A complete rule set for designing symmetry combination materials from protein molecules

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.

Keywords: biomaterials; nanotechnology; protein design; self-assembly; symmetry.

Fig. 1.

Diagrams of symmetric oligomeric building blocks and example two-component SCMs. (A) Illustration of point group symmetries (top row) C2, C3, C4, C5, C6, D2, (bottom row) D3, D4, D6, T, and O with their symmetry axes. (B) A finite assembly with octahedral symmetry constructed by combining a C3 trimer (blue) and a C4 tetramer (orange)—O:{C3}{C4} (Left). An extended p6 2D layer formed by combining a C3 trimer (blue) and a C6 hexamer (orange)—p6:{C3}{C6} (Middle). A P422 3D crystalline array assembled by combining a D2 tetramer (blue) and a D4 octamer (orange)—P422:{D2}{D4} (Right).

Fig. 2.

Procedural construction for rigid body sampling, diagrammed for three example SCMs. The examples illustrate how the rules can be implemented in a stepwise procedure that enables the allowed construction space to be sampled. In each case, the rigid body sampling degrees of freedom are highlighted in red. The examples are all cases where the number of degrees of freedom is 3, although they present in different forms. The orientation setting matrices are given in SI Appendix, Table S1. SI Appendix, Table S3 provides the full listing of degrees of freedom for sampling every construction type (available as Dataset S1). Further construction details are provided for the middle example (F432:{C4}{D2}) in SI Appendix, Fig. S6. Construction protocols for all 124 SCMs are described in pseudocode in SI Appendix, Text (also available as Dataset S2).

Fig. 3.

Illustration of the concept of ring size for three example SCMs. The ring size is the number of oligomers of each type in a closed circuit. I:{C2}{C5}, a finite assembly with icosahedral symmetry constructed by combining a C2 dimer (blue) and a C5 pentamer (orange), has a ring size of 3 (Left). p4:{C2}{C4}, a 2D layer with p4 symmetry formed by combining a C2 dimer (blue) and a C4 tetramer (orange), has a ring size of 4 (Middle). P23:{C2}{T}, a 3D crystalline array with P23 symmetry assembled by combining a C2 dimer (blue) and a T dodecamer (orange), has a ring size of 2 (Right). For each case, a single ring is illustrated with participating oligomers in two shades of red.

Fig. 4.

Potentially privileged SCMs for 3D crystal designs. A P432 crystal constructed by combining a C3 trimer (blue) and a D4 octamer (orange) (Top). A crystal with space group I432 constructed by combining a C4 tetramer (blue) and a D3 hexamer (orange) (Bottom). The SCMs illustrated here have three degrees of freedom available for construction and a ring size of 2. Furthermore, in both cases, the combination of the two symmetry types can only give rise to a single type of 3D space group. The SCMs are illustrated as networks on the Left. A plausible protein packing is modeled for each case (Middle). A single redesigned contact between the hypothetically docked oligomers is necessary and sufficient to hold the respective modeled architectures together (Right). The protein constructions are shown to convey design plausibility. Additional candidates are possible with different choices for the component oligomers; multiple plausible docking modes are sometimes possible with the same oligomers in different orientations. The examples shown are therefore only representatives of a multitude of plausible candidates and were selected based on criteria suitable for interface design (SI Appendix, Text). Other criteria are possible for alternate interfacial modes and connection types (Protein Data Bank ID codes: Top, 2Q0T, blue, and 4B4K, orange; Bottom, 1CUK, blue, and a D3 hexamer, 2V78, orange).