Non-viral protein shells with octahedral and tetrahedral symmetries and a theory of critical density waves

Unlike viral capsids, which protect and transport genetic material, non-viral protein shells perform diverse biological functions, including enzymatic catalysis, iron storage and protein chaperone activity. These shells often exhibit tetrahedral (T) and octahedral (O) point group symmetries, and their structures can include four- and even eightfold capsomers that are absent in viral capsids. Here, to rationalize the design of such nanocages, we develop a phenomenological theory based on the approach of critical density waves (DWs). We consider these spherical proteinaceous structures to be the result of condensation of one or two irreducible spherical DWs with close wavenumbers and place individual proteins (or capsomers) at the maxima of interference patterns generated by these waves. By examining the structures presented in the protein data bank (PDB), we identify 22 structural types of shells well rationalized within the theory and discuss the principles controlling protein order. In particular, we explain why these assemblies often possess local quasicrystalline order and predict the structures of yet unknown shells with cubic symmetries. The results obtained offer valuable insights into the rational design of stable, functional protein nanocages.