Effectively explore metastable states of proteins by adaptive nonequilibrium driving simulations

Nonequilibrium drivings applied in molecular dynamics (MD) simulations can efficiently extend the visiting range of protein conformations, but might compel systems to go far away from equilibrium and thus mainly explore irrelevant conformations. Here we propose a general method, called adaptive nonequilibrium simulation (ANES), to automatically adjust the external driving on the fly, based on the feedback of the short-time average response of system. Thus, the ANES approximately keeps the local equilibrium but efficiently accelerates the global motion. We illustrate the capability of the ANES in highly efficiently exploring metastable conformations in the deca-alanine peptide and find that the $0.2-\mu \mathrm{s}$ ANES approximately captures the important states and folding and unfolding pathways in the HP35 solution by comparing with the result of the recent $398-\mu \mathrm{s}$ equilibrium MD simulation on Anton [S. Piana et al., Proc. Natl. Acad. Sci. USA 109, 17845 (2012)].

Figure 1

(a) Evolution of the RMSD and end-to-end distance $\xi$ of deca-alanine in the FANES. (b) Schematic diagram of deca-alanine controlled by force. (c) End-to-end distance $\xi$ versus the external force. Four distinct branches (gray dots) from top to bottom are, respectively, the coil (around $35\AA$), the $\alpha$ helix (around $15\AA$), and the compact region including the $\beta$-hairpin, $\beta$-bridge, and $\omega$-shaped turns (from $11\AA$ to $4\AA$). The quasistatic force-distance curves in decreasing force from the extended state and increasing force from the $\alpha$ helices are shown as the dashed and solid black lines, respectively. The hysteresis loop does not cover the compact states. (d) Energy landscape in the 2D plane spanned by $\xi$ and the RMSD. The colors label energy levels. Two basins corresponding to the $\alpha$ helices and the compact regions are shown.

Figure 2

(a) Evolution of the RMSD (to the $\alpha$ helix) of deca-alanine in vacuum in the TANES. (b) The RMSD, temperature, and potential energy (in $\mathrm{kcal}/\mathrm{mol}$) of conformations obtained in the TANES. (c) Energy landscape in the 2D space spanned by the RMSD and $\xi$. The corresponding representative structures are shown. (d) Samples around $300\mathrm{K}$ in the TANES are shown in the 2D space. Four $40-\mathrm{ns}$ equilibrium simulations at the temperature starting from different ANES conformations are shown with different symbols and colors, where the red circles are from the $\alpha$ helices and the yellow squares, blue triangles, and orange triangles from others. The inset illustrates the evolution of the RMSD in the four equilibrium trajectories with the same colors. (e) Schematic diagram demonstrating the evolution of structures in the TANES. (f) Comparison of the FANES conformation around zero force (orange circles) with the TANES conformations around $T=300\mathrm{K}$ (blue triangles). The corresponding snapshots are also shown.

Figure 3

(a) Evolution of the RMSD (to the native structure) of the HP35 with implicit solvent (ff03 force field) in two independent TANESs. The horizontal line corresponds to the RMSD equal to $2\AA$. (b) Samples of the first TANES trajectory around $300\mathrm{K}$ in the 2D space spanned by the RMSDs of the segments $A$ and $B$ of the protein to the native structure (see the text). The corresponding presentative structures are also shown. (c) Evolution of the RMSD of the HP35 with implicit solvent (CHARMM27 force field) in two independent FANESs. (d) Samples of the second FANES trajectory around the zero force in the 2D plane.

Figure 4

(a) Schematic diagram of the HP35 in explicit water pulled by the applied force in the FANES. The force is exerted at the $5-C_{\alpha }$ and $32-C_{\alpha }$ (pink beads) atom pair. (b) Two $100-\mathrm{ns}$ FANES trajectories of the HP35 with water at $342\mathrm{K}$ are projected in the 2D plane spanned by RMSDs of the segments $A$ and $B$ of the HP35 (dashed red line and solid green line). The background is the free energy landscape from the $398-\mu \mathrm{s}$ equilibrium MD simulation at $345\mathrm{K}$ on Anton [30]. Three metastable regions, including the native (bottom left), the intermediate (top left), and the unfolded (top right), are visible.

Figure 5

(a) Contour map of the 2D potential $U(x,y)=20x^2+0.7y^2{(y-4)}^2+0.5y$. (b) Motion of a Brownian particle with $\beta =1$ in the potential projected onto the 1D reaction coordinate $y$. (c) Motion of a Brownian particle driven by ANES. It is evident that the ANES significantly enhances the transition between states and focuses on the important conformations. (d) Global equilibrium histogram and the histogram from the simulation using TANES along the $y$ axis plotted as a solid black line and red circle line, respectively. It is clear that the interstate equilibrium is broken by TANES. The insets depict comparisons of the distribution (selected at $\beta =1$) inside local states and the corresponding local equilibrium distributions along the $y$ axis. Obviously, TANES could keep the local equilibrium inside each state.

Figure 6

Evolution of the force-driving process of deca-alanine in vacuum. To clearly understand how the method works, a segmental trajectory (13–17ns) is shown here as a series of successive figures: (a) The extended structure jumps automatically down to the compact region (indicated by the red curve), (b) the compact structure is stretched to the extended structure again, (c) the conformation enters the compact region after going through the $\alpha$ helices and a turn helix (the typical snapshot shown in c), (d) deca-alanine is stretched to the extended structure, (e) the $\alpha$ helices are formed, and (f) the $\alpha$ helices are stretched to the extended coil.

Figure 7

(a) Evolution of the RMSD to the native state and the end-to-end distance $\xi$ of deca-alanine in water solvent. (b) Samples from the trajectory with near-zero forces projected onto the 2D plane spanned by the RMSD and $\xi$. The result differs from the case of deca-alanine in vacuum (Fig. 2) because the $\pi$ helix [the thick helices in (b)] also exists.

Figure 8

(a) Four independent FANES evolutions of RMSDs of the HP35 in a water environment denoted by the red line, black line, green line, and gray line. The RMSDs less than $2\AA$ (dotted lines) stand for the native structures. (b) Some representative samples (red dots) with the near-zero forces from the first trajectory projected on the 2D plane spanned by RMSDs of segments $A$ and $B$. The evolution of the first trajectory in (a) is shown as the gray line, starting from the solid green circle 1 and going through the solid yellow square 2 and the solid blue triangle 3 (in the native state). (c) Four 25-ns independent simulations starting from four different conformations of the trajectory. Within 25 ns all the simulations are limited in their initial conformational regions without transitioning to another. (d) Relationship between the extensions $\xi$ of the HP35 and the applied forces. The data points come from the one of the trajectories in (a).The native and non-native structures shown by blue triangles (lower ones) and orange (upper ones) circles, respectively.

Figure 9

Evolution of the fraction of the $\alpha$ helical structure of all residuals ($y$ axis) of the HP35. (a) Evolution of the fraction of $\alpha$ helical structures of the first trajectory with Amber ff03 in Fig. 3. (b) The $\alpha$ helical fraction of the first force trajectory under CHARMM27 [Fig. 3]. (c) Fraction of $\alpha$ helices of the HP35 in water solvent at 300 $\mathrm{K}$. (d) Fraction of $\alpha$ helices of the HP35 in water at 342 $\mathrm{K}$.