Abstract
Biology is teeming with intricate molecular structures whose geometries are inextricably linked to their function. A prototypical example is the helical bacterial flagellum, a complex curved crystalline assembly of proteins that the bacterium uses to swim. Because synthetic analogues of these and other curved crystalline assemblies could be valuable platforms for nanotechnologies, including drug delivery and plasmonics, controllable synthesis of variable-curvature structures of diverse material systems, from fullerenes to supramolecular assemblies, has been a long-standing goal. Here, we develop and implement a design strategy to program the self-assembly of a complex spectrum of two-periodic curved crystals with variable periodicity, spatial dimension, and topology, spanning from toroids to achiral serpentine tubules to both left- and right-handed helical tubules. Notably, our design strategy exploits a kirigami-based mapping of a modular class of 2D planar tilings to 3D curved crystals that preserves the periodicity, twofold rotational symmetries, and subunit dimensions by modulating the arrangement of disclination defects. We survey the modular geometry of these curved crystals and infer the addressable interactions required to assemble them from triangular subunits. To demonstrate this design strategy in practice, we program the self-assembly of toroids, helical, and serpentine tubules from DNA origami subunits, deriving the distinct kirigami foldings of a single two-periodic planar tiling. A simulation model of the assembly pathways reveals physical considerations for programming the geometric specificity of the intersubunit angles in the curved crystal required to avoid defect-mediated misassembly.