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Abstract Modern experimental methods in programmable self-assembly make it possible to precisely design particle concentrations, shapes and interactions. However, more physical insight is needed before we can take full advantage of this vast design space to assemble nanostructures with complex form and function. Here we show how a substantial part of this design space can be quickly and comprehensively understood by identifying a class of thermodynamic constraints that act on it. These thermodynamic constraints form a high-dimensional convex polyhedron that determines which nanostructures can be assembled at high equilibrium yield and reveals limitations that govern the coexistence of structures. We validate our predictions through detailed, quantitative assembly experiments of nanoscale particles synthesized using DNA origami. Our results uncover physical relationships underpinning many-component programmable self-assembly in equilibrium and form the basis for robust inverse design, applicable to various systems from biological protein complexes to synthetic nanomachines. 

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. 

Programmable self-assembly has recently enabled the creation of complex structures through precise control of the interparticle interactions and the particle geometries. Targeting ever more structurally complex, dynamic, and functional assemblies necessitates going beyond the design of the structure itself, to the measurement and control of the local flexibility of the intersubunit connections and its impact on the collective mechanics of the entire assembly. In this study, we demonstrate a method to infer the mechanical properties of multisubunit assemblies using cryogenic electron microscopy (cryo-EM) and RELION’s multi-body refinement. Specifically, we analyze the fluctuations of pairs of DNA-origami subunits that self-assemble into tubules. By measuring the fluctuations of dimers using cryo-EM, we extract mechanical properties such as the bending modulus and interparticle spring constant. These properties are then applied to elastic models to predict assembly outcomes, which align well with experimental observations. This approach not only provides a deeper understanding of nanoparticle mechanics but also opens pathways to refining subunit designs to achieve precise assembly behavior. This methodology could have broader applications in the study of nanomaterials, including protein assemblies, where understanding the interplay of mechanical properties and subunit geometry is essential for controlling complex self-assembled structures. 

We describe a modular design approach for creating versatile DNA origami subunits that can target diverse self-assembled structures. The subunit consists of a constant “core module” with variable “bond modules” and “angle modules” added to its exterior, controlling interaction specificity, strength, and structural geometry. The design features flexible joints between subunits, implemented by using single-stranded angle modules, whose mechanical properties and possible conformations are characterized by cryogenic electron microscopy and coarse-grained molecular modeling. We demonstrate the design’s versatility through the assembly of structures with different Gaussian curvature, including sheets, spherical shells, and tubes. Our findings suggest that incorporating a judicious amount of flexibility in the bonds provides error tolerance in design and fabrication while maintaining target fidelity. Furthermore, off-target assemblies potentially introduced by flexibility can be counterbalanced by increasing the number of distinct bonds. This approach enables precise targeting of specific structural binding angles across a broad range of configurations by eliminating unfavorable interactions. 

Attaching enzymes to nanostructures has proven useful to the study of enzyme functionality under controlled conditions and has led to new technologies. Often, the utility and interest of enzyme-tethered nanostructures lie in how the enzymatic activity is affected by how the enzymes are arranged in space. Therefore, being able to conjugate enzymes to nanostructures while preserving the enzymatic activity is essential. In this paper, we present a method to conjugate single-stranded DNA to the enzyme urease while maintaining enzymatic activity. We show evidence of successful conjugation and quantify the variables that affect the conjugation yield. We also show that the enzymatic activity is unchanged after conjugation compared to the enzyme in its native state. Finally, we demonstrate the tethering of urease to nanostructures made using DNA origami with high site-specificity. Decorating nanostructures with enzymatically-active urease may prove to be useful in studying, or even utilizing, the functionality of urease in disciplines ranging from biotechnology to soft-matter physics. The techniques we present in this paper will enable researchers across these fields to modify enzymes without disrupting their functionality, thus allowing for more insightful studies into their behavior and utility. 

Programmable self-assembly has seen an explosion in the diversity of synthetic crystalline materials, but developing strategies that target “self-limiting” assemblies has remained a challenge. Among these, self-closing structures, in which the local curvature defines the finite global size, are prone to polymorphism due to thermal bending fluctuations, a problem that worsens with increasing target size. Here, we show that assembly complexity can be used to eliminate this source of polymorphism in the assembly of tubules. Using many distinct components, we prune the local density of off-target geometries, increasing the selectivity of the tubule width and helicity to nearly 100%. We further show that by reducing the design constraints to target either the pitch or the width alone, fewer components are needed to reach complete selectivity. Combining experiments with theory, we reveal an economical limit, which determines the minimum number of components required to create arbitrary assembly sizes with full selectivity.