# Elucidating the Mechanical Energy for Cyclization of a DNA Origami Tile
## 3 Results  
### Experiments and Theory  
Figure 3a,b illustrates two, honeycomb and square, lattices in typical DNA origami designs. In a honeycomb lattice, a crossover is placed every 2/3 turn, which corresponds to exactly 7 bp per crossover. This crossover arrangement is compatible with the intrinsic helicity (~10.5 bp/turn) of right-handed B-form dsDNA. In contrast, a crossover is placed every 3/2 turns in a square lattice, which equals to 15.75 bp per crossover. The closest integer 16 bp per crossover is typically used, thus yielding 10.67 bp/turn. This will cause a right-handed twist in the global structure. Figure 3c shows the conformation of our DNA origami tile from the FEM simulations with 10.5 bp/turn. A slight, right-handed curvature is observed, thus matching with the lattice design. If the tile is cyclized and the upper and lower edges are sealed with a set of linker staples (shown as the purple line), the structure will have all the dsDNA bundles (shown as gray rods) perfectly aligned as illustrated in Figure 3d. It is worth noting that the curvature of the DNA origami tile before cyclization cannot be observed under AFM. During deposition, the DNA origami collapses on the mica surface by the attraction force, thus flattening the tile [27]. To see the curvature clearly, multiple tiles may be polymerized side by side into a long ribbon (>1 um in the polymerized direction). We previously found that the ribbon also flattens under AFM, but the accumulated twisting leads to a multiple folding of the ribbon and parallelogram-shaped kinks emerge at the  
Figure 3: Schematics and FEM simulation results of DNA origami. Comparison of (**a**) honeycomb and (**b**) square lattice designs. The gray cylinder represents a dsDNA bundle from the lattice. The blue and red lines with arrows indicate the scaffold and staple strands, respectively, with their 3’ ends marked as arrows. In a honeycomb lattice, crossovers can be placed every 7 bp which is accounted for 2/3 turns in the axial direction of a dsDNA bundle. This corresponds to 10.5 bp/turn, consistent with the helicity of native B-form DNA. In a square lattice, however, a possible crossover is arranged in every 16 bp or 3/2 turns, yielding 10.67 bp/turn. (**c**) CanDo simulation results of a single-layer DNA origami rectangle in a square lattice (side and front views). With the designed helicity at 10.67 bp/turn, the tile has a minor global twist in the FEM simulations at 10.5 bp/turn. (**d**) Schematics of the cyclized DNA origami tile in both views. The purple line indicates the sealed upper and lower boundaries by linker staples in experiments. folded parts [24]. Moreover, the kinks indicate the handedness of the curvature and their density may change if the DNA helicity is altered. We studied the cyclization of the single-layer DNA origami rectangle designed at 10.67 bp/turn in a series of experiments, as well as theoretical calculations in a previous publication [27]. In the experiments, we added 13 linker staples to the solution containing preformed DNA origami tiles and held at various temperatures (35-50 \({}^{\circ}\)C). Then, we measured the origami conformations after different durations of incubation (0.5 to 4 h). The fractions were determined by counting the origami species based on their conformations (e.g., flat vs. cyclized tiles) in the AFM images. A set of ordinary differential equations (ODEs) was developed to relate the fractions of species as functions of incubation time and temperature. Several kinetic parameters such as reaction rate constants were determined by numerically fitting the experimental data. Along with rate constants, the activation energy of 32.4 \(\pm\) 0.7 kcal/mol was obtained based on the Arrhenius plot [27]. This is the minimum energy needed to overcome barriers and drive the cyclization. The cyclization process was also studied theoretically [27]. Due to the difficulties in the theoretical calculation of cyclization, the structural transformation was conceptually analyzed as a two-step (flattening and flexing) process.