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.

从蛋白质笼子到扩展的二维（2D）和三维（3D）晶体阵列，蛋白质工程领域的各种努力都开始产生新型的对称自组装结构。已经介绍了创建对称蛋白质材料的部分理论框架，但尚未阐明完整的系统。实验仅探索了可能的设计空间的一小部分，部分原因是该空间尚未在理论上进行描述。在这里，我们以乘法表的形式，为可以通过以精确配置组合两个手性低聚成分（例如蛋白质）而创建的材料列出了完整的规则集。描述了一个统一的系统，用于参数化和搜索所有此类对称组合材料（SCM）的构造空间。总共，识别出124种不同类型的SCM，然后通过计算构造对其进行证明。在每种情况下都建立了数学特性，例如最小的环形或电路尺寸，从而可以对潜在的有利设计目标进行战略预测。该研究为快速增长的基于蛋白质的纳米技术领域提供了理论前景和详细的计算处方，并与数学网络和化学材料（如金属有机骨架）建立了许多潜在的联系。