Proteinaceous Nanoshells with Quasicrystalline Local Order

Among various proteinaceous nanocontainers and nanoparticles, the most promising ones for various applications in nano- and medical science appear to be those whose structures differ fundamentally from icosahedral viral capsids described by the paradigmatic Caspar-Klug model. By analyzing such anomalous assemblies represented in the Protein Data Bank, we identify a series of shells with square-triangular local order and find that most of them originate from short-period approximants of a dodecagonal tiling consisting of square and triangular tiles. Examining the nonequilibrium assembly of such packings, we propose a new method for obtaining periodic square-triangle approximants and then construct the simplest models of tetragonal, octahedral, and icosahedral shells based on cubic and icosahedral nets cut from the approximant structures. Since gluing the nets can change the distances between adjacent vertices of the resulting shell, we introduce an effective energy, the minimization of which equalizes these distances. While the obtained spherical polyhedra reproduce the structures of experimentally observed protein shells and nanoparticles, the principles of protein organization that we lay out, and the ensuing structural models, can help to discover and investigate similar systems in the future.