# Algorithm-driven DNA nanostructure design for advanced functionality

Nicolas Levy, Julie Finkel, Allan Mills, Gerrit Wilkens, Anjelica Kucinic, Aurelie Ancelin, Pierre Marcus, Octave Hazard, Joris Picot, Daria Pchelina, et al.

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Nicolas Levy, Julie Finkel, Allan Mills, Gerrit Wilkens, Anjelica Kucinic, et al. Algorithm-driven DNA nanostructure design for advanced functionality. 2025. hal-05410792

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# Algorithm-driven DNA nanostructure design for advanced functionality

Nicolas Levy<fnref n="1" />,†

Julie Finkel<fnref n="2" />,†

Allan Mills<fnref n="2" />

Gerrit David Wilkens<fnref n="2" />

Anjelica Kucinic<fnref n="2" />

Aurélie Ancelin<fnref n="2" />

Pierre Marcus<fnref n="1" />

Octave Hazard<fnref n="1" />

Joris Picot<fnref n="1" />

Daria Pchelina<fnref n="1" />

Gaëtan Bellot<fnref n="2" />,∗

Nicolas Schabanel<fnref n="1" />,<fnref n="3" />,∗

September 15, 2025

**Abstract.** Form and function are intimately linked—a guiding principle across both biology and engineering. While evolution and human ingenuity have shaped this landscape through function-driven form selection, AI-driven generative design expands it beyond natural and intuitive geometries, enabling the exploration of structure-driven functionalities that were previously inaccessible. At the nanoscale, however, realizing such non-intuitive architectures remains a key challenge, limiting the development of multifunctional nanostructures across biology, chemistry, and materials science. To address these constraints, we introduce an algorithmic design paradigm that enables DNA helices to follow non-planar 3D trajectories, thereby supporting the structural and functional outcomes required for advanced capabilities. We implemented this paradigm in ENSnano, a software platform that integrates mathematical models to automate structural design in 3D space without the need of human intervention. This framework allows us to rapidly generate DNA nanostructures with key functional features such as curvature, encapsulation, and hierarchical organization—reminiscent of naturally occurring biological architectures. As a key application, we demonstrate an automated pathway from biological to engineered structures by designing and experimentally assembling Vault-like cages directly derived from the emPDB model of the Vault protein, marking a step forward in biomimetic DNA nanostructure design.

**One-sentence summary.** Algorithms enable accurate design of DNA origami nanostructures with complex 3D curvature and torsion.

## Introduction

The interplay between form and function is a unifying principle across natural and engineered systems. In engineered gears, specific geometries optimize force transmission, while in architectural arches, curvature and torsion enhance structural rigidity and stability<fnref n="1" />,<fnref n="2" />. Many of engineered designs draw inspiration from biology, where evolution has refined structural forms to achieve specific functional outcomes<fnref n="3" />. Advances in artificial intelligence and generative design now enable the creation of structural designs that depart from both evolutionary precedent and human intuition, thereby expanding the accessible form–function landscape⁴.

Translating these functional forms to the nanoscale presents a significant challenge, requiring molecular-level precision to ensure function, an essential step toward the development of smart materials, synthetic biological systems, and precision therapeutic devices. Structural DNA nanotechnology addresses this challenge by redefining nucleic acids as programmable building blocks for encoding functional shapes at the nanoscale⁵.

Among self-assembly strategies, DNA origami⁶ has become the benchmark for constructing accurate nanoscale architectures. It enables precise control over geometry, site-specific chemical functionalization, and dynamic mechanical properties⁷, ⁸, ⁹, ¹⁰, while producing bil-

<footnote n="1">¹ LIP, École Normale Supérieure de Lyon, France, http://perso.ens-lyon.fr/perso/first.lastname</footnote>

<footnote n="2">² CBS, Université de Montpellier, France.</footnote>

<footnote n="3">³ CNRS, IXXI, École Normale Supérieure de Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07, France.</footnote>

<footnote>† These authors contributed equally to this work</footnote>

<footnote>∗ Corresponding authors: nicolas.schabanel@ens-lyon.fr and gaetan.bellot@cbs.cnrs.fr</footnote>
lions of copies in a single-pot reaction. In this approach, a long single-stranded DNA scaffold is folded into a target shape by hundreds of short oligonucleotides through sequence-specific base pairing. This programmable assembly method has facilitated the creation of diverse nanostructures, advancing both biological research and materials science by encoding functional form at the molecular scale<sup>11, 12, 13</sup>.

However, current design strategies remain largely restricted to geometries composed of straight segments<sup>14, 15, 16</sup> and planar arcs<sup>17, 18</sup>. This constraint arises from the conventional DNA origami approach, which models helices as near-rigid cylinders connected by crossovers at fixed spacing. The reliance on fixed crossover spacing cannot accommodate the non-integer helical twist of DNA<sup>19, 20</sup>, which is approximately 10.44 base pairs per turn<sup>21, 22, 23</sup>. Furthermore, classical geometric approaches<sup>24</sup> often introduce unintended helical twist when mapping DNA onto non-planar curves. As a result, current frameworks support only planar arrangements of helices and break down when non-planar trajectories involving both curvature and twist are required. Yet such trajectories are essential for expanding the accessible design space, as many functional structures in biology and materials science exhibit non-planar three-dimensional curves and twisted geometries<sup>1, 2</sup>.

Here, we introduce an algorithm-driven design framework that enables the construction of DNA origami structures with helices arranged along non-planar trajectories in three-dimensional space. This approach allows precise crossover placement and promotes uniform strain distribution across helices while minimizing torsional stress. Our approach integrates three core elements (Fig. 1): a nucleotide positioning algorithm for non-planar 3D curves, automated computation of helix trajectories to cover target shapes, and structural optimization routines to relieve strain in the design. This algorithmic workflow—automating the most time-consuming aspects of curved DNA nanostructure design<sup>18</sup>—is fully integrated into our open-source software, ENSnano<sup>25, 26</sup>. Using this framework, we designed and assembled previously inaccessible nanostructures, including curved helical bundles in three-dimensional space, twist-rich hollow architectures, and nested spherical shells connected by narrow tunnels. These structures demonstrate key functional attributes—such as mechanical robustness, molecular encapsulation, compartmentalized reactivity, and channel-like features that regulate inter-compartmental transport and flow—commonly observed in biological systems.

As a direct application, we designed an artificial nanostructure that mimics the geometry and dimerization behavior of the Vault protein by generating a DNA origami nanocage with geometry automatically derived from the protein's atomic model in the Protein Data Bank (PDB). This biomimetic construct demonstrates how computational control over nanoscale curvature and twist enables the rational design of functional, bioinspired architectures that transcend the geometric constraints of conventional DNA origami.

## Results

**Algorithmic nucleotide positioning along arbitrary 3D curves.** Mapping a DNA helix onto a non-planar 3D curve requires defining a moving frame along the curve. The classical Frenet frame approach<sup>24</sup> fails in this context, as it rotates erratically under the influence of torsion in non-planar curves, introducing unwanted twist into the mapped helix. To address this, we developed a custom algorithm that computes a lazily rotating moving frame along the curve (algorithm SA1). We then use the computed frame to map the DNA helix onto the curve using the best-known structural parameters for DNA<sup>22</sup> (table ST1 and fig. S1). Figure S2 illustrates the algorithm in action, while Figure S3 compares the classical Frenet frame to our approach. The comparison demonstrates that our algorithm avoids unnecessary frame rotation, producing a minimally twisted DNA helix with a more regular twist, closer to its canonical value in the straight conformation.

We validated our DNA model by simulating a nucleosome, a key example of DNA curvature in biology. In the nucleosome, the DNA helix undergoes significant bending, wrapping approximately 1.65 turns around the histone octamer to form a left-handed superhelix<sup>27</sup>. Figure S4 compares crystallographic data (2CV5.PDB) with the output of our algorithm for a DNA helix wound around a nucleosome. Our model accurately reproduces nucleotide positions with an RMSD of 0.54 ± 0.23 nm, and phosphate with an RMSD of 0.44 ± 0.22 nm (figs. S5 and S6).

**DNA helix bundle non-planar curvature programming.** DNA helix bundles are a fundamental component of the DNA origami design toolbox. Early design strategies<sup>14, 18</sup> laid the groundwork for a wide range of applications, including nuclear pore complex mimics<sup>28</sup>, liposome templating<sup>29</sup>, membrane shaping<sup>11</sup>, spring-loaded nanodevices<sup>30</sup>, and targeted drug delivery systems<sup>13</sup>. Here we present an algorithm that
<figure><img src="image_3.png" /><figcaption>Figure 1: Algorithm-driven conceptual framework for DNA nanostructure design. (a) The target shape is defined using Bézier curves and geometric descriptors, such as the revolution surface tool. (b) The algorithm automatically computes helix trajectories that cover the target shape at an appropriate scale for the chosen scaffold strand, positioning nucleotides along their (possibly non-planar) 3D paths. A series of automated optimizations is applied to relieve structural strain. (c) The designer places scaffold and staple strand crossovers between helices. (d) Staple strand sequences are exported for assembly and imaging. Shown here is a negative-stain 2D TEM class average of the resulting self-assembled DNA nanostructure, box size is 80 × 80 nm.</figcaption></figure>

enables the design of curved helix bundles in full 3D space, with continuously varying radii of curvature.

In our approach, a DNA helix bundle is defined by attaching a grid (square or honeycomb) to a Bézier curve $t \mapsto M(t)$. Each helix, located at coordinates $(i, j)$, follows the path $t \mapsto P_{ij}(t)$ of the grid point at coordinates $(i, j)$ in the moving frame $F(t)$ centered on $M(t)$ (fig. S7). This construction ensures that the trajectories of the helices remain uniformly spaced along the entire length of the bundle. DNA helices are then mapped in real time onto their respective paths $P_{ij}(t)$ using the previously described DNA model. A core feature is that, since all helices in the bundle evolve along the common parameter $t$, we introduced an adaptive 2D view of the structure in ENSnano software. In this 2D representation, the width of each nucleotide's cell is automatically adjusted based on $t$, ensuring that all nucleotides at the same cross-section appear vertically aligned (fig. S8). As a result, nucleotides eligible for crossovers are naturally aligned in the 2D view, eliminating the need for manual insertions or deletions. For closed curves, the rise and helicity of each DNA helix are automatically adjusted to ensure an integer number of helical turns. In practice, these adjustments are minimal (only a few %) and remain well within established precision limits. This optimization helps

to evenly distribute mechanical strain along each DNA helix (figs. S9 to S12). All calculations are performed in real time, enabling the designer to interactively fine-tune the curve, either to precisely position functional groups or to ensure an *exact* match with the scaffold length.

Then, we proceed to position the crossovers by first placing a single crossover between adjacent helices. Next, we apply our *auto-roll* algorithm, which rotates neighboring helices to minimize crossover lengths. Once the helices are rotationally aligned, a standard scaffold and staple pattern is applied, with crossover positions visually adjusted in both 2D and 3D views of the design to optimize their length and orientation.

To explore the effectiveness of our method we designed a four-loop DNA origami structure, referred as *hex*-origami. The *hex*-origami consists of a 6-helix bundle programmed with four precise 270-degree looped bends (figs. S13 to S18). The crossover patterns were selected to minimize structural strain by 1) spacing the scaffold crossovers approximately every 100 base pairs, and 2) reducing the occurrence of double crossovers for the staples. The scaffold layout follows a “leaf” pattern (fig. S19), where each domain or “leaf” averages around 86 base pairs in length. Staples follow the classic C-shape (fig. S20), with domains typically ranging from 15 to 20 base pairs in straight regions. To ac-
commodate curvature, staple domains on the in-
ner side of the curve may be shortened by a full
turn while those on the outer side may be ex-
tended by a full turn – adjusting lengths by ap-
proximately 8 to 12 bps depending on curvature.
These modifications are guided by inspection of
the 3D nucleotide mapping in the interactive 3D
view. In our design, staple domains range from
7 to 30 base pairs, with staple strand lengths be-
tween 19 and 60 base pairs. Crossover lengths
range from 0.65 nm and 1.7 nm according to our
model. After optimizing folding conditions via
agarose gel electrophoresis and negative-stain
TEM (ns-TEM) particle analysis, optimal folding
was achieved by mixing the scaffold with sta-
ple oligonucleotides at a 10-fold molar excess, in
the presence of 20 mM MgCl₂, and following a
42-hour temperature annealing protocol (Fig. 2
and figs. S21 to S23). ns-TEM particle analy-
sis yielded 2D class averages that closely match
the intended four looped bends, each forming
a precise 270-degree angle, confirming success-
ful folding of the structure. To further assess
the fidelity of curvature programming conferred
by the global crossover pattern, we introduced
several structural variants: (1) opening a sec-
tion of the 6-helix bundle by splitting the sta-
ple strands joining the scaffold at its termini
(fig. S24), and (2) splitting the vertical staples
linking the top and bottom scaffold domains at
the four crossover points—designated NW, SW,
SE, and NE—to decouple the vertical connection
(fig. S25). Each variant yielded folded struc-
tures consistent with the degrees of freedom in-
troduced by the respective modifications, with
shape outcomes analogous to those achievable by
a flexible plumbing pipe making four 270° turns
(see section S2.5 and figs. S26 to S35).

**Optimization of helix torsion and trajectories in hollow shapes.** Designing DNA origami in capsule-like shapes is crucial for advancing nanotechnology, offering a protective barrier between the internal volume and the external environment. These 3D DNA nanostructures are finding applications in areas such as mimicking natural capsids for antigen presentation, virus trapping, low-volume reactors, templated assembly, molecular transport, and drug delivery³¹,¹³. Optimal functionality in these applications often depends on the creation of curved 3D hollow shapes. Previous methods for designing such structures typically involved slicing them into a stack of parallel concentric circles¹⁸,¹⁷. However, this approach has two main drawbacks¹⁸: First, the angles between adjacent layers are limited to the multiples of ~34°, corresponding to the angle between two consecutive nucleotides. Sec-

ond, the perimeter of each DNA ring must cor-
respond to an integer number of full DNA helix
turns. Moreover, this technique is only applicable
to simple revolution surfaces¹⁷.

