Wang-Landau molecular dynamics technique to search for low-energy conformational space of proteins

Multicanonical molecular dynamics (MD) is a powerful technique for sampling conformations on rugged potential surfaces such as protein. However, it is notoriously difficult to estimate the multicanonical temperature effectively. Wang and Landau developed a convenient method for estimating the density of states based on a multicanonical Monte Carlo method. In their method, the density of states is calculated autonomously during a simulation. In this paper, we develop a set of techniques to effectively apply the Wang-Landau method to MD simulations. In the multicanonical MD, the estimation of the derivative of the density of states is critical. In order to estimate it accurately, we devise two original improvements. First, the correction for the density of states is made smooth by using the Gaussian distribution obtained by a short canonical simulation. Second, an approximation is applied to the derivative, which is based on the Gaussian distribution and the multiple weighted histogram technique. A test of this method was performed with small polypeptides, Met-enkephalin and Trp-cage, and it is demonstrated that Wang-Landau MD is consistent with replica exchange MD but can sample much larger conformational space.

Figure 1

(Color) Conformations of Met-enkephalin sampled by Wang-Landau molecular dynamics (WLMD, upper panel) and replica-exchange molecular dynamics (REMD, lower panel) projected onto the first two principal component axes (PCA). During WLMD 160 000 conformations were saved and 10 000 lowest energy conformations were selected. In the same manner, 10 000 lowest energy conformations were selected for REMD. These 20 000 conformations in total were used for determining the principal axes. Upper panel: 10 000 lowest energy conformations obtained by WLMD are projected. Lower panel: 100 000 lowest energy conformations obtained by REMD are projected. Each pixel is colored according to the cRMS from the PDB structure.

Figure 2

(Color) Comparison of density of states of Met-enkephalin obtained by WLMD (blue lines) and REMD (red line). Eight density of states were obtained from eight WLMD simulations (blue lines), one density of states was calculated from one REMD simulation with eight replicas (red line). The lower panel shows the difference of each density of states of WLMD from that of REMD.

Figure 3

(Color) Density of states of Trp-cage obtained from three independent Wang-Landau molecular dynamics simulations. Each line represents the density of states obtained from one WLMD run.

Figure 4

(Color) Time series of nonbond energy of Trp-cage. Upper panel: Wang-Landau molecular dynamics; the trajectories for the last $1\mathrm{ns}$ are enlarged for clarity. Lower panel: Replica exchange molecular dynamics. Three arbitrarily selected trajectories out of 24 are shown.

Figure 5

(Color) Time series of temperatures of Trp-cage. Upper panel: Three trajectories from Wang-Landau molecular dynamics simulations; the trajectories for the last $1\mathrm{ns}$ are enlarged for clarity. Lower panel: Trajectories of three replicas from replica exchange molecular dynamics simulations. In both figures, each trajectory corresponds to that of Fig. 4.

Figure 6

(Color) Trajectories of coordinate root mean square deviations (cRMS) from the native structure of Trp-cage. Upper panel: WLMD; lower panel: REMD. Each trajectory corresponds to that of Fig. 4.

Figure 7

(Color) Conformations of Trp-cage projected onto the principal space spanned by the first and second principal axes. Upper panel: WLMD; lower panel: REMD. Each pixel is colored according to the cRMS deviation from the native structure of Trp-cage.