Introduction

Nature is replete with functional biomolecular structures, such as viral capsids and microtubules, which are self-assembled with remarkable precision, relying on specific geometries and interactions encoded in molecular subunits1,2,3. Inspired by Nature, scientists have developed synthetic building blocks with programmable geometries and interactions—such as DNA tiles4, DNA-coated colloids5,6,7,8, DNA origami9, DNA bricks10,11, and de novo-designed proteins12,13—to assemble prescribed, functional nanostructures. Among the available platforms, DNA origami stands alone in its ability to simultaneously encode the block geometry, binding angles, and interaction specificity for programming assemblies14,15,16,17,18,19,20,21,22,23,24,25,26,27,28, while enabling precise labeling of functional particles like metallic nanoparticles, quantum dots, and proteins29,30,31,32,33, making it a promising tool in nanotechnological applications, such as photonics34,35, plasmonics36,37, drug delivery and anti-viral therapeutics21,38,39.

Although DNA origami offers a vast design space, the range of accessible structures is still limited, particularly to those with uniform curvatures, due to the high financial and labor cost associated with synthesizing many unique DNA origami subunits. Constructing complex geometries with nonuniform curvature is limited by a few factors. First, each new building block shape requires designing a distinct scaffold routing, and a corresponding set of unique staple strands. Second, all unique bond interactions and angles must be correctly programmed, and the iterative tuning required becomes increasingly difficult as the number of unique bonds grows. To overcome these hurdles, modular design strategies have been developed: a small set of subunits is reused, with only minor staple modifications to encode type specificity (the local rules for which particle types bind to one another)18,26,28,40. While this approach enables the creation of many uniquely interacting particles, both the particle shapes and interface geometries remain immutable. As a consequence, the assembled structures have so far been limited to relatively simple geometries with uniform curvatures, such as sheets19,24,41, 3D crystals34,35,36,37, shells21, and cylindrical tubules22,23.

In contrast, assembly of geometries with nonuniform curvatures remains a challenge. Targeting a unique surface with variation in spatial curvature requires precise control over both type specificity and geometric specificity (the local geometry of the bonds between neighboring subunits) simultaneously in a complex mixture of many different species of building blocks. Recent efforts by Karfusehr et al.42 and us43 demonstrate modular DNA origami design with programmable binding angles encoded by varying the length of single-stranded DNA (ssDNA) spacers. However, flexible ssDNA overhangs are unsuitable for encoding distance between two ends; they sacrifice the bond rigidity and the ability to encode large angles. More importantly, no design platform has yet shown the capability of predictively and reliably assembling arbitrary complex 2D manifolds with fully programmable type and geometric specificity. For detailed comparison of previous modular designs, see the Supplementary Section I.

Here, we introduce a modular DNA origami building block whose interaction specificity and binding angle can be programmed independently to assemble geometrically complex 2D manifolds, predictively and reliably. We design a 50 nm triangular DNA origami building block with extruded DNA overhangs at the edges (Fig. 1A). By designing the part of the overhang sequence, we encode specific interactions (Fig. 1B). Additionally, we encode the binding angles by adjusting the relative length of the tunable, rigid double-stranded DNA (dsDNA) domains (Fig. 1C). We characterize the binding angles of our design through various methods, including cryogenic electron microscopy (cryo-EM) reconstruction, oxDNA simulations, and yield assessment of simple assembly structures, from which we obtain a predictive relationship between the binding angles and the dsDNA overhang lengths. We find that our dsDNA strut design encodes a wider range of angles compared to previous modular designs using ssDNA. Using the design platform, we assemble simple Platonic and non-Platonic shells at high yield. The high fidelity in programming both the interaction and geometric specificity opens doors to large, sophisticated structures that were previously inaccessible. Without any iterative refinement, we assemble a T = 13 icosahedral shell with five unique subunits and seven angles, and a hollow toroid with 12 unique subunits and seven angles (Fig. 1D).

Fig. 1: Modular DNA origami building block with programmable interactions and binding angles.
Fig. 1: Modular DNA origami building block with programmable interactions and binding angles.The alternative text for this image may have been generated using AI.
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A Schematic and cryo-EM reconstruction of the modular DNA origami building block, a hollow right equilateral triangular prism, but lacking up-down symmetry in the bonds. Cylinders represent dsDNA helices and are packed on a square lattice15,16. Top, side, and section views. B ssDNA overhangs protrude from the sides of identical triangles. The symbol ✓ indicates a binding pair and ✗ indicates a non-binding pair. Only 1 of 6 possible bonds forms in this example. Colors denote members of a set of extruded overhangs on a single side. The interaction matrix contains the symbol ∘ when two sides bond and is blank otherwise. C The binding angle between two bound triangles, θ, is determined by the relative length of the two rows of overhangs on a single side. The binding angle can be made positive or negative by changing the location of the overhangs. D A schematic and EM image of a T = 13 icosahedral shell and a toroid. Different colors indicate different species of triangles encoded with a unique set of interactions and binding angles.