One of the most challenging steps in hollow DNA origami design is determining the trajectories of the DNA helices. A small miscalculation in helix placement or torsion can lead to significant deviations in the structure, affecting both its stability and functionality. This task involves solving a multicriteria optimization problem that determines the appropriate number of helices and the scale of the shape, ensuring that the helices are uniformly spaced (with an average interdistance of 2.65 nm⁶,³²), minimally curved or twisted, and that their total length matches the target size of the scaffold. This step was highlighted as the most time-consuming in previous approaches¹⁸. To address this, we introduce a spiral-based algorithm for slicing a revolution surface (possibly twisted) into curved, equally wide stripes. This method overcomes both of the aforementioned limitations of using concentric circles in two key ways, as detailed in the supplementary text (section S3). By allowing the curvature radius to vary continuously along the trajectories, our method enables a continuous range of crossover angles between adjacent helices. Moreover, since only a few long spirals (only 2 in our designs) are needed to assemble the shape, adjusting the helix parameters by a few negligible % allows for continuous rescaling (figs. S36 to S38). This results in an automatic computation of the ideal scale for the target shape, optimizing the use of every single nucleotide in the scaffold. Finally, the use of spirals allows for seamless design of tubular portions with inner radii as small as 4.3 nm within a larger shape.

Our method is integrated in ENSnano software as an automatic pipeline. We proceed by specifying the surface and spirality parameters. The surface is given by its section curve, the number of twists, and its revolution radius. The spiral parameters include the number of spirals covering the surface, the *winding* parameter which dictates the rate at which the spirals coil around the surface and controls their inclination and torsion (figs. S39 and S40), and the target scaffold strand length. Our algorithm calculates the spiral trajectories for the helices using a dedicated spring system which keeps them as straight and evenly spaced as possible, minimizing their torsion along the way, as demonstrated in fig. S41. The algorithm scales the shape (section and revolution radius) until the distance between adjacent helices and the total helix length match the desired crossover distance and target scaffold length. The entire process is completed in a few
<figure><img src="image_4.png" /><figcaption>Figure 2: Spiral design algorithm for intricate hollow shapes. (a) Design workflow for a curved DNA origami composed of a 6-helix bundle with four programmed loops. In ENSnano, the bundle's Bézier path is first drawn and adjusted by the user, followed by algorithmic nucleotide mapping and crossover placement to produce the final assembly. (b) Design workflow for a half-twisted torus, defined in ENSnano as the surface generated by rotating a 2 × 1 ellipse by half a turn around the revolution axis. A spring-based system then optimizes helix trajectories and iteratively scales the surface until the total length matches the target scaffold length (here, 7560 bases), enabling automated nucleotide mapping and crossover placement. (c) Progressive increase in overall twisting complexity from T2 to T6, with schematic representations followed by algorithmic DNA helix routing, nucleotide mapping, and crossover placement. Below each TEM micrograph: Left: local torsion; Right: curvature radius along helices (color scales: 0 nm⁻¹ [pink] to 0.2 nm⁻¹ [dark blue] for torsion; and 4 nm [pink] to 30 nm [dark blue] for curvature). (d) Class averages from single particles observed with cryo-TEM and 3D reconstruction of T6 from 72 684 particles. For each structure, a 1% agarose gel (post-folding) and TEM micrographs are shown, alongside reference-free class averages from single particles (particle sets available in figs. S82, S86 to S88 and S92). White bars represent 200 nm; box sizes for projections and class averages are 112 × 112 nm for the four-loop DNA origami and 99 × 99 nm for the Tori.</figcaption></figure>

seconds.

Inspired by the prevalence of twisted features in nature³³, we investigated the robustness of our spiral routing algorithm by designing a series of six toroidal structures referred as

T1, ..., T6 with progressively increasing torsion. The surface of $T_k$ is obtained by rotating a 2-by-1 ellipse around a revolution axis while applying $k$ half-turn twists to the section, as depicted in Fig. 2. Our algorithm generated, for
each of them, a pair of spiraling helices that are evenly distributed across the surface, and whose total length precisely matches the target scaffold size (7560 nucleotides). By selecting appropriate winding parameters, we minimized the torsion of the resulting helices, yielding smooth and uniform trajectories throughout the structures (figs. S41 and S42). The resulting helices complete 7 (T5 and T6), 8 (T3 and T4), 9 (T2), and 10 coils (T1), while maintaining an interaxis spacing of 2.65 *nm* ± 10% relative to their counterparts.

As for the ᵇ-origami, we optimized the stable and scaffold crossover placements using the 3D and 2D live-editable views provided by our algorithms (figs. S43 to S78).

We studied the assembly of T1 by combining 20 nM scaffold with a 10-fold excess of staple oligonucleotides, varying magnesium concentrations (2–30 *mM*) over a 42-hour folding ramp. Agarose gel electrophoresis (fig. S79) identified 12 mM as the optimal magnesium concentration. ns-TEM particle analysis yielded 2D classifications in close agreement with the T1 structure model (Fig. 2). We optimized the self-assembly protocol for each structure T2 through T6, as detailed in the supplementary text. Agarose gel electrophoresis, ns-TEM particle analysis, and 2D class averaging revealed uniform particle distribution and orientation, consistent with simulated 2D projections, and demonstrating proper folding relative to the predicted shapes (Fig. 2 and figs. S80 to S92). We further validated the T6 design using cryo-TEM imaging, generating a 3D electron density map through single-particle cryo-TEM analysis (Fig. 2 and figs. S84 and S85). The map revealed the characteristic six twists per ring, confirming the structural integrity of the design.

**Algorithmic surface design with curvature inversion and tunnels.** We further tested the robustness of our algorithms by designing an additional structure, double sphere DS, which explores novel architectural possibilities. This design features two nested spheres in a convex-concave configuration, interconnected by narrow channels. The resulting structure includes a hermetic compartment between the inner and outer spheres, as well as an internal open compartment regulated by tunnel diameters.

Our algorithms computed evenly spaced, spiraling helices that span this structure, which were defined as the revolution surfaces generated by rotating a C-shaped section around an axis. DS consists of two twisted DNA helices forming an outer sphere (41 nm diameter) connected by two 10 nm-wide channels to an inner

sphere (29 nm diameter). The design was automatically adjusted to match the scaffold length (8,064 nucleotides) (Fig. 3 and figs. S93 to S98). The helices form 17 coils each. After optimizing folding conditions (see supplementary text and fig. S99), optimal assembly was achieved with 30 mM MgCl₂ over 42 hours, confirmed by agarose gel electrophoresis and ns-TEM analysis. 2D classifications revealed a homogeneous particle distribution, distinct contrast between spheres, and clear channel structures (Fig. 3 and fig. S100).

**Biomimetic DNA origami design.** We explored the bio-inspired applications of our algorithms, by designing a DNA origami that mimicks the shape of the vault protein, a barrel-shaped ribonucleoprotein complex found in the cytoplasm of eukaryotic cells. The vault protein can dimerize to form a closed cage³⁴,³⁵,³⁶. Its unique architecture and potential applications in targeted drug delivery and nanomaterial scaffolding have attracted considerable interest in nanotechnology³⁷.

To create the design, we began by modelling the cross-sectional curve of the vault protein PDB model 4V60 as a Bézier curve and applied our spiraling algorithm to generate the corresponding revolution surface. The shape was scaled automatically by a factor of 1.4 horizontally and 1.3 vertically, to match the target scaffold size of 8064 nucleotides, resulting in a structure 53 nm in width and 42 nm in height. Two inner hems were included at the top and the bottom of the structure to conceal the blunt ends of the helices and stabilize the 8.6 nm-wide tight hole at the top. The resulting two helices make 13 coils each. Additionally, the bottom of the design was equipped with six equally spaced 18-nucleotide anchor strands to facilitate homodimerization. The resulting design, referred as vault origami (figs. S101 to S106), was then validated through direct comparison with the Vault protein model, imported in our software as an STL file generated by Chimera from the PDB file (Fig. 4).

We optimized folding of the vault monomer using agarose gel electrophoresis and ns-TEM, identifying 20 mM MgCl₂ as optimal (fig. S107). As detailed in the supplementary text and figs. S108 to S112, optimal dimerization was achieved by incubating gel-purified monomers at 37°C for 15 hours, yielding dimer structures consistent with the design, as confirmed by ns-TEM and 2D class averaging (Fig. 4 and fig. S112).
<figure><img src="image_5.png" /><figcaption>Figure 3: Spiral-based design for spherical surfaces. (a) Schematic illustration and staple design of DS. (b) Left: curvature radius along the two twisted DNA helices in DS. Right: local torsion. Color scales: 4 nm [pink] to 30 nm [dark blue] for curvature and $0\ \text{nm}^{-1}$ [pink] to $0.2\ \text{nm}^{-1}$ [dark blue] for torsion. Graphs show the evolution of curvature radius (in nm) as a function of the number of turns along each helix. (c) 1% agarose gel after folding and TEM micrographs of the purified band (green star), with representative projections from 3D PDB model and reference-free class averages from single particles (particle sets available in fig. S100). Scale bar represents 200 nm.</figcaption></figure>

Discussion

Detailed analysis of our designs reveals that crossover spacing does not correlate with either local curvature radius or helix torsion, as shown in figs. S113 to S120 and further detailed in Supplementary Section S8. This observation highlights a critical limitation of classical rule-based design strategies$^{19,18}$, which rely on fixed patterning that cannot accommodate the geometric complexity of non-planar curved structures.

As DNA nanostructures evolve to encode increasingly sophisticated forms and functions, existing design constraints call for more advanced algorithmic strategies. Our algorithm-driven paradigm addresses this challenge by enabling non-planar curvature within DNA helix bundles and single-layer architectures, without requiring human intervention, thereby supporting greater

functional intricacy. It also enhances planar designs by reducing reliance on iterative post-experimental adjustments$^{20}$, and paves the way for the scalable assembly of larger functional structures with fewer components than multi-layered composites$^{12}$. This design framework is available through our open-source software, ENSnano$^{38}$, which integrates algorithmic control of helix geometry with intuitive 2D and 3D visualization, real-time editing, and a user-friendly interface. ENSnano is designed for continuous development, with planned extensions to support hybrid architectures—including DNA/RNA and DNA/protein systems—further bridging natural and engineered nanoscale assemblies. Finally, just as the design of a rectangle$^{21}$ in caDNAno allowed refine the pitch of straight helical DNA from 10–11 bp/turn to the now-standard 10.44 bp/turn, our framework provides
<figure><img src="image_6.png" /></figure>

Figure 4: Bio-inspired vault protein-like design. (A) Bézier curve aligned to the cross-sectional profile of the vault protein, PDB model 4V60. (B) Spiral-based method applied to the revolution surface of the vault monomer. (C) Staple design, 1% agarose gel after folding, and TEM micrographs with reference-free class averages from single particles (particle sets available in fig. S109). (D) Schematic illustration of the dimer assembly strategy, 1% agarose gel after dimerization, and TEM micrographs with reference-free class averages from single particles (particle sets available in fig. S112). (E) Curvature radius along helices. Color scale from pink (4 nm) to dark blue (30 nm). White bars represent 200 nm.

a design environment for investigating the phys-
ical principles, structural properties, and func-
tional behavior of curved and twisted DNA he-
lices—both as they occur in biological systems,
with implications for genome organization, gene
regulation, and the structural physics of nucleic
acids in health and disease, and as components
of engineered nanodevices.

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32 Thubagere, A. J. et al. A cargo-sorting dna robot. *Science* **357**, eeaan6558 (2017). URL https://www.science.org/doi/abs/10.1126/science.aan6558. https://www.science.org/doi/pdf/10.1126/science.aan6558.

33 Simien, J. M. & Haglund, E. Topological twists in nature. *Trends in Biochemical Sciences* **46**, 461–471 (2021).

34 Travis, J. The vault guy. *Science* **384**, 1058–1062 (2024).

35 Van Zon, A., Mossink, M. H., Scheper, R. J., Sonneveld, P. & Wiemer, E. A. C. The vault complex. *Cellular and Molecular Life Sciences (CMLS)* **60**, 1828–1837 (2003).

36 Berger, W., Steiner, E., Grusch, M., Elbling, L. & Micksche, M. Vaults and the major vault protein: novel roles in signal pathway regulation and immunity. *Cellular and molecular life sciences* **66**, 43–61 (2009).

37 Buehler, D. C., Toso, D. B., Kickhoefer, V. A., Zhou, Z. H. & Rome, L. H. Vaults engineered for hydrophobic drug delivery. *Small* **7**, 1432–1439 (2011).

38 Levy, N. & Schabanel, N. Ensnano: a 3D modeling software for DNA nanostructures. In *Proc. of DNA27-27th Int. Conf. on DNA Computing and Molecular Programming* (2021). URL https://www.ens-lyon.fr/ensnano/.

39 Rothemund, P. W. Folding DNA to create nanoscale shapes and patterns. *Nature* **440**, 297–302 (2006).
Materials and Methods

**DNA strands** Desalted staple strands were purchased from Integrated DNA Technologies (IDT) in water in 96-well plates at a concentration of 100 µM each. The staple strands were pooled and diluted with water to a working concentration of 400 nM each. The different scaffolds used (M13 8064, M13 7560, M13 7249) were purchased from Tilibit Nanosystems. The scaffold (5 nM to 20 nM) and staples ten times in excess were combined in a one-pot reaction with a buffer of 5 mM Tris-HCl, pH=8.0, 1mM EDTA, different concentrations of MgCl₂ depending on the origami (12 mM to 20 mM). The thermal annealing ramp was spanned over a time-period from 3.5 hours to 63 hours. The scaffold, concentrations of the components and thermal annealing ramps used for each origami (if not otherwise noted) are described in table T1.

**Agarose gel electrophoresis** DNA nanostructures visualized by Electron Microscopy were purified from 1% agarose gels (0.5X TBE, 45mM Tris-borate, 1mM EDTA, pH 8.3) supplemented with 11mM MgCl₂ and 0.5mg·ml⁻¹ Sybr SAFE. Samples were migrated on the gel for 3h with a running buffer of 0.5X TBE, 11mM MgCl₂ 2.85V·cm⁻¹ at room temperature. Bands corresponding to properly folded DNA origami were excised and transferred to a DNA gel extraction spin column (Merck, France) centrifuged at 5,000 g for 5 min at 4°C.

**Transmission Electron Microscopy - Negative staining** Purified origami were visualized by adsorption onto glow-discharged carbon-coated grid (Quantifoil Micro tools GmbH, Germany), stained for 60 s with a 2% (w/vol) aqueous uranyl acetate (Merck, France) solution, and then dried with ashless filter paper (VWR, France). Observations of EM grids were carried out on a JEM-1400Flash Tungsten microscope working at 120kV, equipped with a GATAN One View camera. Two-dimensional class averages were computed using EMAN2.

