Algebra for large macromolecular complexes*

Large macromolecular complexes play central roles in biophysics and biochemistry. Their combinatorial complexity, however, has hindered their theoretical study using the standard methods of statistical physics. To overcome this barrier, we introduce an algebraic formalism for describing classical multi-particle complexes. Using a Fock space comprised of hard-core bosons, this framework allows pre-existing particles to be joined together into large complexes based on algebraically defined assembly rules. Physically interesting quantities, such as Gibbs free energy, can then be computed based on the contributions from individual component particles, pairwise interactions between these particles, and so on. We also introduce diagrammatic techniques that make this algebra visually intuitive and facilitate analytical calculations through a (somewhat surprising) realization of Wick's theorem. Finally, we show how this algebra unifies seemingly distinct notions of coarse graining, a fact we illustrate in the context of a biophysical model of transcriptional regulation. We expect that our Fock space formalism will be useful for mathematical and computational studies of a wide range of combinatorially complex systems in biophysics and biochemistry.