Fast protein folding is governed by memory-dependent friction

Abstract

When described by a low-dimensional reaction coordinate, the folding rates of most proteins are determined by a subtle interplay between free-energy barriers, which separate folded and unfolded states, and friction. While it is commonplace to extract free-energy profiles from molecular trajectories, a direct evaluation of friction is far more elusive and typically relies on fits of measured reaction rates to memoryless reaction-rate theories. Here, using memory-kernel extraction methods founded on a generalized Langevin equation (GLE) formalism, we directly calculate the time-dependent friction acting on the fraction of native contacts reaction coordinate Q, evaluated for eight fast-folding proteins, taken from a published set of large-scale molecular dynamics protein simulations. Our results reveal that, across the diverse range of proteins represented in this dataset, friction is more influential than free-energy barriers in determining protein folding rates. We also show that proteins fold in a regime where the finite decay time of friction significantly reduces the folding times, in some instances by as much as a factor of 10, compared to predictions based on memoryless friction.

Keywords: generalized Langevin equation; molecular friction; non-Markovian dynamics; protein folding; reaction rate theory.

Fig. 1.

The folding and unfolding of eight fast-folding proteins. (A) Native states for all proteins with the number of amino acids indicated. All protein trajectories are taken from ref. . (B) 250 μs trajectory segment for the Q(t) reaction coordinate (α₃D protein). Left magnification: a sequence of folding first-passage events, from the unfolded state Q_u to the barrier topQ_b. Right magnification: an example folding transition path and corresponding transition path time τ_TP. (C) Free-energy profile for the α₃D protein. Configuration snapshots show example unfolded, barrier-top, and folded states. Q_u, Q_b, and Q_f are the reaction coordinate values in the unfolded, barrier-top, and folded states, respectively. For α₃D, the barrier faced by the unfolded protein, $U_0^{\mathrm{u}}=U(Q_{\mathrm{b}})-U(Q_{\mathrm{u}})=1.8k_{\mathrm{B}}T$, is less than the barrier faced by the folded protein $U_0^{\mathrm{f}}=U(Q_{\mathrm{b}})-U(Q_{\mathrm{f}})=3.2k_{\mathrm{B}}T$. The distance from the unfolded state to the barrier top L_u = Q_b − Q_u is greater than the distance from the folded state to the barrier top L_f = Q_f − Q_b. (D) Normalized GLE memory kernel Γ(t), extracted from Q(t) for the α₃D protein. Inset: Running integral G(t) (black line), the limiting total friction γ (dashed line), and an exponential fit (red curve). (E) Total friction γ for each protein, divided by k_BT, plotted as a function of the number of residues N, (power law with exponent ∼2.8 (red line)). (F) Effective mass m, plotted as a function of N (power law with exponent ∼1.5 (red line)). (G) and (H) show the Arrhenius factor ${\mathrm{e}}^{{\mathrm{U}}_0/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}$ and barrier crossing times ${\tau }_{\mathrm{MFP}}^{\mathrm{MD}}$ for folding and unfolding transitions individually.

Fig. 2.

Comparison of relevant time scales for protein folding and unfolding kinetics. (A) The inertia time scales τ_m compared to the diffusion times τ_D, showing that all systems are in the over-damped regime. (B) Memory decay time scales τ_mem, calculated from the first moment of the memory kernel Γ(t), compared to the diffusion times τ_D . The light gray lines indicate the bounding regime for τ_mem between τ_D × 10⁻² and τ_D × 10¹, which is the domain in which memory-induced kinetic acceleration is expected. (C) Memory times τ_mem compared to the folding and unfolding times, expressed as the mean first-passage times ${\tau }_{\mathrm{MFP}}^{\mathrm{MD}}$. (D) Memory times compared to the transition path times τ_MTP leading from the folded and unfolded state minima to the barrier tops. The broken lines in each plot indicate exact equivalence between the respective times.

Fig. 3.

Comparison of simulated protein folding and unfolding times ${\tau }_{\mathrm{MFP}}^{\mathrm{MD}}$ with predictions on different levels of theory. (A) Diffusion times τ_D, according to Eq. 2, which do not depend on free-energy barriers. (B) Markovian predictions ${\tau }_{\mathrm{MFP}}^{\mathrm{Mar}}$, according to Eq. 3, which use the extracted free-energy profile for each protein. (C) Free-energy-dependent factor ξ_U that accounts for the effect of the free energy profile but not for the friction, according to Eq. 3. (D) Non-Markovian predictions ${\tau }_{\mathrm{MFP}}^{\mathrm{noMar}}$, according to Eq. 5. Multiexponential memory kernels are explicitly accounted for. The root-mean-squared deviations of the logarithmically transformed data (RMSLD) provides a measure of relative deviations for each model prediction from the observed data ${\tau }_{\mathrm{MFP}}^{\mathrm{MD}}$ over the combined set of folding and unfolding times. A lower RMSLD indicates that a given model prediction is more accurate, where the dashed lines indicate an exact prediction. In (C), $\tau {\prime }_{\mathrm{D}}$ is calculated using γ = 1.1 × 10⁹unm²μs⁻¹ for all proteins, chosen to minimize the RMSLD. See Fig. 2 for symbol legend.

Fig. 4.

Barrier crossing times indicate memory-induced speed up. (A) Deviations from Markovian barrier crossing kinetics, plotted as a function of rescaled memory times τ_mem/τ_D, for the folding and unfolding of all eight proteins. Rescaled MD simulation values for each protein (${\tau }_{\mathrm{MFP}}^{\mathrm{MD}}/{\tau }_{\mathrm{MFP}}^{\mathrm{Mar}}$, red and black symbols) are compared to multimodal, non-Markovian prediction (ξ_noMar, Eq. 4, blue and yellow symbols). See Fig. 2 for symbol legend. (B) Reaction-time curves generated by scaling all memory times by a common factor α. Vertical dashed lines indicate α = 1, which are different for folding (f) and unfolding (u) due to unique τ_D values. The multimodal non-Markovian prediction (${\xi }_{\mathrm{noMar}}^{\alpha }$ from Eq. 4—folding (blue) and unfolding (yellow)) is compared to the Grote–Hynes prediction (τ^{GH, α} from Eq. M6), rescaled by the memoryless, high-friction limit (${\tau }_{\text{HF}}^{\text{GH}}$ - Eq. M10) for folding (orange) and unfolding (grey). The blue and yellow symbols shown in (A) for the four example proteins coincide with the curve intercepts of the α = 1 lines in (B) for ${\xi }_{\mathrm{noMar}}^{\alpha }$. The red and black symbols are the same as those shown in (A), corresponding to the MD results for each protein, located at α = 1. The solid black lines show quadratic scaling, predicted for ξ_noMar in the long memory-time limit. Dashed horizontal black lines show unity for all plots.