Symmetry-guided inverse design of self-assembling multiscale DNA origami tilings

To circumvent the challenges associated with direct enumeration, we develop a symmetry-based design strategy to generate complex interaction matrices. Each planar tiling can be classified by the symmetry operation that leaves it unchanged. Given their translational symmetry, all 2D planar tilings must fall under one of the 17 Wallpaper groups [35, 36]. Here, we use orbifold notation to describe the symmetries of the Wallpaper groups [37]. In our system, the allowed symmetry operations are constrained by the interaction rules and the geometry of the particles. Since we prohibit triangles from flipping in-plane, reflections, and glide reflections are not allowed. Additionally, 90-degree rotational symmetries are prohibited in a system with triangular subunits. Therefore, we find only four of the available Wallpaper groups: o tilings with only translational symmetry, 2222 tilings with four distinct 2-fold rotational symmetries, 333 tilings with three distinct 3-fold rotational symmetries, and 632 tilings with 6, 3, and 2-fold rotational symmetries (Fig. 2D).

Taking advantage of the Wallpaper symmetries, we develop a tractable approach to designing 2D tilings with arbitrarily large complexity. The core idea is to invert the process of generating tilings: Rather than enumerating every possible interaction matrix and then sifting through them to find those that encode 2D tilings, we directly generate tilings using symmetry and then we infer the interaction matrices that encode the assembly of those tilings. We break down our approach into three steps.

First, we generate a parallelogram that tiles the plane and corresponds to the periodicity of the resulting tiling (Fig. 2E). Specifically, we create a coordinate system $(h,k)$ along two lattice directions with the subunit size being unit length and define a parallelogram using two linearly independent vectors with integer components, $(h_{1},k_{1})$ and $(h_{2},k_{2})$. We call these two vectors the primitive unit vectors, and the corresponding parallelogram, the primitive unit cell (PU cell).

Next, we ‘color’ individual triangles and specify their orientations in each parallelogram while enforcing the symmetry operations prescribed by the Wallpaper groups. The procedure consists of: (1) choosing a blank triangle, (2) coloring it with a new color, (3) coloring all symmetrically invariant triangles with the same color, and (4) repeating this process until every triangle has been colored in (Fig. 2F). We note that because 2-fold symmetries can be placed on either an edge or a vertex, and 3-fold symmetries can be placed on either a vertex or a face, more than one tiling can exist for a given PU cell for 2222 and 333 symmetries (see SI Section III). Then we repeat this procedure to specify the orientations of each of the triangles similarly to how we specified the colors (Fig. 2G). Here, again, an exception exists for tiles with 3-fold symmetry. When the 3-fold symmetry point lies on the face of a triangle, the three edges of the triangle become homologous, and therefore the triangle does not have a deterministic orientation within the tiling.

Finally, we derive the interaction matrix that encodes the tiling deterministically by assigning interactions between unique bond pairs (Fig. 2H). This procedure amounts to setting the interaction matrix equal to the adjacency matrix. Each unique bond pair is recorded as a filled box in the interaction matrix.

This design method allows us to generate 2D tilings with a large number of species using a personal computer. To show the feasibility of our approach, we generate an exhaustive list up to PU cells containing $200$ triangles, which consist of 1628 o tilings, 2826 2222 tilings, 52 333 tilings, and 38 632 tilings, totaling 4544 2D tilings. Figures 3 and S15 show some examples of o, 2222, 333, and 632 tilings. Using a typical PC, computing all the associated interaction matrices takes only a few days, whereas direct enumeration would be impossible (see SI Section II B for details) and other modern inverse-design methods, such as SAT-assembly [38, 39], start to become intractable above roughly 100 species [40].

Inspired by the engineering challenge of assembling programmable 2D arrays of plasmonically functional nanostructures [31, 33], we demonstrate the use of our symmetry-guided design to template the assembly of supercrystals of 10-nm-diameter gold nanoparticles. To demonstrate the principle, we label the centers of specific subsets of triangles, though more sophisticated multispecies labelings can be achieved by exploiting the addressability of DNA origami [41, 8, 19]. Given the symmetry rules of the tiling templates, the supercrystals that one can assemble in this way are required to satisfy a small number of constraints. First, the largest possible lattice parameter of the supercrystal is bounded by the PU cell size, $S$. Second, the symmetry of the supercrystal need not be the same as the symmetry of the underlying tiling. And third, the order of symmetry of the supercrystal, $O_{\textrm{super}}$, cannot be smaller than the order of symmetry of the underlying tiling, $O_{\textrm{tiling}}$, where the order of symmetry, $O$, is the size of the PU cell divided by the size of the fundamental domain. For example, a 2222 tiling ($O_{\textrm{tiling}}=2$) can template the assembly of a 632 supercrystal ($O_{\textrm{super}}=6$), but a 632 tiling ($O_{\textrm{tiling}}=6$) cannot template the assembly of a 2222 supercrystal ($O_{\textrm{super}}=2$).

