Protein Folding Dynamics as Diffusion on a Free Energy Surface: Rate Equation Terms, Transition Paths, and Analysis of Single-Molecule Photon Trajectories

Abstract

The rates of protein (un)folding are often described as diffusion on the projection of a hyperdimensional energy landscape onto a few (ideally one) order parameters. Testing such an approximation by experiment requires resolving the reactive transition paths of individual molecules, which is now becoming feasible with advanced single-molecule spectroscopic techniques. This has also sparked the interest of theorists in better understanding reactive transition paths. Here we focus on these issues aiming to establish (i) practical guidelines for the mechanistic interpretation of transition path times (TPT) and (ii) methods to extract the free energy surface and protein dynamics from the maximum likelihood analysis of photon trajectories (MLA-PT). We represent the (un)folding rates as diffusion on a 1D free energy surface with the FRET efficiency as a reaction coordinate proxy. We then perform diffusive kinetic simulations on surfaces with two minima and a barrier, but with different shapes (curvatures, barrier height, and symmetry), coupled to stochastic simulations of photon emissions that reproduce current SM-FRET experiments. From the analysis of transition paths, we find that the TPT is inversely proportional to the barrier height (difference in free energy between minimum and barrier top) for any given surface shape, and that dividing the TPT into climb and descent segments provides key information about the barrier's symmetry. We also find that the original MLA-PT procedure used to determine the TPT from experiments underestimates its value, particularly for the cases with smaller barriers (e.g., fast folders), and we suggest a simple strategy to correct for this bias. Importantly, we also demonstrate that photon trajectories contain enough information to extract the 1D free energy surface's shape and dynamics (if TPT is >4−5-fold longer than the interphoton time) using the MLA-PT directly implemented with a diffusive free energy surface model. When dealing with real (unknown) experimental data, the comparison between the likelihoods of the free energy surface and discrete kinetic three-state models can be used to evaluate the statistical significance of the estimated free energy surface.

The Effect of Time Resolution on Apparent Transition Path Times Observed in Single-Molecule Studies of Biomolecules

“…The changes in measured properties due to smoothing are significant even when the smoothing time is a small fraction (e.g., 10%) of the mean transition path time and the distortion of the apparent potential of mean force caused by smoothing is negligible. The effect of finite time resolution may be even more complicated when the experimental analysis involves additional data processing steps such as the maximum likelihood/hidden Markov analyses often used in single-molecule FRET experiments. ,,,,,− …”

“…The effect of finite time resolution may be even more complicated when the experimental analysis involves additional data processing steps such as the maximum likelihood/hidden Markov analyses often used in single-molecule FRET experiments. 2 , 3 , 6 , 8 , 23 , 31 − 35 …”

Effects of Hydrodynamic Backflow on the Transmission Coefficient of a Barrier-Crossing Brownian Particle

“…Chemical reactions that take place in condensed-phase media can sometimes be modeled by the stochastic evolution of a single coarse-grained reaction coordinate along a metastable free energy surface, effectively reducing the process to the motion of a colloidal particle over a potential barrier in the presence of thermal fluctuations. This reduced description of chemical dynamics has proved to be a useful theoretical framework for interpreting the results of experiments and simulations, not just of small molecules but even of large biological systems . In particular, it has added considerably to our understanding of dynamic disorder in single enzyme kinetics − and of inertial and memory effects on the transitions of proteins between different conformational substates. − These and other related effects have often been traced to molecular interactions that occur on picosecond and subpicosecond time scales.…”

“…This reduced description of chemical dynamics has proved to be a useful theoretical framework for interpreting the results of experiments and simulations, 1 not just of small molecules but even of large biological systems. 2 In particular, it has added considerably to our understanding of dynamic disorder in single enzyme kinetics 3−6 and of inertial and memory effects on the transitions of proteins between different conformational substates. 7−13 These and other related effects have often been traced to molecular interactions that occur on picosecond and subpicosecond time scales.…”

Microscopic origin of diffusive dynamics in the context of transition path time distributions for protein folding and unfolding

“…These, in turn have triggered many theoretical studies. [21][22][23][24][25][26][27][28][29][30][31] In most theoretical analyses, the dynamics is modeled in terms of a diffusion equation along a free energy surface connecting the stable states. The theoretical studies included a variety of dynamical effects, such as the role of memory [23][24][25]27,29,30,32 inertial contributions, 21,22 multidimensional transition paths 28 and barrier shapes.…”