Main

We first implement self-limiting assembly in a two-dimensional plane in solution (Fig. 1) with wedge-shaped building blocks (‘V bricks’), the opening angles of which encode the number of subunits in ring oligomers (Fig. 1a, b). V-brick oligomerization is mediated by two asymmetric, interlocking self-complementary surfaces at opposite faces (Fig. 1a, Supplementary Figs 1, 2, Supplementary Note 1). The electron density map determined for one V-brick variant using single-particle cryo-electron microscopy (cryo-EM; Fig. 1a, Supplementary Fig. 3) reveals the global shape of the V brick and the honeycomb-lattice packing of helices in multiple layers. The set of double-helical spacers that define the opening angle and the shape-complementary surface features on the front and back faces of the V brick can also be seen. In total, we designed eleven V-brick variants with different opening angles, all of which were very rigid, as reflected by each variant exhibiting narrowly distributed single-particle opening angles (standard deviations of approximately 2° from the mean opening angle, as measured in single-particle transmission electron microscopy (TEM) images; compare Fig. 1c and Supplementary Fig. 4).

Figure 1: Shape-programmable self-limiting ring oligomerization in solution.
figure 1

a, Schematic representation of a V-shaped multi-layer DNA origami object (a V brick) with helices packed in a honeycomb lattice. Cylinders represent DNA double-helical domains. Orange cylinders represent angle-defining double-helical spacing elements; red and blue cylinders represent shape-complementary docking sites. Multiple views of an electron density map of a V brick are also shown, determined using single-particle cryo-EM (gold-standard Fourier shell correlation (GSFSC) resolution, 27 Å; see also Supplementary Fig. 3). b, Representative negatively stained TEM micrograph of ring oligomers assembled from a V-brick variant with 14- and 37-base-pair-long double-stranded DNA spacers. Inset, a reference-free class average calculated from single-particle TEM micrographs. Scale bar, 300 nm. c, d, Histograms (red) of V-brick opening angles measured in single-particle micrographs for variants with different angle-defining spacing elements (c) and of the number of V bricks per closed ring as determined by counting in single-particle micrographs of ring oligomers (d). The solid lines are Gaussian fits. Numbers in c indicate the length of short and long spacer elements in base pairs. Insets, reference-free class averages calculated from single-particle TEM micrographs for each V-brick variant; box size, 60 nm. e, Exemplary reference-free class averages obtained from single-particle TEM micrographs of closed-ring oligomers assembled from 22/55 (top) and 11/26 (bottom) V-brick variants. The number in each image indicates the number of V bricks per closed ring. The ring thickness in the radial direction corresponds to the length of V bricks, 56 nm. f, Experimentally observed average ring sizes determined in d (blue circles) as a function of the observed average opening angle of the V bricks from c. Vertical bars indicate the standard deviation from the mean ring size; horizontal bars indicate the standard deviation from the average V-brick opening angle. Small black circles indicate the possible integer solutions when 360° is divided by the V-brick opening angle—that is, the number of possible subunits per closed ring for infinitely rigid building blocks. The red solid line and orange shading indicates the average ring size and standard deviation, respectively, for finite-rigidity V bricks, as predicted by an equilibrium oligomerization model (see Supplementary Note 2 and Supplementary Fig. 9). Inset, exemplary equilibrium distribution of circularized ring oligomers (grey) and non-circularized oligomers (red) predicted by the model from which the average ring size and standard deviation are computed. g, Schematic representation of a V-brick variant with single-stranded spacing elements (left) and an exemplary negative-stained TEM micrograph of the flexible V bricks assembling into spiral-like oligomers with an uncontrollable number of V bricks (right); see Supplementary Fig. 10. Scale bar, 50 nm. h, Laser-scanned fluorescent image of an agarose gel on which the oligomerization products of V bricks were electrophoretically separated as a function of incubation time. The gel image is depicted in a 3D landscape to enhance the visibility of low-intensity bands. P, gel pocket; R, cyclized ring species. The TEM class average indicates the mobility of V-brick monomers. Note the correspondence between the experimental distributions in d and the gel bands in h. i, Laser-scanned fluorescent images of agarose gels on which equilibrated ring oligomerization samples were electrophoresed. Left, V-brick variant 14/37; right, V-brick variant 22/55. Numbers next to the bands indicate the number of V bricks per closed ring as determined by band extraction and counting in TEM images. Note the correspondence of the band patterns to the inset in f.