**Cryo-EM sample preparation** The purified and concentrated T6 samples were incubated 5 minutes twice to glow-discharged grids with an ultrathin carbon film supported by a 2/2 carbon film on a 200-mesh copper grid (Quantifoil Micro tools GmbH, Germany). Plunge freezing in liquid ethane was performed with a ThermoFisher Mk-IV Vitrobot with a blot time of 3 s, a blot force of 0 at 15°C and 100% humidity.

**Cryo-EM image acquisition** Selected grid of T6 was transferred under cryogenic conditions into the TFS Titan Krios TEM with acceleration voltage of 300 kV. Micrographs were acquired in two sessions on the Falcon4i camera under low-dose condition with a total flux of 40 e-/Å². Data were collected with underfocus values ranging from -0.7 µm to -4.1 µm in nominal resolution of 1.2 Å/px. Automated data acquisition was recorded into 40 individual frames. The forty-frame movies were then globally and locally (5x5 patches, 20% patch overlap) aligned to compensate for drift and beam-induced motion using program motioncor2⁴⁰. The resulting dose-weighted sum of aligned frames was used for subsequent image processing steps, except for estimating the contrast transfer function (CTF) parameters, which were estimated from aligned, non-dose weighted micrographs using the program ctfind4⁴¹, 42, 43.

**Cryo-EM image processing** Data processing was done using SCIPION 3.0.9 package⁴⁴. 15,127 dose-weighted micrographs of T6 were used for the processing. The SPHIRE-crYOLO⁴⁵ protocol was used for particle picking from a set of 3x binned micrographs. The particles were subjected to three rounds of RELION 2D classification⁴⁶, 47, 48, 49 to remove falsely picked grid contaminations and damaged particles, high-quality 2D class averages were selected and duplicated particles with Xmipp3-remove duplicates⁵⁰, 5¹ with 80px radius, leaving a set of 142,198 particles. Relion initial model⁴⁶, 47, 48, 49 was used for generating the initial 3D volume. First, a 3D refinement without any symmetry (and with the initial model filtered down to 120Å) was applied, producing a 3D map at 26Å and confirming the presence of the expected D6 symmetry. Following this initial refinement, a series of 3D refinements with progressively tightening masks and 3D classifications⁴⁶, 47, 48, 49 were applied to get a final set of 72,684 particles, subjected to a final 3D refinement⁴⁶, 47, 48, 49 with D6 symmetry, effectively reaching 13Å with a global mask.

**Dimerization of the Vault-like origami** Each Vault-like monomer was designed to display six anchor staples at its bottom to enable dimerization. Two different versions of the 6 anchor staples A and B were designed with 18-base parts complementary to each other. The sequences of these staples are described in tables T2 and T3. The sequences annealing inside the monomers are in black, the sequences designed to be out of the monomers are in red (two T and the sequence to enable the annealing of staples A
<table><caption>Table T1: Final folding conditions for each origami.</caption><thead><tr><th>Object</th><th>Denaturation phase (15 min) (°C)</th><th>Temperature ramp</th><th>Hold temperature (°C)</th><th>Folding ramp time period (h)</th><th>Magnesium chloride concentration (mM)</th><th>Scaffold</th><th>Scaffold concentration (nM)</th><th>Staples concentration (nM)</th></tr></thead><tbody><tr><td>%</td><td>65</td><td>60°C to 40°C: -1°C/2h</td><td>10</td><td>42</td><td>20</td><td>M13 7249</td><td>20</td><td>200</td></tr><tr><td>T1</td><td>65</td><td>60°C to 40°C: -1°C/2h</td><td>10</td><td>42</td><td>12</td><td>M13 7560</td><td>20</td><td>200</td></tr><tr><td>T2</td><td>65</td><td>1) 60°C to 55°C: -1°C/10min<br>2) 55°C to 25°C: -1°C/1h</td><td>25</td><td>33</td><td>18</td><td>M13 7560</td><td>20</td><td>200</td></tr><tr><td>T3</td><td>65</td><td>1) 60°C to 55°C: -1°C/10min<br>2) 55°C to 25°C: -1°C/1h</td><td>25</td><td>33</td><td>18</td><td>M13 7560</td><td>20</td><td>200</td></tr><tr><td>T4</td><td>65</td><td>1) 60°C to 55°C: -1°C/10min<br>2) 55°C to 25°C: -1°C/1h</td><td>25</td><td>33</td><td>18</td><td>M13 7560</td><td>20</td><td>200</td></tr><tr><td>T5</td><td>65</td><td>60°C to 40°C: -1°C/10min</td><td>10</td><td>3.5</td><td>18</td><td>M13 7560</td><td>5</td><td>50</td></tr><tr><td>T6</td><td>65</td><td>1) 60°C to 55°C: -1°C/10min<br>2) 55°C to 25°C: -1°C/1h</td><td>25</td><td>33</td><td>18</td><td>M13 7560</td><td>20</td><td>200</td></tr><tr><td>DS</td><td>65</td><td>60°C to 40°C: -1°C/2h</td><td>10</td><td>42</td><td>30</td><td>M13 8064</td><td>20</td><td>200</td></tr><tr><td>Vault</td><td>65</td><td>60°C to 40°C: -1°C/3h</td><td>10</td><td>63</td><td>20</td><td>M13 8064</td><td>20</td><td>200</td></tr></tbody></table>

<table><caption>Table T2: Version A of the six staples enabling the dimerization.</caption><tr><td>• Staple no. 1:</td><td>5' TTGTTTAACGTCACATT CATTACAAAATGAAAA TT AGTATATAGATAATTAAA 3'</td></tr><tr><td>• Staple no. 2:</td><td>5' TAACGAGCGTCGATTGCC TT AGTATATAGATAATTAAA 3'</td></tr><tr><td>• Staple no. 3:</td><td>5' CGACATTCAACCGATCACCGTCAC TT AGTATATAGATAATTAAA 3'</td></tr><tr><td>• Staple no. 4:</td><td>5' CGGGAACGGATAACCTCCAGTCACGACGTTGT GCCATTC TT AGTATATAGATAATTAAA 3'</td></tr><tr><td>• Staple no. 5:</td><td>5' CTGGTAATGGGTAAAGGTTTCTTTTTTCGCA TT AGTATATAGATAATTAAA 3'</td></tr><tr><td>• Staple no. 6:</td><td>5' ATGCCAACGGGAAAGGCTGGCCAGCACCGTCG TT AGTATATAGATAATTAAA 3'</td></tr></table>

<table><caption>Table T3: Version B of the staples enabling the dimerization.</caption><tr><td>• Staple no. 1: 5' TTTAATTATCTATATACT TT TAGCAGCAACTGAACACCCTGA 3'</td></tr><tr><td>• Staple no. 2: 5' TTTAATTATCTATATACT TT GTGGTGCGTTACCTGCA 3'</td></tr><tr><td>• Staple no. 3: 5' TTTAATTATCTATATACT TT AGGCTGCACTCCAGCCAGCTTTCCGGAACCCGTCGGATTCT 3'</td></tr><tr><td>• Staple no. 4: 5' TTTAATTATCTATATACT TT GTTCCGGCAAACGCGGTGTGTACATCGACATAA 3'</td></tr><tr><td>• Staple no. 5: 5' TTTAATTATCTATATACT TT CGACTTGACCGTAATCAGTAGCGACAGAATCA 3'</td></tr><tr><td>• Staple no. 6: 5' TTTAATTATCTATATACT TT CTCAATCCGCCG GAGCACATCCTCATAACAGCGGATCA 3'</td></tr></table>

with staples B, leading to the dimerization of the vault-like origami).

The monomers of the Vault-like origami with staples A and staples B were folded according to the conditions detailed in Table T1. Then, they were purified by agarose gel electrophoresis. Finally, they were assembled in equal concentration for 15 hours at 37°C with 100 mM MgCl₂.

## References for Material and Methods

³⁹ Mastronarde, D. SerialEM: A Program for Automated Tilt Series Acquisition on Tecnai Microscopes Using Prediction of Specimen Position. Microscopy and Microanalysis, 9(S02), 1182-1183. doi:10.1017/S1431927603445911 (2003)

⁴⁰ Zheng, S. Q. et al. MotionCor2: anisotropic correction of beam-induced motion for improved cryo-electron microscopy. Nat. Methods 14, 331–332 (2017)

⁴¹ Mindell, J. K.; Grigorieff, N. Accurate determination of local defocus and specimen tilt

in electron microscopy. J. Struct. Biol. 142/3, 334–347 (2003)

⁴² Rohou, A.; Grigorieff, N. CTFFIND4: Fast and accurate defocus estimation from electron micrographs. J. Struct. Biol. 192/2, 216–221 (2015)

⁴³ Grant, T.; Rohou, A.; Grigorieff, N. (2018) cistTEM, user-friendly software for single-particle image processing. eLife 7:e35383

⁴⁴ De la Rosa-Trevín, J. M. et al. Scipion: A software framework toward integration, reproducibility and validation in 3D electron microscopy. J. Struct. Biol. 195, 93–99 (2016)

⁴⁵ Wagner, T. et al. SPHIRE-crYOLO is a fast and accurate fully automated particle picker for cryo-EM. Commun Biol 2, 218 (2019)

⁴⁶ Zivanov, J. et al., New tools for automated high-resolution cryo-EM structure determination in RELION-3. Elife 7, (2018)

⁴⁷ Sjors H.W. Scheres, RELION: Implementation of a Bayesian approach to cryo-EM structure determination, Journal of Structural Biology,
Volume 180, Issue 3, 2012, Pages 519-530,
ISSN 1047-8477

48 Sjors H.W. Scheres, A Bayesian View on Cryo-EM Structure Determination, Journal of Molecular Biology, Volume 415, Issue 2, 2012, Pages 406-418, ISSN 0022-2836

49 D. Kimanius, B. O. Forsberg, S. HW. Scheres, E. Lindahl, Accelerated cryo-EM structure

50 De la Rosa-Trevín et al. Xmipp 3.0: An improved software suite for image processing in electron microscopy. J. Struct. Biol. 184 issue 2, 321-328 (2013)

51 Sorzano, C.O. et al. (2013). Semiautomatic, High-Throughput, High-Resolution Protocol for Three-Dimensional Reconstruction of Single Particles in Electron Microscopy. In: Sousa, A., Kruhlak, M. (eds) Nanoimaging. Methods in Molecular Biology, vol 950. Humana Press, Totowa, NJ. https://doi.org/10.1007/978-1-62703-137-0_11

### Acknowledgements:

**Fundings:** French National Research Agency grant : ANR-22-CE09-0034 (GB); French National Research Agency grant : ANR-22-CE45-0020 (NS); French National Research Agency grant : ANR-19-CE14-0035 (GB); French Centre National de la Recherche Scientifique: INS2I (NS); French Centre National de la Recherche Scientifique: MITI

(NS); French Infrastructure for Integrated Structural Biology (GB); French École Normal Supérieure de Lyon (NS); French IXXI Complex System Institute (NS).

**Author contributions:** Conceptualization: NS, NL, GB, JF, AM; Software: NL, NS, PM, JP, OH, DP; Algorithms: NS, NL; User Interface design: NS, NL, JP; Origami design: NS, NL, GB, JF, AM, GW; Experimental work: JF, AM, AK, GW, AA; Electron microscopy: JF, AM, AK, GW, AA; Data analysis: GB, JF, AM, AK, GW, AA; Visualization: JF, GB; Funding acquisition: GB, NS; Project administration: NS, GB; Writing – original draft: NL, JF, NS, GB; Writing – review & editing: NS, GB, JF, NL, AK, GW.

**Competing interests:** The authors have no competing interest.

**Code and data availability:** ENSnano source code is available in the public Github repository²⁶; the .ens and .xlsex files for the designs will be available on an open-source website at the time of publication.

**Thanks:** The authors thank Olivier Gandrillon at the LBMC lab (ÉNS de Lyon) for granting access to the lab bench, and Catherine Koerding and Élodie Vallin for experimental guidance. We thank Josephine Lai Kee Him for help with TEM measurements. We thank the Institut Pasteur in Paris and Eyen for access to TEM facilities and data acquisition.
Supplementary Materials for

Algorithm-driven DNA nanostructure design for advanced functionality

Nicolas Levy & Julie Finkel,

Allan Mills, Gerrit David Wilkens, Anjelica Kucinic, Aurélie Ancelin,

Pierre Marcus, Octave Hazard, Joris Picot, Daria Pchelina,

Gaëtan Bellot* and Nicolas Schabanel*

*Corresponding authors: nicolas.schabanel@ens-lyon.fr and gaetan.bellot@cbs.cnrs.fr
# Supplementary Information

## Table of Contents

S1 ENSnano geometric model to design curved DNA helix bundles, Table ST1, algorithm SA1, and figs. S1 to S12 S3

S1.1 Default DNA parameters used in ENSnano S3

S1.2 Curvature and torsion of a 3D curve S4

S1.3 Curved DNA model in ENSnano S5

S1.4 Adaptive moving frame algorithm S5

S1.5 Adaptive moving frame along a curve S7

S1.6 Comparison of ENSnano curved DNA helix model with the human nucleosome core particle 2CV5 crystallographic data S8

S1.7 Curved DNA bundles S11

S1.8 2D embedding of curved DNA bundles S12

S1.9 DNA parameters adjustment for closed curves S13

S2 A four-loops bundle, Figures S13 to S35 S18

S2.1 General design of the four-loops bundle S18

S2.2 Patterning of scaffold and staples S24

S2.3 Experimental validation of the design of the four-loops bundle S26

S2.4 Open/close and junctions design variants S29

S2.5 Study of different conformations of the four-loops bundle depending on the design S31

S3 Twisted revolution surfaces: model, parameters and helix routing algorithm, Figures S36 to S42 S41

S3.1 Initial helix spiraling route S41

S3.2 Helix routing and scaling algorithm S43

S3.3 The winding parameter S45

S3.4 Torsion of the resulting helices as a function of the winding parameter S46

S3.5 Impact on the helix trajectory torsion of the helix routing and scaling algorithm S47

S3.6 Torsion of the helix trajectories for all designs S48

S4 Experimental validation of spiral-based design for hollow shaped toroidal surfaces, Figures S43 to S92 S49

S4.1 Tori designs S49

S4.2 Tori experimental validation S85

S5 Double-sphere, Figures S93 to S100 S99

S5.1 Double-sphere DS design S99

S5.2 Double-sphere DS experimental validation S105

S6 Vault-like origami, Figures S101 to S112 S107

S6.1 Vault-like origami design S107

S6.2 Vault-like origami experimental validation S113

S7 Summary of all the designs, Figure S113 S119

S8 Distribution of staple strand crossover interdistances, Figures S114 to S121 S120
# S1 ENSnano geometric model to design curved DNA helix bundles, Table ST1, algorithm SA1, and figs. S1 to S12

## S1.1 Default DNA parameters used in ENSnano

The 3D positions of Nucleotides are computed in ENSnano using six parameters as shown in fig. S1: the helix radius *R*, the rise *Δ*, the inclination *I*, the helicity *χ*, the minor groove angle *α* and the inter-helices axis distance *H*. The default value of these parameters in table ST1 are based on three papers: the crystallographic data from²² for *R*, *Δ*, *I*, and *α*; data from²¹ for *α* and *χ*; and data from³² for *H*. Phosphate positions parameters from²² are used to compute the nucleotides positions parameters by placing the nucleotides in the middle of two consecutive phosphates: as a consequence, the nucleotide inclination is set to $I_N = I_P - 2 \times (\Delta/2) = 0.375 - 0.332 = 0.043$ nm and the minor groove angle is set to $\beta_N = \beta_P - 2 \times (\alpha/2) = 170.4^{\circ} - 34.48^{\circ} = 135.92^{\circ}$. The inter-helix axis distance is set to $H = 8\Delta = 2.65$ nm based on the double-layer origami data from³² as the inter-helix distance from one layer corresponds to the distance spanned by 8 bases pair in the other layer. This value is moreover consistent with the average inter-helix distance reported in³⁹.

