Modular programming of interaction and geometric specificity enables assembly of complex DNA origami nanostructures

Domain: physics
Published: 2025-12-11
Editorial outcome: minor_revision

Abstract
Self-assembly of nanoscale building blocks with programmable geometries and interactions offers a powerful route to engineer materials that mimic the complexity of biological structures. DNA origami provides an exceptional platform for this purpose, enabling precise control over subunit shape, binding angles, and interaction specificity. Here we present a modular DNA origami design approach to address the challenges of assembling geometrically complex nanoscale structures, including those with nonuniform curvatures. This approach features a core structure that completely conserves the scaffold routing across different designs and preserves more than 70% of the DNA staples between designs, dramatically reducing both cost and effort, while enabling precise and independent programming of subunit interactions and binding angles through adjustable overhang lengths and sequences. Using cryogenic electron microscopy, gel electrophoresis, and coarse-grained simulations, we validate a set of robust design rules and demonstrate the assembly of diverse self-limiting structures, including anisotropic shells, a T=13 icosahedral shell, and a toroid with globally varying curvature. This modular strategy provides an efficient and cost-effective framework for the synthetic fabrication of complex nanostructures.

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 subunits1–3. Inspired by Nature, scientists have developed synthetic building blocks with programmable geometries and interactions—such as DNA tiles4, DNA-coated colloids5–8, 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 assemblies14–28, while enabling precise labeling of functional particles like metallic nanoparticles, quantum dots, and proteins29–33, 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 crystals34–37, 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).

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).

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 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 indicated different species of triangles encoded with a unique set of interactions and binding angles.

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:

θ=2arcsin(0:34nm×nδ / 2×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 σEM 298K = σEM 136K * sqrt(298K/136K).

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 of 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.

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.

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 whereas 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.

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 Nanosprint 5 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 20 °C 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 timestep 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.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript at hand, titled “Modular programming of interaction and geometric specificity enables assembly of complex DNA origami nanostructures,” by Saha et al. describes the assembly of multi-component DNA origami structures. The manuscript is well-written, and the results are certainly of interest to the DNA nanotech community.\n\nThe authors use DNA origami with a triangular shape, which they connect via hybridization of overhanging ssDNA strands on the sides. A certain angle of the connections between the DNA origami is introduced by varying the length of one of the connection sides. The respective angles varied from 0° to >100°. Subsequently, they used these connections to assemble several different multi-component systems, analyzed the yield of the structures and the interconnection of designed angle and resulting structures.\n\nThe manuscript is highly polished, and the text, as well as the figures, are easily understandable. The authors use oxDNA simulations and cryoEM to analyze the binding angles between the DNA origami with great detail. They further analyze the yield of the multi-component structures with regard to the final geometry of the structures (presumably with TEM) and yield, by means of AGE band intensity measurements. The applied methods are fully appropriate for the analysis of their structures. However, to some extend, the work appears incremental, when compared to the group's other published research. In general, publication in Nat.Com. would be recommended after some rework and additions.\n---\nMajor points:\nThe central criticism of the presented manuscript is the large similarity of this manuscript with previously reported structures and prior methods of higher-order DNA origami assembly:\nThis very research group used variations of the same DNA origami structure already in Sigl et al. 2021, and subsequent publications (Hayakawa et al. 2022, Wei et al. 2024, etc.). ssDNA handles for connections of DNA origami were also used with this very origami before (Videbaek et al. 2024, Videbaek et al. preprint). Angled connections with a slightly different approach on how the angles are induced are also presented in a different manuscript from the same lab (preprint).\nUsing ssDNA overhangs/handles to connect DNA origami is ubiquitous in semantomorphic DNA nanotechnology. The method of using angled connections, made by DNA connections of different lengths, was also already introduced almost a decade ago (Funke & Dietz, 2016, Wagenbauer et al. 2017). Angled connections between DNA origami made with different length ssDNA handles were also already explored by Karfusehr et al. (preprint) and Koehler et al. (preprint) for repetitive higher-order structures, as also cited in the manuscript at hand.\nThus, the novelty in the manuscript is the construction of higher-order structures with defined geometries from different building blocks (concerning the connection site angles). It should therefore be compared to the higher-order structures of Luu et al. 2024. The structures presented here do not generally have a better yield or are more complex than those of Luu et al.. The presented torus structure (Fig. 1D, Fig. 4) is indeed much larger than the structures from Luu et al., but Fig. S16 reveals a nonexistent yield. This reviewer thinks that the authors’ approach of creating higher-order structures is very promising for the creation of large, higher-order structures; it would just be good to show this in the manuscript.\nThis reviewer appreciates the amount of detail with which the structures are analyzed and the large effort with which the data is presented. This speaks for the scientific excellence of the manuscript. However, for publication in Nat.Com. the authors should show explicitly how their approach can lead to better yields and larger structures than, for example, Luu et al., who were using massive amounts of unique building blocks and connections.\n---\nMinor points:\n• Main text, line 61: misspelled “Karfusehr”\n• Main text: a short introduction to the term “gaussian curvature” would be good, especially for the broad audience of Nat.Com.\n• Main text, lines 85 and following: exchanging 30 % of staples to change the connection sites (thus 10 % per connection site) is quite a lot of DNA, and thus not really a great strength of the presented connection approach and shouldn’t be presented as such. (similar approaches were also implemented in a manifold of other works and thus are not novel)\n• Each connection uses a manifold of unique handle sequences that can and will interact non-specifically. The authors claim specificity by high off-rates and display this in an interaction matrix. The authors should quantify this, maybe with NUPACK.\n• Can the authors to comment on the choice of DNA hybridization connection setup: The chosen setup looks somewhat suboptimal, so why not use a three-strand setup? It would also render these kinds of connections changeable via toehold-mediated strand displacement, which would make the higher-order structure more versatile and thus more attractive for any downstream applications.\n• The yield of the structures is analyzed by AGE band intensities. TEM images, however, reveal many broken structures. The authors should at least comment on this and discuss the implications for the use of structures in downstream applications.\n• The authors should further discuss for which nanotechnological applications their structures and the presented angled connections can be used, and what advantages and drawbacks over other approaches they have.", "reviewer_2": "The work of Saha et al introduces a cost effective modular method to assemble self limiting structures.