Figure 4
(Color online) (a) Space filling optimal helix, with a pitch to radius ratio
(drawn using MATHEMATICA). As explained in Appendix , this optimal value is determined by requiring that the radius of curvature of the helical curve is equal to half the minimum distance of closest approach between different turns of the helix. The corresponding tube (that can be thought of as being inflated uniformly around the curve) is optimally space filling since it stops growing when reaching its maximum thickness both locally (the radius of curvature) and nonlocally (half the minimum distance of closest approach between different turns) at the same time (see Appendixes ). Such an optimality criterion is shared by some of the conformations selected as ground states in our simulations in the marginally compact phase such as helices or planar hairpin and sheets shown in Fig. 3, when it is properly translated for the case of a discrete chain [see below and Eq. (
A5) in Appendix ]. It can be shown [
21] that the planarity of hairpins and sheets is a consequence of this optimal space filling criterion. The same geometrical feature is strikingly found to hold, within 3%, for
helices occurring in the native state of natural proteins [
38]. (b) Plot of the ratio
of the nonlocal radius of curvature
(with
) over the radius of curvature
as a function of the residue index
for the native state structure of sperm whale myoglobin (Protein Data Bank code 1mbn), where
refers to the spatial coordinates of the
atom of the
th residue,
[see Appendix for the definition of the triplet radius
]. In correspondence with the eight
helices present in the myoglobin fold, shown as the solid (red) parts in the plot, the values of
oscillate around unity, demonstrating that helices in natural proteins are optimally space filling in the sense described above.
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