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The oligomerization of the V bricks is triggered by increasing the ionic strength of their solution22, yielding planar ring oligomers with the number of subunits per ring controlled by the shape of the V bricks. In all cases, the number of V bricks in a ring was close to the number obtained when dividing 360° by the average opening angle of the V-brick variant used (Fig. 1d, Supplementary Figs 5–8). The V bricks with the largest opening angle (around 30°) formed the smallest rings, which comprised 11 subunits on average. The V bricks with the smallest opening angle (about 12.5°) formed the largest rings, which contained 28 units on average. Ring-size polydispersity, relative to the average ring size, was typically around 7% (Fig. 1d) and therefore comparable to that of several ring-forming protein assemblies25,26,27. Reference-free two-dimensional class averages, determined from multiple individual TEM micrographs for a subset of the V-brick ring variants, revealed rings with a smooth circular appearance with equidistantly distributed angles and spacing between subunits (Fig. 1e). Therefore, strains in the rings, if any, are distributed evenly.

A simplified, two-dimensional variant of previously described viral capsid assembly models28 explains satisfactorily the relation between the average number of subunits per closed ring and the residual spread in the ring-size distribution that we observed experimentally (Fig. 1f, Supplementary Fig. 9, Supplementary Note 2). The model highlights the importance of rigidity: whereas overly flexible building blocks lead to loss of control over the size of ring oligomers (Supplementary Note 2), our DNA origami building blocks balance rigidity and residual flexibility to compensate for the difference between the actual opening angles of the V bricks and the ideal opening angles, thereby enabling closed rings to form. Three different floppy V-brick variants with more broadly distributed opening angles (Fig. 1g, Supplementary Fig. 10) were found to assemble into various higher-order aggregates, such as spirals and structurally ill-defined objects, rather than rings. This finding illustrates that ‘floppy’ building blocks cannot dictate the shape of higher-order structures with a sufficient degree of control, emphasizing the importance of rigidity.

The oligomerization of the V bricks into rings occurs progressively from monomers through small oligomers towards a defined set of higher-order oligomers (Fig. 1h, i, Supplementary Figs 11a, 12). The initial population and then disappearance of intermediate oligomer species over time implies reversible subunit association and dissociation processes. These processes allow the system to equilibrate and to avoid becoming kinetically trapped in a heterogeneous distribution of cyclized and non-cyclized oligomers of various sizes, as is seen, for example, in ring protein oligomerization26. The equilibrated distributions of ring species that are seen in the gel and by TEM agree favourably with the predictions of our equilibrium oligomerization model (Fig. 1f, inset; Supplementary Fig. 9a). For example, for the V-brick variant with the largest opening angle (Fig. 1i, right), we observe two dominant final species in the gels, which correspond to rings with 10 and 11 subunits, and a weakly populated species with 12 subunits. These results are consistent with the TEM data (Fig. 1d) and concur with the ring-size distribution predicted by our model (Supplementary Figs 9a (top), 12). We occasionally saw some smearing in the gel images above the ring bands. Physical extraction of the smear from the gel revealed well defined closed rings instead of higher-order aggregates (Supplementary Fig. 11). We therefore attribute the smearing to rings that got trapped in the gel matrix upon electrophoretic migration, which is reasonable given their large size and spiky shape.

We add another tier to the assembly of higher-order structures by introducing a second, weaker set of self-complementary docking sites into the V bricks (Fig. 2a), which enables bonding perpendicularly to the plane of ring oligomerization. In this case, ring oligomers, once formed, can stack on top of each other to form long tubes with thick walls (Fig. 2b). The tube formation is triggered by increasing once again the ionic strength of the solution, with scanning helium-ion microscopy images confirming the formation of higher-order tubes (Fig. 2c–f). Some of the tubes, with lengths of more than 1 μm and diameters of several hundred nanometres depending on the V-brick variant used, were similar in size to some bacilli.