Note that the most important parameters for DNA origami design are *Δ*, *α*, *χ* and *H* as any error on them will accumulated over the span of the design (as opposed to *R*, *β*, and *I*). As a fortunate consequence, they are thus the easiest to adjust if needed, as any error on them would be easily detected in experimental data. Finally, recall that it is always possible to impose your own DNA parameters in the ENSnano .ens file, as these may change with your own experimental setting (buffer,...).

<table><caption>Table ST1: DNA double-helix parameters used in ENSnano: the nucleotides positions are deduced from the phosphate positions as measured in²² and the DNA parameters used in²¹,³².</caption><thead><tr><th></th><th>Phosphate positions<sup>22</sup></th><th>Nucleotide positions</th></tr></thead><tbody><tr><td>Radius</td><td colspan="2">R = 0.93 nm</td></tr><tr><td>Rise</td><td colspan="2">Δ = 0.332 nm</td></tr><tr><td>Inclination</td><td>I = 0.375 nm</td><td>I = 0.043 nm</td></tr><tr><td>Twist (α) and Helicity (χ)</td><td colspan="2">α = <sup>360°</sup>⁄<sub>χ</sub> = 34.48° with χ = 10.44 bp/turn</td></tr><tr><td>Minor groove angle</td><td>β = 170.4°</td><td>β = 135.92°</td></tr><tr><td>Inter-helices axis distance</td><td colspan="2">H = 2.65 nm</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>Figure S1: Illustration of DNA double helix parameters.</figcaption></figure>
937 S1.2 Curvature and torsion of a 3D curve

Given a 3D curve $t \mapsto M(t)$, we denote by $\dot{M}(t)$, $\ddot{M}(t)$ and $\dddot{M}(t)$ the 1st, 2nd and 3rd derivatives of $M(t)$ with respect to $t$. The curvature at $M(t)$ is defined as:

$$ \gamma(t) = \frac{\|\dot{M}(t) \wedge \ddot{M}(t)\|}{\|\dot{M}(t)\|^3} $$

Its inverse $R(t) = 1/\gamma(t)$ is the curvature radius at $M(t)$, that is the radius of its osculating circle, which is the circle best approximating the curve at $M(t)$. The torsion $\tau(t)$ at $M(t)$ measures how fast the curve takes off from the plane of its osculating circle. It is given by the formula:

$$ \tau(t) = \frac{\det[\dot{M}(t), \ddot{M}(t), \dddot{M}(t)]}{\|\dot{M}(t) \wedge \ddot{M}(t)\|^2} $$

938 It is measured in $\text{nm}^{-1}$.

939 Curvature and torsion at $M(t)$ characterizes the osculating helix, that is the helix best matching the curve locally at $M(t)$. Torsion is thus a natural parameter to measure when dealing with DNA helices. Refer to$^{24}$ for full details.

940
941
942
943
944
945 As DNA helices are not continuous mathematical curves but discrete, made of nucleotides organised in helicoidal configuration, we use the following discrete version of the torsion. If $M(i)$ be the positions of consecutive nucleotides in a DNA strand, we define the 1st, 2nd, and 3rd discrete derivatives at $M(i)$ as:

$$ \Delta M(i) = M(i + 1) - M(i), $$

$$ \Delta^2 M(i) = \Delta M(i + 1) - \Delta M(i) = M(i + 2) - 2M(i + 1) + M(i), $$

$$ \text{and } \Delta^3 M(i) = \Delta^2 M(i + 1) - \Delta^2 M(i) = M(i + 3) - 3M(i + 2) + 3M(i + 1) - M(i). $$

The discrete torsion is then defined as:

$$ \frac{\Delta \tau(i)}{\|\Delta M(i) \wedge \Delta^2 M(i)\|^2} = \frac{\det[\Delta M(i), \Delta^2 M(i), \Delta^3 M(i)]}{\|\Delta M(i) \wedge \Delta^2 M(i)\|^2} $$

The discrete torsion of a straight DNA helix is constant to:

$$ \frac{\Delta}{\tau_{\text{straight DNA helix}}} \approx 0.4851 \text{ nm}^{-1} $$

946
947

Next, we will use discrete torsion to evaluate the quality of our algorithms mapping DNA onto arbitrary 3D curves.
### S1.3 Curved DNA model in ENSnano

Our geometric curved DNA helix model relies on the definition of a moving frame $F(t) = (\mathbf{X}(t), \mathbf{Y}(t), \mathbf{Z}(t))$ that adaptively follows the intended curve $t \mapsto M(t)$ (Fig. 1A). The i-th nucleotides from the forward and backward strands are placed around the curve using the frame $F(t)$. Their positions $N_i^f$ and $N_i^b$ are computed from the helix axis points $M(t_i^f)$ and $M(t_i^b)$ located at curvilinear distances $i \Delta$ and $(I + i \Delta)$ from the origin of the curve, where $\Delta$ and $I$ are the rise and inclination of the DNA B-form helix$^{22}$ using the following formula:

$$N_i^f = M(t_i^f) + R \cos(\rho + i \alpha) \mathbf{X}(t_i^f) + R \sin(\rho + i \alpha) \mathbf{Y}(t_i^f)$$

$$N_i^b = M(t_i^b) + R \cos(\rho + i \alpha + \beta) \mathbf{X}(t_i^b) + R \sin(\rho + i \alpha + \beta) \mathbf{Y}(t_i^b)$$

where $s(t) = \int_{0}^{t} ||\dot{M}(t)|| dt$ is the curvilinear abscissa (with $\dot{M}(t) = \frac{dM(t)}{dt}$),

and $t_i^f = s^{-1}(i \Delta)$ and $t_i^b = s^{-1}(I + i \Delta)$ are the parameters $t$ for the projections of the i-th nucleotides (forward and backward) on the curve,

and $R$, $\alpha$, $\beta$, and $\rho$ are respectively the radius, the helical twist, the narrow groove angle, and the roll of the DNA helix (table ST1 and fig. S1).

Several approaches exist for defining a frame $F(t)$ along a curve. The classically used is the Frenet frame, which captures important geometrical properties of the curve such as the tangent, the normal, the osculating plane, etc. However, the Frenet frame tends to undergo sudden rotations in order to adapt to the osculating plane$^{24}$. Using this frame leads to unrealistically twisted DNA helices, ignoring the prevalence of the base-pair stacking interactions which tends to enforce a constant twist around the DNA helix axis (fig. S3).

We thus introduce a lazily adaptive frame $F(t) = (\mathbf{X}(t), \mathbf{Y}(t), \mathbf{Z}(t))$ that rotates as little as possible along the curve. $\mathbf{Z}(t)$ is the normalized tangent to the curve at $M(t)$ for all $t$. $\mathbf{X}(t)$ and $\mathbf{Y}(t)$ are built incrementally along the curve at each of the required locations $t_i^f, t_i^b$ according to algorithm SA1.

### S1.4 Adaptive moving frame algorithm

We denote by Normalized($U$) = $U/||U||$ the normalized vector of a non-zero vector $U$. For two non-colinear vectors $u$ and $v$, we denote by rot$_{u \to v}$ the rotation mapping $u$ to $v$ around the axis perpendicular to both $u$ and $v$ (i.e., around the cross product $u \wedge v$ of $u$ and $v$).

Algorithm SA1 constructs the adaptive frames used to embed DNA helices in 3D along a given curve in ENSnano.

<caption>**Algorithm SA1:** Lazily adaptive frame along a parameterized curve $M(t)$</caption>

**Input:** A parameterized curve $t \mapsto M(t)$ and a set $T = \{t_i^f\} \cup \{t_i^b\}$ of the parameters where we need a frame
**Output:** An adaptive right-handed frame $F(t) = (\mathbf{X}(t), \mathbf{Y}(t), \mathbf{Z}(t))$ for $t \in T$.

/* Initialization */
sort $T = \{t_1 < \dots < t_{2n}\}$
forall $k = 1..2n$ do
| set $\mathbf{Z}(t_k) := \text{Normalized}(\dot{M}(t_k))$ /* $\mathbf{Z}(t_k)$ is the normalized tangent */
$\mathbf{X}(t_1)$ and $\mathbf{Y}(t_1)$ are set arbitrarily so that $(\mathbf{X}(t_1), \mathbf{Y}(t_1), \mathbf{Z}(t_1))$ is a right-handed frame, e.g.: choose
$U \in \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ such that $U$ is not colinear to $\mathbf{Z}(t_1)$, and set
$\mathbf{Y}(t_1) := \text{Normalized}(\mathbf{Z}(t_1) \wedge U)$ and $\mathbf{X}(t_1) := \mathbf{Y}(t_1) \wedge \mathbf{Z}(t_1)$
/* Adaptive frame computation */
for $k = 2, 3, \dots, 2n$ do
| set $\rho_k = \text{rot}_{\mathbf{Z}(t_{k-1}) \to \mathbf{Z}(t_k)}$ /* the rotation mapping $\mathbf{Z}(t_{k-1})$ to $\mathbf{Z}(t_k)$ */
set $\mathbf{X}(t_k) := \rho_k(\mathbf{X}(t_{k-1})$ and $\mathbf{Y}(t_k) := \rho_k(\mathbf{Y}(t_{k-1})$ /* rotate the frame according to $\mathbf{Z}(t_k)$ */
<figure><img src="image_8.png" /><figcaption>Figure S2: The moving frames $\mathbf{F}_k = (\mathbf{X}_k, \mathbf{Y}_k, \mathbf{Z}_k)$ is computed iteratively along the curve (yellow dots) such that $\mathbf{Z}_k$ is tangent to the curve. The nucleotides of the DNA helix following this curve are then positioned according to this moving frames.</figcaption></figure>
S1.5 Adaptive moving frame along a curve

As shown in fig. S3, our algorithm SA1 produces a smoother embedding of the DNA double helix em-
bedding than the classic Frenet frame. Indeed the Frenet frame is very sensitive to any local change in
curvature and tends to rotate erratically as can be observed in fig. S3(left), inducing unwanted torsion in
the DNA helix mapping, while our adaptive frame keeps a consistent orientation along the whole curve,
minimizing the overall torsion along the DNA strand as can be observed in the lower part of the figure.

<figure><img src="image_9.png" /><figcaption>Figure S3: Frenet vs adaptive moving frame along a curve: the curve is shown in red and the computed frames at each nucleotide position are shown in green (X) and cyan (Y) (Z is not shown for clarity as it is always tangent to the curve). At the bottom: the values of the discrete torsion of the two forward and backward DNA strands in both models; in bold, the moving average taken over a full turn of each DNA strand shows that the iterative frame produces a smoother torsion whose value is closer to the torsion of a straight DNA helix.</figcaption></figure>
S1.6 Comparison of ENSnano curved DNA helix model with the human nucleosome core particle 2CV5 crystallographic data

Figures S5 and S6 evaluate the discrepancy between the DNA strands 1) as placed in space by ENSnano algorithm using the trajectory of the axis of the DNA helix in the human nucleosome core particle crystallographic data PDB-2CV5, and 2) as placed in the PDB file. Figure S6 compares the positions of the phosphates in both ENSnano model and crystallographic data. Figure S6 compares the positions of the nucleotides in both ENSnano model and crystallographic data. We observe a discrepancy of 0.444±0.219 nm for the phosphates and 0.540 ± 0.231 nm for the nucleotides. Note that the DNA helix in the nucleosome core particle is constrained to adopt its shape by its interactions with the histones at its core, interactions which are not taken into account by ENSnano which relies on the sole trajectory of the helix to place the DNA strands.

<figure><img src="image_10.png" /><figcaption>Figure S4: Curved DNA helix model generated by ENSnano from the crystallographic data of nucleosome. A) Crystallographic data PDB-2CV5; B) DNA helix axis trajectory *T* (in pink) deduced from the PDB file; C) Curved DNA helix generated by ENSnano from the trajectory *T* compared to the crystallographic data PDB-2CV5.</figcaption></figure>
Nucleotides positions in ENSnano and PDB files
(Nucleotide=(C3'+C4')/2) (roll = 151.00°) (RMSD: 0.540 ± 0.231 nm)

<figure><img src="image_11.png" /></figure>

Distance in nm between nucleotides in ENSnano and PDB files
(Nucleotide=(C3'+C4')/2) (roll = 151.00°) (RMSD: 0.540 ± 0.231 nm)

<figure><img src="image_12.png" /><figcaption>Figure S5: Our curved DNA helix model vs crystallographic data of nucleosome.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Distance in nm between phosphates in ENSnano and PDB files
(roll = 134.50°) (RMSD: 0.444 ± 0.219 nm)</figcaption></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

Figure S6: Our curved DNA helix model vs crystallographic data of nucleosome.
S1.7 Curved DNA bundles

Curved DNA bundle are defined in ENSnano by attaching a grid to a Bezier curve as shown in fig. S7. The frames computed along the curve by algorithm SA1 are used to orient the grid. The grid can be either squared or honeycomb and has edges of length *H* corresponding the target inter-helix axis distance (2.65 nm by default). The trajectory of each helix in the bundle simply follows the trajectory of the corresponding point in the grid along the curve (fig. S7B). To avoid self-collision of the origami design, each grid can be moved vertically at the control points of the Bézier curve as we did in our ℏ-origami (fig. S7D).