Results

Modular design concept

In our design, an 8064-nucleotide scaffold strand is folded by two types of staple strands: universal ‘core staples’ in the interior of the triangular prism and ‘interface staples’ that have portions that protrude out of the triangle sides at assigned locations. The 148 unique core staples remain constant across all triangle designs; the 60 unique interface staples, 20 per side, vary from design to design and program the interaction specificity and binding angles. Each unique design utilizes two rows of interface staples per side (Fig. 2A). We note that once initial sets of staples are purchased, only eight additional staples are required per unique edge, significantly lowering the cost for multicomponent assemblies (see Supplementary section II for details).

Fig. 2: Design and characterization of binding angles.
Fig. 2: Design and characterization of binding angles.The alternative text for this image may have been generated using AI.
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A Front view of the DNA origami building block. B Side view of bonded pair at binding angle θ, with magnified views of extruded overhangs. C Left: cryo-EM reconstruction of +10δ dimers. Right: binding angle distributions for +6δ (blue) and +10δ (red) dimers. Dashed lines are measurements assumed to be equilibrated at 136 K, the solid line is rescaled to 298 K. Two extreme conformations are shown. D Top row: (left) oxDNA-generated mean configuration of +10δ dimers at 298 K, (right) binding angle distributions for +6δ (blue) and +10δ (red). Dashed lines are for 136 K, solid lines are for 298 K. For clarity, histograms are displayed only for 136 K. Two extreme conformations at 136 K are shown. Bottom row: oxDNA mean configuration of +15δ dimers with zoomed-in views of overhangs and two-thymine spacers. E Anisotropic monomers with directional labeling: top (dotted) and bottom (solid) orientations. F Monomers with side 1 binding to side 2 assemble into vertices. G TEM images of vertices. H Selectivity of vertices of different sizes as angle-domain length varies, obtained from gel electrophoresis. Each color signifies a unique vertex. I Monomers with sides 1 and 2 both binding to themselves assemble into strips or rings. J TEM images of rings. K Selectivity of rings of different sizes as angle-domain length varies, derived from gel electrophoresis. L Summary of binding angles from geometrical prediction (•, black), oxDNA dimer simulation (×), cryo-EM multibody analysis (•, blue), and vertex (□) and ring (♢) gel electrophoresis assembly analysis (red for positive and green for negative angles). Solid line indicates the linear fit to the vertex and ring assembly results. Error bars indicate standard deviations. All values are either obtained or scaled to 298 K.

The interface staples are conceptually divided into two parts: one part binds to the scaffold and remains invariant from design to design, while the second binds to other interface staples and can vary between designs. This variable part contains two domains, a ‘bond’ domain that programs the interaction specificity and an ‘angle’ domain that programs the binding angles (Fig. 2B).

Although the triangular prism has top-bottom symmetry, we introduce asymmetric bonding sites that distinguish up from down, which is essential for forming structures with a well-defined inside and outside, such as capsids. The bond domain of a single staple is a five-nucleotide-long sequence located at the end of an interface staple. Four pairs of bonds are located on each of the three sides of a triangle. The eight bonds that bind two triangles together are designed to have a high off-rate to promote rapid equilibration. We further design the interactions between two triangles to be highly specific and orthogonal, taking advantage of the sequence and positional specificity of the overhangs. See Supplementary Section IV for details on the sequence design.

The angle domain, formed by hybridizing adjacent interface staples, is a double-stranded region that controls the binding angle θ between neighboring triangles and can be tuned from 0 to 15 base pairs (Fig. 2A, B). We program the binding angle by choosing the difference in lengths of the angle domains in two different rows of interface staples. To create positive angles, we set one angle domain to have zero length, in which case only one staple extrudes from the core, while the other angle domain requires two adjacent staples extruding from the core (Fig. 2B). The sign of the angle is controlled by the relative location of the shorter angle domains with respect to the longer angle domains. We classify angle domains as +nδ for positive and −nδ for negative angles, where nδ is the length difference of the angle domain in units of the number of base pairs. All interface staples have a two-thymine-long spacer where the overhang protrudes from the building block. The design details are discussed in Supplementary Section III.

Designing and characterizing the binding angle

We start by developing a simple geometrical model to build an intuition for the relationship between the difference in lengths of the angle domains and the resultant binding angle. By assuming that the bridges linking neighboring triangles are rigid rods, we derive an expression relating the binding angle θ to the length difference of the angle domains nδ:

$$\theta=2\arcsin \frac{0.34\,{{{\rm{nm}}}}\times n\delta }{2\times d},$$
(1)

where d = 2.6 nm is the interhelical distance16,44, assuming the helices are close-packed (Fig. 2B). We also assume that the bridges approximate rigid rods because their lengths are much shorter than the persistence length of dsDNA45. The factor of 0.34 nm/bp is the length of dsDNA per base pair46. For small angles, θ is approximately linear with nδ (Fig. 2L).

