Accurate Protein-Folding Transition-Path Statistics from a Simple Free-Energy Landscape

Abstract

A central goal of protein-folding theory is to predict the stochastic dynamics of transition paths-the rare trajectories that transit between the folded and unfolded ensembles-using only thermodynamic information, such as a low-dimensional equilibrium free-energy landscape. However, commonly used one-dimensional landscapes typically fall short of this aim, because an empirical coordinate-dependent diffusion coefficient has to be fit to transition-path trajectory data in order to reproduce the transition-path dynamics. We show that an alternative, first-principles free-energy landscape predicts transition-path statistics that agree well with simulations and single-molecule experiments without requiring dynamical data as an input. This "topological configuration" model assumes that distinct, native-like substructures assemble on a time scale that is slower than native-contact formation but faster than the folding of the entire protein. Using only equilibrium simulation data to determine the free energies of these coarse-grained intermediate states, we predict a broad distribution of transition-path transit times that agrees well with the transition-path durations observed in simulations. We further show that both the distribution of finite-time displacements on a one-dimensional order parameter and the ensemble of transition-path trajectories generated by the model are consistent with the simulated transition paths. These results indicate that a landscape based on transient folding intermediates, which are often hidden by one-dimensional projections, can form the basis of a predictive model of protein-folding transition-path dynamics.

Barrier Crossing Dynamics from Single-Molecule Measurements

“…Another open question is how to quantify the heterogeneity of the transition path ensembles suggested by experimental , and theoretical ,, studies. For simple systems, heterogeneity is easy to understand intuitively.…”

“…Heterogeneity of transition path ensembles has recently been suggested by experimental studies 43,44 and is also evident in simulations. 32,45 Unfortunately, this is a property that is somewhat fuzzy and hard to quantify. The coefficient of variation of the distribution p TP (t) appears to provide a useful proxy for the ensemble heterogeneity, but whether better, more direct measures of this property exist remains an open question.…”

“…This question remains open, but given the renewed interest in non-Markov processes reflected in recent publications in mathematics, physics, and chemistry (see, e.g., refs33, 66, and 67), other mathematical models of non-Markov dynamics may soon emerge as better descriptions of experimental observations. Another open question is how to quantify the heterogeneity of the transition path ensembles suggested by experimental 43,44 and theoretical 32,45,68 studies. For simple systems, heterogeneity is easy to understand intuitively.…”

Out-of-Equilibrium Biophysical Chemistry: The Case for Multidimensional, Integrated Single-Molecule Approaches

“…Despite the proven utility of 1-D energy landscapes, several simplifications inherent to this conceptualization make functionally important biomolecular dynamics less clear and may lead to the misinterpretation of experimental results, among other potential problems. For example, experiments typically yield data on experimentally tractable observable coordinates , such as end-to-end distance, radius of gyration, bond distances, bond angles, or affinities between the interacting molecules, rather than actual reaction coordinates. , Only in the most simple systems, a slip-bond type receptor–ligand interaction, for example, do the reaction coordinate and the observable coordinate coincide . Therefore, one must carefully consider how data collected along an observable coordinate projects onto the reaction coordinate to make a genuinely quantitative analysis using the 1-D energy landscape paradigm, as discussed for folding energy landscapes. ,− …”

“…For decades, structural biology’s goal has been to find and present representative structures corresponding to local minima (i.e., macrostates) in the functional energy landscape, , and X-ray crystallography and cryo-electron microscopy (cryo-EM) have been widely successful at doing so. We have learned much about biological macromolecular mechanisms by determining structures in various conditions (e.g., ligand-bound and unbound states) and inferring dynamics between these structures.…”