To demonstrate the power of our design approach, we develop a system based on DNA origami to construct 2D tilings from a large number of distinct species, on which supercrystals of gold nanoparticles can be assembled. Specifically, we make triangular subunits that are roughly 50 nm in edge length and encode specific interparticle interactions using DNA hybridization of sticky ends that protrude from the edges of the subunits (Fig. 1A). In this way, we can program the complex interaction matrices that we generate above by exploiting Watson-Crick base pairing. We then assemble DNA origami tilings isothermally at the temperature at which monomers and assemblies coexist, typically around 34 ^∘C for the 6-nucleotide sticky ends that we use. Subsequently, we add DNA-grafted gold nanoparticles at a small stoichiometric excess at room temperature to assemble the supercrystals, which we then image using negative-stain transmission electron microscopy (TEM) (Fig. 3A–B). We find that the labeling efficiency of gold nanoparticles is roughly 75% at this stoichiometric ratio.

First, we show that micrometer-sized tilings assemble for the simplest interaction matrix possible. We encode all three sides of a single species of triangle to be homologous and self-complementary. We predict that the triangles in the resulting tiling will have no orientational order, and will correspond to the simplest 632 tiling. Under TEM, we observe 2D sheets spanning micrometers in size—containing over 1,000 subunits—that have the anticipated symmetry (Fig. 3B).

We further demonstrate the utility of our approach by making supercrystals from binary tilings of three different symmetries. By using multiple species with specific interactions, we encode more sophisticated patterns with larger PU cells. First, we assemble representative 2222, 333, and 632 tilings from two species of triangles at the stoichiometric ratios of the tilings (1:1 for 2222 and 333; and 1:3 for 632), as shown as red and blue in Fig. 3C, left. The resulting supercrystals, overlayed with false color, are shown in Fig. 3C, right. Interestingly, whereas the 2222 and 632 tilings lead to supercrystals with the same symmetry as the parent tiling, the 333 tiling produces a supercrystal with 632 symmetry. This observation highlights the importance of choosing the position of the gold nanoparticle in determining the symmetry of the supercrystal: labeling the center of a triangle yields 632 crystals whereas labeling the edge of a triangle yields 333 crystals, matching the symmetry of the parent.

Encouraged by the success of our two-species experiments, we assemble more complex supercrystals with larger PU cells, including supercrystals with periodicities comparable to the wavelength of visible light. Figure 3D shows examples of 2222, 333, and 632 tilings assembled from 12 species of triangles (see Fig. S16 for examples of 6-species tilings and SI Section VI for programmed interactions). We label two triangles for all of the 12-species tilings to maintain an appreciable density of gold nanoparticles. In all cases, we find that the supercrystals are consistent with the underlying tilings, have the same symmetries as their parent tilings, and have low defect densities, indicating that the programmed interactions are truly orthogonal. As anticipated, the size of the PU cell increases as the number of species of triangles increases, leading to larger distances between the gold nanoparticles. This is further seen by the increasing complexity and low-spatial-frequency signal from the Fourier transforms of the gold-particle positions (see SI Section VII and Fig. S14 for details). Additionally, we observe that the PU cell size increases with the order of the symmetry of the tiling for a fixed number of species from 175 nm to 200 nm to 300 nm for the 2222, 333, and 632 symmetry tilings, respectively. This final observation hints at the possibility that some tiling patterns might be more useful than others in templating nanoparticle supercrystals.

From the perspective of inverse design, we ask the following questions: What is the ‘cost’ of assembling a given tiling, and are some tilings more ‘economical’ than others [42]? In the experiment, it is natural to define the cost as the number of species of triangles $N$ that need to be synthesized because each species requires unique DNA staple sequences and must be folded and purified separately. We must also define the ‘value’ of a tiling. Here, we choose the size of the PU cell, $S$, as the value of the tiling, since this parameter controls the spacing of the periodic patterns. Combining these two definitions, the ‘species economy’, $E_{\textrm{S}}$, of a tiling can be calculated by taking the ratio of the value to the cost, or $E_{\textrm{S}}=S/N$.

We find that the species economy has a maximum value that is determined by the order of symmetry of the tilings, $O_{\textrm{tiling}}$ (Fig. 4A). We observe that there is an upper limit to the species economy that can be achieved depending on the symmetry of tilings, which is 1, 2, 3, and 6 for o, 2222, 333, and 632 tilings, respectively. For o and 2222, the species economy for all tilings is constant, while for 333 and 632 tilings, some tilings do not reach the maximum species economy.

