PENTAGON PACKING MODELS FOR ALL-PENTAMER VIRUS STRUCTURES

A connection is made between 1) the observed structures of virus capsi ds whose capsomers are all pentamers and 2) the mathematical problem o f determination of the largest size of a given number of equal regular spherical pentagons that can be packed on the surface of the unit sph ere 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 additiona l insight into a mathematical problem. At the same time this mathemati cal problem enables prediction of an octahedral form of papillomavirus particles consisting of 24 pentamers. It is also found that the vario us tubular and spherical ''all-pentamer'' virus structures identified so far can be represented by closest-packing arrangements of equal mor phological units composed of equal regular pentagons on a cylinder and on a sphere.