Complete maps of molecular‐loop conformational spaces

Abstract

It has come to our attention that in the online version of the above article the figures that were published were not the final version approved by the author. All of the final, approved figures for the article are shown on the pages that follow. We apologize for any confusion or inconvenience that was created by this error. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 (a) A six-atom ring. (b) A general six-torsion loop on a long molecular chain with fixed end-points. Within the shaded subchains, the relative positions of all atoms are known and they can hence be viewed as a single rigid body. Having fixed positions in 3-space, atoms 1, 2, 11, and 12 also define a rigid body. Free torsion angles are indicated with circular arrows. (c) and (d) Body-and-hinge models of the two loops, with the bodies labelled B1,…,B6. (e) and (f) Their distance models. Translation of bond-bending (a), torsion-angle (b), and rigid-body (c) constraints. Bars between atoms indicate fixed distances. The mainly studied molecular loops (left) together with their distance models (center) and equivalent robots (right). For each loop, we cite the main applicable techniques under its distance model. All loops can attain, at most, 16 conformations. The convex-hull and subdivision properties illustrated for a planar curve of parametric form (x, f(x)) = Σ vibi,3. Left: for x ∈ [0, 1] the curve is inside the convex hull of the control points v0,…,v3. Right: the control points vi′ corresponding to a subinterval [a, b] ⊂ [0, 1] can be computed from the vi′s using the subdivision property. The vi′s are closer to (x, f(x)) than the vi's. Some of the analyzed loops and their distance models. (a) The disulfide bond. (b) Cyclohexane. (c) Bicyclohexane. (d) Adamantane. In all models, a line joining two atoms indicates that the distance between them is fixed. Thick lines correspond to covalent bonds. Except Ω in (a), all torsion angles are a priori unknown. Four stages of the root search performed on the disulfide-bond loop. The x and y axes correspond to r2,5 and r3,7, respectively. The 18 conformations of the chosen disulfide bond loop, for Ω = ±90°. The bottom row shows a stereogram of all conformations overlaid. The conformational space of cyclohexane has one isolated point and a cyclic one-dimensional path, corresponding to the chair and skew boat conformations, respectively. The intervals of the shown bounding box are [6.1, 9.3] Å2 in all dimensions. Top and middle: The conformational space of cycloheptane obtained at different precisions. Red and yellow boxes correspond to the chair and boat pseudo-rotation paths. Bottom: Box splittings and reductions performed by the algorithm to obtain the rightmost plot of middle row. Top: Boxes approximating the conformational space of cyclooctane, plotted for their r1,4, r2,5, and r3,6 dimensions. The approximation contains 273,626 boxes, which are here shown with semitransparent walls. Bottom: Two slices of such space, corresponding to clipping the surface with the planes π1 and π2 shown above. The left slice corresponds to fixing r3,6 = 8.5 Å2, and the right one to r3,6 = 11.5 Å2. (a) Boxes approximating the conformational space of bicyclohexane. Red boxes correspond to rigid conformations. They are somewhat enlarged to appreciate them. Yellow and blue boxes correspond to the two mobile boat–boat forms. Two bifurcation points exist on the yellow path, corresponding to the shown twisted boat–boat conformations. (b) The actual topology of the space. (c) The fifteen rigid conformations, with atoms 1, 2, and 3 held in a fixed position.