\nAmongst classic platonic solid examples, presented are also complex shells and a toroid structure.\nThe modular framework is characterized in great detail allowing to utilize this approach across other DNA assemblies.\n- The data is presented in a clear way, however I would ask to restructure the supplement to better follow the text flow.\nExample the Results section opens up with \"Figs. 1A, 2A and S25\" and next \"Fig. 1B. See 166 SI Section VIII\" this continues throughout\nthe text.\n- There's a small typo \"Karfuscher et al. [36]\" should be Karfusehr.\n- For the oxDNA simulations the number of repetitions for each tried condition should be explicitly stated in the methods section.\n- SI Section IV Provided traces for +6 and +10 are not converged for 298K and in every case an angle close to 50 degrees\nwas chosen for the subsequent 136 K run.\nAdditional analysis is needed to sample the complete Free energy landscape of the angles, which will make the\ndiscrepancy to CryoEM discussion stronger,\nand it would be interesting to see how the model would equilibrate from more extreme starting angles for both +6 and + 10\ncases.\nI recommend acceptance once addressed.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "This manuscript presents a modular strategy for programming DNA origami subunits with independent control over inter-subunit binding specificity and binding angles. The authors propose that this framework overcomes key challenges in assembling geometrically complex nanostructures, particularly those with nonuniform Gaussian curvature. While the work is clearly described and technically sound, its overall novelty and impact should be more carefully contextualized. Below are my main comments:\nCost-effective design is one of the central contributions of this work, it lies more in an incremental improvement than in conceptual novelty. The modular design, which combines the separation of geometric control and interaction specificity via overhang domains, builds directly on prior efforts (e.g., references [22], [23], [35], [36]) that have already demonstrated programmable angles and modular subunit design. The present study refines these approaches by preserving scaffold routing and 70% of staple sequences across different geometries, offering cost and design efficiency benefits. However, the ability to build large assemblies of triangular DNA origami has already been demonstrated by the same research group (e.g., Videbæk et al., Science Advances, 2024 and arXiv:2411.09801 submitted on 14 Nov 2024). Similarly, cost-effectiveness and modularity have been addressed in recent work (e.g., Weck & Heuer-Jungemann, Nat. Commun. 2025) that enabled fully addressable superstructures using single modular DNA origami units.\nThe construction of a toroidal structure is presented as a key achievement. However, the reported yield is very low, and the design requires 12 unique triangle species. While this is more economical than designing 12 completely distinct origamis, the approach is far from trivial. For applications that demand dozens or even more of distinct subunits, the strategy may not scale efficiently. Despite the claim that this framework enables precise control of curvature and structure, the toroid example suggests that achieving complex, deterministic topologies remains a substantial challenge. Is it possible to have wide-field images of well folded toroids under TEM ? Given these limitations, to what extent can this modular framework be realistically extended to more functionally diverse assemblies?\nAlthough the authors emphasize the programmability of binding angles and interaction matrices, the system does not reliably produce single, well-defined structures. Multiple polymorphic forms are observed, particularly at higher angle-domain lengths. This outcome is well documented in the paper but it highlights a fundamental limitation: the current design rules do not fully constrain the assembly pathway to a unique structure, especially in more complex or crowded mixtures even with program type-specific interactions. The results raise a central question: does the approach provide enough design specificity to be predictive and controllable at scale?\nAdditional Comments\nCryo-EM validation of the dimer angles is central to the paper’s claims, but representative images of the DNA origami dimers are missing. The number of cryo-EM movies acquired, and any classification statistics or images are missing.\nFigure 4C shows a cross-section from a TEM tomography reconstruction of the toroid and is cited as evidence for agreement with the predicted geometry. However, I didn’t find mention of the tomography acquisition or reconstruction methods in the main text or Methods section, nor are tomogram datasets like different slices of the entire tomogram from bottom to top should be included in the supplementary information."}, "author_response": "We thank the reviewers and Editor for their careful reading of the manuscript and their insightful, constructive comments. We have addressed all the comments as described below.\nReviewer #1\nComment #1: The manuscript at hand, titled “Modular programming of interaction and geometric specificity enables assembly of complex DNA origami nanostructures,” by Saha et al. describes the assembly of multi-component DNA origami structures. The manuscript is well-written, and the results are certainly of interest to the DNA nanotech community.\nThe authors use DNA origami with a triangular shape, which they connect via hybridization of overhanging ssDNA strands on the sides. A certain angle of the connections between the DNA origami is introduced by varying the length of one of the connection sides. The respective angles varied from 0° to >100°. Subsequently, they used these connections to assemble several different multi-component systems, analyzed the yield of the structures and the interconnection of designed angle and resulting structures.\nThe manuscript is highly polished, and the text, as well as the figures, are easily understandable. The authors use oxDNA simulations and cryoEM to analyze the binding angles between the DNA origami with great detail. They further analyze the yield of the multi-component structures with regard to the final geometry of the structures (presumably with TEM) and yield, by means of AGE band intensity measurements. The applied methods are fully appropriate for the analysis of their structures. However, to some extent, the work appears incremental, when compared to the group's other published research. In general, publication in Nat.Com. would be recommended after some rework and additions.\nResponse #1: We thank the reviewer for their positive overall assessment of our manuscript and their recommendation to publish it in Nature Communications after addressing their comments.\nComment #2:\nMajor points:\nThe central criticism of the presented manuscript is the large similarity of this manuscript with previously reported structures and prior methods of higher-order DNA origami assembly:\nThis very research group used variations of the same DNA origami structure already in Sigl et al. 2021, and subsequent publications (Hayakawa et al. 2022, Wei et al. 2024, etc.). ssDNA handles for connections of DNA origami were also used with this very origami before (Videbaek et al. 2024, Videbaek et al. preprint). Angled connections with a slightly different approach on how the angles are induced are also presented in a different manuscript from the same lab (preprint).\nUsing ssDNA overhangs/handles to connect DNA origami is ubiquitous in semantomorphic DNA nanotechnology. The method of using angled connections, made by DNA connections of different lengths, was also already introduced almost a decade ago (Funke & Dietz, 2016, Wagenbauer et al. 2017). Angled connections between DNA origami made with different length ssDNA handles were also already explored by Karfusehr et al. (preprint) and Koehler et al. (preprint) for repetitive higher-order structures, as also cited in the manuscript at hand.\nThus, the novelty in the manuscript is the construction of higher-order structures with defined geometries from different building blocks (concerning the connection site angles). It should therefore be compared to the higher-order structures of Luu et al. 2024. The structures presented here do not generally have a better yield or are more complex than those of Luu et al.. The presented torus structure (Fig. 1D, Fig. 4) is indeed much larger than the structures from Luu et al., but Fig. S16 reveals a nonexistent yield. This reviewer thinks that the authors’ approach of creating higher-order structures is very promising for the creation of large, higher-order structures; it would just be good to show this in the manuscript.