We test the geometric relationship between binding angle (θ) and angle-domain length (nδ) using three complementary approaches. First, we experimentally assemble dimers and use cryo-EM reconstruction to characterize the binding angle distributions as a function of nδ (Fig. 2C). Second, we experimentally assemble two classes of self-closing assemblies, vertices and rings, and compare the yields of different polymorphs, whose sizes are governed by the binding angles, bending rigidity, and effective binding free energy, to a thermodynamic model. Third, we use the oxDNA coarse-grained model to represent the dimers and carry out molecular dynamics simulations to predict their binding angle distributions (Fig. 2D)47,48.

We obtain 3D structures of dimers of particles using single-particle cryo-EM. The average reconstructions for three designs with angle-domain lengths of +6δ, +10δ and −7δ are shown in Fig. 2C and Supplementary Fig. S9. From these average reconstructions, we fit the separate triangular bodies and find the opening angle between them. The measured average angle is in excellent agreement with the geometric model (Fig. 2L). To quantify the fluctuations in the system, we also perform a multibody analysis49,50 on the dimer, assuming that each triangular monomer is a rigid body (Fig. 2C). In cryo-EM, samples are quenched to the vitrification temperature of water, 136 K, in about 10−4 seconds, which we assume is enough time for the bond angle distribution to equilibrate as the rotational diffusion time of a monomer is much less than the quench time. Furthermore, we assume that the physical properties of DNA are independent of the temperature to relate the fluctuations that we measure at the vitrification temperature to the fluctuations we would expect at room temperature. Thus, in Fig. 2C we assume the measured bond angle distribution is equilibrated at 136 K and then, to obtain the distribution at 298 K, we scale the standard deviation by temperature as \({\sigma }_{298{{{\rm{K}}}}}^{{{{\rm{EM}}}}}/{\sigma }_{136{{{\rm{K}}}}}^{{{{\rm{EM}}}}}=\sqrt{298\,{{{\rm{K}}}}/136\,{{{\rm{K}}}}}\). More details are in Supplementary Section V.

To test the assumption that the physical properties of dsDNA are independent of temperature, ranging from room temperature to the vitrification temperature of water, we design an experiment to infer the binding angles experimentally at 298 K from the assembly of self-closing structures whose polymorphs are predictable. For example, vertices ranging from hexamers to trimers self-assemble when side 1 binds to side 2 of the triangle (with side 3 inert), depending on the angle-domain lengths (Fig. 2E–H). Rings form when side 1 binds to side 1 and side 2 binds to side 2 (with side 3 inert) (Fig. 2I–K). We conduct these experiments using both positive and negative binding angles. See Supplementary Figs. S30, S31 for the wide-field-of-view transmission electron microscopy (TEM) images of vertex and ring assemblies, respectively.

We measure the equilibrium room temperature distributions of the vertex and ring sizes using gel electrophoresis, and infer the average binding angle for each angle-domain length from a thermodynamic model. By analyzing the band intensities from gel electrophoresis, we obtain the relative yields of all of the self-closing structures that assemble (Fig. 2H, K). We find a monotonic decrease in the most probable structure size with increasing angle-domain lengths. This trend is expected because increasing nδ should increase θ, resulting in closed structures with a smaller radius of curvature and fewer subunits. To infer the preferred binding angles from the experiments, we fit the selectivity of the various self-closing structures, defined as the relative yield of a given polymorph to that of all possible self-closing structures, to a theoretical selectivity estimated by considering the energies of all allowed polymorphs in a constant-temperature (canonical) ensemble (see Supplementary Section VI)20. The absolute values of the binding angles deduced from electrophoresis experiments are systematically smaller than the predictions of the geometrical model, with the deviation increasing for longer angle-domain lengths (Fig. 2L). Next, we analyze the binding angles computed from oxDNA simulations (Fig. 2D). We first run simulations to determine the equilibrium distribution of binding angles at room temperature, 298 K, which is close to the temperature at which assembly experiments are performed, and then we quench the system to 136 K, which is the temperature at which we presume the dimers are equilibrated in the cryo-EM measurements. The average angles obtained from oxDNA at 298 K are consistent with the vertex and ring measurements. The average angles vary with temperature, with the bond angle being larger at low temperature. Notably, as shown in Fig. 2C, D, the standard deviation of the distributions determined by cryo-EM at 136 K are about two-times greater than predicted by oxDNA. Consequently, oxDNA predicts a roughly 4-fold greater bending modulus than obtained by cryo-EM. For discussion of the oxDNA results and possible explanations of the discrepancy with cryo-EM, see Supplementary Sections VII and VIII.

Finally, we derive a design rule for encoding angles via a linear fit to our data points from the assembly-size experiments. For positive and negative angles, we obtain +6.65° and −6.28° per base pair (Fig. 2L), which are roughly 15% and 22% smaller than one would predict from the geometrical model over its linear regime, respectively, but are in good agreement with cryo-EM measurements, vertex and ring assembly experiments, and oxDNA predictions. Going forward, we use this linear relationship to program the angles necessary to assemble geometrically complex structures.

