Topology in Molecular Biology

8 The Structure of Collagen 159

The overlap structure is much more symmetrical than the gap. It has:

(a) Orthogonal mirror axes that coincide with the diagonals of the rhombi.
(b) Many points are aligned and regularly spaced, on the direction of the
diagonal of the square, which is also direction (2,−1) in the unit cell grid.
(c) It has two unit cells, a rhombus-shaped unit cell (with four vertices and
an area of 90 elementary triangles)^8 , and a smaller, rectangular unit cell
made of mirror planes (two vertices).
(d) It exists at two different scales (Fig. 8.6b, c) with the same convergents.

One can constitute a “square” cell in the overlap structure, of sides (2,−1)

and (−1,−2), containing 20 points (or 5 unit cells, arranged as in Fig. 8.7). A

square of 2×2 unit cells of the gap structure contains 16 points. The two cells

fit almost exactly (side length:

√

196 and

√

192 (gap);

√

189 (overlap)). The

rotation tan−^1 (

√

3 /6) + tan−^1 (

√

3 /9) = tan−^1 (5

√

3 /17) = 26. 996 ◦=3π/ 20

is the metric equivalent of the topological rotation of tan−^1 (1/2) of Bouli-

gand. The small orthorhombic distortions are different in the gap and in the

overlap. This is why one can observe two superposed, different X-ray diffrac-

tions patterns, corresponding to gap and overlap distances. Experimentally,

the distortion is 2% on average [1].

8.8 Transverse Structure; Coincidence Lattice of Two

Orthogonal, Triangular Lattices; Approximants of

√

3

The inflation multiplier for the collagen molecule is 1 +

√

3=[2,1].Itis

expected to dominate also the transverse structure of collagen, that should

be based on the coincidence points of two perpendicular triangular lattices,

with coordinates corresponding to rational convergents of

√

3. The superposed
   triangular lattices have a non-crystallographic, 12-fold rotation symmetry. The
   resulting lattice of near-coincidence points is, topologically, (two copies of) the
   Archimedean lattice 3^2. 4. 3. 4 .It is the gap structure, with unit cell shown in
   Fig. 8.6a. It has nearly square symmetry, with a slight orthorhombic distortion
   of 1.01.^9
   A triangular lattice is the set of points (1/2)ai+(

√

3 /2)bj, witha,bin-

tegers, both odd or even, whereiandjare orthogonal unit vectors. The 12

axes of symmetry are given bya=±bor bya=0orb= 0, and permutation

ofiandj. If the origin is one coincidence point, another lies:

(^8) The areas of the primitive cells match: overlap: (5/4)90 = 112.5, gap: 112 ele-
mentary triangles.
(^9) Two triangular lattices, rotated byθand superposed, give moir ́e patterns in
general, except forθ=π/3 (exact superposition) andθ=π/2(mod.π/3) (two
Archimedean lattices 3^2. 4. 3 .4).