\nThis reviewer appreciates the amount of detail with which the structures are analyzed and the large effort with which the data is presented. This speaks for the scientific excellence of the manuscript. However, for publication in Nat.Com. the authors should show explicitly how their approach can lead to better yields and larger structures than, for example, Luu et al., who were using massive amounts of unique building blocks and connections.\nResponse #2: We thank the reviewer for their careful assessment and their recognition of the scientific rigor, polished presentation, and potential of our approach. We appreciate the reviewer’s concern regarding the novelty of our method in light of previous work, particularly within our research group and others in the field. We respectfully submit that our current manuscript presents a conceptual and technical advance beyond prior approaches, which we now clarify more explicitly in the revised version of the manuscript.\nTo address the reviewer’s question regarding the novelty, we consider the contributions of the current work from four different perspectives:\n1. Modularity with scaffold-conserved design (comparison to our previous works):\nOur current work represents a significant advance through a modular design strategy that enables programmable control of both interaction specificity and binding angles using DNA overhangs. Unlike our pre-2025 approaches [21–23, 25], where altering binding angles required rerouting the entire scaffold and generating a completely new staple set, the modular method preserves the scaffold routing and only requires replacing a small fraction of staples for each monomer design.\nThis decoupling brings two key advantages:\n1. Cost-efficiency and scalability: Multicomponent assemblies like the toroid, requiring 12 components can be built using only one core subunit with fully conserved scaffold-routing and preserving at least 70% of the staples per variation, significantly reducing synthesis cost.\n2. Predictive and programmable angle control: Binding angles can be tuned and reassigned by modifying only the overhangs, enabling rapid success in assembling complex structures with fewer iterations in binding angle. On the contrary, our previous approaches required a more time-consuming iteration process to obtain an accurate binding angle, involving re-routing the scaffold design and subsequent characterization using cryo-EM.\nTo clarify this we have edited the second paragraph of the Introduction section.\n“Although DNA origami offers a vast design space, the range of accessible structures are 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.”\n[21] C. Sigl et al., Programmable icosahedral shell system for virus trapping, Nature Materials 20, 1281 (2021).\n[22] D. Hayakawa et al., Geometrically programmed self-limited assembly of tubules using DNA origami colloids, PNAS 119, e2207902119 (2022)\n[23] T. Videbæk et al., Economical routes to size-specific assembly of self-closing structures, Science Advances 10, eado5979 (2024).\n[25] W.-S. Wei et al., Hierarchical assembly is more robust than egalitarian assembly in synthetic capsids, Proceedings of the National Academy of Sciences 121, e2312775121 (2024).\n2. Comparison to previous angled connection strategies\nWe appreciate the reviewer for highlighting the previously published modular approaches including various overhang-mediated angled connections for building higher order structures. Modularity has only recently been introduced to the DNA origami field, beginning in 2024. Our approach shares features with others in the field, but there are also important distinctions.We have edited the third paragraph of the Introduction section, including the following:\n“Recent efforts by Karfusehr et al.[42] and us [43] 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 SI section I.”\n[42] C. Karfusehr, M. Eder, and F. C. Simmel, Self-assembled cell-scale containers made from DNA origami membranes, bioRxiv 10.1101/2024.02.09.579479 (2024).\n[43] W.-S. Wei et al., Economical and versatile subunit design principles for self-assembled DNA origami structures, ACS Nano 19, 30889 (2025).\nAdditionally, we have added a new supplementary section (Section I) that compares our design with prior modular DNA origami platforms with programmable angles.\nFIG. S2. Different DNA overhang connection setups. (A) Direct hybridization of single-stranded DNA overhangs with poly-thymine spacers [9, 13, 14]. (B) Double-stranded DNA overhang connection, as discussed in this article. (C) Linker mediated 3-strand connection setup [16]. (D) Overhang connection [7]. (E) Double stranded overhang connection [10].\n[7] M. T. Luu et al., Reconfigurable nanomaterials folded from multicomponent chains of DNA origami voxels, Science Robotics 9, eadp2309 (2024).\n[9] J. J. Funke and H. Dietz, Placing molecules with Bohr radius resolution using DNA origami, Nature Nanotechnology 11, 47 (2016).\n[10] K. F. Wagenbauer, C. Sigl, and H. Dietz, Gigadalton-scale shape-programmable DNA assemblies, Nature 552, 78 (2017).\n[13] C. Karfusehr, M. Eder, and F. C. Simmel, Self-assembled cell-scale containers made from DNA origami membranes, bioRxiv 10.1101/2024.02.09.579479 (2024).\n[14] W.-S. Wei et al., Economical and versatile subunit design principles for self-assembled DNA origami structures, ACS Nano 19, 30889 (2025).\n[16] J. Lowensohn et. al., Self-assembly and crystallization of DNA-coated colloids via linker-encoded interactions, Langmuir 36, 7100 (2020).\n3. Comparison to Luu et al (2024)\nWe agree with the reviewer that Luu et al. (2024) present an elegant voxel-based modular origami platform capable of assembling 2D and 3D reconfigurable self-limited nanostructures using up to 12 distinct components achieving impressive yields in the range of 20% - 40%. Their design also demonstrates dynamic reconfiguration between target shapes using external linker strands.\nOur work differs both in objective and design philosophy. Luu et al. grow architectures in a Cartesian (voxel) framework that is naturally suited to rectilinear forms (boxes and bricks) but is not suitable for closed spherical surfaces or manifolds with spatially varying curvature, which require a prescribed field of dihedral angles. By contrast, our framework is based on a triangular framework, chosen because mathematical theorems prove that any surface can be triangulated. Additionally, our equilateral triangle block has three interfaces that can be simply, accurately and arbitrarily programmed to define bond angles with high accuracy and thereby tile surfaces with nonuniform curvature.\nBoth approaches leverage modular platforms for multicomponent assembly where the interface configuration plays a key role in achieving the design objective. While Luu et al’s interface configuration excels in dynamic reconfiguration; ours excels at angle programming. The two approaches are complementary, and the appropriate choice depends on the design goals.\nRegarding yield, we caution against direct comparison with assemblies shown by Luu et al. (2024). The apparent low yield of toroids is difficult to quantify:\n● Toroids are very large (~300 nm), preventing yield analysis by Agarose gel electrophoresis, and often fracture during TEM grid preparation.\n● Self-closing assemblies are prone to kinetic deadlock (Deeds et al. 2012), where incomplete intermediates consume monomers and prevent ring closure. Our relatively fast annealing likely exacerbates this, consistent with the partial toroids observed in TEM.\n● Addressing these kinetic challenges requires further studies, which are beyond the scope of the present work.\nGiven the above reasons, we think that the simple comparison of yield between our toroid assemblies and structures in Luu et al.’s work may be misleading. Rather, a fair comparison would be that between similar structures such as capsids designed by us, Sigl et al. (2021) and Wei et al. (2024, 2025). However, in our current work we have not optimized the best condition for capsid assembly. For example, we find the yield of T=1 capsid to be 30% (using +4𝛿 design), compared to 60% in Wei et al. (2025) and 70% in Sigl et al. (2021). Further improvements in the yield can be possible by optimizing annealing rate, assembly temperature and ionic strength.\n4. New addition of a T=13 icosahedral shell assembly\nWe appreciate the reviewer’s concern that our work may appear similar to previously reported DNA origami assembly. While we have taken care to highlight the novelty of our approach relative to prior studies, we recognize that the distinction may not be immediately evident to a broader audience. Hence, we have included additional experimental results i.e., the assembly of a T=13 icosahedral shell, the largest Caspar–Klug icosahedral DNA origami shell ever achieved, realized at marginal additional cost, demonstrating the potential of our work.