152 N. Rivier and J.-F. Sadoc
Fig. 8.3.The single collagen chain PPII (Gly−X−Y) is a left-handed helix,
that can be drawn on a right-handed B–C helix with half the vertical spacing. The
helix is represented on a flat, rectangular strip by identification of the two long
sides of the rectangle. In the Hopf fibration of polytope{ 3 , 3 , 5 },fiveaminoacids
Gly,(orX,orY, resp.) lie on a fibre that is a great circle with ten vertices. Note
that the axisGly−Xof helix 7/2 is orthogonal to the base, so that a stack of
three segments 7/2 gives one period of the collagen chain in Euclidean space, the
left-handed helix 21/6, with a fibre of sevenGlywinding around once. The winding
number of the underlying right-handed B–C helix is 41/15, the next convergent,
after 30/11 (Fig. 8.2c), of 1 +
√- There are 14 vertices in the fibre. The collagen
chain is periodic and the embedding space is completely decurved
8.4 Decurving
The structure of collagen in Euclidean space is obtained by decurving space,
i.e. by iteratively increasing the radius of the polytope, in a way that keeps con-
stant the rate of intertwining of the chains, and the connectivity and symmetry
of the base space. Decurving increases the radius and area of the base space
and lengthens the fibres. In a fibration, the base must remain an Archimedean
polyhedron or honeycomb, a tiling of triangles and squares (decurved trian-
gles) with vertex connectivityz=5.^5 This suggests a decurving of the base
(^5) If the individual helices are represented by triangles in base space, as in Fig. 8.4b,
decurving must not involve any physical distance. The triangle that changed into
a square is neither the core of threeGly, nor the triangleGly−X−Yrepresenting
one polypeptide chain, nor even the triangleGly−(Gly−X), because it involves
the horizontal hydrogen bond between theGlyof one chain and the X of another.
It can only be the triangle (Y−Gly)=X, with the distance between theXof
one chain and theYof the other stretched to become the diagonal of the square.
The snub cube contains then the projection of two complete molecules. If the
triangle representing the polypeptide chain was changed into a square, one would
have obtained an alpha helix [4].