Boerdijk-Coxeter helix and biological helices

Abstract:

Helices and dense packing of spherical objects are two closely related problems. For instance, the Boerdijk-Coxeter helix, which is obtained as a linear packing of regular tetrahedra, is a very efficient solution to some close-packing problems. The shapes of biological helices result from various kinds of interaction forces, including steric repulsion. Thus, the search for a maximum density can lead to structures related to the Boerdijk-Coxeter helix. Examples are presented for the -helix structure in proteins and for the structure of the protein collagen, but there are other examples of helical packings at different scales in biology. Models based on packing efficiency related to the Boerdijk-Coxeter helix, explain, mainly from topological arguments, why the number of amino acids per turn is close to 3.6 in -helices and 2.7 in collagen.