\nWe have updated the Figure 1 and Figure 4 (below) in the main text that now includes the T=13 shell assembly results. We have discussed the assembly results (presented below) in detail in the section D of the main text. Additionally, we have discussed the geometry of the T=13 shell in detail in SI section X and analysis of the TEM micrographs in SI section XII.\nFIG 4: Self-assembly of T13 and toroids using specific interaction and binding angle. (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-stained 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. The geometry of the toroid with the locations of all disclination defects (left), unique rotational symmetries of the toroid (middle), and the species and binding angles between triangles (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.\n“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.\nGeometrically, T13 can be conceived as a large icosahedron, whose triangular face is subdivided into 13 smaller triangles [1] (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 degrees. 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 SI section X.\nAlthough 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 kinetics [19, 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 (section I). 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 (SI 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 7.03×10^4 nm^2, respectively, the mean of the measured edge lengths and areas are 188±19 nm and 7.44±1.16× 10^4 nm^2, respectively, corresponding to roughly 15% expansion (Fig.4C and SI section XII). We attribute the size increase to the flattening of the structures during sample preparation. See SI 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 section I). 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.”\n[1] D. L. D. Caspar and A. Klug, Physical Principles in the Construction of Regular Viruses, Cold Spring Harbor Symposia on Quantitative Biology 27, 1 (1962).\n[19] C. M. Wintersinger et al., Multi-micron crisscross structures grown from DNA-origami slats, Nature Nanotechnology 18, 281 (2023).\n[23] T. Videbæk et al., Economical routes to size-specific assembly of self-closing structures, Science Advances 10, eado5979 (2024).\n[24] D. Hayakawa et al., Symmetry-guided inverse design of self-assembling multiscale DNA origami tilings, ACS Nano 18, 19169 (2024)\nMinor points:\nComment #3: Main text, line 61: misspelled “Karfusehr”\nResponse #3: We thank the reviewer for pointing out our mistake. We have fixed the spelling in the Introduction of the main text.\n“Recent efforts by Karfusehr et al.[42] and us [43] demonstrate modular DNA origami design with programmable binding angles …”\nComment #4: Main text: a short introduction to the term “gaussian curvature” would be good, especially for the broad audience of Nat.Com.\nResponse #4: We appreciate the comment of the reviewer. As now we have demonstrated the assembly of T13 that does not have varying Gaussian curvature, we have removed the term “Gaussian” from the main text for better readability and streamlining the message.\nComment #5: Main text, lines 85 and following: exchanging 30 % of staples to change the connection sites (thus 10 % per connection site) is quite a lot of DNA, and thus not really a great strength of the presented connection approach and shouldn’t be presented as such. (similar approaches were also implemented in a manifold of other works and thus are not novel)\nResponse #5: We acknowledge that similar concepts of modularity have appeared in recent studies (Wei et al. (2025), Karfusehr et al. (2024), Luu et al. (2024)). We agree with the reviewer’s comment that the reuse efficiency of 71% is not impressively high, compared to Luu et al.'s design of 78% (72 out of 324 staples) or our previous design of 88% (Wei et al. (2025) (table I).\nImportantly, the effective reuse of staples is higher than the reported 71%, lowering the expected cost for staple purchases. Among the 20 overhangs that are variable per side, 12 strands can extrude on the 3’-end and contain the bond domain while the remaining eight that extrude on the 5’-end contain only the angle domain. The eight strands with an angle domain always have the same sequences, and thus, do not scale with the number of unique monomers. Essentially, they can be considered a part of initial investment, similar to the conserved core staples. The 12 strands are unique to the specific interaction, and must be ordered for every new interaction. However, since we need only 8 out of 12 staples to extrude of the triangle face at a time, we only need 8 staples per interaction. Assuming all three sides of the triangle are unique, we need 24 new staples per monomer, which is similar to Wei et al. (2025) design with 88% reuse efficiency. The main difference is an initial investment in 168 angle-domain staples, which can then be reused across different assemblies.\nAlthough the additional cost for 168 staples may not be rationalized for assemblies with a few unique subunits, our designs become more economical for assemblies with large numbers of unique subunits. In terms of staple usage, a toroid would cost $37,000 using the shape-complementary approach while using our method, it will cost $10,000 (FIG. S3) and once the overhangs are purchased can be reused across a myriad of other structures such as helical tubules demonstrated in a separate study [51].\n[51] M. Price et al., From toroids to helical tubules: Kirigami-inspired programmable assembly of two-periodic curved crystals, arXiv:2506.16403 (2025).\nWe have added the following in the first paragraph of Section A of the main text,\n“We note that once initial sets of staples are purchased, only eight additional staples are required per unique edge, significantly lowering the cost for multicomponent assemblies (see SI Section II for details).”\nWe have written a new section (section II) in the supplementary text analysing the cost of our approach compared to the shape-complementary approach.\nFIG. S3. Number of unique staples required and the corresponding cost. Number of unique staples (Left Y-axis) required for multispecies assembly calculated with increasing number of species for two different approaches: shape-complementary method and our modular platform. 15$/staple is assumed to be the cost (right Y-axis).\nComment #6: Each connection uses a manifold of unique handle sequences that can and will interact non-specifically. The authors claim specificity by high off-rates and display this in an interaction matrix. The authors should quantify this, maybe with NUPACK.\nResponse #6: We characterize the interaction strength between different edges using an in-house code. Assuming that strands can only bind with each other when they are located on the complementary positions, we estimate the binding energies of two edges by summing the free energy contributions from the eight positions. Free energy of hybridization between two strands is estimated using nearest-neighbor model from SantaLucia et al. (2004) at 1 M NaCl concentration. The analysis shows a clear separation in free energies between complementary and non-complementary edges, suggesting high orthogonality of the assembly system (Fig. S6). Specifically, we find a gap of about 10 kcal/mol between the weakest complementary edge pairs and strongest non-complementary edge pairs.\nTo clarify this orthogonality and our design procedure of the sequences, we have added a new SI section (Section IV), as well as the following sentences in the ‘Modular design concept’ section of the main text:\n“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 SI Section IV for details on the sequence design.”\nFig. S4. Side-side interaction energy. (A) Binding free energy between different sets of bond-domain sequences, shown in Table II. The free energy of an edge is calculated as the sum of all the free energy contributions from each strand on the edge. The binding energy between individual strands is estimated at 1 M NaCl concentration using values from [3]. (B) To highlight the weaker binding energy of the off-target interactions, we set all intended binding interaction energies to 0 kcal/mol and rescaled the colorbar.\n[3] J. SantaLucia, Jr. and D. Hicks, The thermodynamics of DNA structural motifs, Annu. Rev. Biophys., 415 (2004)\nComment #7: Can the authors to comment on the choice of DNA hybridization connection setup: The chosen setup looks somewhat suboptimal, so why not use a three-strand setup? It would also render these kinds of connections changeable via toehold-mediated strand displacement, which would make the higher-order structure more versatile and thus more attractive for any downstream applications.