Topological defect engineering enables size and shape control in self-assembly

TL;DR

This work demonstrates that size and shape in self-assembled structures can be controlled using defect engineering with a single subunit type. By favoring energetically favorable grain boundaries over perfect crystalline order, the authors induce finite, tunable assembly radii governed by the defect-to-crystal energy balance, with a characteristic size scaling set by the ratio of defect energy to crystalline energy. They validate the concept experimentally using DNA origami nanocylinders to form size-controlled 2D vortex and fiber-like assemblies, and they extend the approach to 3D with rhombic dodecahedral subunits. The work combines analytical theory, Monte Carlo simulations, and DNA-based experiments to establish a general framework for defect-driven self-assembly with broad potential design-space opportunities. Overall, the study provides a versatile route to engineer finite-size, shape-specific nanostructures using a single subunit type and tunable, topologically constrained defect interactions.

Abstract

The self-assembly of complex structures from engineered subunits is a major goal of nanotechnology, but controlling their size becomes increasingly difficult in larger assemblies. Existing strategies present significant challenges, among which the use of multiple subunit types or the precise control of their shape and mechanics. Here we introduce an alternative approach based on identical subunits whose interactions promote crystals, but also favor crystalline defects. We theoretically show that topological restrictions on the scope of these defects in large assemblies imply that the assembly size is controlled by the magnitude of the defect-inducing interaction. Using DNA origami, we experimentally demonstrate both size and shape control in two-dimensional disk- and fiber-like assemblies. Our basic concept of defect engineering could be generalized well beyond these simple examples, and thus provide a broadly applicable scheme to control self-assembly.

Figures (21)

• Figure 1: Crystals with energetically favorable defects may not grow to arbitrarily large sizes. (A) Our proposed subunits form crystals with binding free energy $\tilde{\epsilon}_c <0$, but also favor defective grain boundaries with $\tilde{\epsilon}_d < \tilde{\epsilon}_c$. Singular points at the end of these defects generically incur a cost $\tilde{\epsilon}_p>0$ related to $\tilde{\epsilon}_d$ and $\tilde{\epsilon}_c$. Denoting the typical size scale of the assembly by $r$, the associated free energies respectively scale as $\tilde{\epsilon}_c r^2$, $\tilde{\epsilon}_d r^1$ and $\tilde{\epsilon}_p r^0$. (B) Because these quantities scale differently with $r$, small assemblies are dominated by energetically costly singular points and large ones by crystalline interactions. The favorable defect interactions may dominate in assemblies with intermediate sizes, making them the most stable. Expressing $r$ in units of the subunit size, this qualitative result is recovered by minimizing the free energy per subunit $\approx -|\tilde{\epsilon}_c|-|\tilde{\epsilon}_d|/r+|\tilde{\epsilon}_p|/r^2$, yielding a preferred radius $r^*\approx|\tilde{\epsilon}_p/\tilde{\epsilon}_d|$.
• Figure 2: Modeling demonstrates size control in randomly assembling subunits. (A) We consider hexagonal subunits with anisotropic two-body interactions visualized by colored patches. All relative binding orientations not pictured in this panel are strongly penalized in our model. (B) Analytical phase diagram derived from Eqs. \ref{['eq:cam_and_crystal_energies']}. Colored symbols (triangles, square, hexagon, star, circle) link to the nomenclature defined in panel C. Inset: vortex assembly structure considered in our analytical calculations. (C) Final structures of simulated assemblies for the parameter values marked by symbols in panel B. See Tab. \ref{['tab:simu_parameters']} for detailed parameters. Long protrusions can form at the corner of the smallest assemblies, but are increasingly penalized for smaller values of $\epsilon_c/\epsilon_d$ as discussed in the Supplementary Text and Fig. \ref{['fig:branches_explanation']}. (D) Distributions of final assembly sizes at the end of the simulations (symbols) methods and comparison with an ideal-gas-of-cluster theory applied to Eq. \ref{['eq:camembert_energy']} (solid lines, see Supplementary Text). The moderate deviations observed for the largest assemblies are likely due to equilibrium entropic effects as well as nucleation kinetics (see Fig. \ref{['fig:rate_study']}).
• Figure 3: DNA origami nanocylinders form size-controlled assemblies. (A) OxDNA molecular model of our nanocylinders. Light (dark) blue linkers provide crystalline (defect) interactions. (B) Top row: In a crystalline interaction, complementary crystalline linkers on opposite nanocylinder sides (e.g., 2 and 5) hybridize. Defect linkers remain unhybridized. Bottom row: In a defect interaction (e.g., between sides 2 and 4) two pairs of defect linkers hybridize. Sides 5 and 6 never form defect interactions and do not carry defect linkers. (C) TEM micrographs of the assemblies formed by our nanocylinders illustrate the dependence of their radius on $\epsilon_c/\epsilon_d$. White arrow: a defect-interaction-induced appendage reminiscent of our simulations (Fig. \ref{['fig:theory']}C; more examples in Fig. \ref{['fig:protrusions']}) (D) Labeling of side 3 with gold nanoparticles reveals subunit orientations consistent with our vortex assembly design (red arrows). Apparent defect interaction sites are outlined in dark blue ($\epsilon_c/\epsilon_d= 0.5$). (E) The empirical distribution of orientations closely resembles an ideal $r=3$ vortex assembly structure with noise added ($\epsilon_c/\epsilon_d= 0.5$). (F) The assembly size distribution is robust to variations in incubation time. Here $[\text{MgCl}_2]=$5${\mathrm{m}\mathrm{M}}$, $\epsilon_c/\epsilon_d= 0.5$. (G) The assembly size distribution is robust to variations in salt concentration ($\epsilon_c/\epsilon_d= 0.5$). H. The assembly radius depends on the binding free energies as predicted by theory. See methods for statistical information.
• Figure 4: Defect engineering provides a rich design space. Here subunits with identical orientations form crystalline interactions with energy $\epsilon_c$ throughout. (A) Defect interactions and conceptual schematic for fiber-like assemblies. (B) Corresponding phase diagram derived from Eqs. \ref{['eq:crystal_energy']} and \ref{['eq:fiber']}. It differs from Fig. \ref{['fig:theory']}B only through the slopes of the diagonal lines. Inset: Structure of an ideal fiber. (C) Simulations conducted as in Fig. \ref{['fig:theory']}C. (D) TEM micrograph of a fiber formed as in Fig. \ref{['fig:exp1']} but with the new defect interactions. (E) Patterned bulk design. (F) Rectangular size-limited design where entropic effects control the aspect ratio of the assembly. (G) A non-size-controlled design. (H) Three-dimensional size control. The subunits are rhombic dodecahedral whose orientation is indicated by a dark grey arrowhead. These arrows point away from a hedgehog topological defect at the center of the assembly. See Tab. \ref{['tab:simu_parameters']} for simulation parameters and Supplementary Text for the predictions of the optimal assembly sizes $r^*$
• Figure S1: We measure the assembly size $N$ and surface $S$ in the simulation and deduce the vortex assembly radius $r$ and branch size $b$. (A) $N$ is the number of subunits in the assembly and $S$ is the number of subunit edges in contact with an empty site outside of the assembly. (B) The vortex radius $r$ is the size of a triangular crystalline domain (in dark gray) and the total branch length $b$ is the number of subunits within a branch (in dark red).