Self-assembly of deltahedral shells

To test our design scheme, we start by assembling triangles with identical angle-domain lengths and self-complementary sequences on all three sides. Because all the sides bind to each other and the average binding angles of all three sides are the same, we expect to assemble symmetric structures with constant binding angles between neighboring subunits, e.g., 2D sheets, icosahedra, octahedra, and tetrahedra, with binding angles of 0°, 41.8°, 70.5°, and 109.5°, respectively (Fig. 3A). Given the empirical design relationship between binding angle, θ, and the difference in the number of bases in the angle domain, nδ (Fig. 2L), we expect these structures to emerge for +0δ, +6δ, +11δ, and +16δ, respectively. However, as each bond has a distribution of binding angles, other structures could emerge as well.

Fig. 3: Shell assembly by tuning the angle-domain length.
Fig. 3: Shell assembly by tuning the angle-domain length.The alternative text for this image may have been generated using AI.
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A Every side of the monomer is encoded with identical, self-complementary interaction and angle-domain length. By changing the angle-domain length, monomers assemble into various structures from 2D sheets to self-closed Platonic solids, shown in TEM micrographs. Note that the bottom face of the triangle always points outward. B Laser-scanned image of gel electrophoresis where the monomers are assembled while varying the relative angle-domain length from 0 bp and 3 bp to 15 bp. Stars indicate angle-domain lengths whose corresponding binding angles are predicted to match closely with icosahedron, octahedron, and tetrahedron. C TEM micrographs of various non-Platonic shells that are observed in TEM at different angle-domain lengths. D Programmable self-assembly of non-Platonic shells by controlling the interaction specificity and binding angles, using up to two species of triangles. Blue arrows indicate interacting sides. For each assembly, we show a representative TEM micrograph. E The yield of specific and non-specific assemblies of non-Platonic shells is compared side-by-side, accompanied by the corresponding gel scan. The arrows indicate the location of the target assemblies. For non-specific assembly, we reference the angle-domain length with the highest yield for the target structure from (B).

By varying the angle-domain length from 0 to 15 base pairs, we observe the formation of four distinct structures, which we characterize using gel electrophoresis and TEM micrographs (see Supplementary Section IX). Platonic solids, such as icosahedra, octahedra, and tetrahedra, consistently dominate across all angle-domain lengths, with yields of 30% at +4δ, 35% at +12δ, and 26% at +14δ (Fig. 3B). Each solid forms over a range of angle-domain lengths, but the distributions are shifted toward shorter lengths than predicted by their designed nδ values. We hypothesize that this shift arises from the finite width of the dimer angle distribution, which allows closed shells to form at sizes larger or smaller than their target angles. Furthermore, the higher stability of closed structures compared to open ones may favor their formation. Smaller closed structures are also likely thermodynamically and kinetically favored compared to larger ones, as they reduce the entropic penalty and form more readily. For the longest angle domain (+15δ), monomers aggregate into large, irregular structures and rarely form stable, self-closed assemblies (Supplementary Fig. S29).

The finite bending rigidity of inter-particle bonds allows the system to assemble non-Platonic polymorphs, whose binding angles vary throughout the structure (Fig. 3C and Supplementary Fig. S5). We observe the formation of at least eight distinct non-Platonic shells, though these structures appear at lower yields compared to the Platonic forms. This behavior highlights the inherent flexibility of the system, which can accommodate a range of curvatures and structural arrangements. To address the observed polymorphism and further enhance the yield of targeted structures, we next incorporate both interaction and geometric specificity into our design strategy.

The ability to program type-specific interactions and inter-particle binding angles enables the targeted assembly of asymmetric, non-Platonic shells with increased yield. We select four such shells—triangular bipyramid, pentagonal bipyramid, gyroelongated square bipyramid, and hexagonal antiprism (Fig. 3D)—classified into two groups. The first two are formed using a single triangle species with differing binding angles, while the latter two use two triangle species with four unique interactions and varying binding angles. Details on the symmetry-guided inverse design and the required specific interaction are in the supplementary information (Supplementary Sections III and XI).

Optimizing type and geometric specificity enhances yields and minimizes polymorphism in non-Platonic shells. The triangular and pentagonal bipyramids achieve yields of 54% and 36%, up from 12% and 6%, respectively (Fig. 3E). Off-target octahedra account for 3% during pentagonal bipyramid assembly. Similarly, the gyroelongated square bipyramids and hexagonal antiprisms reach 39% and 30% yields, compared to 8% and 7% previously. Further improvements may be possible by optimizing annealing rates, assembly temperature, and ionic strength.

Programmable self-assembly of complex, self-limiting structures

To test the limits of our design strategy, we assemble a larger shell, specifically, a left-handed T = 13 icosahedral shell assembled from equilateral triangles, which we refer to as T13. Geometrically, T13 can be conceived as a large icosahedron, whose triangular face is subdivided into 13 smaller triangles1 (Fig. 4A). The resulting T13 consists of 260 equilateral triangles, comprising five unique triangles, 13 unique edge interactions, and seven unique binding angles, which range from −9.3° to 42°. Since the T = 13 icosahedron is inherently chiral, we designed our T13 to be left-handed, as illustrated in Fig. 4A. Using our 50 nm DNA origami triangle, the distance between the five-fold vertices of T13 is predicted to be around 180 nm with the maximum diameter of 340 nm. For the detailed geometry of T13, see Supplementary Section X.