\nResponse #7: We appreciate the reviewer’s suggestion regarding a three-strand connection setup (such as in Fig. S2C and E) and agree that such a design could, in principle, enable dynamic reconfigurability through toehold-mediated strand displacement. However, we identified several concerns with this approach. First, since the binding strength and binding angles are both programmed by the bond domain length, the two values are coupled. Second, the rigidity of binding angles are impacted by how many linkers are properly bound, which may be stochastic. In addition, there are concerns specific to each configuration:\n1. Scheme C (two identical overhangs linked by a connector): The design requires one overhang to bend in-plane, which introduces additional mechanical stress. The binding angles may not be as programmable as our dsDNA strut design.\n2. Scheme E (connector linking 3′ and 5′ overhangs): Since 5’-ends of synthetic oligos are more frequently truncated due to synthesis errors,, the hybridization yield may be lower.\nOur choice of the current design (scheme B) was motivated both by legacy considerations and by the need to robustly encode spatially varying curvature in complex architectures. That said, we acknowledge that the three-strand design (scheme E) shares similarities with our current approach and indeed offers the attractive feature of dynamic reconfigurability. We agree this is an exciting future direction, though careful evaluation of the mechanical and geometric trade-offs will be essential. In SI section I, we rationalize our design choice as follows:\nBuilding on these works, the current paper proposes using dsDNA struts to control the binding angles more effectively and predictably. Although there are many candidate overhang designs, as shown in Fig. S2, we choose our current dsDNA configuration (Fig. S2B) for a few reasons. First, the distance between the two particles is set by a single, rigid double helix, which makes encoding of the angles intuitive and reliable for a wide range. Second, the 3’-end of DNA (arrow end) extrudes out of the origami in our design. DNA synthesis progresses from 3’-end to 5’-end, making the 5’-end more susceptible to truncations. Especially for longer DNA staples, such as the overhang staples, there is a larger chance for 5’-end to be truncated, preventing hybridization. By adopting a design that extrudes on the 3’-end, we can minimize the impact of staple impurities. Finally, the binding strength in our design is decoupled from the binding angles. As mentioned previously, poly-T spacer designs (Fig. S2A) suffer from weaker interactions for increasing poly-T length. In the case of linkers (Fig. S2C, D, E), the distance between two particles is set by the linking domain length, which directly impacts the binding energies. Nevertheless, although we justify the use of dsDNA overhang configuration for our purposes, we acknowledge the potential utility of other overhang designs over our designs. For example, linker designs naturally allow for hierarchical assemblies and reconfigurations by adding linkers or removing linkers via strand displacement [7]. Also, the interaction strength can be tuned partially via linker concentrations. In-depth characterization of advantages and disadvantages for each design in Fig. S2, as well as reasonable hybrid designs that combine them, will provide helpful insight toward a rational interface in the future.\nFIG. S2. Different DNA overhang connection setups. (A) Direct hybridization of single-stranded DNA overhangs with poly-thymine spacers [9, 13, 14]. (B) Double-stranded DNA overhang connection, as discussed in this article. (C) Linker mediated 3-strand connection setup [16]. (D) Overhang connection [7]. (E) Double stranded overhang connection [10].\n[7] M. T. Luu et al., Reconfigurable nanomaterials folded from multicomponent chains of DNA origami voxels, Science Robotics 9, eadp2309 (2024).\n[9] J. J. Funke and H. Dietz, Placing molecules with Bohr radius resolution using DNA origami, Nature Nanotechnology 11, 47 (2016).\n[10] K. F. Wagenbauer, C. Sigl, and H. Dietz, Gigadalton-scale shape-programmable DNA assemblies, Nature 552, 78 (2017).\n[13] C. Karfusehr, M. Eder, and F. C. Simmel, Self-assembled cell-scale containers made from DNA origami membranes, bioRxiv 10.1101/2024.02.09.579479 (2024).\n[14] W.-S. Wei et al., Economical and versatile subunit design principles for self-assembled DNA origami structures, ACS Nano 19, 30889 (2025).\n[16] J. Lowensohn et. al., Self-assembly and crystallization of DNA-coated colloids via linker-encoded interactions, Langmuir 36, 7100 (2020).\nComment #8: The yield of the structures is analyzed by AGE band intensities. TEM images, however, reveal many broken structures. The authors should at least comment on this and discuss the implications for the use of structures in downstream applications.\nResponse #8: We appreciate the reviewer’s observation regarding the broken structures observed in the TEM images, particularly in the case of the toroid assemblies. Indeed, while our smaller shell structures (~100 nm in diameter) remain largely intact during sample preparation, the larger assemblies such as the toroid (~300 nm) and T13 (~340 nm) are more frequently observed in broken or aggregated state. We hypothesize that the negative stain sample preparation process for TEM damages the large hollow structures.\nWe have included this in the second paragraph of section D in the main text.\n“..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 (SI Fig. S24).”\nTEM images highlight the importance of characterization while maintaining structural integrity of the assembled structures for various downstream applications and we appreciate the reviewer pointing that out. We have included the following in the second paragraph of the Conclusion section of the main text.\n“New 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 coating [52] or covalent linking [53], should be explored and standardized.”\n[52] L. Nguyen et al., DNA-origami-templated silica growth by sol-gel chemistry, Angewandte Chemie International Edition 58, 912 (2019).\n[53] T. Gerling et al., Sequence-programmable covalent bonding of designed DNA assemblies, Science Advances 4, eaau1157 (2018).\nComment #9: The authors should further discuss for which nanotechnological applications their structures and the presented angled connections can be used, and what advantages and drawbacks over other approaches they have.\nResponse #9: We agree with the reviewer and appreciate their comment. We have added a paragraph (below) in the conclusion section of our main text discussing the applications of our modular framework.\n“Leveraging precise control over binding angles, symmetries, and periodicities in a multicomponent environment, our method supports the assemblies of self-limited structures [54, 55] and surface-based crystals [56, 57] that are orders of magnitude larger than individual subunits. These structures can help organize nanoscale particles into specific alignments, unlocking the ability to engineer macroscale material functionalities such as photonic properties [34-37, 58], or enhanced enzyme activities [59].\nIn addition, DNA origami shells have shown promise in antiviral therapeutics [21, 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, offering new possibilities in the design of synthetic biological defenses.”\n[21] C. Sigl et al., Programmable icosahedral shell system for virus trapping, Nature Materials 20, 1281 (2021).\n[34] G. Posnjak et al., Diamond-lattice photonic crystals assembled from DNA origami, Science 384, 781 (2024).\n[35] H. Liu et al., Inverse design of a pyrochlore lattice of DNA origami through model-driven experiments, Science 384, 776 (2024).\n[36] P. Wang et al., Programming self-assembly of DNA origami honeycomb two-dimensional lattices and plasmonic metamaterials, JACS 138, 7733 (2016).\n[37] J. S. Kahn et al., Encoding hierarchical 3D architecture through inverse design of programmable bonds, Nature Materials 24, 1273 (2025).\n[39] A. Monferrer et al, Broad-spectrum virus trapping with heparan sulfate-modified DNA origami shells, ACS Nano 16, 20002 (2022).\n[54] L. E. Perotti et al., Kirigami and the Caspar-Klug construction for viral shells with negative Gauss curvature, Phys. Rev. E. 99, 022413 (2019).\n[55] C. Chuang et al., Systematics of toroidal, helically-coiled carbon nanotubes, high-genus fullernens, and other exotic graphite materials, Procedia Engineering 14, 2372 (2011).