrated the presence of conformational dynamics in a protein. Another direction of our research is understanding the role of diffusion on the kinetics of a complex networks of bimolecular reactions in which the molecules have multiple binding sites or can undergo conformational changes that alter their reactivity. In such cases new processes occur that have not been incorporated into ordinary chemical kinetics. The treatment of such effects appeared to make the formalism prohibitively complicated and the solution of the problem eluded us for many years. Recently, we discovered that these difficulties disappeared when we focused not on individual binding and dissociation events but rather on the net flux associated with each reaction. In this way, we were able to develop a completely general but surprisingly simple way of treating networks of diffusion- influenced reactions (reference 2). Our key result is a set of non-Markovian rate equations involving stoichiometric matrices and net reaction rates (fluxes), in which these rates are coupled by a time-dependent pair association flux matrix, whose elements have a simple physical interpretation. Specifically, each element is the probability density that an isolated pair of reactants irreversibly associates at time t via one reaction channel on the condition that it started out with the dissociation products of another (or the same) channel. In the Markovian limit, the coupling of the chemical rates is described by committors (or splitting/capture probabilities). The committor is the probability that an isolated pair of reactants formed by dissociation at one site will irreversibly associate at another site rather than diffuse apart. We illustrate the use of our formalism by considering three reversible reaction schemes: (1) binding to a single site, (2) binding to two inequivalent sites, and (3) binding to a site whose reactivity fluctuates. In the first example, we recover the results published earlier, while in the second one we show that a new reaction channel appears, which directly connects the two bound states. The third example is particularly interesting because all species become coupled and an exchange-type bimolecular reaction appears. In the Markovian limit, some of the diffusion-modified rate constants that describe new transitions become negative, indicating that memory effects cannot be ignored. As an application of our general theory, we considered the influence of diffusion on the kinetics of ligand binding to a macromolecule with two sites (reference 3) where, in the reaction-controlled limit, there is no cooperativity and hence the sites are independent. We show that the rate constants of chemical kinetics cannot just be renormalized. Rather a new reaction channel, which connects the two singly occupied states, must be introduced. The rate constants of this new channel depend on the committor or capture probability that a ligand that just dissociated from one site rebinds to the other. This result is rederived in an elementary way using the encounter complex model. Illustrative calculations are presented where the kinetics of the fractional saturation of one site is compared with that of a macromolecule that has only this site. If all sites are initially empty, then the second site slows down binding to the first due to competition between the sites. On the other hand, if the second site is initially occupied, the binding of the first site speeds up because of the direct diffusion-induced transitions between the two singly bound states. One of the great challenges of molecular simulations is to determine times for processes such as barrier crossing that would take forever to calculate by brute force. Milestoning is a procedure that can string together many short trajectories in a clever way to solve this problem. One starts by choosing a set of points, called milestones, and then initiate short trajectories from each milestone, that are terminated when they reach an adjacent milestone for the first time. From the average duration of these trajectories and the probabilities of where they terminate, a rate matrix can be constructed and then used to calculate the mean first-passage time (MFPT) between any two milestones. All these MFPT's turn out to be exact even when they are extremely long because the milestones are separated by a large barrier. Here we adopt a point of view from which this remarkable result is not unexpected. In addition, we clarify the nature of the states whose interconversion is described by the rate matrix constructed using information obtained from short trajectories and provide a microscopic expression for the equilibrium population of these states in terms of equilibrium averages of the committor.
| Cossio, Pilar; Hummer, Gerhard; Szabo, Attila (2018) Transition paths in single-molecule force spectroscopy. J Chem Phys 148:123309 |
| Chung, Hoi Sung; Meng, Fanjie; Kim, Jae-Yeol et al. (2017) Oligomerization of the tetramerization domain of p53 probed by two- and three-color single-molecule FRET. Proc Natl Acad Sci U S A 114:E6812-E6821 |
| Berezhkovskii, Alexander M; Szabo, Attila; Rotbart, T et al. (2017) Dependence of the Enzymatic Velocity on the Substrate Dissociation Rate. J Phys Chem B 121:3437-3442 |
| Hummer, Gerhard; Szabo, Attila (2017) Dynamics of the Orientational Factor in Fluorescence Resonance Energy Transfer. J Phys Chem B 121:3331-3339 |
| Gopich, Irina V; Szabo, Attila (2016) Reversible Stochastically Gated Diffusion-Influenced Reactions. J Phys Chem B 120:8080-9 |
| Cossio, Pilar; Hummer, Gerhard; Szabo, Attila (2016) Kinetic Ductility and Force-Spike Resistance of Proteins from Single-Molecule Force Spectroscopy. Biophys J 111:832-840 |
| Chung, Hoi Sung; Louis, John M; Gopich, Irina V (2016) Analysis of Fluorescence Lifetime and Energy Transfer Efficiency in Single-Molecule Photon Trajectories of Fast-Folding Proteins. J Phys Chem B 120:680-99 |
| Berezhkovskii, Alexander M; Szabo, Attila (2016) Theory of Crowding Effects on Bimolecular Reaction Rates. J Phys Chem B 120:5998-6002 |
| Gopich, Irina V; Szabo, Attila (2016) Influence of diffusion on the kinetics of multisite phosphorylation. Protein Sci 25:244-54 |
| Cossio, Pilar; Hummer, Gerhard; Szabo, Attila (2015) On artifacts in single-molecule force spectroscopy. Proc Natl Acad Sci U S A 112:14248-53 |
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