Fig 1.
Comparison of Hausdorff and Fréchet distance. Two paths P (green) and Q (cyan) begin at state c0 and end at state cf with directionality indicated by the arrows.
The Fréchet distance δF and Hausdorff distance δH are given by the lengths of the purple and orange lines, respectively. The purple lines are the same length and correspond to the minimally stretched Fréchet “leash”; the orange line spans a pair of points separated by the Hausdorff distance (only one is shown because in this case there are infinitely many pairs of points with the same δH). Due to the backtracking of path P toward state A, combined with the monotonicity (no-backward-movement) constraint of the Fréchet metric, δF > δH.
Fig 2.
Macromolecular transitions studied with PSA.
(A) The closed ↔ open transition for adenylate kinase (AdK) involves the hinge-like motion of the LID (green) and NMP (cyan) domains about the relatively stable CORE (gray). (B) Diphtheria toxin (DT) can exist in two different crystallographic conformations that are connected by a closed ↔ open transition involving the unravelling and swinging-open of the Receptor-binding (R or C-terminal) domain (gray), about the Catalytic (C or N-terminal; cyan) and the Translocation (T or middle; green) domains.
Fig 3.
Comparison with a linearly interpolated path. A hypothetical transition pathway P (cyan line) in a 3D configuration space composed of a discrete number of conformer snapshots (cyan circles) connects an initial state (green circle), c0, and final state (red diamond), cf.
The reference path R (black line) is represented by LinInt. Each snapshot pk is associated with its projection, rk, on R; the progress, ζ(k), is the rmsd between rk and cf (dashed purple line along R) and the displacement, ρ(k), is the rmsd between pk and rk (dashed purple line perpendicular to R).
Fig 4.
The toy model consists of a cluster of connected particles moving in a double-well potential along the z-axis under the influence of a linear ramp potential (not shown).
In the cluster for N = 8, each particle (red) is connected to every other particle with a harmonic spring (blue) of equilibrium length 0 (cluster not shown to scale.) The potential landscape for constant z forms a “double barrel”—red (blue) regions correspond to high (low) energies—is parabolic along the x-direction (cyan line), and has a double-well shape in the y-direction (purple line), which produces a central barrier separating two “barrels” (gray crosshatching). A saddle point is located at the intersection of the cyan and purple lines. Motion in this landscape is biased toward either of the low-energy barrels, but transitions between barrels are possible at finite temperatures.
Table 1.
Modeling of energetics in tested path-sampling methods.
Table 2.
Approach to generating paths in tested path-sampling methods.
Table 3.
Summary of simulations, calculations, and analyses.
Fig 5.
Double-barrel potential energy landscape projected onto the xy-plane and yz-plane.
Groups of point masses (clusters) mutually connected by harmonic springs move under the influence of a transition-inducing ramp potential in the positive z direction and the two low-energy minima of the “barrels” at y = ±0.8. Colored lines depict the center of mass trajectories for each cluster. (A–C) trajectories at T = 0 K. (D–F) trajectories at T = 250 K. (A, D) Projection of paths onto the xy-plane together with the double-barrel potential. (B, E) Projection of paths onto the yz-plane. (C, E) Clustered heat maps summarize the Fréchet distances for all pairs of trajectories; dendrograms record cluster distances according to the Ward criterion. Trajectory colors in each row match the corresponding path(s) in the dendrogram. The trajectory-averaged radius of gyration for clusters at finite temperature is 0.35 (black circles).
Fig 6.
Path similarity analysis of trajectories generated by different path-sampling methods.
The AdK closed → open transition was sampled three times (except LinInt) with different methods (see text). Smaller distances indicate transition paths with greater similarity. The dendrogram depicts a hierarchy of clusters where smaller node heights of parent clusters indicate greater similarity between child clusters. Fréchet distances δF are in Å and correspond to a structural rmsd in accordance with the rmsd point metric. See text for a description of the methods. S5 Fig contains the same data annotated with numerical values of δF.
Fig 7.
Projections of trajectory 2 of the AdK closed → open transition from each path-sampling method onto low-dimensional collective variables.
The location of the initial structure is shown in each plot by the green circle, while the final structure is represented by the red diamond. (A) Projection of all pathways from the various path-sampling methods onto NC space. The horizontal axis corresponds to the percentage of contacts (of a transition snapshot) shared with the initial 1AKE:A structure (green circle) and the percentage of contacts in common with the final 4AKE:A structure (red diamond) is displayed on the vertical axis. The top-left legend identifies EN-based methods; the other methods are listed in the bottom legend. The LinInt path is shown for reference as a broken black curve. (B) Projection on NMP angle (θNMP) vs LID angle (θLID). In B and C, trajectories generated by the dynamical methods (DIMS, rTMD, FRODA, MDdMD, GOdMD) are plotted with diamonds and non-dynamical method trajectories with circles. (C) ζ-ρ space projection using LinInt as the reference path. Trajectory progress in ζ-ρ space is from left to right from higher to lower values of the progress variable ζ. MDdMD terminates at 1.5 Å Cα rmsd from 4AKE (red diamond); DIMS MD terminates at 0.5 Å heavy atom rmsd.
Fig 8.
Clustered heat map comparing ensembles of diphtheria toxin (1MDT to 1DDT) transition pathways produced by DIMS (blue bars) and FRODA (red bars) using the Fréchet distance δF.
All clusterings are produced using the Ward’s criterion in ascending distance order; incomplete trajectories were filtered and not displayed (see text).
Fig 9.
“Hausdorff pairs” (δH-pairs) analysis using 200 DIMS (cyan) and 200 FRODA (light green) trajectories projected into AA space.
Hausdorff distances were computed for all unique path pairs. (A) Conformer pairs—corresponding to the δH-pairs with the median and maximum Hausdorff distances (solid and dashed lines, respectively)—are projected onto the domain angle space for the following comparisons: DIMS–FRODA (purple), DIMS–DIMS (red), and FRODA–FRODA (blue). Experimental crystal structures, including some intermediates, are shown as stars [99], with further details available in S1 Table. Insets: Two heavy-atom representations are shown for the median δH-pair between a DIMS path and FRODA path, corresponding to snapshots from the respective trajectories. The magnitude of the displacement vector between the two conformations is projected onto each atom. Color bar units for atomic displacement are in Ångström. The initial and final conformations (green circle and red diamond, respectively) are shown along with the linear interpolation path LinInt—black dashed line) for reference. (B,C) Salt bridges in the DIMS and FRODA conformers from the DIMS-FRODA median Hausdorff pair. Three LID-NMP salt bridges (R156-D33, D158-R36, and K157-D54) and a CORE-NMP salt bridge (E170-K57) are intact in the DIMS structure (B) that are broken in the FRODA structure (C). The residues responsible for these salt bridges tug on the NMP domain more substantially than their counterparts in the LID domain, which are located toward the base of the LID.