Non-Markovian modeling of protein folding

Abstract

We extract the folding free energy landscape and the time-dependent friction function, the two ingredients of the generalized Langevin equation (GLE), from explicit-water molecular dynamics (MD) simulations of the α-helix forming polypeptide [Formula: see text] for a one-dimensional reaction coordinate based on the sum of the native H-bond distances. Folding and unfolding times from numerical integration of the GLE agree accurately with MD results, which demonstrate the robustness of our GLE-based non-Markovian model. In contrast, Markovian models do not accurately describe the peptide kinetics and in particular, cannot reproduce the folding and unfolding kinetics simultaneously, even if a spatially dependent friction profile is used. Analysis of the GLE demonstrates that memory effects in the friction significantly speed up peptide folding and unfolding kinetics, as predicted by the Grote-Hynes theory, and are the cause of anomalous diffusion in configuration space. Our methods are applicable to any reaction coordinate and in principle, also to experimental trajectories from single-molecule experiments. Our results demonstrate that a consistent description of protein-folding dynamics must account for memory friction effects.

Keywords: generalized Langevin equation; mean first-passage times; memory effects; non-Markovian processes; protein folding.

Fig. 1.

(A) The free energy $U\left(q\right)$ for the mean hydrogen-bond distance reaction coordinate of ${\mathrm{A}\mathrm{l}\mathrm{a}}_9$ for different simulation lengths; representative snapshots of the polypeptide backbone in all local minima are shown. The barrier used for the calculation of unfolding and folding times is positioned at $q_B=0.54\hspace{0.17em}$ nm. (B) A $200\hspace{0.17em}$-ns-long segment of the trajectory is shown.

Fig. 2.

(A) Running integral $G\left(t\right)$ over the memory function; Inset shows a lin-log plot. The horizontal dashed line denotes the total friction coefficient $\overline{\gamma }$. (B) Memory function $\mathrm{\Gamma }\left(t\right)$; Inset includes short times. Gray lines correspond to the numerical data; red lines correspond to the multiexponential fit according to Eq. 3. (C) Mean-square displacement of the reaction coordinate; MD (blue line) and GLE (orange broken line) simulation results agree perfectly and exhibit superdiffusion for times up to 0.1 ps and subdiffusion up to 1 ns. Underdamped (underd.; red line) and overdamped (overd.; green line, underneath the red line) Markovian Langevin simulations agree perfectly with each other but miss the anomalous diffusion.

Fig. 3.

(A) Comparison of unfolding and folding MFPTs ${\tau }_{\mathrm{M}\mathrm{F}\mathrm{P}\mathrm{T}}\left(q_S,q_F\right)$ from MD (blue) and GLE (orange) simulations as a function of the final position $q_F$ for start positions $q_S=q_L=0.32\hspace{0.17em}$ nm (solid lines) and $q_S=q_R=0.99\hspace{0.17em}$ nm (broken lines). The gray curve shows the folding free energy $U\left(q\right)$. (B) Dependence of different MFPTs from GLE simulations on the memory time rescaling factor $\alpha$; the corresponding start and final positions are illustrated in C, Inset. Open and filled circles correspond to open and filled arrows, respectively, in C, Inset. The colored horizontal lines denote corresponding results for the overdamped Markov limit from Eq. 10. (C) Ratios of the MFPTs shown in B. Ratios of reciprocal MFPTs do not depend on $\alpha$ (red, green, and blue lines that connect colored circles); only the ratio of the folding and unfolding times to the barrier top, ${\tau }_{\mathrm{M}\mathrm{F}\mathrm{P}\mathrm{T}}\left(q_R,q_B\right)/{\tau }_{\mathrm{M}\mathrm{F}\mathrm{P}\mathrm{T}}\left(q_L,q_B\right)$ (open green and filled red spheres), depends on $\alpha$.

Fig. 4.

(A) Friction coefficient profiles $\gamma \left(q\right)$ from KMC analysis for different lag times $\mathrm{\Delta }t$ (different colors) for the underdamped (underd.) Langevin model, Eq. 8, from MD (filled circles) and GLE simulations (open circles) and for the overdamped (overd.) Langevin model, Eq. 9, from MD (solid lines) and GLE simulations (broken lines). The gray horizontal line shows the total friction coefficient $\overline{\gamma }$ extracted from MD simulations. (B) Friction profiles computed from the MD MFPT profiles in Fig. 3A using Eq. 11. ${\gamma }_{\mathrm{u}\mathrm{n}\mathrm{f}}\left(q_F\right)$ follows from the unfolding MFPTs for start position $q_S=0.32\hspace{0.17em}$ nm, and ${\gamma }_{\mathrm{f}\mathrm{o}\mathrm{l}}\left(q_F\right)$ follows from folding MFPTs for $q_S=0.99\hspace{0.17em}$ nm. The gray horizontal line denotes the friction coefficient $\overline{\gamma }$ extracted from MD simulations. The gray curve in the background shows the folding free energy $U\left(q\right)$. (C) MFPTs from MD and GLE simulations are compared with overd. Markovian predictions according to Eq. 10 using ${\gamma }_{\mathrm{u}\mathrm{n}\mathrm{f}}\left(q_F\right)$ and ${\gamma }_{\mathrm{f}\mathrm{o}\mathrm{l}}\left(q_F\right)$ from B.