\n[56] H. Tanaka et al., Programmable self-assembly of nanoplates into bicontinuous nanostructures, ACS Nano 17, 15371 (2023).\n[57] C. M. Duque et al., Limits of economy and fidelity for programmable assembly of size-controlled triply periodic polyhedra, PNAS 121, e2315648121 (2024).\n[58] A. Kuzyk et al., DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response, Nature 483, 311 (2012).\n[59] J. S. Khan et al., Cascaded enzyme reactions over a three-dimensional, wire-frame DNA origami scaffold, JACS Au 2, 357 (2022).\nReviewer #2\nComment #1: The work of Saha et al introduces a cost effective modular method to assemble self limiting structures.\nAmongst classic platonic solid examples, presented are also complex shells and a toroid structure.\nThe modular framework is characterized in great detail allowing to utilize this approach across other DNA assemblies.\n- The data is presented in a clear way, however I would ask to restructure the supplement to better follow the text flow.\nExample the Results section opens up with \"Figs. 1A, 2A and S25\" and next \"Fig. 1B. See 166 SI Section VIII\" this continues throughout the text.\nResponse #1: We thank the reviewer for their positive overall assessment of our manuscript. In response to the suggestion to improve the alignment between the main text and the Supplementary Information, we have reorganized the Supplementary file to better follow the flow of the main text. In particular, figures and sections in the Supplementary Information are now referenced more sequentially and grouped according to the order in which they are discussed in the Results section, to enhance clarity and ease of navigation.\nComment #2: There's a small typo \"Karfuscher et al. [36]\" should be Karfusehr.\nResponse #2: We thank the reviewer for pointing out our mistake. We have fixed the spelling in the Introduction of the main text.\n“Recent efforts by Karfusehr et al.[42] and us [43] demonstrate modular DNA origami design with programmable binding angles… ”\nComment #3: For the oxDNA simulations the number of repetitions for each tried condition should be explicitly stated in the methods section.\nResponse #3: We thank the reviewer for bringing this to our attention. We have incorporated this information into the methods (section J) section of the main text to reflect that one independent simulation was carried out for each of the 32 studied angle-domain lengths at 298 K and three additional independent simulations were carried out for both +6δ and +10δ, where the initial configurations were simulated at 298K for an extended time, followed by an immediate quenching to 136 K.\n“... The John thermostat with diffusion coefficient and Newtonian step settings of 2.5 and 103 was used to maintain a constant temperature of 27℃. 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 scripts [62].\nAdditional 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 (∆t) 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 (∆t) of 9.09 fs. C …”\n[62] J. Bohlin, M. Matthies, E. Poppleton, J. Procyk, A. Mallya, H. Yan, and P. Šulc, Design and simulation of DNA, RNA and hybrid protein-nucleic acid nanostructures with oxView, Nature Protocols 17, 1762 (2022).\nComment #4: SI Section IV Provided traces for +6 and +10 are not converged for 298K and in every case an angle close to 50 degrees was chosen for the subsequent 136 K run.\nAdditional analysis is needed to sample the complete Free energy landscape of the angles, which will make the discrepancy to CryoEM discussion stronger,\nand it would be interesting to see how the model would equilibrate from more extreme starting angles for both +6 and + 10 cases.\nResponse #4: We thank the reviewer for bringing this to our attention. Previously reported 298 K simulations were 10 times shorter than those at 136 K because our focus was to obtain plausible structures for low temperature simulations. Thus, the binding angle traces at 298 K showed a finer resolution and represented lesser equilibrated systems, as the reviewer noted. We have now performed extended simulations with a total simulated time of 9.09 μs (3,000 μs of physical time) at each temperature, shown in the figure below. The new traces for 298 K show an improved convergence about the mean binding angles. Importantly, the new mean binding angles for both temperatures are consistent with the ones reported previously. Furthermore, the ratios of the standard deviations between 298K and 136K simulations approach the theoretical value of ~1.48, consistent with the scaling of experimental distributions used in this study. We also note that the supporting running average and autocorrelation function analyses suggest that the extended simulations at 298 K have reached equilibrium.\nFinally, we note that the last recorded structures at 298 K (reported inset the top of each panel in blue) differ in binding angle across the replicates and in some cases deviate from the mean, providing a wide range of starting configurations for the subsequent 136K simulations for both +6δ and +10δ. Nonetheless, the dimers at 136 K appear to equilibrate to comparable binding angles regardless of the starting angle (e.g., panels B and C show dimers equilibrating to ~87° from ~60° and ~73°, independently).\nWe have replaced Figure S11 in the supplementary text with the following figure below (now Figure S17 in the revised SI).\nFig. S17: OxDNA simulations were equilibrated at 298~K and subsequently quenched to 136K.\nPanels (A–C) show three repetitions of +10𝛿, while (D–F) correspond to +6𝛿. The average binding angles at both temperatures and the ratio of their standard deviations (𝜎) are reported. Each panel contains the supporting running average and autocorrelation function analyses, suggesting that simulations at 298 K and 136 K temperatures have reached equilibrium.\nComment #5: I recommend acceptance once addressed.\nResponse #5: Again, we thank the reviewer for their positive overall assessment of our manuscript and their recommendation to publish it in Nature Communications following minor revisions.\nReviewer #3\nComment: I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.\nResponse: We thank the reviewer for evaluating our manuscript.\nReviewer #4\nComment #1: This manuscript presents a modular strategy for programming DNA origami subunits with independent control over inter-subunit binding specificity and binding angles. The authors propose that this framework overcomes key challenges in assembling geometrically complex nanostructures, particularly those with nonuniform Gaussian curvature. While the work is clearly described and technically sound, its overall novelty and impact should be more carefully contextualized. Below are my main comments:\nCost-effective design is one of the central contributions of this work, it lies more in an incremental improvement than in conceptual novelty. The modular design, which combines the separation of geometric control and interaction specificity via overhang domains, builds directly on prior efforts (e.g., references [22], [23], [35], [36]) that have already demonstrated programmable angles and modular subunit design. The present study refines these approaches by preserving scaffold routing and 70% of staple sequences across different geometries, offering cost and design efficiency benefits. However, the ability to build large assemblies of triangular DNA origami has already been demonstrated by the same research group (e.g., Videbæk et al., Science Advances, 2024 and arXiv:2411.09801 submitted on 14 Nov 2024). Similarly, cost-effectiveness and modularity have been addressed in recent work (e.g., Weck & Heuer-Jungemann, Nat. Commun. 2025) that enabled fully addressable superstructures using single modular DNA origami units.\nResponse #1: We thank the reviewer for raising this important point. The novelty of our work lies in applying modular origami design principles to program surfaces with nonuniform curvature, which has not been the focus of prior modular frameworks as shown by Luu et al. (2024) and Weck et al. (2025).\nPrior assemblies—including those in our earlier work and by others—primarily targeted structures with uniform curvature [Sigl et al. (2021) , Hayakawa et al. (2022), Videbaek et al. (2024)] or fully addressable bulk 3D architectures [Luu et al. (2024)].\nIn contrast, our current approach is designed to apply modular design principles to geometrically complex 2D manifolds, where each bond encodes both specificity and inter-subunit angle. As a proof of concept, we assembled a toroid with varying Gaussian curvature, transitioning from negative to zero to positive curvature. Achieving this required precise control over both interaction specificity and geometry, implemented through 12 unique subunit species and 7 distinct binding angles. The architectural complexity of this target demonstrates the utility of our framework beyond previous applications of modularity in DNA origami.