Computing ensembles of transitions from stable states: Dynamic importance sampling

Abstract

There is an increasing dataset of solved biomolecular structures in more than one conformation and increasing evidence that large-scale conformational change is critical for biomolecular function. In this article, we present our implementation of a dynamic importance sampling (DIMS) algorithm that is directed toward improving our understanding of important intermediate states between experimentally defined starting and ending points. This complements traditional molecular dynamics methods where most of the sampling time is spent in the stable free energy wells defined by these initial and final points. As such, the algorithm creates a candidate set of transitions that provide insights for the much slower and probably most important, functionally relevant degrees of freedom. The method is implemented in the program CHARMM and is tested on six systems of growing size and complexity. These systems, the folding of Protein A and of Protein G, the conformational changes in the calcium sensor S100A6, the glucose-galactose-binding protein, maltodextrin, and lactoferrin, are also compared against other approaches that have been suggested in the literature. The results suggest good sampling on a diverse set of intermediates for all six systems with an ability to control the bias and thus to sample distributions of trajectories for the analysis of intermediate states.

Figure 1:

Initial and final conformational states for the systems: a) Protein A, b) Protein G, c) Calcium Sensor (S100A6), d) Glucose-galactose binding protein, e) Maltodextrin and, f) Lactoferrin.

Figure 2:

Proteing-G folding transition going from unfolded to folded. Helix is in purple and, β-strand is in yellow. Hydrophobic residues are highlighted.

Figure 3:

RMS and Energy changes along transitions generated by DIMS and targeted molecular dynamics for the folding transition of Protein A, a three helix bundle.

Figure 4:

Histogram analysis of the population of path sampling in the two-dimensional projected coordinate space of the folding transition of protein A (three helix bundle). The two axes are number of hydrogen bonds in the structure (representing the formation of regular secondary structure) and the percent of native contacts.

Figure 5:

Two different strategies to increase diversity, in both cases the same final structure is used for protein G. a) Generating sets of ten trajectories using six different unfolded states as the initial structure. b) Generating 60 trajectories going from a single unfolded state.

Figure 6:

Pairwise comparison between DIMS trajectories. Figure a) shows, via a color scale, the RMS differences between all conformations along two different DIMS trajectories, in white the line that represents the shortest path in RMSD space. Note that the two sets of conformations have a large range of RMS differences relative to each other. Figure b) shows the RMS values for the shortest path for several trajectories, it can be seen that the structures between two any given trajectories have a high RMS distance that ranges from 3 Å up to 5 Å for the intermediate states.

Figure 7:

Relative distribution of OM scores is shown. The set reflects on the diversity of sampled transitions.

Figure 8:

Normal mode self avoidance selection for Calcium Sensor S100A6 over 1000 trajectories. In red is shown the progress variable. Several modes are selected along the transition allowing the algorithm to sample contributions from different modes.

Figure 9:

a) The angle probability distribution in time, for the hinge opening movement of the glucose-galactose binding protein. b) Force distribution along the hinge angle.

Figure 10:

a) Angle probability distribution in time, for the hinge bending of maltodextrin. b) Force distribution along the hinge angle.

Figure 11:

Total root mean square distance traveled along the trajectory per residue for three different maltodextrin trajectories (a),b) and c)). In black the distance for the initial part, while in red the final regions of the trajectory. Residues in the catalytic site are labeled with blue bullets, residues in the N-terminal domain with green triangles. d) The same color scheme is used to illustrate the location of the residues in the structure. The C-terminal domain is highlighted in orange.

Figure 12:

Probability of acceptance (equation (9)) as a function of RMS distance to the target, along the transitions for maltodextrin.

Figure 13:

a) The angle probability distribution in time, for the hinge opening movement of lactoferrin. b) The force distribution along the hinge angle.

Figure 14:

Comparison between MolMovDB, TMD and DIMS. a) Angle displacement as a function of the time step. b) Pulling velocity.