Lessons on structure from the structure of viruses
Richard D. James
University of Minnesota

Abstract:  As the most primitive organisms, occupying the gray area between
the living and nonliving, viruses are the least complex biological system.
One can begin to think about them in a quantitative way, while still being at
some level faithful to biochemical processes. We make some observations about
their structure, formalizing in mathematical terms some rules-of-construction
discovered by Watson and Crick and Caspar and Klug. We call the resulting
structures objective structures. It is then seen that objective structures
include many of the most important structures studied in science today:
carbon nanotubes, the capsids, necks, tails and other parts of many viruses,
the cilia of some bacteria, DNA octahedra, buckyballs, actin and collagen and
many other common proteins, and certain severely bent and twisted beams. The
rules defining them relate to the basic invariance group of quantum
mechanics.  We develop a methodology for computing such structures.  Some of
the nonperiodic structures revealed by the formulas exhibit beautifully
subtle relations of symmetry.  This common mathematical structure paves the
way toward many interesting calculations for such structures: the possibility
of unusual electromagnetic and other collective properties, simplified
schemes for exact molecular dynamics of such structures, phase
transformations between them, defects and failure, nonlinear elastic
properties, and their growth by self-assembly.