The generic geometry of helices and their close-packed structures

The formation of helices is an ubiquitous phenomenon for molecular structures whether they are biological, organic, or inorganic, in nature. Helical structures have geometrical constraints analogous to close-packing of three-dimensional crystal structures. For helical packing the geometrical constraints involve parameters such as the radius of the helical cylinder, the helical pitch angle, and the helical tube radius. In this communication, the geometrical constraints for single helix, double helix, and for double helices with minor and major grooves are calculated. The results are compared with values from the literature for helical polypeptide backbone structures, the α-, π-, 3₁₀-, and γ-helices. The α-helices are close to being optimally packed in the sense of efficient use of space, i.e. close-packed. They are also more densely packed than the other three types of helices. For double helices comparisons are made to the A, B, and Z forms of DNA. The helical geometry of the A form is nearly close-packed. The packing density for the B and Z forms of DNA are found to be approximately equal to each other.