Fig. 4: Self-assembly of T13 and toroids using specific interaction and binding angle.
Fig. 4: Self-assembly of T13 and toroids using specific interaction and binding angle.The alternative text for this image may have been generated using AI.
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A The design of T13 with five unique species of building blocks and seven unique binding angles. We note that the two angles shown in yellow are very close in values, and thus, are represented with one color. B Negative-stain TEM micrograph of T13. C Distribution of the lengths of hexagon edges observed under TEM. D Mass-weighted size distribution of T13 assemblies and monomers measured using dynamic light scattering. E The design of toroids with 12 unique species of triangles and 7 unique angles. Geometry of the toroid showing disclination locations (left), rotational symmetries (middle), and triangle species with binding angles (right). F TEM micrograph of the toroid. False color is overlaid to guide the eye. G Tomography reconstruction of a toroid scanning through the middle of the toroid.

Although the mesoscopic size of T13 makes it difficult to characterize the structure, we find indications of successful assembly using both TEM and dynamic light scattering (DLS). Since multispecies assemblies generally have slower kinetics19,23,24, we increase the concentration of monomers by about three times and reduce the annealing rate by about half compared to assemblies previously demonstrated (Methods section). Because of the hollow architecture of T13 and the nature of negative-stain TEM sample preparation, few shells remain structurally intact, except in regions of thicker stains where assemblies tend to aggregate (Supplementary Fig. S24). While individual triangles cannot be observed through the thick stain, we find hexagonal structures with sizes comparable to our T13 (Fig. 4B). To characterize the assemblies, we measure the edge lengths and sizes of these hexagons. In comparison to the predicted mean edge lengths and area obtained by projecting structurally intact T13, which are 165 nm and 7.03 × 104 nm2, respectively, the mean of the measured edge lengths and areas are 188 ± 19 nm and 7.44 ± 1.16 × 104 nm2, respectively, corresponding to roughly 15% expansion (Fig. 4C and Supplementary Section XII). We attribute the size increase to the flattening of the structures during sample preparation. See Supplementary Section XII for the detailed analysis of micrographs. We further assess the assemblies with DLS, which characterizes the size of structures in solution (details in Methods). The particle size distribution is multimodal: We find three distinct peaks at a hydrodynamic radius of 2.2, 30, and 254 nm (Fig. 4D). The first peak at 2.2 nm is likely due to dissociation of ssDNA staples during assembly. The second peak aligns in size with the distribution from the passivated monomer control solution, suggesting the presence of monomers and small assemblies. Finally, the third peak at 254 nm, which is roughly equal to the radius of T13, indicates the presence of large assemblies that are absent in the monomer solution. The relatively narrow range of distribution and the apparent separation from the monomer peak suggest the presence of stable self-limited structures that are similar in size.

To further demonstrate the potential of our design framework, we target another self-closing structure: a hollow toroid with more complexity in interactions and binding angles. A toroidal geometry contains spatially varying Gaussian curvature including both positive and negative Gaussian curvature. In our design, we use eight 5-fold vertices to encode positively curved exterior and four 8-fold vertices for negatively curved inner regions. Overall, our target toroid consists of 96 equilateral triangles arranged in 422 symmetry (Fig. 4E)51. Using this symmetry, the number of unique species of triangles required to assemble the toroid can be reduced to 12, containing 36 unique interactions and 7 unique binding angles ranging from 109.5° to −49.1°, which covers a much wider range of angles than any other previous assemblies. The details of this design are described in Supplementary Section XI.

Next, we demonstrate the assembly of complete toroids under TEM, thereby validating our design approach for assembling complex 2D manifolds with spatially varying curvature (Fig. 4F). Figure 4G shows the equatorial cross-section from a TEM tomography reconstruction, which clearly indicates the hollow interior and four-fold symmetry of the toroid (as also seen in Supplementary Movie 1). Interestingly, we find that 21% of all toroids that we encounter are 5-fold toroids made of 120 subunits (Supplementary Fig. S28). We speculate the differences in the interparticle distance, that arise from the encoded angle-domain lengths may impact the global structure, which warrants further study for future designs. See Supplementary Section XIII for detailed results for toroid self-assembly.

Discussion

In conclusion, we present a modular design strategy for assembling DNA origami subunits into complex 2D manifolds with independently programmable interaction specificity and binding angles. Using dsDNA as spacers to program binding angles, our modular particle can access a wide range of angles. In addition, through a detailed characterization of the angles from cryo-EM measurements, oxDNA simulations, and vertex and ring assemblies, designs can be realized from predicted target geometries, with the minimal use of trial and error. Our modular approach enables the rational design of 2D manifolds, reducing reliance on empirical screening. In our first attempt, we successfully assembled T13 and toroids, and, in a separate study, helical tubules51, all of which require many unique particle species and binding angles. We emphasize that conventional iterative designs for such complex geometries would be unfeasible due to cost, irrespective of the use of modular particles. We also apply a previously developed symmetry-guided inverse design method towards curved surfaces, which derives the minimal number of subunits required to assemble the target geometry24. Taken together, the theoretical and experimental developments described in this work open doors to fully predictive, rational design of self-assembling systems.

