Tubular packing of spheres in biological fine structure

The symmetrical arrangements of monomers into such cylindrical structures as microfilaments of actin, flagella of bacteria, microtubules of many organisms, and the protein coats of viruses can be specified by citing the index numbers of two or three sets of contact parastichies, or helical ranks of monomers, as has been done in classical studies of phyllotaxis. This specification has the form k(m, n) or k(m, n, m+n), where m, n, and (m+n) are parastichy numbers specifying screw displacements, and k is the jugacy, or frequency of rotational symmetry. For simple structures, k = 1. This notation has the advantage of terseness and of indicating the basic isometries of these helically symmetrical structures. Theoretical models of the packing of spheres whose centers lie on the surface of a cylinder have been investigated geometrically. Their symmetry properties are discussed. Parameters of these models, such as the angular divergence, alpha, the longitudinal displacement between successive spheres, h, the radius of the cylinder, and the angles of inclination of the parastichies, have been computed for representative patterns. The ultrastructural symmetry of several biological structures of this sort has been inferred by comparison with these models. Actin, for example, has the symmetry (1, 2), Salmonella flagella, 2(2, 3, 5), the tobacco mosaic virus, (1, 16, 17) and the microtubules of many higher organisms, (6, 7, 13).