Combinatorial Model of Viral Assembly and Binding

What physical conditions constrain how viruses assemble and bind to their target receptors? Such conditions should exist since, for simple viruses, the viral capsid can self-assemble and then attach itself to new host cells through a process entirely driven by physical laws. Moreover, the two processes of assembly and binding must work in concert for a virus to propagate its genetic material successfully: The individual proteins must prefer assembly over individual existence, and the assembled structure must select the correct receptors to bind. Finally, the entire process needs to navigate the combinatorial sea of the many ways individual proteins can form an assembled structure and the many ways assembled structures can bind to available receptors.

Motivated by this question, we use analytic combinatorics and statistical physics to build a model that reflects the combinatorial properties inherent to biomolecular assembly and binding. Combining combinatorial models of integer partitions and generalized derangements, we write the general partition function for assembly and binding and then approximate the result to obtain thermal conditions for the two processes. We find that assembly and binding processes can exist in multiple phases, only one of which ensures that an assembled structure can bind to a correct set of receptors. Within this phase, there is a binding-assembly temperature that scales as the inverse logarithm of the number of binding sites. In all, the constructed model provides an archetype for viral assembly and a theoretical foundation for building synthetic agents that can replicate viral functions for purposes beneficial to cells.