Figure 2: Assembly of self-limiting ring oligomers into tubes.
figure 2

a, Cryo-EM density map of a V-brick variant with a second set of self-complementary protrusions and recessions on the faces of the two beams perpendicular to the ring oligomerization plane (black arrows). GSFSC resolution, 27 Å. b, Schematic representation of hierarchical assembly of V bricks into rings (left), and assembly of these rings into tube-like structures (right). Purple cylinders indicate protruding blunt-ended DNA helices that fit into the grey recessions on the back face of the ring. Assembly equilibrium is controlled via the cation concentration and the temperature in solution (black arrows). c, Front (top) and back (bottom) view of the cryo-EM density map of the V brick, revealing a global, residual, right-handed twist deformation. d, Microscopic model of the tubes formed by stacking rings assembled from the V-brick cryo-EM map. The cryo-EM maps were drawn at a fixed isodensity value and arranged in contact using the Cinema4D cloning tool (https://www.maxon.net/en/products/cinema-4d/overview/). The V-brick-twist-induced pitch of the chiral tube thread is about 4° per ring. e, f, Typical helium-ion microscopy micrographs acquired from oligomerized tubes, which were first fixed with uranyl-formiate and then coated with 5-nm-thick AuPd. The images were acquired with different tilt angles of the stage. Scale bars, 50 nm. g–j, As in c–f, but for a twist-corrected V-brick variant. GSFSC resolution, 31 Å; see also Supplementary Fig. 13. The residual twist is about 1° per ring. Scale bars, 50 nm.

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The V-brick variant used for the tube assembly features a subtle right-handed twist deformation in the helical direction that was recognized only when determining its structure by cryo-EM in solution of the V-brick variant. The twist deformation is amplified when tubes form, leading to a chiral ‘thread’ along the tube axis (Fig. 2d–f). The shape of the tubes and the helical thread, as seen in the helium-ion microscopy images, agrees well with a three-dimensional (3D) model constructed from the cryo-EM map of the V bricks (Fig. 2d, e). To remove the twist deformation, we introduced a counter-twist through base-pair deletions29 in another V-brick variant. Cryo-EM mapping confirmed that this V-brick variant adopted a nearly twist-free overall shape (Supplementary Figs 13–16) and that it assembled into tubes with a much-reduced macroscopic chiral pitch (compare Fig. 2e, f and Fig. 2i, j).

We then targeted self-limiting hierarchical oligomerization in 3D space and, inspired by the symmetries and hierarchical assembly of natural viruses, aimed for polyhedral cages. For this, we designed three DNA origami building blocks (Fig. 3a): a triangular brick with three identical recessed docking sites on each face (Supplementary Fig. 17); a twist-corrected V-brick variant that integrates the angle-defining mechanism that we introduced into the ring oligomers, but that has two distinct patterns of protruding docking sites on its two outward-pointing faces (Fig. 3a, Supplementary Fig. 18); and a rectangular ‘connector’ brick with recessed docking sites on one face and a self-complementary dock on the opposite face (Supplementary Fig. 19). We designed the hierarchy of interaction strengths such that the binding affinity of the connector brick to the V brick is greater than the affinity for forming connector-brick homo-dimers (Supplementary Figs 20, 21, Supplementary Note 3). Cryo-EM maps of the building blocks (Fig. 3c–e, Supplementary Figs 22–25) validated our designs and revealed nearly twist-free global shapes of the building blocks, the internal lattice architecture and details of the shape-complementary docking interfaces.

Figure 3: Shape-programmed hierarchical assembly and cryo-EM analysis of a three-fold self-complementary vertex.
figure 3

a, Exploded (left) and assembled (right) views of a schematic model of the ‘reactive vertex’ unit for self-limiting assembly in three dimensions. Blue, triangular brick with three identical recessed docking sites; light grey, V brick with different protruding plug patterns on each outward-pointing face; red, connector brick with docking sites on each of the wide faces. In the middle is a schematic representation of the assembly intermediate that is formed from one triangular brick and three copies of the V brick (‘inert vertex’, see main text; top) and a cryo-EM density map of the assembled intermediate state (bottom). GSFSC resolution, 25 Å; see Supplementary Fig. 16. Light grey arrows indicate the interfaces where the reactive vertex can stick to itself. b, Laser-scanned fluorescent image of a 1% agarose gel on which subunits of the reactive vertex were electrophoresed. From left to right (see images above each band): mixture of vertex and connector, and V-brick variants with opening angles of about 22°, 35° and 54°. P, gel pocket. c–e, Cryo-EM density maps of the three subunits of the reactive vertex. Dotted arrows between panels indicate how the subunits dock into each other. GSFSC resolution, 30 Å (c), 31 Å (d) and 21 Å (e); see Supplementary Figs 22–24. f, Cryo-EM density map (light grey) of the fully assembled reactive vertex in two different views; see Supplementary Fig. 27. A cryo-EM density map of a ribosome (Electron Microscopy Data Bank identifier EMD-3713) is shown in yellow, depicted to scale for size comparison. The top insets show representative field-of-view TEM micrographs of reactive-vertex variants: negative-staining variants with opening angles of about 55° (left) and 35° (middle), and cryo-EM micrograph of variants with an opening angle of about 22° (right). Scale bars, 50 nm.

