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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 study the ground state thermodynamics of a model class of geometrically frustrated assemblies, known as warped-jigsaw particles. While it is known that frustration in soft matter assemblies has the ability to propagate up to mesoscopic, multi-particle size scales, notably through the selection of the self-limiting domain, little is understood about how the symmetry of shape-misfit at the particle scale influences emergent morphologies at the mesoscale. Here we show that polarity in the shape-misfit of warped-jigsaw puzzles manifests at a larger scale in the morphology and thermodynamics of the ground-state assembly of self-limiting domains. We use a combination of continuum theory and discrete particle simulations to show that the polar misfit gives rise to two mesoscopically distinct polar, self-limiting ribbon domains. Thermodynamic selection between the two ribbon morphologies is controlled by a combination of the binding anisotropy along distinct neighbor directions and the orientation of polar shape-misfit. These predictions are valuable as design features for ongoing efforts to program self-limiting assemblies through the synthesis of intentionally frustrated particles, further suggesting a generic classification of frustrated assembly behavior in terms of the relative symmetries of shape-misfit and the underlying long-range inter-particle order it frustrates. 

In geometrically frustrated assemblies local intersubunit misfits propagate to intra-assembly strain gradients, giving rise to anomalous self-limiting assembly thermodynamics. Here we use theory and coarse-grained simulation to study a recently developed class of “curvamer” particles, flexible shell-like particles that exhibit self-limiting assembly due to the build up of curvature deformation in cohesive stacks. To address a generic, yet poorly understood aspect of frustrated assembly, we introduce a model of curvamer assembly that incorporates both intraparticle shape deformation as well as compliance of interparticle cohesive gaps, an effect we can attribute to a finite range of attraction between particles. We show that the ratio of intraparticle (bending elasticity) to interparticle stiffness not only controls the regimes of self-limitation but also the nature of frustration propagation through curvamer stacks. We find a transition from uniformly bound, curvature-focusing stacks at small size to gap opened, uniformly curved stacks at large size is controlled by a dimensionless measure of inter- versus intracurvamer stiffness. The finite range of interparticle attraction determines the range of cohesion in stacks that are self-limiting, a prediction which is in strong agreement with numerical studies of our coarse-grained colloidal model. These predictions provide critical guidance for experimental realizations of frustrated particle systems designed to exhibit self-limitation at especially large multiparticle scales. 

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. 

We propose and investigate an extension of the Caspar–Klug symmetry principles for viral capsid assembly to the programmable assembly of size-controlled triply periodic polyhedra, discrete variants of the Primitive, Diamond, and Gyroid cubic minimal surfaces. Inspired by a recent class of programmable DNA origami colloids, we demonstrate that the economy of design in these crystalline assemblies—in terms of the growth of the number of distinct particle species required with the increased size-scale (e.g., periodicity)—is comparable to viral shells. We further test the role of geometric specificity in these assemblies via dynamical assembly simulations, which show that conditions for simultaneously efficient and high-fidelity assembly require an intermediate degree of flexibility of local angles and lengths in programmed assembly. Off-target misassembly occurs via incorporation of a variant of disclination defects, generalized to the case of hyperbolic crystals. The possibility of these topological defects is a direct consequence of the very same symmetry principles that underlie the economical design, exposing a basic tradeoff between design economy and fidelity of programmable, size controlled assembly. 

In self-assembling systems, geometric frustration leads to complex states characterized by internal gradients of shape misfit. Frustrated assemblies have drawn recent interest due to the unique possibility that their thermodynamics can sense and select the finite size of assembly at length scales much larger than constituent building blocks or their interactions. At present, self-limitation is chiefly understood to derive from zero-temperature considerations, specifically the competition between cohesion and scale-dependent elastic costs of frustration. While effects of entropy and finite-temperature fluctuations are necessarily significant for self-assembling systems, their impact on the self-limiting states of frustrated assemblies is not known. We introduce a generic, minimal model of frustrated assembly and establish its finite-temperature and concentration-dependent thermodynamics by way of simulation and continuum theory. The phase diagram is marked by three distinct states of translation order: a dispersed vapor, a defect-riddled condensate, and the self-limiting aggregate state. We show that, at finite temperature, the self-limiting state is stable at intermediate frustration. Furthermore, in contrast to the prevailing picture, its thermodynamic boundaries with the macroscopic disperse and bulk states are temperature controlled, pointing to the essential importance of translational and conformational entropy in their formation.