From toroids to helical tubules: Kirigami-inspired programmable assembly of two-periodic curved crystals

Biology is full of intricate molecular structures whose geometries are inextricably linked to their function. Many of these structures exhibit varying curvature, such as the helical structure of the bacterial flagellum, which is critical for their motility. Because synthetic analogues of these shapes could be valuable platforms for nanotechnologies, including drug delivery and plasmonics, controllable synthesis of variable-curvature structures of various material systems, from fullerenes to supramolecular assemblies, has been a long-standing goal. Like two-dimensional crystals, these structures have a two-periodic symmetry, but unlike standard two-dimensional crystals, they are embedded in three dimensions with complex, spatially-varying curvatures that cause the structures to close upon themselves in one or more dimensions. 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. Our design strategy uses a kirigami-based mapping of 2D planar tilings to 3D curved crystals that preserves the periodicity, two-fold rotational symmetries, and subunit dimensions via the arrangement of disclination defects. We survey the modular geometry of these curved crystals and infer the addressable subunit interactions required to assemble them from triangular subunits. To demonstrate this design strategy, we program the self-assembly of toroids, helical- and serpentine-tubules from DNA origami subunits. A simulation model of the assembly pathways reveals physical considerations for programming the geometric specificity of angular folds in the curved crystal required to avoid defect-mediated misassembly.