Although our modular DNA origami framework demonstrates the programmable assembly of nanoscale curved surfaces, achieving robust assembly of larger and more complex structures will be crucial for future applications. Despite our success in assembling T13 and toroids, there are a few unaddressed questions and issues beyond the scope of this paper. First, the yield of larger structures is difficult to characterize, preventing further iterative refinement on assembly conditions. Not only does the size prevent characterization by gel electrophoresis, but also the structures are not well preserved through the intrusive TEM sample preparation procedure. Other techniques for characterizing and purifying larger structures, such as rate-zonal centrifugation or asymmetric flow field flow fractionation, or reinforcing the structural integrity of the assemblies, such as silica coating52 or covalent linking53, should be explored and standardized. Second, the interface design is still underexplored. The interface design directly impacts various properties of assemblies, including kinetics, binding angles, and bending moduli. In the future, thorough characterization and comparison between different interface designs, including ssDNA42,43 and our dsDNA strut designs, are required for rational interface designs. We also highlight that DNA strand displacement reactions at the interface grant reconfigurable assemblies, which is attractive for designing dynamic, stimuli-responsive structures7,26. Finally, the field will benefit from a more holistic strategy for high-yield assembly of large, specific structures. So far, we have focused on encoding interaction and binding specificity accurately to assemble the target structure. However, in reality, the imperfect specificity and bond angle fluctuations lead to polymorphisms that lower the yield of assemblies. It is essential to be able to systematically predict when and how polymorphisms arise, prior to intensive simulation or experimental testing. Additionally, preventing kinetic deadlocks through designing kinetic pathways may be helpful54. Controlling kinetic pathways through hierarchical assemblies, programmed annealing protocols, or triggered activation and deactivation of interactions may all be beneficial for high-yield assemblies, especially for large structures.

Leveraging precise control over binding angles, symmetries, and periodicities in a multicomponent environment, our method supports the assemblies of self-limited structures55,56 and surface-based crystals57,58 that are orders of magnitude larger than the individual subunits. These structures can help organize nanoscale particles into specific alignments, unlocking the ability to engineer macroscale material functionalities such as photonic properties34,35,36,37,59, or enhancing enzyme activities60. In addition, DNA origami shells have shown promise in antiviral therapeutics21,39, by encapsulating viruses and preventing their interaction with host cells. Our approach extends this envelopment strategy to accommodate a broader range of viral shapes and sizes, opening possibilities for the rational design of synthetic biological defenses.

Methods

Folding DNA origami

Each DNA origami particle is folded by mixing 50 nM of p8064 scaffold DNA (Tilibit) and 200 nM each of staple strands with folding buffer and annealed through a temperature ramp starting at 65 °C for 15 min, then 58 to 50 °C, −1 °C per hour. Our folding buffer, contains 5 mM Tris Base, 1 mM EDTA, 5 mM NaCl, and 15 mM MgCl2. We use a Tetrad (Bio-Rad) thermocycler for annealing the solutions.

Agarose gel electrophoresis

To assess the outcome of folding, we separate the folding mixture using agarose gel electrophoresis. Gel electrophoresis requires the preparation of the gel and the buffer. The gel is prepared by heating a solution of 1.5% w/w agarose, 0.5× TBE to boiling in a microwave. The solution is cooled to 60 °C. At this point, we add MgCl2 solution and SYBR-safe (Invitrogen) to adjust the concentration of the gel to 5.5 mM MgCl2 and 0.5× SYBR-safe. The solution is then quickly cast into an Owl B2 gel cast, and further cooled to room temperature. The buffer solution contains 0.5× TBE and 5.5 mM MgCl2, and is heated to 45 °C before use. Agarose gel electrophoresis is performed at 90 V for 1.5 h in room temperature. The gel is then scanned with a Typhoon FLA 9500 laser scanner (GE Healthcare).

Gel purification and resuspension

After folding, DNA-origami particles are purified to remove all excess staples and misfolded aggregates using gel purification. The folded particles are run through an agarose gel (now at a 1× SYBR-safe concentration for visualization) using a custom gel comb, which can hold around 4 mL of solution per gel. We use a blue fluorescent table to identify the gel band containing the monomers. The monomer band is then extracted using a razor blade, which is further crushed into smaller pieces by passing through a syringe. We place the gel pieces into a Freeze ’N Squeeze spin column (Bio-Rad), freeze it in a −80 °C freezer for 30 min, thaw at room temperature, and then spin the solution down for 5 min at 13,000 × g.