<figure><img src="image_16.png" /><figcaption>Figure S7: Grids-based routing of helices in curved DNA bundles: A) the helices' positions on the honeycomb grid; B) the trajectories of each helices is deduced by translating every point of the curve according to the grid; C) the resulting DNA bundle; D) grid can be moved vertically at Bézier control points, for instance to avoid self-collision.</figcaption></figure>
S1.8 2D embedding of curved DNA bundles

In ENSnano, DNA bundle helices trajectories are computed by translating the points of a Bézier curve according to the grid. It follows that we can use the grid to determine which nucleotides belongs to the same section of the DNA bundle and use this information to automatically align them in the 2D view as shown in fig. S8: even if the helices have very different lengths, their nucleotides are aligned across them without the need for the user to add insertion nor deletion as one would do in caDNAo. ENSnano proceeds by automatically adapting the size of the cell of each nucleotide in the 2D view: enlarging/shortening it when the helix is shorter/longer respectively.

<figure><img src="image_17.png" /><figcaption>Figure S8: Automatic insertion- and deletion-less 2D embedding of curved DNA bundles: the circles mark nucleotides on helices 2 and 4 and their automatic positioning in the 2D view. Observe that even if helices 2 and 4 have very different lengths, their nucleotides belonging to the same section of the bundle are placed in the same section in the 2D view.</figcaption></figure>
S1.9 DNA parameters adjustment for closed curves

Closed (curved) DNA helices require special care as their strands need to wrap upon themselves, see fig. S9. DNA bundles require furthermore adjustment and will be handled in the next section. Single closed DNA helices are handled in a three steps process that aims at spreading the constraints evenly over the whole curve:

1. *Tune the rise*: the first step consists in tuning minimally the rise so that the length of curve corresponds to an integral number of base pairs. Let $L$ be the length of the Bézier curve. Let $N = \lfloor L/\Delta \rfloor$ be the rounded number of base pairs fitting into the curve. We set $\tilde{\Delta} = L/N$ be the adjusted rise of DNA for this curve. Note that $\tilde{\Delta}$ differs from $\Delta$ by at most $\Delta/2N$ which is a few ‰ as soon as $N \gtrsim 200$.

2. *Compute the frames*: Use algorithm SA1 to compute the frames at every position and map the DNA strands along the curve accordingly.

3. *Tune the helicity*: the last step consists in aligning the angle of both ends of the strands, see fig. S11. Let $\Theta \in [-\pi, \pi)$ denotes the angle that the virtual $N+1$-th nucleotide makes with the 1st of the strand. We set $\tilde{\alpha} = \alpha - \Theta/N$ so that the angle difference gets compensated little by little over the whole length of the strand bringing both ends together. Note that, again, the relative difference between $\tilde{\alpha}$ and $\alpha$ is less that $\pi/N$ which accounts for a few ‰ as soon as $N$ is larger than a few hundred. Equivalently, we set $\tilde{\chi} = 1/(1/\chi - \Theta/N)$ so that $2\pi N/\tilde{\chi} = 2\pi N/\tilde{\chi} - \Theta$.

This process is motivated by the fact that we believe that the constraints on the rise (optimized first) are tighter than the constraints on the twist (optimized second) which can be absorbed by the DNA flexibility as was demonstrated by the variety of origamis which were designed an assembled successfully with helicity ranging from 10.2 to 10.8 base pairs per turn. Note that these adjustments are below the currently known precision on these parameters as soon as $N \gtrsim 500$, that is for strands of $\gtrsim 150$ nucleotides-long.
<figure><img src="image_18.png" /><figcaption>Figure S9: Rounding closed curve's length, twist so that both of its ends match</figcaption></figure>
DNA helix bundles require furthermore adjustments of the parameters because in order for both ends of shifted curves to match, we need to align also the initial and final frames as shown in fig. S10. We proceed by modify Step 2 as follows:

2'. *Compute and adjust the frames for the Bézier curve supporting the DNA bundle:* Use algorithm SA1 to compute the frames at every position. Let $\Theta \in [-\pi, \pi)$ be the angle between the virtual $N+1$-th frame and the 1st. For $i = 1..N$, rotate the $i$-th frame $F_i$ by $-i \times \Theta/N$ around its Z axis to absorb $\Theta$ (see fig. S10). Interpolate using piecewise Bézier curves or Chebyshev polynomials the trajectories of the grid-points by which passes the helices of the bundle, as illustrated in fig. S10 (bottom). Finally, run the three steps in the section above for each of these trajectories as shown in fig. S11.

Note that each helix of the closed-curved bundle has its own adjusted rise and twist (or helicity). The resulting helix 3D embedding is shown in fig. S12.

<figure><img src="image_19.png" /><figcaption>Figure S10: The case of closed curve bundle: 1) Adapt the DNA helix' rise so that the curve's length corresponds to an integer number of base pairs, then compute the iterative frames and rotate them progressively and minimally so that both of initial and final frames match</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Figure S11: The case of closed curve bundle: 2) Adapt the helicity parameter minimally so that the ends of both strands align.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>Figure S12: The case of closed curve bundle: 3) Finally, compute the frames and adapt the helix parameters as before for each shifted helix independently, resulting in closed DNA helices respecting the desired inter distance everywhere.</figcaption></figure>
## S2 A four-loops bundle, Figures S13 to S35

### S2.1 General design of the four-loops bundle

<figure><img src="image_22.png" /><figcaption>Figure S13: 3D view of the ℏ-origami design, line representation. A: Back view of the full design and side views of the four junctions (North-West, South-West, North-East, South-East). The staples involved in the junctions are displayed in helix representation (in black for North-West, red for South-West, green for North-East and blue for South-East) ; B: Bottom view. Each line represents a double helix of DNA. Transversal bars represent crossovers. The staples shown here are the core staples of the origami, the staples involved in the four junctions (NW, NE, SW, SE) and the staples of the origami in the close state.</figcaption></figure>
A

Back

<figure><img src="image_23.png" /></figure>

C

<figure><img src="image_24.png" /></figure>

B

Bottom

<figure><img src="image_25.png" /></figure>

89 nm

<figure><img src="image_26.png" /></figure>

14 nm

<caption>Figure S14: 3D view of the ℮-origami design, helix representation. A: Back view; B: Bottom view. Each ball represents a deoxynucleotide. The staples shown here are the core staples of the origami, the staples involved in the four junctions (NW, NE, SW, SE) and the staples of the origami in the close state; C: Dimensions of the ℮-origami. Top view (up), side view (bottom). Diameter of 89 nm (red circle), height of 14 nm.</caption>
<figure><img src="image_27.png" /><figcaption>Figure S15: 3D view of the ℏ-origami design, helix (left) and line (right) representations. Bottom views of the six helices trajectories (top), the scaffold and staples together (middle up), the scaffold alone (middle down) and the staples alone (bottom). Transversal bars on the line representation represents a crossover. The staples shown here are the core staples of the origami, the staples involved in the four junctions (NW, NE, SW, SE) and the staples of the origami in the close join state.</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>Figure S16: 2D view of the ℥-origami design. Scaffold and staples. The staples shown here are the core staples of the origami, the staples involved in the four junctions (NW, NE, SW, SE) and the staples of the origami in the close state.</figcaption></figure>
<figure><img src="image_29.png" /><figcaption>Figure S17: 2D view of the $\mathcal{G}$-origami design. Scaffold only.</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>Figure S18: 2D view of the $\mathcal{G}$-origami design. Staples only. The staples shown here are the core staples of the origami, the staples involved in the four junctions (NW, NE, SW, SE) and the staples of the origami in the close state.</figcaption></figure>
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## S2.2 Patterning of scaffold and staples

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In the ℥-origami design, we used the classic "leaf-shaped" scaffold consisting of several copies of the same leaf-pattern as illustrated in fig. S19

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<figure><img src="image_31.png" /><figcaption>Figure S19: Leaf-shaped scaffold principle used in the ℥-origami.</figcaption></figure>
In all our designs, we favored the C-shaped staples pattern as illustrated in fig. S20.

<figure><img src="image_32.png" /><figcaption>Figure S20: C-shaped staples principle used in every one of our origamis (illustrated here with the ℥-origami).</figcaption></figure>
S2.3 Experimental validation of the design of the four-loops bundle

<figure><img src="image_33.png" /><figcaption>Figure S21: Gel analysis of the influence of MgCl₂ concentration and different folding ramps on folding quality of the 𝄽-origami. The foldings were analyzed by agarose gel electrophoresis (1%). L: 1kb DNA ladder.</figcaption></figure>
<figure><img src="image_34.png" /><figcaption>Figure S22: Transmission Electron Microscope (TEM) micrographs of purified ℥-origami folded with 20 mM MgCl₂ with different folding ramps. A: ℥-origami folded with the following folding ramp: 65°C during 15 minutes, 60°C to 40°C -1°C/10 minutes, 10°C Hold; B: ℥-origami folded with the following folding ramp: 65°C during 15 minutes, 60°C to 40°C -1°C/2h, 10°C Hold. Particles not folded in the full ℥-origami are circled in black. Scale bar is 200 nm.</figcaption></figure>
<figure><img src="image_35.png" /><figcaption>Figure S23: Transmission Electron Microscope (TEM) micrographs of typical particles of the ℘-origami in optimal folding conditions. Each box is 132 nm x 132 nm.</figcaption></figure>
S2.4 Open/close and junctions design variants

<figure><img src="image_36.png" /><figcaption>Figure S24: 2D view of the ℏ-origami design. View of the close state design (left) and the open state design (right). The open-T state design is the same as the open state design with extra poly-Ts added to prevent stacking. The scaffold is represented in black, staples in colors. For each representation, a horizontal bold black line splits the 2D view into two parts corresponding to the two ends of the bundle.</figcaption></figure>
<figure><img src="image_37.png" /><figcaption>Figure S25: 3D and 2D views of the junctions of the ℥-origami design. A: 3D side views of the four junctions with line representation (North-West, South-West, North-East, South-East). The staples involved in the junctions are displayed in helix representation (in black for North-West, red for South-West, green for North-East and blue for South-East); B: 2D views of the junctions' designs. The staples in bold red are the staples involved in the junctions. For each representation, a horizontal bold black line splits the 2D view into two parts corresponding to the two layers of the ℥-origami that we can see in the 3D view; C: 2D views of the design of the junctions' locations when the junction is not made. The staples in bold red are the staples involved in the junctions, without making crossovers between the layers in this case. For each representation, a bold black line splits the 2D view into two parts corresponding to the two layers of the ℥-origami that we can see in the 3D view.</figcaption></figure>
S2.5 Study of different conformations of the four-loops bundle depending on the design

<figure><img src="image_38.png" /><figcaption>Figure S26: Gel analysis of the influence of the presence of junctions on the folding of the ᵃ-origami. The foldings were analyzed by agarose gel electrophoresis (1%). L: 1kb DNA ladder ; CnoJ: ᵃ-origami folded in the close state without the junctions NW, NE, SW, SE ; NoSW: ᵃ-origami folded in the close state without the junction SW but with the junctions NW, NE, SE ; NoSE: ᵃ-origami folded in the close state without the junction SE but with the junctions NW, NE, SW ; NoNW: ᵃ-origami folded in the close state without the junction NW but with the junctions SW, NE, SE ; NoNE: ᵃ-origami folded in the close state without the junction NE but with the junctions NW, SW, SE ; CJ: ᵃ-origami folded in the close state with the junctions SW, NW, NE, SE.</figcaption></figure>
<figure><img src="image_39.png" /><figcaption>Figure S27: Various structural conformations expected using plumber pipes linked with 3/4-turns corners.</figcaption></figure>
<figure><img src="image_40.png" /><figcaption>Figure S28: Realization of various shapes observed in Transmission Electron Microscope (TEM) micrographs of purified ✠-origami using plumber pipes linked with 3/4-turns corners. ✠-origami folded in the close state, without the junctions NE, SE, NW, SW. Box size of the TEM micrographs = 100 nm.</figcaption></figure>

Interestingly, we observed that some origami folded in the full ✠-shape even though only one of the four junctions was present, with a better yield when using the staples involved in the NW junction alone than with the other junctions (figs. S29 and S30).

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<figure><img src="image_41.png" /><figcaption>Figure S29: Gel analysis of the influence of the presence of junctions on the folding of the 𝄽-origami. The foldings were analyzed by agarose gel electrophoresis (1%). L: 1kb DNA ladder ; None: 𝄽-origami folded in the close state without the junctions NW, NE, SW, SE ; All: 𝄽-origami folded in the close state with the junctions SW, NW, NE, SE ; NW: 𝄽-origami folded in the close state with the junction NW and without the junctions SW, NE, SE ; NE: 𝄽-origami folded in the close state with the junction NE and without the junctions SW, NW, SE ; SW: 𝄽-origami folded in the close state with the junction SW and without the junctions NW, NE, SE ; SE: 𝄽-origami folded in the close state with the junction SE and without the junctions SW, NE, NW.</figcaption></figure>
<figure><img src="image_42.png" /><figcaption>Figure S30: Transmission Electron Microscope (TEM) micrographs of purified 𝄽-origami folded in the close state with only one junction out of four. A: 𝄽-origami folded with the junction NE and without the junctions SE, NW, SW; B: 𝄽-origami folded with the junction SE and without the junctions NE, NW, SW; C: 𝄽-origami folded with the junction SW and without the junctions SE, NW, NE; D: 𝄽-origami folded with the junction NW and without the junctions SE, NE, SW. Particles folded in the full 𝄽-origami are circled in black. Scale bar is 200 nm.</figcaption></figure>

In the same way, we observed on agarose gel electrophoresis as well as with negative-stain transmission electron microscopy that the proportion of particles folded in the full 𝄽-origami with the SE, SW and NE but not the NW junctions is less than when the origami is folded using any other triplet of junctions (figs. S26 and S31). These results together suggest that the NW junction staples stabilize the curved-shape better than the other junctions and is enough to stabilize the curvature in the full 𝄽-shape.

