How the diffusivity profile reduces the arbitrariness of protein folding free energies

Abstract

The concept of a protein diffusing in its free energy folding landscape has been fruitful for both theory and experiment. Yet the choice of the reaction coordinate (RC) introduces an undesirable degree of arbitrariness into the problem. We analyze extensive simulation data of an α-helix in explicit water solvent as it stochastically folds and unfolds. The free energy profiles for different RCs exhibit significant variation, some having an activation barrier, others not. We show that this variation has little effect on the predicted folding kinetics if the diffusivity profiles are properly taken into account. This kinetic quasi-universality is rationalized by an RC rescaling, which, due to the reparameterization invariance of the Fokker-Planck equation, allows the combination of free energy and diffusivity effects into a single function, the rescaled free energy profile. This rescaled free energy indeed shows less variation among different RCs than the bare free energy and diffusivity profiles separately do, if we properly distinguish between RCs that contain knowledge of the native state and those that are purely geometric in nature. Our method for extracting diffusivity profiles is easily applied to experimental single molecule time series data and might help to reconcile conflicts that arise when comparing results from different experimental probes for the same protein.

Effect of viscogens on the kinetic response of a photoperturbed allosteric protein

“…It has therefore often been stated that a non-exponential response is related to diffusion in multidimensions. 58,59 That is to say that the coordinate along which we perturb and/or observe the system (i.e., the width of the binding groove in our concrete example) is not a "good" reaction coordinate, in the sense that a free energy projected onto that coordinate cannot describe the dynamics of the system with the help of a memory-free Langevin or diffusion equation. In the Appendix, we present a simple harmonic model with additional hidden coordinates.…”

Solvent fluctuations in hydrophobic cavity–ligand binding kinetics

“…Theoretically, binding rates are often calculated using implicit approaches, where explicit solvent effects are neglected or only coarsely described (21)(22)(23)(24). In the typical view of diffusive encounter of the two reactants, binding kinetics is then governed not only by the free energy (potential of mean force) along the reaction path but also the local diffusivity (or friction) profile (25)(26)(27)(28)(29), which, however, is a priori unknown due to missing microscopic insights. In fact, wet/dry hydration fluctuations in the pocket confinement may propagate a new time scale to binding, which can range from a few tens of picoseconds to hundreds of nanoseconds sensitively depending on the size and geometry of the nanocontainer (11,15,(30)(31)(32)(33)(34).…”

Communication: Local structure-mobility relationships of confined fluids reverse upon supercooling

“…43 as it traverses an arbitrary equilibrium free-energy landscape βF(z); in turn, both D z (z) and βF(z) govern the Markovian propagator G(z,t 0 + ∆t|z ′ ,t 0 ) that characterizes its temporal displacements. 12,15 Equation (1) is known to self-consistently model single-particle dynamics in dense inhomogeneous fluids given the PMF, βF(z) = − ln{ ρ(z)} + C (where ρ(z) is local density and C is an arbitrary constant), which encodes the influence of all other particles, walls, etc., on the free energy landscape of the tracer. 13 To calculate D z (z) profiles for each species from simulation data, we use a mean-first passage times (MFPTs) method 15,45,46 applicable for the steadystate (i.e., ∂G/∂t = 0) limit of Eq.…”

“…As a result, the static properties of confined fluids, such as local one-body density ρ(z), are now well-understood in terms of physical intuition (e.g., emergence of particle layering near boundaries to relieve packing frustration 1,2 ) and can be predicted using microscopic approaches like density functional theory. 3,4 However, much less is understood about what controls the dynamics of inhomogeneous fluids, and only recently have efforts broadened to include developing theories [5][6][7][8][9][10][11] and other tools [12][13][14][15][16] for characterizing particle dynamics both on a spatially averaged basis and as a function of position.…”