Since the concentration of particles obtained after gel purification is typically not high enough for assembly, we concentrate the solution through ultrafiltration61. First, a 0.5 mL Amicon 100 kDA ultrafiltration spin column is equilibrated by centrifuging down 0.5 mL of the folding buffer at 5000 × g for 7 min. Then, the DNA origami solution is added up to 0.5 mL and centrifuged at 14,000 × g for 15 min. Finally, we flip the filter upside down into a new Amicon tube and spin down the solution at 1000 × g for 2 min. The concentration of the DNA origami particles is measured using a Nanodrop (Thermofisher), assuming that the solution consists only of monomers, where each monomer has 8064 base pairs.

Assembly experiment

All assembly experiments are conducted at a DNA-origami particle concentration of 10 nM. For two-species assembly, the total DNA origami concentration is 10 nM, whereas for T13 shells and toroids, the total concentration is 25 nM and 36 nM respectively, and each triangular species is mixed in a stoichiometric ratio of the target structure. By mixing the concentrated DNA-origami solution after purification with buffer solution, we make 50 μL of 10 nM DNA origami (e.g., for the deltahedral shell assembly experiment) at 20–30 mM MgCl2, depending on the sample. The solution is carefully pipetted into 0.2 mL strip tubes (Thermo Scientific) and annealed from 40 °C to 25 °C at −0.1 °C per 20 min, using a Tetrad (Bio-Rad) thermocycler. For assembly of T13 and toroids, the annealing rate is set to −0.1 °C per 45 min.

Negative stain TEM

We first prepare a solution of uranyl formate (UFo). ddH2O is boiled to deoxygenate it and then mixed with uranyl formate powder to create a 2% w/w UFo solution. The solution is covered with aluminum foil to avoid light exposure, then vortexed vigorously for 20 min. The solution is filtered using a 0.2 μm filter. The solution is divided into 0.2 mL aliquots, which are stored in a −80 °C freezer until further use.

Prior to each negative-stain TEM experiment, a 0.2 mL aliquot is taken out from the freezer to thaw at room temperature. We add 4 μL of 1 M NaOH and vortex the solution vigorously for 15 s. The solution is centrifuged at 4 °C and 16,000 × g for 8 min. We extract 170 μL of the supernatant for staining and discard the rest.

The EM samples are prepared using FCF400-Cu grids (Electron Microscopy Sciences). We glow-discharge the grid prior to use at −20 mA for 30 s at 0.1 mbar, using a Quorum Emitech K100X glow-discharger. We place 4 μL of the sample on the grid for 1 min to allow adsorption of the sample to the grid. During this time 5 μL and 18 μL droplets of UFo solution are placed on a piece of parafilm. After the adsorption period, the remaining sample solution is blotted on a Whatman filter paper. We then touch the carbon side of the grid to the 5 μL drop and blot it away immediately to wash away any buffer solution from the grid. This step is followed by picking up the 18 μL UFo drop onto the carbon side of the grid and letting it rest for 30 s to deposit the stain. The UFo solution is then blotted to remove excess fluid. Grids are allowed to dry for a minimum of 15 min before insertion into the TEM.

We image the grids using an FEI Morgagni TEM operated at 80 kV with a Nanosprint5 CMOS camera (AMT). The microscope is operated at 80 kV and images are acquired between ×8000 to ×28,000. The images are high-pass filtered and the contrast is adjusted using ImageJ.

TEM tomography

To obtain a tilt-series, we use an FEI F20 equipped with a Gatan Ultrascane 4kx4k CCD camera, operated at 200 kV. The grid is observed at ×29,000 magnification from −50° to 50° in 2° increments. The data is analyzed and the z-stack is reconstructed using IMOD62.

Cryo-electron microscopy

Higher concentrations of DNA origami are used for cryo-EM grids than for assembly experiments. To avoid assembly and aggregation of the subunits, we removed ssDNA strands protruding from the faces of the DNA origami. To prepare samples we fold 2 mL of the folding mixture, gel purify it, and concentrate the sample by ultrafiltration, as described above, targeting a concentration of 300 nM of DNA origami. EM samples are prepared on glow-discharged C-flat 1.2/1.3 400 mesh grids (Protochip). Plunge-freezing of grids in liquid ethane is performed with an FEI Vitrobot IV with sample volumes of 3 μL, blot times of 16 s, a blot force of −1, and a drain time of 0 s at 20C and 100% humidity.

Cryo-EM images for the modular block DNA origami were acquired with a Tecnai F30 TEM with a field emission gun electron source operated at 300 kV and Compustage, equipped with an FEI Falcon II direct electron detector at a magnification of ×39,000. Particle acquisition is performed with SerialEM. The defocus is set to −2 μm for all acquisitions with a pixel size of 2.87 Angstrom.

Cryo-EM images for DNA origami dimers were acquired with a Tecnai F20 TEM with a field emission gun electron source operated at 200 kV and Compustage, equipped with a Gatan Oneview CMOS camera at a magnification of ×29,000. Particle acquisition is performed with SerialEM. The defocus is set between −1.5 and −4 μm for all acquisitions with a pixel size of 3.757 Angstrom.