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A

<figure><img src="image_43.png" /></figure>

B

<figure><img src="image_44.png" /></figure>

C

<figure><img src="image_45.png" /></figure>

D

<figure><img src="image_46.png" /></figure>

Figure S31: Transmission Electron Microscope (TEM) micrographs of purified 𝄽-origami folded in the close state with three junctions out of four. A: 𝄽-origami folded without the junction NE and with the junctions SE, NW, SW ; B: 𝄽-origami folded without the junction SE and with the junctions NE, NW, SW ; C: 𝄽-origami folded without the junction SW and with the junctions SE, NW, NE ; D: 𝄽-origami folded without the junction NW and with the junctions SE, NE, SW. Scale bar is 200 nm.
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Additionally, when trying to fold the 𝄽-origami with only two junctions out of four, we observe that the 𝄽-origami can reach its full 𝄽-shape, with a better yield when using opposite junctions (e.g. NW and SE or NE and SW) rather than adjacent ones (e.g. NW and NE or NW and SW) (figs. S32 and S33). This can be logically explained by the fact that opposite junctions are in contact with the four sides while adjacent ones are only in contact with three.

<figure><img src="image_47.png" /><figcaption>Figure S32: Gel analysis of the influence of the presence of junctions on the folding of the 𝄽-origami. All the 𝄽-origami were folded in the close state. The foldings were analyzed by agarose gel electrophoresis (1%). L: 1kb DNA ladder.</figcaption></figure>
<figure><img src="image_48.png" /></figure>

Figure S33: Transmission Electron Microscope (TEM) micrographs of purified ✠-origami folded in the close state with two junctions out of four. A and B displays ✠-origami with two adjacent junctions whereas C and D displays ✠-origami with two opposite junctions. A: ✠-origami folded with the junction NW and SW and without the junctions NE and SE; B: ✠-origami folded with the junction NW and NE and without the junctions SW and SE; C: ✠-origami folded with the junction NE and SW and without the junctions NW and SE; D: ✠-origami folded with the junction NW and SE and without the junctions NE and SW. Particles folded in the full ✠-origami are circled in black. Scale bar is 200 nm.
<figure><img src="image_49.png" /><figcaption>Figure S34: Gel analysis of the influence of the presence of junctions and open, open-T or close state on the folding quality of the ℮-origami. The foldings were analyzed by agarose gel electrophoresis (1%). L: 1kb DNA ladder. CJ: ℮-origami folded in the close state with the four junctions: NW, NE, SW, SE ; OJ: ℮-origami folded in the open state with the four junctions: NW, NE, SW, SE ; OTJ: ℮-origami folded in the open-T state with the four junctions: NW, NE, SW, SE ; CNJ: ℮-origami folded in the close state without any of the four junctions: NW, NE, SW, SE ; ONJ: ℮-origami folded in the open state without any of the four junctions: NW, NE, SW, SE ; OTNJ: ℮-origami folded in the open-T state without any of the four junctions: NW, NE, SW, SE.</figcaption></figure>
<figure><img src="image_50.png" /></figure>

Figure S35: Transmission Electron Microscope (TEM) micrographs of purified 🟧-origami folded in different states with the four junctions NW, NE, SW and SE. A: 🟧-origami folded in the open state ; B: 🟧-origami folded in the open-T state. Particles not folded in the full 🟧-origami are circled in black. Scale bar is 200 nm.
## S3 Twisted revolution surfaces: model, parameters and helix routing algorithm, Figures S36 to S42

### S3.1 Initial helix spiraling route

Consider a continuously differentiable closed curve $S: u \in [0, 1] \mapsto S(u)$ in the plane $(XZ)$. Let us denote by $\text{rot}_\theta^U$ denotes the rotation of angle $\theta$ around the U-axis. Consider a twist $\tau \in \mathbb{N}$. Let $S_\tau^R(t, u) = \text{rot}_{2\pi t}^Z(R \cdot X + \text{rot}_{\pi\tau t}^Y(S(u)))$ denotes the image of point $S(u)$ rotated by $\tau t$ half-turns around the Y-axis, then translated by $R$ along the X-axis and finally rotated by $t$ turn around the Z-axis. The $\tau$-twisted revolution surface $S_\tau^R$ of $S$ with revolution radius $R$ is defined as the set of points, see fig. S36:

$$S_{\tau}^{R} = \{S_{\tau}^{R}(t, u) : t, u \in [0, 1]\}$$

When $\tau = 0$, $S_0^R$ is simply the (untwisted) revolution surface with revolution radius $R$. When $\tau$ is even or when $\tau$ is odd and $S$ has a half-turn symmetry, $S_\tau^R$ is closed and continuously differentiable. The section $S_\tau^R(t)$ of the twisted revolution surface at $t$ turn (or, equivalently, at angle $2\pi t$) refers to the set:

$$S_{\tau}^{R}(t) = \{S_{\tau}^{R}(t, u) : u \in [0, 1]\}$$

The designs considered in this paper correspond to:

• For the tori T1, . . . , T6: $S(u) = (2 \cos(2\pi u), 0, \sin(2\pi u))$ is the (2, 1)-ellipse and $\tau$ ranges from 1 to 6;

• For the double-sphere DS: $S$ corresponds to the $\mathbb{S}$ curve and $\tau = 0$;

• For the vault: $S$ corresponds to the profile of the vault protein and $\tau = 0$.

Let $P$ denotes the perimeter of $S$:

$$P = \int_{u} ||\dot{S}(u)|| du, \text{ where } \dot{S}(u) = dS(u)/du$$

We denote by $s: \mathbb{R} \to \mathbb{R}$ the curvilinear abscissa of $S$:

$$s(x) = \int_{u=0}^{x} ||\dot{S}(u)|| du, \text{ such that } P = s(1)$$

<figure><img src="image_51.png" /><figcaption>Figure S36: Initial helices routing.</figcaption></figure>
and by $\sigma = s^{-1}$ its reciprocal, such that $\sigma(P) = 1$.

The initial spiral helix routing consists in routing $2n$ equally-spread helices per section according to a spiraling pattern. The number of helices is chosen to be even for allowing Given a revolution radius $R$, a section scale $\gamma > 0$, a winding parameter $w \in \mathbb{Z}$, the initial trajectories $(\Gamma_i)$ passing through $2n$ helices per section are defined:

$$ \Gamma_i(t) = \gamma \cdot S_{\tau}^{(R/\gamma)}(t, \sigma((i + wt)P/2n)), \text{ for } t \in \mathbb{R} \text{ and } i = 0, 1, \dots $$

The principle is the following: on the initial section ($t = 0$ turn) the $2n$ helices are equally spread along the section, separated by $P/2n$ measured along the section (before $\gamma$-rescaling). Then the distance measured along the section between them is kept constant but their position within the current section is shifted linearly as they rotate around the revolution axis, so that when a full turn is completed ($t = 1$), they have moved by an integral number of $P/2n$ (namely $(i + w)$ times $P/2n$), so that they connect to one of the initial helix, resulting into a set of spirals covering the whole target surface. The way the helices are connected together depends on the winding parameter $w$. Precisely: when $\tau$ is even, Helix $i$ connects to Helix $(i + w) \mod 2n$ after a full turn; whereas Helix $i$ connects to Helix $(i + n + w) \mod 2n$ after a full turn when $\tau$ is odd. It follows that Spiral $i$ is the concatenation of all one-turn helix segment indexed in $(i + w\mathbb{Z}) \mod 2n$ when $\tau$ is even, and by $(i + (n + w)\mathbb{Z}) \mod 2n$ when $\tau$ is odd. If $k = \gcd(2n, w)$ when $\tau$ is even, and $k = \gcd(2n, n + w)$ when $\tau$ is odd,<fnref n="1" /> then there are exactly $k$ spirals making each exactly $2n/k$ turns around the revolution axis. Note that if we want $k$ to be even, for instance to ensure the existence of a proper scaffold routing, then we need $w$ to be even if $\tau$ is even, and $w$ to have the same parity as $n$ if $\tau$ is odd.

The winding parameter not only influences the number of spirals and their number of turns around the revolution axis, but also the curvatures of the helices and their fit to the surface. This influence will be explored in the next section. Finally, one could think that the section scale $\gamma$ should be set to $2nH/P$ so that the inter-spiral distance $\gamma P/2n = H = 2.65$ nm equals the desired inter-DNA helix axis distance. However, as this inter-spiral distance is measured along the section, it does not reflect to the real distance between the neighboring spirals and it needs to be adjusted to the angle taken by the spirals against the sections, which in turns depends on $\gamma$ and $w$. This is why we need a special rescaling algorithm detailed in the following section S3.2.

<footnote n="1">¹where $\gcd(a, b)$ denotes the greatest common divisor of integers $a$ and $b$.</footnote>
## S3.2 Helix routing and scaling algorithm

As mentioned earlier, when the surface goes through abrupt changes of curvature, the helices hit the section with a tight angle and their distance along the section must be adjusted. This is for instance the case at the entrance of the tunnels in the double sphere designs or in the most twisted tori (T3 and above). As shown in fig. S38, the distance between consecutive helices measured along the sections must be increased when the helices hit the section with a tight angle. As a consequence, we need to scale up the sections accordingly. This in turn requires to adjust the revolution radius, until a proper set of parameter (section scale, revolution radius) is found so that the total helix length matches the target scaffold length. We have designed a special algorithm to solve this non-linear optimization system. Our algorithm relies on a coarse-grained finite element system mapped according to *m* equally spaced sections, typically *m* ~ 20–200, see fig. S37. Each of the *k* helices, each making *q* = 2*n*/*k* turns, is broken into *qm* segments joining as many masses sliding along the *m* sections. Within each section, the masses are linked by springs whose empty lengths ($l_{ij}^0$) are adaptively adjusted so that the segments corresponding to the helices are equally spaced at distance *H* (by default *H* = 2.65 nm) from each other as explained next. In order to ensure that the helices stay as straight as possible, we add an angular spring around each mass so that the angle between its two adjacent helix segments ($\beta$ in fig. S37) is pushed toward 180°.

<figure><img src="image_52.png" /><figcaption>Figure S37: The coarse-grained finite element system used to optimize the helix trajectories.</figcaption></figure>

**Our algorithm** proceeds as illustrated in fig. S38. It consists in two alternating phases: section scaling and revolution radius scaling, see fig. S38-C. During the section scaling, the revolution radius is fixed and the section are rescaled until the helices are *H* appart from each other. In order to do so, we let the masses converge to an equilibrium while adaptively adjusting the empty lengths of the springs to take into account the current incident angles of both segments of helices with the section: using the notations in fig. S38-C2, we want the distance *d* between two neighboring masses to verify that *d* cos α₁ ≈ *H* ≈ *d* cos α₂; at every step of the simulation, we thus adjust the empty length of each spring to $l_{ij}^0 := (H/\cos α₁ + H/\cos α₂)/2$ and rescale the sections accordingly so that their perimeters are on average equal to the sum of the empty length of their springs. When the equilibrium is reached, we turn to the revolution radius phase: as illustrated in fig. S38-C2, the algorithm uses Chebyshev interpolation to compute a continuous trajectory for each of the *k* helices, compute their lengths in base pairs (rounding to the nearest integer) If the total length equals to the target scaffold length, the process ends. Otherwise the revolution radius is adjusted (reduced if the total length is higher, increased otherwise) and the algorithm goes back to the section rescaling phase.

As the process may not converge to the exact length of the scaffold, the user may interrupt the process at any time and the algorithm enforce the total length to the length of the scaffold by tuning *a* *minima*
the rise of the *k* helices. This is useful without any harm when the process bounces between two integers
surrounding the target value (e.g. between 7248 and 7250 when the target length is 7249).

<figure><img src="image_53.png" /><figcaption>Figure S38: Helix routing and scaling algorithm</figcaption></figure>
S3.3 The winding parameter

The winding parameter controls the coiling and super-coiling of the spirals. Indeed, when Helix *i* completes a full turn, it completes $τ/2 + w/2n$ turns (super-coils) around the section axis. As shown in fig. S39, adjusting this parameter allows to select a set of initial trajectories that fits the surface while minimizes their torsion and curvature, and thus hopefully maximizing their foldability. ENSnano allows to tune this parameter to find the best winding parameter suited for your target surface.

<figure><img src="image_54.png" /><figcaption>Figure S39: Influence of the winding parameter on the coiling, supercoiling and curvature of the helix spiraling trajectories in a virtual design with $12 = 2n$ helices per section. Setting $w = +3$ produces the smoothest trajectories. Curvature color ranges from 5 nm to 20 nm.</figcaption></figure>
S3.4 Torsion of the resulting helices as a function of the winding parameter

Figure S40 shows the average absolute value of the torsion measured along the resulting helix trajectories for the designs DS, and T1 to T6 for different values of the winding parameters (the closest possible around the chosen value for our designs). Observe that the winding parameter chosen for the origami design (marked with a red star) is the one minimizing the torsion along the helices. Observe as well that the torsion increases in designs T1 to T6 allowing to challenge our design paradigm with increasing curvature and torsion complexities.

<figure><img src="image_55.png" /><figcaption>Figure S40: Average absolute torsion measured in nm⁻¹ along the different helix trajectories as a function of the winding parameter *w* for the designs DS, and T1 to T6. The value of the winding parameter is indicated on top of each point. The chosen winding parameter is indicated by a red star for each design.</figcaption></figure>
1154

1155

## S3.5 Impact on the helix trajectory torsion of the helix routing and scaling algorithm

1156

1157

1158

1159

1160

1161

Figure S41 shows the evolution of the average absolute torsion (in nm⁻¹) of the helices trajectories computed during the execution of the helix routing and scaling algorithm (fig. S38). We observe that the angular springs (fig. S37) allow to reduce significantly the torsion of the helix trajectories during the first steps of the algorithm during which the inter-helix distances are balanced. The torsion then stabilizes while the algorithm finely tunes the shape scale to match the target scaffold length in the remaining steps.

<figure><img src="image_56.png" /></figure>

Figure S41: Evolution of the average torsion along the helices during the execution of the helix routing and scaling algorithm (fig. S38) for the designs DS and T1 to T6 with the chosen winding parameters. For every design, the torsion drops during the first stages of the algorithm during which the inter-helix distance are balanced and the trajectories are "rectified" and then the torsion remains quasi-constant during the second phase of the algorithm where the shape scale is finely tuned to reach the target scaffold length.
S3.6 Torsion of the helix trajectories for all designs

Figure S42 shows the absolute value of the torsion measured along each helix trajectory for all the designs. Observe that the torsion in the vault and double-sphere DS reaches its peak in the tunnels where the helices sustain the highest slope change. Observe that the torsion increases significantly from T1 to T6, demonstrating that DNA helices sustains more and more strain from T1 to T6 (mostly in the inner circle).

