Implementation of Telescoping Boxes in Adaptive Steered Molecular Dynamics

Abstract

Long-time dynamical processes, such as those involving protein unfolding and ligand interactions, can be accelerated and realized through steered molecular dynamics (SMD). The challenge has been the extraction of information from such simulations that generalize for complex nonequilibrium processes. The use of Jarzynski's equality opened the possibility of determining the free energy along the steered coordinate, but sampling over the nonequilibrium trajectories is slow to converge. Adaptive steered molecular dynamics (ASMD) and other related techniques have been introduced to overcome this challenge through the use of stages. Here, we take advantage of these stages to address the numerical cost that arises from the required use of very large solvent boxes. We introduce telescoping box schemes within adaptive steered molecular dynamics (ASMD) in which we adjust the solvent box between stages and thereby vary (and optimize) the required number of solvent molecules. We have benchmarked the method on a relatively long α-helical peptide, Ala₃₀, with respect to the potential of mean force and hydrogen bonds. We show that the use of telescoping boxes introduces little numerical error while significantly reducing the computational cost.

Figure 1

Two possible schemes for the implementation of telescoping boxes in ASMD: (a) constant cross section in which one adjusts—viz. increases or pulls—only the length between the end points of the protein which defines an axis labeled as z or (b) constant volume in which one adjusts all three widths to preserve the volume by increasing along z by a factor of f and along x and y each by a factor of .

Figure 2

An illustration of the progression of end-to-end distances of the ensemble of Ala₃₀ structures along the stretching in and between two characteristic stages. The average end-to-end distance r_ee is shown in red, and the ensemble of trajectories is overlapped in gray. For specificity, the data shown here comes from the pulling of Ala₃₀ for the three stages starting from the initial pull—viz. i = 2 in the notation of the x-axis labels.

Figure 3

Work values of the ensemble of trajectories in naïve ASMD along the unfolding direction of Ala₃₀ employing three different schemes: (a) constant (and large) solvation box, (b) telescoping boxes with constant volume scheme, and (c) telescoping boxes with constant cross section. The black curves represent the work values for all generated trajectories, while the red dashed curves are the PMF. All PMF profiles have been obtained using 100 tps at 1 Å/ns. The naïve ASMD results for case (a) are the same as reported and benchmarked in ref (98).

Figure 4

Comparison of the energetics of Ala₃₀ among different ASMD variants. The blue curve corresponds to naïve ASMD with a constant solvent box from ref (98), while the orange and green curves correspond to ASMD with the telescoping box schemes noted in the legend. The PMFs have been obtained using 100 tps at 1 Å/ns.

Figure 5

RMSE analysis for two types of telescoping box schemes compared to naïve ASMD with a constant box (dashed black curve) from ref (98).

Figure 6

Hydrogen bond profiles for different schemes of ASMD: constant box (blue), constant volume (orange), and constant cross section (green). The top panel corresponds to the intrapeptide hydrogen bond interaction, while the bottom panel corresponds to the number of the peptide-water hydrogen bonds for the water molecules within 10 Å of the peptide. The results for naïve ASMD with a constant box was reanalyzed with MDAnalysis 2.0 version in consideration of the capped atoms of the peptide, resulting in slightly different but improved estimates of the hydrogen bonding.

Figure 7

Classification of hydrogen bonds, including 3₁₀- (top) and α-helical (bottom) for each ASMD method for the three schemes of constraining the box size as labeled in Figure 6. The π-helical contacts are not shown because they were essentially zero along the process.