Measuring the average shape of transition paths during the folding of a single biological molecule

Abstract

Transition paths represent the parts of a reaction where the energy barrier separating products and reactants is crossed. They are essential to understanding reaction mechanisms, yet many of their properties remain unstudied. Here, we report measurements of the average shape of transition paths, studying the folding of DNA hairpins as a model system for folding reactions. Individual transition paths were detected in the folding trajectories of hairpins with different sequences held under tension in optical tweezers, and path shapes were computed by averaging all transitions in the time domain, 〈t(x)〉, or by averaging transitions of a given duration in the extension domain, 〈x(tjτ)〉 τ . Whereas 〈t(x)〉 was close to straight, with only a subtle curvature, 〈x(tjτ)〉 τ had more pronounced curvature that fit well to theoretical expectations for the dominant transition path, returning diffusion coefficients similar to values obtained previously from independent methods. Simulations suggested that 〈t(x)〉 provided a less reliable representation of the path shape than 〈x(tjτ)〉 τ , because it was far more sensitive to the effects of coupling the molecule to the experimental force probe. Intriguingly, the path shape variance was larger for some hairpins than others, indicating sequence-dependent changes in the diversity of transition paths reflective of differences in the character of the energy barriers, such as the width of the barrier saddle-point or the presence of parallel paths through multiple barriers between the folded and unfolded states. These studies of average path shapes point the way forward for probing the rich information contained in path shape fluctuations. energy landscapes | diffusive reactions | DNA hairpins | optical tweezers T ransition paths involve those segments of a reaction during which the energy barrier between reactants and products is crossed (1, 2). They represent the most interesting part of any reaction because they exclude the nonproductive fluctuations, focusing only on the productive portions of the trajectories (Fig. 1). In particular, the high-energy states occupied along the transition paths dominate the reaction kinetics and effectively encode the reaction mechanism. Transition paths are especially interesting for understanding the folding of biological molecules like proteins and nucleic acids, because of the variety and complexity of possible mechanisms. They are technically challenging to measure in folding reactions, however, because they can only be observed in single molecules and have a very brief duration. As a result, it has only recently become possible to measure transition path properties directly (3). Work to date using fluorescence and force spectroscopy has probed properties such as the average transition path time for proteins and nucleic acids (4-8), the variations in transit times for individual transitions (9, 10), the occupancy statistics within transition paths (11,12), the distribution of velocities along the transition paths (13), and the agreement between expe...

Broad distributions of transition-path times are fingerprints of multidimensionality of the underlying free energy landscapes

“…Our analysis of toy 2D diffusion models and of the MSM of a protein suggests that broader-than-single-exponential distributions of the transition path time arise as a result of multiple parallel reaction pathways. Consistent with this proposal, recent experimental (49,50) and computational (18) observations have already indicated heterogeneity of the ensembles of the biomolecular transition paths, but such heterogeneity is a property that is not easy to quantify. In particular, Hoffer et al (50) invoke energy landscapes with multiple saddles or with broad saddles (as in SI Appendix, Fig.…”

Time-dependent communication between multiple amino acids during protein folding

“…It was demonstrated for dielectric and density uctuations in glass-forming systems that c 4 (t) 2 exhibits essentially the same dynamics as c(t). 34 Since the average function F(t) is intimately related to the "shape" function of the transition path that is now experimentally accessible, 35 measuring c 4 (t) by varying experimental conditions will provide experimental evidence of the microscopic cooperativity in protein folding.…”

Disordered proteins follow diverse transition paths as they fold and bind to a partner

“…Although the transition path contains all of the structural information about the process, it has been experimentally studied only recently using single-molecule spectroscopy because barrier crossing is stochastic and thus cannot be synchronized for ensemble measurements and because transition path times are very short (1)(2)(3). To date, transition paths have been studied for folding of proteins (1,3) and nucleic acids (4,5) and binding of disordered proteins (6, 7) by using either two-color single-molecule fluorescence or single-molecule force spectroscopy, both of which provide information along only a one-dimensional (1D) reaction coordinate (single distance). Theory and atomistic molecular dynamics simulations, however, predict that individual transition paths for macromolecular conformational changes such as protein folding are very different from reaction pathways for small-molecule chemical reactions (8)(9)(10)(11)(12), even though the kinetics have been successfully described on a 1D free-energy surface (9,13,14).…”

“…The diversity of protein folding and unfolding pathways is a central feature of the energy landscape theory and has been studied for several naturally evolved proteins (31)(32)(33)(34) and indirectly observed for DNA hairpins through variance in transition path shape (5). The diverse pathways on an energy landscape funneled toward the native structure enable fast and efficient protein folding-because it would take a long time to find a single, well-defined folding pathway (8,12,35,36)-but have not been experimentally measured.…”