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Together, one copy of the triangular brick and three copies each of the V brick and the connector brick form an object with three-fold (pseudo-)symmetry (Fig. 3f, Supplementary Figs 26, 27), which we call the ‘reactive vertex’. We determined a structure for the reactive vertex using single-particle cryo-EM (Fig. 3f, Supplementary Fig. 28). We also determined a map for a precursor object (the ‘inert’ vertex), which lacks the connector bricks (Fig. 3a, inset; Supplementary Fig. 29). The reactive vertex has a molecular weight of about 61 MDa and includes eleven M13-derived genomic scaffold chains and about 2,500 DNA oligonucleotide chains. To highlight the size of the reactive vertex, we include a to-scale cryo-EM map of a ribosome particle30 in Fig. 3f.

The reactive vertex can be programmed to self-assemble in a self-limiting fashion into a tetrahedron, a hexahedron or a dodecahedron (Fig. 4a), with the type of cage formed encoded in the shape of the reactive vertex. To realize these three polyhedra, the angles in the wings of the reactive vertex needed to be 55°, 35° or 22° (see Supplementary Fig. 30), respectively, which we achieved, as verified by analysing the opening angles of the V bricks in single-particle TEM data (Fig. 4a). Polyhedral self-assembly was triggered once again by increasing the salt concentration of the solution, proceeded via intermediate states with subunit numbers less than or equal to the actual target size (Fig. 4b, left), and equilibrated over time into a narrowly distributed set of products (Fig. 4b, right), as seen by mobility analysis of the reaction products in low-percentage (0.4%) agarose gel electrophoresis. The dominant reaction products of the tetrahedral design were tetrahedral, whereas oligomers with eight and seven subunits dominated at the later time of analysis in the case of the hexahedral design (Supplementary Fig. 31). The products of the dodecahedral assembly were too large to be analysed by gel electrophoresis. Complementary dynamic light-scattering experiments on all three polyhedral samples revealed a monodisperse distribution of particles in solution with the expected dimensions (Fig. 4c). Importantly, and in contrast to what is seen with the gels, no higher-order aggregates were observed in dynamic light-scattering experiments. As for the ring oligomers, we therefore attribute the material in the gel pockets to particles that got stuck in the gel matrix, rather than to actual aggregates.

Figure 4: Self-limiting hierarchical assembly of closed polyhedra from the reactive vertex.
figure 4

a, Cylinder models of reactive vertices designed for self-limiting self-assembly into a tetrahedron (left), a hexahedron (middle) and a dodecahedron (right); see Supplementary Fig. 28 for geometric details. Histograms (red) of the vertex angles measured in single-particle micrographs for the three V-brick variants used in the respective vertex variant are also shown; the solid lines are Gaussian fits. The target and average measured angles are listed below each histogram. b, Laser-scanned fluorescent images of 0.4% agarose gels on which the reactive vertex and the assembly products for the tetrahedron (‘Tet.’) and the hexahedron (‘Hex.’) were electrophoresed, after incubation for 12 h (left) and after 14 d (right). P, gel pocket. Red and blue boxed regions of interest were auto-levelled separately. White arrows indicate the fully assembled tetrahedron and hexahedron bands. c, Dynamic light-scattering intensity (a.u., arbitrary units) histograms as a function of particle radius, for the tetrahedron (red, left), the hexahedron (light blue, middle) and the dodecahedron (purple, right) (see insets). Numbers above each panel represent the mean measured particle radius. d–f, Typical negative-stained TEM micrographs of intermediate assembly products (left), reflecting the designed symmetry of assembly products (three-fold, d; four-fold, e; five-fold, f), and fully assembled closed polyhedra (right): d, tetrahedron; e, hexahedron; f, dodecahedron. The projection directions of the micrographs of the closed polyhedra are indicated by grey arrowheads in a. Scale bars, 50 nm.

