Exploring the Paths of (Virus) Assembly

Assembly of viruses that have hundreds of subunits or folding of proteins that have hundreds of amino acids complex biological reactions-are often spontaneous and rapid. Here, we examine the complete set of intermediates available for the assembly of a hypothetical viruslike particle and the connectivity between these intermediates in a graph-theory-inspired study. Using a build-up procedure, assuming ideal geometry, we enumerated the complete set of 2,423,313 species for formation of an icosahedron from 30 dimeric subunits. Stability of each n-subunit intermediate was defined by the number of contacts between subunits. The probability of forming an intermediate was based on the number of paths to it from its precedecessors. When defining population subsets predicted to have the greatest impact on assembly, both stability- and probability-based criteria select a small group of compact and degenerate species; ergo, only a few hundred intermediates make a measurable contribution to assembly. Though the number of possible intermediates grows combinatorially with the number of subunits in the capsid, the number of intermediates that make a significant contribution to the reaction grows by a much smaller function, a result that may contribute to our understanding of assembly and folding reactions.