Pentagon packing models for "all-pentamer" virus structures.

A connection is made between 1) the observed structures of virus capsids whose capsomers are all pentamers and 2) the mathematical problem of determination of the largest size of a given number of equal regular spherical pentagons that can be packed on the surface of the unit sphere without overlapping. It is found that papillomaviruses provide the conjectured solution to the spherical pentagon packing problem for 72 pentagons. Thus, a study of some virus structures has given additional insight into a mathematical problem. At the same time this mathematical problem enables prediction of an octahedral form of papillomavirus particles consisting of 24 pentamers. It is also found that the various tubular and spherical "all-pentamer" virus structures identified so far can be represented by closet-packing arrangements of equal morphological units composed of equal regular pentagons on a cylinder and on a sphere.