Single-particle reconstruction

Image processing is performed using RELION-463. Contrast-transfer-function (CTF) estimation is performed using CTFFIND4.164. After picking single particles we performed a reference-free 2D classification from which the best 2D class averages are selected for processing, estimated by visual inspection. The particles in these 2D class averages are used to calculate an initial 3D model. A single round of 3D classification is used to remove heterogeneous monomers and the remaining particles are used for 3D auto-refinement and post-processing. Our reconstructions of the dimers use 2838, 3465, and 1936 particles with resolutions of 35.9 Å, 36.7 Å, and 36.7 Å for +6δ, +10δ and −7δ respectively, Supplementary Fig. S8. Our reconstruction of the monomer uses 2650 particles and has a resolution of 21.3 Å (Supplementary Fig. S32).

Cryo-EM multibody analysis

Fluctuations of subunits were processed using RELION-4’s63 multibody refinement49,50. After getting a postprocessed reconstruction of a dimer using single-particle reconstruction, we create masks around the two triangular bodies using the eraser tool in ChimeraX65. These were used in the “3D multi-body" job in RELION 4 to get the set of fluctuations in translation and rotation of the two bodies in the dimer.

oxDNA simulation

TacoxDNA tools and an in-house script were used for generating the topology and configuration files for the DNA-origami equilateral triangle from the caDNAno design66. Rigid-body dynamics in oxView were carried out to align the DNA bundle subunits into a conformation more representative of the correct global structure67,68. Overhangs matching the angle and bond domains were created protruding from a single outside face of the triangle. The structure was then duplicated in oxView to form a dimer67,68. This process of creating overhangs and dimers was performed for all explored angle-domain lengths to generate the initial configurations needed for carrying out subsequent molecular dynamics simulations using the oxDNA2 package47.

To prepare equilibrated structures for the production stage of the simulation, mutual traps between paired scaffold and staple bases were applied, and the structures were subjected to 10,000 steps of gradient descent minimization followed by dynamic relaxation. The initial stages of dynamic relaxation involved substituting the DNA backbone potential with linear springs while applying mutual traps. The maximum applied spring force was gradually increased over 1.52 ns to a force value of 57.09 nN/nm. Subsequently, the backbone spring potential was maintained at 57.09 nN/nm while the time step was gradually increased from Δt = 0.0303 fs to Δt= 9.09 fs over 321.18 ns while preserving mutual traps. For the final stage of dynamic relaxation, the full finitely extensible nonlinear elastic (FENE) potential was enforced by removing the spring force on the backbone while maintaining mutual traps on the base pairs for 90.9 ns at Δt = 9.09 fs.

Dynamic relaxation was followed by a production stage in which mutual traps were removed, and the simulation continued for 0.909 μs at Δt = 9.09 fs at a monovalent salt concentration of 1 M. It is important to note that this simulation time does not directly correspond to physical time, due to the implicit treatment of the solvent and the coarsened resolution of the model, which effectively smoothens the energy landscape48. By applying a previously derived scaling factor48 to estimate the physical time of oxDNA simulations with α ≈ 330, the simulations in this study correspond to ~300 μs. The John thermostat with diffusion coefficient and Newtonian step settings of 2.5 and 103 was used to maintain a constant temperature of 27 °C. One independent simulation was carried out for each of the 32 angle-domain lengths. Coordinates were stored every 0.909 ns of simulated time into a trajectory file containing 1000 frames. The generated trajectory files were used for subsequent analyses, which were conducted using a combination of oxDNA analysis tools and in-house scripts67.

Additional simulations were performed for +6δ and +10δ at 136 K using the same thermostat settings applied for the simulation at 298 K temperature. Three independent dimers for +6δ and +10δ were subjected to dynamic relaxation according to the protocol described above, followed by extended production at 298 K for 9.09 μs with a time step (Δt) of 9.09 fs. These dimers equilibrated at 298 K were used as the starting configurations for the simulations at 136 K. The dimers were then simulated under production conditions at 136 K for 9.09 μs with a time step (Δt) of 9.09 fs. Coordinates were saved every 9.09 ns into trajectory files for further analysis of the developed binding angles.

The binding angle between the triangular origami subunits was computed using a custom script. This script takes as input the list of indices of the nucleotides on each face of the side from which the overhangs are extended, along with two additional nucleotides belonging to one of the triangle faces, selected to establish a consistent positive vector pointing from helix 3 to helix 1. The index selection was carried out in oxView67. The first two principal components of each face were computed, with their corresponding eigenvectors defining the directions along the width and height of each face, respectively. From these directions, the normal vectors to each face were derived, and their directions were adjusted to ensure a consistent definition of the binding angle. The binding angle was then calculated using the dot product of the normal vectors, with the sign corrected based on the predefined positive vector pointing from helix 3 to helix 1. This process was repeated for every frame in the trajectory file to obtain the distribution of binding angles for each simulated system.

Dynamic light scattering

The Dynamic Light Scattering (DLS) measurements were performed using a Wyatt Technology DynaPro Nanostar II instrument. Light scattering signals from DNA origami samples were monitored at a 90° scattering angle. The reported hydrodynamic radius and mass-weighted size distribution for the T13 sample was averaged from 1000 rounds of 5-second measurements at 30 °C.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.