\nWe also note that Karfusehr et al. (2024) and us (Wei et al. 2025) in a separate study had shown the angle programmability using single-stranded DNA (ssDNA) overhangs in a modular DNA origami building block. While in current work we implemented double-stranded DNA (dsDNA) overhangs to encode rigid bonds across a wide range of binding angles both reliably and predictably, we expect that it will be challenging to achieve the same using ssDNA overhangs. Flexibility of ssDNA overhangs sacrifices the ability to encode rigid bonds and distance between two ends, making them unsuitable to program large angles. Notably, in our work we have characterized -90° to +110° degrees binding angles and implemented successfully in the assembled structures including toroid.\nWe thank the reviewer for prompting this clarification and we have revised the manuscript to clearly articulate this distinction in the Introduction section.\n“Although DNA origami offers a vast design space, the range of accessible structures are 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 particle shape and interface geometries remain immutable. As a consequence, the assembled structures have so far been limited to relatively simple geometries with [19, 24, 41], 3D crystals [34-37], shells [21], and cylindrical tubules [22, 23].\nIn 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.\nRecent efforts by Karfusehr et al. [42] and us [43] 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 SI section I“\n[18] G. Tikhomirov et al., Programmable disorder in random DNA tilings, Nature nanotechnology 12, 251 (2017).\n[19] C. M. Wintersinger et al., Multi-micron crisscross structures grown from DNA-origami slats, Nature Nanotechnology 18, 281 (2023).\n[21] C. Sigl et al., Programmable icosahedral shell system for virus trapping, Nature Materials 20, 1281 (2021).\n[22] D. Hayakawa et al., Geometrically programmed self-limited assembly of tubules using DNA origami colloids, PNAS 119, e2207902119 (2022)\n[23] T. Videbæk et al., Economical routes to size-specific assembly of self-closing structures, Science Advances 10, eado5979 (2024).\n[24] D. Hayakawa et al., Symmetry-guided inverse design of self-assembling multiscale DNA origami tilings, ACS Nano 18, 19169 (2024)\n[26] M. T. Luu et al., Reconfigurable nanomaterials folded from multicomponent chains of DNA origami voxels, Science Robotics 9, eadp2309 (2024).\n[28] J. M. Weck and A. Heuer-Jungemann, Fully addressable design superstructures assembled from one single modular DNA origami, Nature Communications 16, 1556 (2025).\n[34] G. Posnjak et al., Diamond-lattice photonic crystals assembled from DNA origami, Science 384, 781 (2024).\n[35] H. Liu et al., Inverse design of a pyrochlore lattice of DNA origami through model-driven experiments, Science 384, 776 (2024).\n[36] P. Wang et al., Programming self-assembly of DNA origami honeycomb two-dimensional lattices and plasmonic metamaterials, JACS 138, 7733 (2016).\n[37] J. S. Kahn et al., Encoding hierarchical 3D architecture through inverse design of programmable bonds, Nature Materials 24, 1273 (2025).\n[40] W. liu et al., Self-organized architectures from assorted DNA-framed nanoparticles, Nature Chemistry 8, 867 (2016).\n[41] G. Tikhomirov et al., Fractal assembly of micrometer-scale DNA origami arrays with arbitrary patterns, Nature 552, 67 (2017).\nAdditionally, we have added a new supplementary section (Section I) that compares our design with prior modular DNA origami platforms with programmable angles.\nFIG. S2. Different DNA overhang connection setups. (A) Direct hybridization of single-stranded DNA overhangs with poly-thymine spacers [9, 13, 14]. (B) Double-stranded DNA overhang connection, as discussed in this article. (C) Linker mediated 3-strand connection setup [16]. (D) Overhang connection [7]. (E) Double stranded overhang connection [10].\n[7] M. T. Luu et al., Reconfigurable nanomaterials folded from multicomponent chains of DNA origami voxels, Science Robotics 9, eadp2309 (2024).\n[9] J. J. Funke and H. Dietz, Placing molecules with Bohr radius resolution using DNA origami, Nature Nanotechnology 11, 47 (2016).\n[10] K. F. Wagenbauer, C. Sigl, and H. Dietz, Gigadalton-scale shape-programmable DNA assemblies, Nature 552, 78 (2017).\n[13] C. Karfusehr, M. Eder, and F. C. Simmel, Self-assembled cell-scale containers made from DNA origami membranes, bioRxiv 10.1101/2024.02.09.579479 (2024).\n[14] W.-S. Wei et al., Economical and versatile subunit design principles for self-assembled DNA origami structures, ACS Nano 19, 30889 (2025).\n[16] J. Lowensohn et. al., Self-assembly and crystallization of DNA-coated colloids via linker-encoded interactions, Langmuir 36, 7100 (2020).\nComment #2: The construction of a toroidal structure is presented as a key achievement. However, the reported yield is very low, and the design requires 12 unique triangle species. While this is more economical than designing 12 completely distinct origamis, the approach is far from trivial. For applications that demand dozens or even more of distinct subunits, the strategy may not scale efficiently. Despite the claim that this framework enables precise control of curvature and structure, the toroid example suggests that achieving complex, deterministic topologies remains a substantial challenge. Is it possible to have wide-field images of well folded toroids under TEM ? Given these limitations, to what extent can this modular framework be realistically extended to more functionally diverse assemblies?\nResponse #2-1: Although we cannot measure the exact yield of the fully assembled toroids due to their size, we understand the reviewer’s concern that the yield of toroids might be low. We have clarified our limitation to measure the yield of toroid in the revised supplementary section (section XIII).\n“Although we have succeeded in assembling toroids, we are unable to measure the yield of toroids using conventional methods such as gel densitometry and TEM. The size of the toroid is fairly large (~300 nm) limiting us to measure the assembly yield using gel densitometry.\nAdditionally, we observe several partially grown toroids and off-target assembly, including five-fold symmetric structures [Fig. S26]. Unfortunately, due to low frequency of fully formed toroids and their fragility under TEM staining and blotting, we were unable to obtain wide-field images showing many well-folded toroids in a single field. However, the representative tomogram confirms that the designed topology can be successfully realized [Fig. S27]. We find about 50 fully assembled toroids under TEM within 2 hours, which we use to characterize the polymorphism. ”\nTo give a better sense of the toroid assembly, we provide a few TEM micrographs with a wider field of view (FIG. S26).\nFIG. S26. Representative, wide-field images of toroids under TEM.\nResponse #2-2: We appreciate the reviewer’s concern about the scalability of our modular framework to more functionally diverse assemblies. Hence, to show the potential of our work, we have now included additional experimental results which is the assembly of a T=13 icosahedral shell (T13), the largest Caspar–Klug icosahedral DNA origami shell ever achieved. Compared to the toroid that comprises 96 building blocks with 12 unique species, the T13 is made of 260 building blocks with 5 unique species, demonstrating the scalability of our modular framework in both the assembly size and design complexity.\nWe have updated the Figure 1 and Figure 4 (attached below) in the main text that now includes the T13 assembly results. We have discussed the assembly results (presented below) in detail in the section D of the main text. Additionally, we have discussed the geometry of T13 in detail in SI section X and analysis of the TEM micrographs in SI section XII.\nFIG 4: Self-assembly of T13 and toroids using specific interaction and binding angle. (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-stained 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. The geometry of the toroid with the locations of all disclination defects (left), unique rotational symmetries of the toroid (middle), and the species and binding angles between triangles (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.\n“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.\nGeometrically, T13 can be conceived as a large icosahedron, whose triangular face is subdivided into 13 smaller triangles [1] (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 degrees. 