<figure><img src="image_57.png" /><figcaption>Figure S42: Absolute value of the torsion along each helix trajectory for all designs. Scale bar are uniform and range from 0.0 nm⁻¹ (blue) to 0.2 nm⁻¹ (purple).</figcaption></figure>
# S4 Experimental validation of spiral-based design for hollow shaped toroidal surfaces, Figures S43 to S92

## S4.1 Tori designs

### S4.1.1 Torus I design

<figure><img src="image_58.png" /><figcaption>Figure S43: 3D view of the T1 design, line representation. A: Top view and cross-section; B: side view. Each line represents a double helix of DNA. Transversal bars represent crossovers.</figcaption></figure>
<figure><img src="image_59.png" /><figcaption>Figure S44: 3D view of the T1 design, helix representation. A: Top view and cross-section; B: side view. Each ball represents a deoxynucleotide; C: Dimensions of the T1 origami. Top view (up), side view (bottom). External diameter of 57 nm (red circle), internal diameter of 20 nm (yellow circle), height of 24 nm.</figcaption></figure>
<figure><img src="image_60.png" /><figcaption>Figure S45: 3D view of the T1 design, helix (left) and line (right) representations. Views of the two spirals (top), the scaffold and staples (middle up), the scaffold (middle down) and the staples (bottom). Transversal bars on the line representation represents a crossover.</figcaption></figure>
<figure><img src="image_61.png" /><figcaption>Figure S46: 2D view of the T1 design. Scaffold and staples.</figcaption></figure>
<figure><img src="image_62.png" /><figcaption>Figure S47: 2D view of the T1 design. Scaffold only.</figcaption></figure>
<figure><img src="image_63.png" /><figcaption>Figure S48: 2D view of the T1 design. Staples only.</figcaption></figure>
S4.1.2 Torus 2 design

<figure><img src="image_64.png" /><figcaption>Figure S49: 3D view of the Torus 2 design, line representation. A: Front view and cross-section; B: side view. Each line represents a double helix of DNA. Transversal bars represent crossovers.</figcaption></figure>
A

<figure><img src="image_65.png" /></figure>

C

<figure><img src="image_66.png" /></figure>

External diameter 1: 65 nm

External diameter 2: 54 nm

Internal diameter 1: 30 nm

Internal diameter 2: 20 nm

B

<figure><img src="image_67.png" /></figure>

<figure><img src="image_68.png" /></figure>

Figure S50: 3D view of the Torus 2 design, helix representation. A: Front view and cross-section; B: side view. Each ball represents a deoxynucleotide; C: Dimensions of the T2 origami. Front view (up), side view (bottom). External diameter 1 of 65 nm (red circle), external diameter 2 of 54 nm (yellow circle), internal diameter 1 of 30 nm (green circle), internal diameter 2 of 20 nm (blue circle), height of 22 nm.
<figure><img src="image_69.png" /><figcaption>Figure S51: 3D view of the T2 design, helix (left) and line (right) representations. Views of the two spirals (top), the scaffold and staples together (middle up), the scaffold (middle down) and the staples (bottom). Transversal bars on the line representation represents a crossover.</figcaption></figure>
<figure><img src="image_70.png" /><figcaption>Figure S52: 2D view of the Torus 2 design. Scaffold and staples.</figcaption></figure>
<figure><img src="image_71.png" /><figcaption>Figure S53: 2D view of the Torus 2 design. Scaffold only.</figcaption></figure>
<figure><img src="image_72.png" /><figcaption>Figure S54: 2D view of the Torus 2 design. Staples only.</figcaption></figure>
<figure><img src="image_73.png" /><figcaption>Figure S55: 3D view of the Torus 3 design, line representation. A: Front view and cross-section; B: side view. Each line represents a double helix of DNA. Transversal bars represent crossovers.</figcaption></figure>
<figure><img src="image_74.png" /><figcaption>Figure S56: 3D view of the Torus 3 design, helix representation. A: Front view and cross-section; B: side view. Each ball represents a deoxynucleotide; C: Dimensions of the T3 origami. Front view (up), side view (bottom). External diameter 1 of 67 nm (red circle), external diameter 2 of 58 nm (yellow circle), internal diameter 1 of 36 nm (green circle), internal diameter 2 of 27 nm (blue circle), height of 20 nm.</figcaption></figure>
<figure><img src="image_75.png" /><figcaption>Figure S57: 3D view of the T3 design, helix (left) and line (right) representations. Views of the two spirals (top), the scaffold and staples (middle up), the scaffold (middle down) and the staples (bottom). Transversal bars on the line representation represents a crossover.</figcaption></figure>
<figure><img src="image_76.png" /><figcaption>Figure S58: 2D view of the Torus 3 design. Scaffold and staples.</figcaption></figure>
<figure><img src="image_77.png" /><figcaption>Figure S59: 2D view of the Torus 3 design. Scaffold only.</figcaption></figure>
<figure><img src="image_78.png" /><figcaption>Figure S60: 2D view of the Torus 3 design. Staples only.</figcaption></figure>

<figure><img src="image_79.png" /></figure>
<figure><img src="image_80.png" /><figcaption>Figure S61: 3D view of the Torus 4 design, line representation. A: Front view and cross-section; B: side view. Each line represents a double helix of DNA. Transversal bars represent crossovers.</figcaption></figure>
<figure><img src="image_81.png" /><figcaption>Figure S62: 3D view of the Torus 4 design, helix representation. A: Front view and cross-section; B: side view. Each ball represents a deoxynucleotide; C: Dimensions of the T4 origami. Front view (up), side view (bottom). External diameter 1 of 66 nm (red circle), external diameter 2 of 56 nm (yellow circle), internal diameter 1 of 33 nm (green circle), internal diameter 2 of 23 nm (blue circle), height of 21 nm.</figcaption></figure>
<figure><img src="image_82.png" /><figcaption>Figure S63: 3D view of the T4 design, helix (left) and line (right) representations. Views of the two spirals (top), the scaffold and staples (middle up), the scaffold (middle down) and the staples (bottom). Transversal bars on the line representation represent a crossover.</figcaption></figure>
<figure><img src="image_83.png" /><figcaption>Figure S64: 2D view of the Torus 4 design. Scaffold and staples.</figcaption></figure>
<figure><img src="image_84.png" /><figcaption>Figure S65: 2D view of the Torus 4 design. Scaffold only.</figcaption></figure>
<figure><img src="image_85.png" /><figcaption>Figure S66: 2D view of the Torus 4 design. Staples only.</figcaption></figure>
S4.1.5 Torus 5 design

<figure><img src="image_86.png" /><figcaption>Figure S67: 3D view of the Torus 5 design, line representation. A: Front view and cross-section; B: side view. Each line represents a double helix of DNA. Transversal bars represent crossovers.</figcaption></figure>
<figure><img src="image_87.png" /><figcaption>Figure S68: 3D view of the Torus 5 design, helix representation. A: Front view and cross-section; B: side view. Each ball represents a deoxynucleotide; C: Dimensions of the T5 origami. Front view (up), side view (bottom). External diameter 1 of 68 nm (red circle), external diameter 2 of 59 nm (yellow circle), internal diameter 1 of 38 nm (green circle), internal diameter 2 of 29 nm (blue circle), height of 19 nm.</figcaption></figure>
<figure><img src="image_88.png" /><figcaption>Figure S69: 3D view of the T5 design, helix (left) and line (right) representations. Views of the two spirals (top), the scaffold and staples (middle up), the scaffold (middle down) and the staples (bottom). Transversal bars on the line representation represents a crossover.</figcaption></figure>
<figure><img src="image_89.png" /><figcaption>Figure S70: 2D view of the Torus 5 design. Scaffold and staples.</figcaption></figure>
<figure><img src="image_90.png" /><figcaption>Figure S71: 2D view of the Torus 5 design. Scaffold only.</figcaption></figure>
<figure><img src="image_91.png" /><figcaption>Figure S72: 2D view of the Torus 5 design. Staples only.</figcaption></figure>
<figure><img src="image_92.png" /><figcaption>Figure S73: 3D view of the Torus 6 design, line representation. A: Front view and cross-section; B: side view. Each line represents a double helix of DNA. Transversal bars represent crossovers.</figcaption></figure>
<figure><img src="image_93.png" /><figcaption>Figure S74: 3D view of the Torus 6 design, helix representation. A: Front view and cross-section; B: side view. Each ball represents a deoxynucleotide; C: Dimensions of the T6 origami. Front view (up), side view (bottom). External diameter 1 of 68 nm (red circle), external diameter 2 of 59 nm (yellow circle), internal diameter 1 of 37 nm (green circle), internal diameter 2 of 28 nm (blue circle), height of 20 nm.</figcaption></figure>
<figure><img src="image_94.png" /><figcaption>Figure S75: 3D view of the Torus 6 design, helix (left) and line (right) representations. Front views of the two spirals (top), the scaffold and staples (middle up), the scaffold (middle down) and the staples (bottom). Transversal bars on the line representation represents a crossover.</figcaption></figure>
<figure><img src="image_95.png" /><figcaption>Figure S76: 2D view of the Torus 6 design. Scaffold and staples.</figcaption></figure>
<figure><img src="image_96.png" /><figcaption>Figure S77: 2D view of the Torus 6 design. Scaffold only.</figcaption></figure>
<figure><img src="image_97.png" /><figcaption>Figure S78: 2D view of the Torus 6 design. Staples only.</figcaption></figure>
1177

## S4.2 Tori experimental validation

1178

### S4.2.1 Torus I experimental validation

<figure><img src="image_98.png" /></figure>

Figure S79: Gel analysis of the influence of MgCl₂ concentration on folding quality of the T1 origami. The foldings were analyzed by agarose gel electrophoresis (1%). L: 1kb DNA ladder; Folding ramp: 65°C during 15 min, 60°C to 40°C -1°C/2h, 10°C Hold.
S4.2.2 Tore VI experimental optimization and validation

To determine good self-assembly conditions, we first focused on optimizing nanostructure T6. Agarose gel electrophoresis analysis of the assembly conditions revealed an optimal assembly at 18 mM with an annealing ramp over 33 hours (figs. S80 and S81). ns-TEM analysis of the gel purified T6 origami and two-dimensional class averages confirmed the fully formed origami according to the design and 2D projection simulations (figs. S82 and S83). We validated the accuracy of the designed structure by generating a 3D electron density map through single-particle cryo-TEM (Fig. 2 and figs. S84 and S85) which revealed the distinctive six twists per ring features within the resolution obtained.

<figure><img src="image_99.png" /><figcaption>Figure S80: Gel analysis of the influence of MgCl₂ concentration on folding quality of the Tore 6. The foldings were analyzed by agarose gel electrophoresis (1%). L: 1kb DNA ladder; a: 65°C during 15 min, 60°C to 40°C -1°C/10minutes, 10°C Hold; b: 65°C during 15 min, 60°C to 40°C -1°C/1h, 10°C Hold.</figcaption></figure>
<figure><img src="image_100.png" /></figure>

Figure S81: Gel analysis of the influence of different folding ramps on folding quality of the Tore 6. The foldings were analyzed by agarose gel electrophoresis (1%). L: 1kb DNA ladder; a: 65°C during 15 min, 60°C to 40°C -1°C/5minutes, 10°C Hold; b: 65°C during 15 min, 60°C to 40°C -1°C/10minutes, 10°C Hold; c: 65°C during 15 min, 65°C to 55°C -1°C/10minutes, 55°C to 25°C -1°C/1h, 25°C Hold; d: 65°C during 15 min, 65°C to 50°C -1°C/10minutes, 50°C to 25°C -1°C/1h, 25°C Hold; e: 80°C during 15 min, 80°C to 55°C -1°C/10minutes, 55°C to 25°C -1°C/1h, 25°C Hold; f: 80°C during 15 min, 80°C to 25°C -1°C/10minutes, 25°C Hold.
<figure><img src="image_101.png" /><figcaption>Figure S82: Transmission Electron Microscope (TEM) micrographs of typical particles of the Tore 6. Each box is 99 nm x 99 nm.</figcaption></figure>
<figure><img src="image_102.png" /><figcaption>Figure S83: Tore 6 projections from 3D PDB model. Observed particles and representative projections from 3D PDB model of the Tore 6 showing different orientations from TEM micrographs. Each box is 99 nm x 99 nm.</figcaption></figure>
<figure><img src="image_103.png" /><figcaption>Figure S84: Cryo-EM images analysis for Tore 6. A: Exemplary micrograph of a total of 15,127 movie stacks. Scale bar = 100 nm. B: Representative 2D Class averages.</figcaption></figure>
<figure><img src="image_104.png" /><figcaption>Figure S85: Cryo-EM map determination for Tore 6. A: Histogram representing the angular distribution of particles. B: FSC plot.</figcaption></figure>
1187 S4.2.3 Tore II to Tore IV experimental validation

1188 This optimized self-assembly protocol was employed for three additional structures (T2 to T4) (figs. S86

1189 to S88 and had to be optimized for T5 (figs. S89 to S92).

<figure><img src="image_105.png" /><figcaption>Figure S86: Transmission Electron Microscope (TEM) micrographs of typical particules of the Tore 2. Each box is 99 nm x 99 nm.</figcaption></figure>
<figure><img src="image_106.png" /><figcaption>Figure S87: Transmission Electron Microscope (TEM) micrographs of typical particules of the Tore 3. Each box is 99 nm x 99 nm.</figcaption></figure>
<figure><img src="image_107.png" /><figcaption>Figure S88: Transmission Electron Microscope (TEM) micrographs of typical particles of the Tore 4. Each box is 99 nm x 99 nm.</figcaption></figure>
1190 **S4.2.4 T5: optimization of folding conditions and experimental validation**

1191 **Optimization of the folding conditions of the T5 origami (figs. S89 to S92).** Despite four different tested folding ramps with a scaffold concentration of 20 nM and staples oligonucleotides in a 10-fold excess, the gel analysis of the T5 showed some aggregates in the well, with different MgCl₂ concentrations and folding ramps (figs. S89 to S92).

1192
1193
1194

<figure><img src="image_108.png" /><figcaption>Figure S89: Gel analysis of the influence of MgCl₂ concentration on folding quality of the Tore 5. The foldings were analyzed by agarose gel electrophoresis (1%). L: 1kb DNA ladder ; a: 65°C during 15 min, 60°C to 40°C -1°C/10minutes, 10°C Hold ; b: 65°C during 15 min, 60°C to 40°C -1°C/1h, 10°C Hold.</figcaption></figure>
<figure><img src="image_109.png" /></figure>

Figure S90: Gel analysis of the influence of the folding ramp on folding quality of the Tore 5 with 18 MgCl₂. The foldings were analyzed by agarose gel electrophoresis (1%). L: 1kb DNA ladder; a: 65°C during 15 min, 60°C to 40°C -1°C/2minutes, 10°C Hold; b: 65°C during 15 min, 60°C to 40°C -1°C/5minutes, 10°C Hold; c: 65°C during 15 min, 60°C to 40°C -1°C/10minutes, 10°C Hold.
1195

1196

1197

1198

1199

To improve the folding's yield, we tested different concentrations of scaffold and staples oligonucleotides. Agarose gel electrophoresis analysis showed that while we still had a well defined monomeric band with 5 nM of scaffold, we had less aggregates in the well, indicating a better yield of the origami at this scaffold concentration. These results were especially observed with a folding ramp over a 3.5 hours period rather than with one over 33 hours (fig. S91).

