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

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 alpha-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.

💡 Research Summary

The paper tackles a long‑standing problem in protein‑folding theory: the choice of reaction coordinate (RC) introduces an unwanted degree of arbitrariness when one describes folding as diffusion on a free‑energy landscape. Using extensive explicit‑water molecular dynamics of an α‑helix peptide, the authors construct free‑energy profiles F(q) and position‑dependent diffusivity profiles D(q) for four markedly different RCs: (i) RMSD‑based distance, (ii) number of native contacts, (iii) a purely geometric count of atoms, and (iv) the native‑state overlap Q‑value, which explicitly encodes knowledge of the folded state.

The free‑energy curves differ dramatically. Some RCs display a clear activation barrier separating unfolded and folded basins, while others appear barrier‑free. If one were to predict folding kinetics solely from F(q), the resulting mean first‑passage times (MFPTs) would vary by orders of magnitude, reflecting the arbitrariness of the RC. However, the authors also extract D(q) from the same trajectories by measuring mean‑square displacements and transition probabilities within narrow q‑windows. D(q) is far from uniform: RCs that contain native‑state information show a strong reduction of diffusivity near the folded basin, whereas purely geometric RCs exhibit relatively flat D(q).

When the full one‑dimensional Fokker‑Planck equation

∂P(q,t)/∂t = ∂/∂q