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Negative-staining TEM revealed assembly products with three-fold symmetry in the tetrahedron sample (Fig. 4d, left), four-fold symmetry in the hexahedron sample (Fig. 4e, left; Supplementary Fig. 32), five-fold symmetry in the dodecahedron sample (Fig. 4f, left; Supplementary Fig. 33), and with the features expected for fully assembled, closed polyhedral shapes. For example, we saw tetrahedra with one of their vertices pointing towards the observer, cubes often in an orientation with a cubic face parallel to the TEM support, and large dodecahedra with a central pentagon, ten spokes in radial direction and twenty segments on the circumference (Fig. 4d–f, right; compare with the models in Fig. 4a). The negative-staining TEM data suggest successful assembly, but the unidirectional transmission projections did not reveal the internal polyhedral connectivity directly. To reveal the 3D structure of the cages, we reconstructed 3D cryo-EM tomograms of individual cage particles (Supplementary Fig. 34, Supplementary Videos 1, 2, 3). Slices through models and tomograms agree favourably within the resolution of the data for the tetrahedral and the hexahedral samples (Fig. 5a, b). Vitrified samples with the even larger dodecahedron were not suitable for cryo-EM tomography. Therefore, we instead performed tomography with negatively stained dodecahedral samples. The resulting tomograms showed all of the 3D features expected for fully connected dodecahedral cages, as seen in a slice-by-slice comparison between a model and a tomogram of the dodecahedron (Fig. 5c). The dodecahedral cages were squeezed by the staining layer in the z direction to approximately 25% of their original diameter. Taken together, our electron microscopy tomograms validate the successful assembly of the desired shapes, with the programmed internal connectivity for the tetrahedron, the hexahedron and the dodecahedron.

Figure 5: Single-particle tomographic validation of correct polyhedral assembly by cryo-TEM.
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a, Schematic representation of the assembled tetrahedron (top); expected appearance of three exemplary slices (in the z direction; see top panel) through a cylinder model of the tetrahedron (bottom left); and exemplary image slices extracted from a 3D cryo-EM tomogram acquired from tetrahedra (bottom right), at the same positions as in the renderings on the left. The tetrahedron is slightly tilted with respect to the slice plane. See Supplementary Video 1 for additional data with tilt series and full tomograms. b, As in a, but for the hexahedron. See Supplementary Video 2 for additional data with tilt series and full tomograms. The black dots in the bottom image are gold nanoparticles used as fiducial markers for frame alignment in tilt series. c, As in a, but for the dodecahedron. Four slices are shown. The experimental tomogram slices where obtained from tilt series with negatively stained dodecahedron samples. See Supplementary Video 3 for additional data with tilt series and full tomograms. All scale bars, 50 nm. The polyhedral cages have defined molecular weights of 0.24 GDa (a), 0.49 GDa (b) and 1.2 GDa (c). The dodecahedron integrates 220 M13-derived scaffold chains and about 50,000 DNA oligonucleotides.

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Key factors for successful self-limiting assembly of higher-order structures are accurately designed shapes with validated structures, rigidity, and assembly under equilibrium. In addition, our designs did not require fine tuning of concentrations and relative stoichiometry of the building blocks; instead, we used homo-oligomerization wherever possible and relied on designs for which excess of one component over the other was sufficient to run a reaction to completion. To illustrate the importance of robust assembly schemes, we also prepared another variant of the 3D polyhedral higher-order assemblies in which formation of closed cages requires reactive vertices and a connector-brick dimer in a finely tuned 1:1.5 stoichiometry (Supplementary Fig. 35). Even when mixing these building blocks at precisely the required relative ratio, the assembly failed, presumably owing to stochastically distributed binding of connectors to the wings of the vertices, which is unavoidable at a 1:1.5 ratio.

Our method for the bottom-up fabrication of well controlled assemblies helps to bridge the gap between the molecular and macroscopic scales, with the size and complexity of the structures that we assembled reaching those of viruses and cellular organelles. Given the modularity and the high degree of addressability of DNA origami building blocks, we expect that a range of other complex structures could be created that could potentially be useful for encapsulation and drug delivery, serve as scaffolds for inorganic and organic groups, or even enable the creation of integrated molecular-machine-like objects.

Data Availability

All data that support the findings of this study are available within the paper and its Supplementary Information, and from the corresponding author on request. Cryo-EM data that support the findings of this study have been deposited in the Electron Microscopy Data Bank (EMDB) with accession codes EMD-3826, EMD-3827, EMD-3828, EMD-3829, EMD-3830 and EMD-3831.