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 SI section X.\nAlthough 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 kinetics [19, 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 (section I). 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 (SI 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 7.03×10^4 nm^2, respectively, the mean of the measured edge lengths and areas are 188±19 nm and 7.44±1.16× 10^4 nm^2, respectively, corresponding to roughly 15% expansion (Fig.4C and SI section XII). We attribute the size increase to the flattening of the structures during sample preparation. See SI 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 section I). 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.”\n[1] D. L. D. Caspar and A. Klug, Physical Principles in the Construction of Regular Viruses, Cold Spring Harbor Symposia on Quantitative Biology 27, 1 (1962).\nResponse #2-3: In addition to that, we would like to emphasize two key advantages of using our design framework. Firstly, due to the detailed characterization of the binding angle in our study (Section B), we have achieved rapid success in assembling complex structures such as the T13 (first attempt) and the toroid. These structures require the programming of seven unique binding angles, which otherwise would have been challenging with the existing frameworks because of the lack of in-depth characterization of binding angles. Secondly, a key advantage of our approach is that the DNA strands from one structure can be repurposed to assemble a diverse range of shapes such as helical tubules, with a negligible additional cost [51]. This flexibility significantly enhances the cost-efficiency of our method, as an initial investment in a library of interaction motifs and binding angles grants access to a broad spectrum of possible nanostructures.\nWe have added the following in the first paragraph of the Conclusion section.\n“…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 tubules [51], all of which require many unique particle species and binding angles.”\n[51] M. Price et al., From toroids to helical tubules: Kirigami-inspired programmable assembly of two-periodic curved crystals, arXiv preprint, arXiv:2506:16403 (2025).\nComment #3: Although the authors emphasize the programmability of binding angles and interaction matrices, the system does not reliably produce single, well-defined structures. Multiple polymorphic forms are observed, particularly at higher angle-domain lengths. This outcome is well documented in the paper but it highlights a fundamental limitation: the current design rules do not fully constrain the assembly pathway to a unique structure, especially in more complex or crowded mixtures even with program type-specific interactions. The results raise a central question: does the approach provide enough design specificity to be predictive and controllable at scale?\nResponse #3: We appreciate the reviewer’s insightful comment regarding the observed polymorphism, particularly at larger angle-domain lengths, and the broader question of whether our framework provides sufficient design specificity for scalable, predictable assembly.\nThe case the reviewer refers to is the deltahedral shell (Fig. 3B), where every side of a triangular subunit can bind to itself and to other sides. In our systematic study of angle-domain lengths (0–15 bp), we found that polymorphism indeed increases at larger lengths. This outcome is expected because longer angle domains introduce greater angular fluctuations, which destabilize the unique geometry. Importantly, we show that this limitation can be mitigated: by introducing type-specific interaction rules in multicomponent systems (Fig. 3E), we achieve higher yields of the intended structures and significantly reduce polymorphism.\nMore broadly, our goal is to advance the field toward predictive self-assembly—where one can anticipate assembly outcomes directly from subunit design and interaction rules. While absolute yield prediction remains beyond current capabilities, we believe our framework can enable prediction of the relative likelihoods of competing structures. Recent studies [25, 51, 69, 70] demonstrate that relative free energies can be estimated from known bond strengths, preferred binding angles, and bending moduli. Because our building blocks are modular and geometrically well-characterized, they provide the necessary inputs for such predictive models.\nIn this way, even if polymorphism cannot yet be eliminated entirely, our system helps identify which polymorphs are likely to dominate under given design rules. This reduces reliance on purely empirical trial-and-error and provides a more rational pathway to designing complex nanostructures.\n[25] W.-S. Wei et al., Hierarchical assembly is more robust than egalitarian assembly in synthetic capsids, Proceedings of the National Academy of Sciences 121, e2312775121 (2024).\n[51] M. Price et al., From toroids to helical tubules: Kirigami-inspired programmable assembly of two-periodic curved crystals, arXiv:2506.16403 (2025).\n[69] M. C. Hübl et al., The polyhedral structure underlying programmable self-assembly (2025), arXiv: 2501.16107.\n[70] B. Tyukodi et al., Magic sizes enable minimal-complexity, high-fidelity assembly of programmable shells (2025), Phys. Rev. Lett. 135, 118203.\nComment #4: Cryo-EM validation of the dimer angles is central to the paper’s claims, but representative images of the DNA origami dimers are missing. The number of cryo-EM movies acquired, and any classification statistics or images are missing.\nResponse #4: We have added a table to the SI (Table V) that contains more detailed information about the data collection for our cryo-EM reconstructions, which can also be found in the EMDB pages for the associated structures. We have also added both representative cryo-EM images and 2D classes used in the reconstruction for the dimer and monomer structures (Figures S8 and S32).\nTABLE V: Details of image collection and reconstruction for the dimers and monomers.\nFIG. S8. Cryo-EM reconstruction of dimers with different binding angles. (A,C,E) Images from top- and side-views of the dimer reconstructions for angle-domain lengths of +6𝛿, +10𝛿 and -7𝛿 respectively. (B,D,F) Plots of the FSC curves used to estimate the resolution of the dimer reconstructions.\nFIG. S32. Cryo-EM single particle reconstruction of the modular block. (A) Top-down view of the reconstruction. (B) Plot of the FSC curves used to estimate the resolution of the reconstruction.\nComment #5: Figure 4C shows a cross-section from a TEM tomography reconstruction of the toroid and is cited as evidence for agreement with the predicted geometry. However, I didn’t find mention of the tomography acquisition or reconstruction methods in the main text or Methods section, nor are tomogram datasets like different slices of the entire tomogram from bottom to top should be included in the supplementary information.\nResponse #5: We appreciate you pointing out the missing details regarding the TEM tomography acquisition and reconstruction. In response, we have now added a description of the tomography reconstruction protocol to the Methods section of the manuscript.\n“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 x29000 magnification from -50 degrees to 50 degrees in 2-degree increments. The data is analyzed and the z-stack is reconstructed using IMOD [62].”\n[62] J. R. Kremer, D. N. Mastronarde, and J. R. McIntosh, Computer visualization of three-dimensional image data using IMOD, Journal of Structural Biology 116, 71 (1996).\nTo further support the interpretation of Figure 4C, we have included a new supplementary figure (Fig S27) showing representative slices from the top, middle, and bottom sections of the toroid. Additionally, we have provided a movie of the full tomogram in the supplementary materials to give a more complete view of the 3D, hollow structure.\nFIG S27. Tomographic reconstruction of a toroid. Top (A), middle (B) and bottom (C) section of the tomographic reconstruction."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have adequately addressed all of this reviewer's concerns and substantially improved the manuscript. It is a very nice piece of work which should now be accepted for publication in Nat. Commun.", "reviewer_2": "The authors have addressed my concerns in sufficient detail and I recommend publication now.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "Dear all,\nI have reviewed the revised version of the manuscript. The authors have addressed all my comments and concerns in their revision. The changes made have improved the clarity and rigor of the study."}, "author_response": ""}]

Decision letter