Topology in Molecular Biology

8 The Structure of Collagen 149

(c) Fibrils of arbitrary length
(d) Intertwining of the chains within the molecule at a constant rate; this rate
of intertwining does not decrease as the molecule gets longer
Points (c) and (d) suggest a pattern of molecules along the fibril, and of
amino acids within the collagen molecule, that is either periodic or inflationary
(quasicrystalline).
Diffraction, especially that of crystalline [Gly−Pro−Pro] 10 , indicates
that the pattern is periodic [1–3], but a chain of amino acids packs naturally
as a Boedijk–Coxeter helix, that contains several approximants of 1 +

√

3 [4]

(see Sect. 8.2). These observations are not incompatible. The (longitudinal

and transverse) structure of collagen is ultimately periodic, but its unit cell

exhibits many successive approximants.

8.2 The Boerdijk–Coxeter Helix and its Approximants

Helices and densely packed spherical objects are two closely related geomet-

rical problems. The simplest means of packing tightly a chain of connected

spheres (representing amino acids) of arbitrary length is the Boerdijk–Coxeter

(B–C) helix, represented in Fig. 8.2. It is a stacking along one direction of reg-

ular tetrahedra, the elementary unit of four close-packed spheres. Figure 8.1c

represents the helix as a two-dimensional graph on triangular lattice covering

Fig. 8.2.Boerdijk–Coxeter helix (left-handed) obtained from a necklace of tetrahe-

dra (a) or as a packing of spheres (b). In (c), a right-handed B–C helix is represented

on a flat strip tiled with equilateral triangles, that constitutes a cylinder with the

vertical grey lines identified. This is a 30/11 helix (30 vertices for 11 turns, a con-

vergent of 1 +

√

3). Note that the axis of the helix (vertical grey line) is not exactly

perpendicular to its base (horizontal grey line). When the horizontal grey lines are

also identified, one obtains one of the four tori that make up the Hopf fibration of

polytope{ 3 , 3 , 5 }, a discrete representation of the hypersphereS^3. There are three

fibres–the steepest lines of ten neighbouring vertices, great circles inS^3 winding 1:1

around the torus