(A) Crystal structure of the swapped dimer, pdb-id 1k3e [66]. The domains of each chain are packed against the complementary domains in the other chain. The presumed hinge region between the two domains of each chain is marked in space-fill representation. (B) The pseudo-monomer consists of the C-terminal domain from chain A (blue), packed against the N-terminal domain from chain B (red). Note the β-sheet in the interface between the two domains.

Domain A (cyan) is connected to Domain B (blue) by a hinge (red). (A) The RBP structure in its open and closed form, pdb-ids 1urp [83] and 2dri [84] respectively. (B) The architecture of RBP–each domain consists of discontinuous segments of residues. The two domains are connected by three hinges that must move in a coordinated way to maintain domain integrity. Domain boundaries are rough estimates. (C) The sequence of RBP showing the discontinuous domains and the secondary structures. This illustration was taken from 1urp Protein Data Bank entry at http://www.rcsb.org/pdb/ [5]; and domain assignments are from [85].

We used the Rosetta soft-repulsive energy score for this analysis, in order to dampen repulsive fluctuations that are due to mild sterical clashes (see text, note that the simulation itself was conducted with the Rosetta score-12 energy function, with a classical vdW potential). Note that the y-axis shows the total energy score, and the specific energy terms are shifted by a constant number of units, for convenient comparison with the other terms. (A) Energy plot for simulation with restricted dofs (central hinge is free to move, and all other dofs can fluctuate by ±30°). (B) Energy comparison for the aligned sections of the restricted (left) and free (right) simulations. In the free simulation, all dofs are free to move. The reaction coordinates of the two simulation were aligned by a string matching algorithm [39] based on structural similarity between conformations (see text, Figure S3 and Video S8).

Here, we aim to find collision-free paths for a point robot in 2D-space starting from a source configuration. (A) The basic Single-RRT algorithm provides fast but rough coverage of unexplored regions, and the target is often missed (red star, top left). During the run, the tree grows in feasible space (white) among obstacles (orange rectangles). In biological examples, these obstacles are high-energy conformations. Each point stands for a two dimensional conformation, and the tree grows from a source conformation (violet star, middle of figure), towards random directions (see Methods). (B) In the Partial-RRT variant, we use partial information to truncate branches that do not grow towards the target (like the truncated branch in the grey ellipse, compare to the branch in the magenta ellipse). The search is more confined to relevant regions, at the expense of overall coverage of the search space. (C, D) Comparison between the basic Single-RRT algorithm and RRT with partial information (Partial-RRT), for the toy example in a and b. The partial information used here is the distance to the target. In SingleRRT-t50 and PartialRRT-t50 the target is also used as an explicit direction of growth once in 50 iterations, in case the tree reaches the proximity of the target but not its exact location. This test follows a common assumption that RRT running time is dominated by the number of collision tests. We compare the Euclidean distance of the RRT node that is closest to the target (y-axis) as a function of the number of collision tests (x-axis) throughout the run. Results are the average distance (in c) or the minimum distance (in d) over 50 independent runs. PartialRRT performed better than SingleRRT, especially for a lower number of collision checks. Better performance is achieved in less running time. As the number of collision checks grows. All methods converge. Note that this is only a toy example for illustrative purposes; in many biological examples, the target conformation might not be given explicitly, and the number of dofs is in general much higher.

(A) The final conformation along the motion pathway of CesT (cyan) is shown for five different examples of predicates (see Table 3). We show the best scoring run with respect to the specified predicate (out of 50 independent runs). The orientation of the N-terminal domain of the native pseudo-monomer is shown in red for comparison. (B) The RMS distance of the final structure, for simulations with different predicates, is plotted relative to SigE (the homologue that was used to guide the motion, blue) and the native pseudo-monomer of CesT (black). Even though the homologue was the reference for biasing the motion, the simulations reached the correct conformation with a better level of accuracy for several predicates. (C) Biasing the motion by combining several distance constraints (see Table 3 for details about the constraints): the results are shown as the fraction among 50 independent simulations that reached given RMSD thresholds (to the native pseudo-monomer).

All runs use the Partially-Restricted (M) set of dofs (Table 2), where the central hinge is allowed free motion, and all other residues can rotate by ±30°. (A) In each run, each residue is scored by how often it tends to be in hinge regions. Hinges are extracted by structural comparison between conformations along the motion sequence. They connect sub-domains that remain rigid during the simulation. Resolution parameter ρ states the RMSD threshold used for rigid alignments. Mild hinges appear only at higher resolutions (low value of ρ), and salient hinges appear at low resolutions (see Figure S2 for a detailed protocol). (B) Pearson's Correlation between the hinge scores of three independent simulations, for different values of ρ. In blue, the correlation over all residues, including the central hinge (residues 48–55). In green, the correlation when excluding the central hinge. (C) Hinge scores for each residue in three independent simulations, for ρ = 1.5 Å and ρ = 3.5 Å. The y-axis denotes how often each residue appears in hinge regions (see Figure S2 for more details). Secondary structures (according to DSSP [86]) are plotted along the x-axis. Observe that milder hinges disappear at lower resolution (3.5 Å). R and R* are the average Pearson's correlations between runs with and without the central hinge region, respectively. For each plot, the crystal structure of Cyanovirin-N is colored according to the corresponding hinge score, with warm colors indicating higher scores. (D) The weighted average of the hinge scores for different values of ρ (see Results). Since higher resolutions contain milder hinges, they were assigned a lower weight.

The hinge score for each residue is plotted for (A) restricted simulations, where only the central hinge (residues 48–55) is free to fully rotate, and the other residues are restricted to ±30° deviation from the initial conformation, and (B) a free simulation (magenta) where all backbone degrees of freedom are free to rotate (see Table 2). Hinge scores are plotted for resolution parameter values ρ = 4 Å and ρ = 1.5 Å. The central hinge is the most salient feature in the free simulation, and therefore it appears even in low-resolution plots. Milder hinges are less robust to the restriction of dofs (see text for more details).