<figure><img src="image_110.png" /></figure>

Figure S91: Gel analysis of the influence of scaffold M13 7560 and staples concentration on folding quality of the Tore 5. The foldings were analyzed by agarose gel electrophoresis (1%).. L: 1kb DNA ladder; a: 65°C during 15 min, 60°C to 40°C -1°C/10minutes, 10°C Hold; b: 65°C during 15 min, 65°C to 55°C -1°C/10minutes, 55°C to 25°C -1°C/1h, 25°C Hold.
<figure><img src="image_111.png" /><figcaption>Figure S92: Transmission Electron Microscope (TEM) micrographs of typical particles of the Tore 5. Each box is 99 nm x 99 nm.</figcaption></figure>
1200

S5 Double-sphere, Figures S93 to S100

1201

S5.1 Double-sphere DS design

<figure><img src="image_112.png" /><figcaption>Figure S93: 3D view of the DS design, line representation. A: Front view ; B: side view. Each line represents a double helix of DNA. Transversal bars represent crossovers.</figcaption></figure>
<figure><img src="image_113.png" /><figcaption>Figure S94: 3D view of the DS design, helix representation. A: Front view ; B: side view. Each ball represents a deoxynucleotide ; C: Dimensions of the DS origami. Front view (up), side view (bottom). External diameter of 41 nm (red circle), internal diameter of 29 nm (yellow circle), tunnel diameter of 10 nm (green circle).</figcaption></figure>
<figure><img src="image_114.png" /><figcaption>Figure S95: 3D view of the DS design, helix (left) and line (right) representations. Views of the two spirals (top), the scaffold and staples together (middle up), the scaffold (middle down) and the staples (bottom). Transversal bars on the line representation represents a crossover.</figcaption></figure>
<figure><img src="image_115.png" /><figcaption>Figure S96: 2D view of the DS design. Scaffold and staples.</figcaption></figure>
<figure><img src="image_116.png" /><figcaption>Figure S97: 2D view of the DS design. Scaffold only.</figcaption></figure>
<figure><img src="image_117.png" /><figcaption>Figure S98: 2D view of the DS design. Staples only.</figcaption></figure>
1202

S5.2 Double-sphere DS experimental validation

1203

1204

1205

1206

1207

**Optimization of the folding conditions of the DS origami (fig. S99)** Three different folding ramps and three different magnesium concentrations were tested and analyzed by agarose gel electrophoresis to optimize the folding conditions. This analysis revealed a brighter monomeric band when the origami was folded over a long period of time (65°C during 15 min, 60°C to 20°C -1°C/2h, 10°C Hold, fig. S99) rather than over shorter folding ramps.

<figure><img src="image_118.png" /><figcaption>Figure S99: Gel analysis of the influence of *MgCl*₂ concentration on folding quality of the DS. The foldings were analyzed by agarose gel electrophoresis (1%). L: 1kb DNA ladder. 65°C during 15 min, 60°C to 20°C -1°C/30min, 10°C Hold ; 65°C during 15 min, 60°C to 20°C -1°C/1h, 10°C Hold ; 65°C during 15 min, 60°C to 20°C -1°C/2h, 10°C Hold.</figcaption></figure>
<figure><img src="image_119.png" /></figure>

Figure S100: Transmission Electron Microscope (TEM) micrographs of typical particles of the DS. A: Selected side view particles. B: Selected Top view particles. Scale bar = 200 nm.
1208 S6 Vault-like origami, Figures S101 to S112

1209 S6.1 Vault-like origami design

A

<figure><img src="image_120.png" /></figure>

B

<figure><img src="image_121.png" /><figcaption>Figure S101: 3D view of the Vault-like design, line representation. A: Front view; B: Bottom view. Each line represents a double helix of DNA. Transversal bars represent crossovers. Bold lines represent the anchor staples (6 pairs) used to enable the homodimerization.</figcaption></figure>
A

<figure><img src="image_122.png" /></figure>

B

<figure><img src="image_123.png" /></figure>

C

<figure><img src="image_124.png" /></figure>

<figure><img src="image_125.png" /></figure>

External diameter: 53 nm

Top diameter: 34 nm

Tunnel diameter: 9 nm

Figure S102: 3D view of the Vault-like design, helix representation. A: Front view; B: Bottom view. Each ball represents a deoxynucleotide. Bold part of the helices represent the anchor staples (6 pairs) used to enable the homodimerization; C: Dimensions of the Vault-like origami. Front view (up), bottom view (bottom). External diameter of 53 nm (red circle), top diameter of 34 nm (yellow circle), tunnel diameter of 9 nm (green circle), height of 42 nm.
<figure><img src="image_126.png" /><figcaption>Figure S103: 3D view of the Vault-like design, helix (left) and line (right) representations. Views of the two spirals (top), the scaffold and staples together (middle up), the scaffold (middle down) and the staples (bottom). Transversal bars on the line representation represents a crossover. Bold strands represent the anchor staples (6 pairs) used to enable the homodimerization.</figcaption></figure>
<figure><img src="image_127.png" /><figcaption>Figure S104: 2D view of the Vault-like design. Scaffold and staples.</figcaption></figure>

<caption>S110</caption>
<figure><img src="image_128.png" /><figcaption>Figure S105: 2D view of the Vault-like design. Scaffold only.</figcaption></figure>
<figure><img src="image_129.png" /><figcaption>Figure S106: 2D view of the Vault-like design. Staples only.</figcaption></figure>
1210

## S6.2 Vault-like origami experimental validation

1211

1212

1213

1214

1215

**Optimization of the folding conditions of the Vault origami (figs. S107 to S112)** Three different folding ramps (over 3.5 hours, 33 hours and 42 hours) were first tested and analyzed by agarose gel electrophoresis, with six different magnesium concentrations (10 to 20 mM) and a scaffold concentration of 20 nM with staples oligonucleotides in a 10-fold excess (fig. S107). The gel showed that the yield was better with higher magnesium concentrations and longer folding ramps.

<figure><img src="image_130.png" /><figcaption>Figure S107: Gel analysis of the influence of MgCl₂ concentration and folding ramp on folding quality of the Vault. The foldings were analyzed by agarose gel electrophoresis (1%). L: 1kb DNA ladder.</figcaption></figure>
We then folded the vault with 20 mM of magnesium over a 63-hour time period and compared it to the foldings of the vault over a 42-hour and over a 3.5-hour time period thanks to agarose gel electrophoresis (fig. S108). For all the conditions, we observed a monomeric lower band, with a better yield with the longest folding ramp (63 hours).

<figure><img src="image_131.png" /><figcaption>Figure S108: Gel analysis of the influence of different folding ramps on the folding quality of the Vault. For each folding ramp, the two versions of the Vault monomer (with 5'-anchors: "Begin Anchors" and with 3'-anchors: "End Anchors") designed to enable the dimerization were folded. The foldings were analyzed by agarose gel electrophoresis (1%). L: 1kb DNA ladder.</figcaption></figure>
We analyzed the purified monomers of the Vault folded with these three conditions with negative staining electron-microscopy. The micrographs, galleries and 2D-class averaging displayed anticipated shapes of the Vault-like origami even though the 3.5-hour and 42-hour long folding ramps showed monomers which looked more flattened than the ones observed with the 63-hour long folding ramp (Figure 4, fig. S109).

<figure><img src="image_132.png" /><figcaption>Figure S109: ns-TEM analysis of purified Vault monomers folded with different folding ramps. For each condition, one TEM micrograph, a 2D-class averaging and a gallery of typical particles observed on micrographs are depicted. A: 65°C during 15 min, 60°C to 40°C -1°C/10min, 10°C Hold; B: 65°C during 15 min, 60°C to 40°C -1°C/2h, 10°C Hold; C: 65°C during 15 min, 60°C to 40°C -1°C/3h, 10°C Hold. Scale bar = 200 nm. Box size = 112 nm x 112 nm.</figcaption></figure>
1225

1226

1227

1228

After optimization of the dimerization conditions (fig. S110), we incubated the Vault monomers obtained with each of the three previously mentioned folding ramps with 100 mM magnesium at 37°C over a 15-hour time period. We then analyzed the dimers formation by agarose-gel electrophoresis (Figure 4, fig. S111) and ns-TEM (Figure 4, fig. S112).

<figure><img src="image_133.png" /><figcaption>Figure S110: Gel analysis of the influence of different MgCl₂ concentrations on the dimerization of the vault-origami. The dimerization quality was analyzed by agarose gel electrophoresis (1%). The monomers “Begin” and “End” correspond to the vault monomers with 5'-anchors and 3'-anchors respectively, after a 42-hour folding and gel purification. L: 1kb DNA ladder. The monomers “Begin” and “End” were incubated with different MgCl₂ concentrations over a 15-hour time period at 37°C to enable dimerization. We observe retarded bands for the dimers compared to the monomers, suggesting some oligomerization (possibly dimerization). At 100 mM MgCl₂, we observe a slightly retarded band compared to the other ones, suggesting a better dimerization of the monomers.</figcaption></figure>
<figure><img src="image_134.png" /><figcaption>Figure S111: Gel analysis of the influence of the folding ramp of the vault monomers on their dimerization. The dimerization quality was analyzed by agarose gel electrophoresis (1%). B: purified vault monomers with 5'-anchors; E: purified vault monomers with 3'-anchors; D: obtained dimer by incubating B and E with 100 mM MgCl₂ over a 15-hour time period at 37°C; L: 1kb DNA ladder. We observe that for all the folding ramps used to fold the monomers, retarded bands were observed after incubation of the monomers together (D), suggesting some oligomerization (possibly dimerization).</figcaption></figure>
The observed dimers formed from monomers folded over a 63-hour time period appeared to be longer than the ones formed from monomers folded with the two other folding ramps (fig. S112).

Hence, we concluded that the optimal folding ramp for the folding of the vault origami was the 63-hour long folding ramp.

<figure><img src="image_135.png" /><figcaption>Figure S112: ns-TEM analysis of purified Vault dimers obtained from vault monomers folded with different folding ramps. For each condition, one TEM micrograph, a 2D-class averaging and a gallery of typical particles observed on micrographs are depicted. A: 65°C during 15 min, 60°C to 40°C -1°C/10min, 10°C Hold; B: 65°C during 15 min, 60°C to 40°C -1°C/2h, 10°C Hold; C: 65°C during 15 min, 60°C to 40°C -1°C/3h, 10°C Hold. Scale bar = 200 nm. Box size = 132 nm x 132 nm.</figcaption></figure>
S7 Summary of all the designs, Figure S113

<figure><img src="image_136.png" /><figcaption>Figure S113: All the designs and their expected sizes</figcaption></figure>
S8 Distribution of staple strand crossover interdistances, Figures S114 to S121

Figures S114 to S121 show the distribution of staple crossover interdistances for the designs ♀, T2, T3, T4, T5, T6, DS, and a classic flat square design. The colors of the bars indicate the average curvature radius (main color scale: 4 nm [red] to 40 nm [blue]) and average torsion (inset color scale: 0/nm [blue] to 0.2/nm [red]) at the crossover locations. As opposed to the classic flat design, where interdistances are sharply centered at 16 and 31 nucleotides (fig. S121), the crossover interdistances in our curved and twisted designs are broadly distributed, forming three main peaks centered around 6–8, 15, and 25 nucleotides. The spike at 1 nucleotide corresponds to double crossovers.

Note that the different curvature radii and torsion values are similarly represented within the two peaks centered at 15 and 25 nucleotides across all designs. However, the peak centered at 4-6 nucleotides appears to include crossovers located at regions of smaller curvature radius (e.g., fig. S120). This observation may serve as a design guideline: regions with smaller curvature radii may require more closely spaced crossovers.

Finally, these graphs suggest that accommodating curvature and torsion requires a broader distribution of crossover interdistances—an outcome made possible by our spiral design paradigm. Moreover, no apparent patterns emerge, as no correlation with either curvature radius or torsion is observed. This supports the conclusion that rule-based patterning is likely insufficient and underscores the value of our algorithmic 3D nucleotide positioning strategy for accurate crossover placement.

<figure><img src="image_137.png" /><figcaption>Figure S114: Distribution of staple strand crossover interdistances correlated (in colors) with local curvature radius and torsion in the ♀-origami.</figcaption></figure>
<figure><img src="image_138.png" /><figcaption>T2: Distribution of xover interdistances in function of the curvature radius (main color) and torsion (inlay color)</figcaption></figure>

Figure S115: Distribution of staple strand crossover interdistances correlated (in colors) with local curvature radius and torsion in the T2-origami.
<figure><img src="image_139.png" /><figcaption>T3: Distribution of xover interdistances in function of the curvature radius (main color) and torsion (inlay color)</figcaption></figure>

Figure S116: Distribution of staple strand crossover interdistances correlated (in colors) with local curvature radius and torsion in the T3-origami.
<figure><img src="image_140.png" /><figcaption>Figure S117: Distribution of staple strand crossover interdistances in the T4-origami.</figcaption></figure>
<figure><img src="image_141.png" /><figcaption>Figure S118: Distribution of staple strand crossover interdistances in the T5-origami.</figcaption></figure>
<figure><img src="image_142.png" /><figcaption>Figure S119: Distribution of staple strand crossover interdistances correlated (in colors) with local curvature radius and torsion in the T6-origami.</figcaption></figure>
DS: Distribution of xover interdistances in function of the curvature radius (main color) and torsion (inlay color)

<figure><img src="image_143.png" /><figcaption>Figure S120: Distribution of staple strand crossover interdistances correlated (in colors) with local curvature radius and torsion in the DS-origami.</figcaption></figure>
<figure><img src="image_144.png" /></figure>

square: Distribution of xover interdistances in function of the curvature radius (main color) and torsion (inlay color)

<figure><img src="image_145.png" /><figcaption>Figure S121: Distribution of staple strand crossover interdistances correlated (in colors) with local curvature radius and torsion in a classic 30-helix square origami (7560-nt scaffold), corresponding to the design shown above.</figcaption></figure>