These observations can be rationalized by considering the fundamental domain of the tilings. The size of the fundamental domain is given by $S/O_{\textrm{tiling}}$, where $O_{\textrm{tiling}}=1,2,3,6$ for o, 2222, 333, and 632, respectively. As a consequence, the fundamental domain contains all unique triangle species and edges that appear in a tiling. If we simply assume that every unique triangle has to appear in the fundamental domain once, we obtain $S/O_{\textrm{tiling}}=N$. Combining this expression with our definition of species economy, we obtain $E_{\textrm{S}}=O_{\textrm{tiling}}$, which places an upper limit on the economy of any given tiling.

The loss of species economy in some 333 and 632 tilings can be attributed to triangles having homologous interactions and, therefore, not maximally exploiting all of the information that can be encoded in every subunit. In Fig. 4B, we show two examples of 632 tilings, one whose species economy is smaller than the order of symmetry and another whose species economy matches the order of symmetry. The major difference between the two tilings is the location of the 3-fold rotational symmetry point; while the tiling with the higher species economy has a 3-fold symmetry point located at the vertex of the tiling, the one with the smaller species economy has a 3-fold symmetry point located in the middle of a triangle. In the lower species economy case, since the 3-fold rotational symmetry is located in the middle of the triangle, the three sides of that triangle must have the same interactions. Therefore, the interactions encoded in that triangle are redundant, decreasing the overall species economy. In general, whenever there is a 3-fold rotational symmetry placed in the middle of the face of a triangle, the triangle has homologous interactions on three sides and results in a loss of species economy. This feature explains why only 333 and 632 tilings encounter this decrease in the species economy since they are the only tilings in which the symmetry points can have the same symmetry as the 3-fold-symmetric triangular subunits.

Instead of focusing on the number of species of triangles, the ‘cost’ could instead be defined as the number of unique interactions, $N_{\textrm{i}}$. Indeed, in some systems, the number of unique, orthogonal interactions might be the limiting factor in realizing more and more complex assemblies, such as systems in which the interactions are specified by DNA sequences, magnetic dipoles, or geometric shapes [43, 7, 44, 45]. Surprisingly, within this definition of economy, which we refer to as the interaction economy, $E_{\textrm{i}}=S/N_{\textrm{i}}$, every planar tiling discussed in this paper follows a simple equation:

irrespective of the specific locations of the symmetry points. This relationship can be understood as a consequence of the fact that the number of unique edges contained in the fundamental domain is always $N_{\textrm{i}}=3S/O_{\textrm{tiling}}$. We note that the number of unique species and interactions are just two potential metrics of the ‘cost’, and other strategies for quantifying the information content of complex assemblies might lead to other definitions of economy [46].

We conclude by considering a specific class of supercrystals, in which the periodic distance between rows of nanoparticles is the only relevant design constraint. Figures 4C and D show two example tilings that yield the same supercrystal, with nanoparticles labeled in lines that are periodic in one of the other two lattice directions. In one case, the tiling requires four unique triangles, one of which is labeled to construct the supercrystal (Fig. 4C). In the other case, it takes 16 unique triangles, of which four are labeled (Fig. 4D). Because we are only concerned with the maximum periodic distance, $L_{\textrm{M}}$, defined as the length of the PU cell perpendicular to the short axis, we define another economy, linear size economy, as the number of unique particles it takes to program the maximum periodic distance, or $E_{\textrm{L}}=L_{\textrm{M}}/N$.

We find that the linear size economy of supercrystals depends on a combination of the symmetry group and the aspect ratio of the PU cell, with highly anisotropic PU cells of high order being the most economical. Figure 4E shows the maximum periodic distance of o and 2222 tilings that have linear and rhombic PU cell shapes. Here, we define a linear PU cell to have length $L_{1}=1$ and a rhombic PU cell to have $L_{1}=L_{2}$, where $L_{1}$ and $L_{2}$ are the lengths of short and long edges of the PU cell, respectively. For o and 2222 tilings, the values of $L_{\textrm{M}}$ can be predicted using a simple geometrical argument. Specifically, for 2222 tilings, the periodicity is given by $L_{\textrm{M}}=\sqrt{3}N/2$ for linear PU cells and ${L_{\textrm{M}}}^{2}=3N/4$ for rhombic PU cells. As before, o tilings are less economical, as the PU cell is half the size of the 2222 PU cell for the same number of subunit species. These results show the importance of the shape of the PU cell for the linear size economy: Tilings with linear PU cells are more economical than rhombic ones.