Theoretical Studies On The Dynamic Aspects Of Macromolecular Function

National Institute Of Diabetes And Digestive And Kidney Diseases

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Abstract

Single-molecule FRET (smFRET) has become a mainstream technique for studying biomolecular structural dynamics. 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 inter-dye 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. The distances between fluorescence labels attached to a molecule fluctuate due to conformational dynamics on a wide range of time scales. The rapid and wide adoption of smFRET experiments by an ever-increasing number of groups has generated significant progress in sample preparation, measurement procedures, data analysis, algorithms and documentation. The paper that describes the current state of the art from different perspectives that was described in the last report has now appeared in print (eLife 2021; 10:e60416, see below for DOI). This year we have been working on the analysis of single-molecule experiments to get molecule's diffusion coefficient and brightness. This work is being done in collaboration with Dr. Hoi Sung Chung from LCP, who studies oligomer formation in amyloid beta (A) aggregation on a single-molecule level. In these experiments, A-peptides are labeled by donor or acceptor fluorophores and mixed together. At the early stage of aggregation, oligomers start to form. The oligomers freely diffuse in solution and occasionally diffuse into the laser confocal spot. Inside the spot, the donor fluorophore is excited and can emit a donor photon or it can transfer the energy to the acceptor, which can emit an acceptor photon. Since the monomers can only emit donor photons, the bursts of donor and acceptor photons emitted by the oligomers can be differentiated by FRET efficiency. The number of oligomers is much smaller than that of the monomers. The question is what can we learn about oligomers from a small number of bursts of photons. Informative parameters would be the brightness and the diffusion coefficient of the molecules, which depend on the oligomer size. The standard FCS correlation function, which is commonly used to get the diffusion coefficient, is not helpful in this case, since the most intense signal comes from the monomers. Moreover, we would also like to use our knowledge of the molecules brightness, which is not included in the correlation function. To characterize the oligomers, we developed a new method that allows one to get both the diffusion coefficient and the photon count rate from a small number of selected bursts of photons. The selection criteria include thresholds for the inter-photon times, the number of photons in a burst, and the FRET efficiency. In this approach, we construct a likelihood function which accounts for these selection criteria and optimize it with respect to the parameters. To avoid complexity and to decrease the computational effort, the elongated laser spot is replaced by an effective isotropic spot with the size of the spot calculated from the condition of the best fit. We have been using the Maximum Likelihood method previously to get FRET efficiencies of single molecules, which are related to the distances between fluorophore labels, as well as the transition rates in protein folding and binding. However, including translational diffusion into analysis happened to be very challenging since burst selection modifies photon statistics and the likelihood function, and could not be circumvented. The new method has been tested by comparison with simulated photon trajectories in the elongated laser spot, and the results are promising. The diffusion coefficient and the count rate can be accurately determined from bursts with small numbers of photons. The work is still in progress, and we expect successful application of the new method to a number of important molecular systems, including protein folding and aggregation of A-peptides. In collaboration with Dr. A.M. Berezhkovskii, we published a paper which just appeared online (doi.org/10.1021/acs.jpcb.2c03757), in which we derive remarkable relations among the numbers of transitions between two boundaries (fluxes), the probability of reaching one boundary before the other (committors), and the time it takes to get from one boundary to the other (first passage times). These boundaries specify what is a reactant and a product in a chemical reaction, so the above-mentioned quantities play central role in describing the kinetics (how concentrations change with time) of reactions where one molecule is transformed to another. For example, the first passage time is just the inverse of the rate and transition states are those conformations for which the committor is equal to one-half. To appreciate the importance of our work, some background is needed. Statistical mechanics that allows us to understand transformations of matter in microscopic terms (i.e., on a molecular level) is more than a century old. Starting in the beginning of this century, a new approach has emerged stimulated largely by computer simulations of the dynamics of molecules, where a trajectory (i.e., the position of every atom in a molecule as a function of time) is obtained. The newly emerging statistical mechanics of trajectories, which focuses on analyzing trajectories, has lead to a number of new and unexpected theoretical results that surprisingly were not discovered before. The relations that are the focus of our paper are arguably the most striking of these. Our algebraic derivation of them provides new insight into their nature and opens the door to using them to speed up simulations in order to study chemical reactions that are too slow to be studied in a brute-force way.