Kramers developed the theory on how chemical reaction rates are influenced by the viscosity of the medium1,2. At the viscosity of water, the kinetics of unimolecular reactions are described by diffusion of a Brownian particle over a free-energy barrier separating reactants and products. For reactions in solution this famous theory extended Eyring’s transition state theory, and is widely applied in physics, chemistry and biology, including to reactions as complex as protein folding3,4. Because the diffusion coefficient of Kramers’ theory is determined by the dynamics in the sparsely populated region of the barrier top, its properties have not been directly measured for any molecular system. Here we show that the Kramers diffusion coefficient and free-energy barrier can be characterized by measuring the temperature- and viscosity-dependence of the transition path time for protein folding. The transition path is the small fraction of an equilibrium trajectory for a single molecule when the free-energy barrier separating two states is actually crossed. Its duration, the transition path time, can now be determined from photon trajectories for single protein molecules undergoing folding/unfolding transitions5. Our finding of a long transition path time with an unusually small solvent viscosity dependence suggests that internal friction as well as solvent friction determine the Kramers diffusion coefficient for α-helical proteins, as opposed to a breakdown of his theory, which occurs for many small-molecule reactions2. It is noteworthy that the new and fundamental information concerning Kramers’ theory and the dynamics of barrier crossings obtained here come from experiments on a protein rather than a much simpler chemical or physical system.
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We are particularly indebted to J. M. Louis for the preparation, dye labelling and purification of the protein used in this work, with technical assistance from A. Aniana. We also thank R. Best, G. Hummer and A. Szabo for discussions and comments on the manuscript, and D.E. Shaw Research for providing access to their molecular dynamics trajectories for the calculations by R. Best. This work was supported by the Intramural Research Program of the National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health.
The authors declare no competing financial interests.
Extended data figures and tables
Dyes were attached to the cysteine residues (red) and a biotin molecule was attached to the lysine residue (blue) in the AviTag sequence.
a, The definition of photon indices and time interval of a photon trajectory with a folding transition. b, c, Photon trajectories were analysed using the two-state model to determine kinetics parameters (b) or the three-state model to determine the average transition path times (tTP = 1/2kS) (c).
Extended Data Figure 3 FRET efficiency histograms of α3D in 2.25 M GdmCl solution at different temperatures.
The FRET efficiency histograms were constructed from 1-ms bins in the trajectories with the mean photon count rate >40 ms−1. Wide and narrow bars are the experimental histograms and the histograms constructed from re-coloured photon trajectories using the parameters obtained from the maximum likelihood method with the two-state model (Extended Data Table 1), respectively. The agreement between the two histograms validates the description of α3D as a two-state folder8. The similar ratio of the integral of the folded (high FRET) and the unfolded (low FRET) distributions indicates that the equilibrium constant is unchanged over the temperature range of the measurement, as shown more precisely in the maximum likelihood analysis. At high temperature and low pH, where the 11 glutamates and 1 aspartate are protonated, more than two states are observed37,38.
Black solid lines are exponential functions that best fit the data. The fitting parameters are listed in Extended Data Table 1.
The FRET efficiency histograms were constructed from 1-ms bins in the trajectories with the mean photon count rate >50 ms−1 for 2.25 M and 3.2 M GdmCl and from 2-ms bins in the trajectories with the mean photon count rate >30 ms−1 for 4.6 M, 4.3 M and 3.8 M GdmCl concentrations. At the relative viscosity (η/η0) 1, 10 and 38, the higher concentration of GdmCl was used to counteract the stabilization of proteins by glycerol to maintain the ratio of folded to unfolded molecules as close to unity as practically possible. The similar ratio of the integral of the folded (high FRET) and the unfolded (low FRET) distributions indicates that the equilibrium constant is unchanged at these conditions, as shown more precisely in the maximum likelihood analysis.
Extended Data Figure 6 Donor–acceptor cross-correlation of the segments of the fluorescence trajectories corresponding to the unfolded state (ref. 24).
a, Black solid lines are exponential functions that best fit the data. The fitting parameters are listed in Extended Data Table 2. b, The unfolded state dynamics are slowed approximately linearly by the solvent viscosity as previously observed at high denaturant concentrations24. The relaxation time at η/ η0 = 1 (aqueous solution) is too fast to be measured by this method.
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Chung, H., Eaton, W. Single-molecule fluorescence probes dynamics of barrier crossing. Nature 502, 685–688 (2013). https://doi.org/10.1038/nature12649
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