An application of the contact pressure model of phyllotaxis to the close packing of spheres around a cylinder in biological fine structure

Abstract

Equations derived from a contact pressure model of phyllotaxis are relevant to close packing of equal spheres on a cylindrical surface. They provide a general method of calculating the parameters for microscopic biological structures that are assembled in helical arrangements of protein monomers. In the triple-contact case of hexagonal packing, with km, kn and k( m+ n) contact parastichies respectively, where m and n are relatively prime, the divergence angle d and the normalized internode distance r on the k fundamental spirals are completely determined by m, n and k, and formulas are given for calculating them. In the double-contact case of rhombic packing, with km and kn contact parastichies respectively, where m < n, d is a function of r, and the domain of r is the interval between the value of r determined for the triple-contact case of km, kn and k( m+ n) parastichies and the value of r determined for the triple-contact case of k( n- m), km and kn parastichies. Here r and d can be determined from the measured ratio of the radius of the cylindrical surface to the radius of the spheres.