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.