Abstract
Diverse efforts in protein engineering are beginning to produce novel kinds of symmetric self-assembling architectures, from protein cages to extended two-dimensional (2D) and three-dimensional (3D) crystalline arrays. Partial theoretical frameworks for creating symmetric protein materials have been introduced, but no complete system has been articulated. Only a minute fraction of the possible design space has been explored experimentally, in part because that space has not yet been described in theory. Here, in the form of a multiplication table, we lay out a complete rule set for materials that can be created by combining two chiral oligomeric components (e.g., proteins) in precise configurations. A unified system is described for parameterizing and searching the construction space for all such symmetry-combination materials (SCMs). In total, 124 distinct types of SCMs are identified, and then proven by computational construction. Mathematical properties, such as minimal ring or circuit size, are established for each case, enabling strategic predictions about potentially favorable design targets. The study lays out the theoretical landscape and detailed computational prescriptions for a rapidly growing area of protein-based nanotechnology, with numerous underlying connections to mathematical networks and chemical materials such as metal organic frameworks.