Minimal folding pathways for coarse-grained biopolymer fragments

Abstract

The minimal folding pathway or trajectory for a biopolymer can be defined as the transformation that minimizes the total distance traveled between a folded and an unfolded structure. This involves generalizing the usual Euclidean distance from points to one-dimensional objects such as a polymer. We apply this distance here to find minimal folding pathways for several candidate protein fragments, including the helix, the beta-hairpin, and a nonplanar structure where chain noncrossing is important. Comparing the distances traveled with root mean-squared distance and mean root-squared distance, we show that chain noncrossing can have large effects on the kinetic proximity of apparently similar conformations. Structures that are aligned to the beta-hairpin by minimizing mean root-squared distance, a quantity that closely approximates the true distance for long chains, show globally different orientation than structures aligned by minimizing root mean-squared distance.

FIGURE 1

Residues 99–153 in regulatory chain B of Aspartate Carbamoyltransferase (47) (PDB code 1AT1) are chosen for analysis. From this domain we select three fragments for investigation. Two are outlined in dashed boxes: β-hairpin residues 126–137, and α-helix residues 147–151. The strand 1-turn-strand-2 tertiary motif, residues 101–130, is also used investigate the importance of noncrossing.

FIGURE 2

(a) β-hairpin fragment, with all-atom and coarse-grained C_α representations superposed. (b) The extended initial state.

FIGURE 3

(a) Residues 101–130 of Aspartate Carbamoyltransferase can be taken as an example of an overpass/underpass problem where chain noncrossing is important. (b) Conformation of the segment in panel a with the β-sheet unformed. Both initial and final structures (with opposite over/under sense) are superposed in this stereo view. (c) A simplified model to capture the essence of the underpass-overpass problem. Both initial and final states are shown as viewed from above. Residues 1–8 must transform to residues 1′−8′, but cannot pass through the obstacle marked with a circled X, representing a long piece of polymer normal to the plane of the figure.

FIGURE 4

Illustration of the general recipe for obtaining minimal pathways (see Methods).

FIGURE 5

Minimal transformations to the β-hairpin. Distances are given in Table 1. (a) Folding pathway in which one strand of the hairpin can be thought of as peeling away by rotations of the links to various critical angles, which are then followed by subsequent translations into their final positions. (b) A minimal pathway that can be thought of as involving kink propagation or peeling away from the extended strand, followed by translation of the links into their final positions in the β-hairpin. (c) A zippering mechanism, in which we have aligned the middle link of the hairpin and sought the minimal distance transformation. The distance here is somewhat larger than the distance for the transformations in panels a and b. (d) The extended strand is aligned to the β-hairpin by minimizing RMSD (blue), or minimizing MRSD (teal). (e) Idealized version of the extended strand and β-hairpin. The extended strand is again aligned to the β-hairpin by minimizing RMSD (blue), or minimizing MRSD (teal). (f) Transformation for the idealized β-hairpin, for RMSD-aligned structures. Initial state is blue, final state is red, and intermediate state is in green. (g) Transformation for the idealized β-hairpin, for MRSD-aligned structures. (h) RMSD-aligned transformation between the extended strand (blue) and β-hairpin (red). An intermediate state is shown in green. (i) MRSD-aligned transformation between the extended strand (blue) and β-hairpin (red). An intermediate state is shown in green. The small arrow points to a link with an somewhat unconventional transformation, which is discussed in the Appendix.

FIGURE 6

(a) Single α-helix of five residues 147–151 taken from PDB 1AT1. (b) Minimal pathway to fold the α-helix (red), from a straight line initial state which has been aligned by minimizing MRSD (shown in blue, see text for description). A conformation partway though the transition is shown in green. (c) Minimal pathway to fold the helix from a straight-line initial conformation with its second link directly aligned to the second link of the helix. Distances for both transformations are given in Table 1.

FIGURE 7

Various steps in a minimal pathway obeying noncrossing. Two conformations are drawn for each step. By convention, we number residues in the conformation that is leading in the transformation. (See text for a description of the transformation.)

FIGURE 8

(a) Extremal trajectories for an inequality constraint problem. In this case, a path that is a minimal distance from point A at (x_A, y_A) = (−1.5, 0) to point B at (x_B, y_B) = (+1.5, 0) is sought subject to the constraint that the path must remain outside a circle of unit radius. Both positive and negative solutions are shown. (b) Lagrange multiplier λ and excess parameter ε for the above problem. If ε ≠ 0, λ = 0, and if ε = 0, λ ≠ 0.

FIGURE 9

(a) Extremal trajectory for a one-link transformation subject to inequality constraints. The link moves from configuration AB to A′B′ in the presence of an obstructing sphere. The link length is conserved during this process. The distance traveled by the end-points A and B of the link is minimized by the transformation shown, which involves straight-line motion of A to A′, and straight-line motion of B along a trajectory tangent to the sphere. Point B traces out a great circle on the surface of the sphere before continuing to B′ on another trajectory tangent to the sphere. (b) When the sphere in panel a is compressed to form a two-dimensional disk of the same radius, the minimal transformation takes the form shown, with a discontinuity in the trajectory of B at point B1. Moreover, the piecewise solution must still retain rotations and is not purely piecewise straight lines. (c) Transformation from AB to AB′, in the presence of an intervening infinite strip. The minimal transformation consists of two piecewise rotations with a corner violation between them: the link rotates from B to B_c, then from B_c to B′.

FIGURE 10

An uncommon one-link transformation that was nevertheless found for one of the links in the MRSD-aligned β-hairpin transformation in Fig. 5 i (link denoted by small arrow). The transformation obeys EL Eqs. 5a–5c and the corner conditions in Eq. 6. The initial and final conditions in this case are links AB and A′B′, respectively. Note the link has a direction—so, for example, A must transform to A′ and not B′. The transformation proceeds by rotating link AB about point c to configuration A″B″. Bead A″ then rotates about B″ to position A′, and B″ rotates about A′ to position B′.