# Elucidating the Mechanical Energy for Cyclization of a DNA Origami Tile
## 2 Materials and Methods
### MD Simulations with OxDNA  
MD computation is a method for analyzing the physical movements of atoms and molecules (particles). All-atom models include all the related atoms in the simulations for better accuracy, while coarse-grained models use a pseudo-atom to represent a group of atoms to save simulation time. The particles are subject under interactions for a fixed duration of time, giving a view of the dynamic evolution of the system. The trajectories of the particles are typically determined by numerically solving [PERSON]'s equations of motion for the particles. The MD simulations based on a coarse-grained model were performed with oxDNA [54] to study equilibrium conformations of the DNA origami tile and their gradual structural deformations in series during cyclization and flattening by external loading. Specifically, the oxDNA2 model was used with the temperature at 300 K. In experiments, Mg\({}^{2+}\) is typically used to mitigate the repulsion between negatively charged DNA strands. However, there is no Mg\({}^{2+}\) concentration in the oxDNA simulations. Instead, the salt concentration was set at [Na\({}^{+}\)] = 0.5 M [54; 55; 56; 57]. The caDNAno2 file of the origami tile was converted into topology and configuration files as the initial conformation in oxDNA with actual sequence information. To simplify the computation, all the ss-regions that do not serve any structural purposes (e.g., 8-nt toeholds on linker staples and free scaffold loops) were not included in the initial conformation. Since these regions were barely seen interfering with dsDNA bundles under AFM, it is reasonable to exclude them. As discussed in the 'Origami Tile Design' section, the linker staples were cut into two halves so that the upper and lower boundaries were separate before structural deformation. The configuration was relaxed by MD simulations without any external forces for 2 \(\times\) 10\({}^{6}\) steps to obtain the initial equilibrium structure. In the cyclization process, external linear spring forces were introduced by mutual traps in oxDNA (see oxDNA documents for details) and applied on the cut sites of the linkers (13 pairs of black dots located at the upper and lower boundaries in Figure 1) on origami tile to induce a cyclization through a series of deformations. Similar spring forces were used to study origami deformations in other reports [55; 58]. The potential energy associated with the spring forces was determined by considering the conformation in each state. Figure 2a shows the loading on a pair of nucleotides. A hypothetical spring that connects nucleotide A and B has a spring force constant \(k\) and an equilibrium length \(l_{0}\). Given the distance of A and B is \(l\), the forces can be expressed following the [PERSON]'s law. \[F_{A}=F_{B}=k(l-l_{0}) \tag{1}\]Assuming the potential energy is 0 at \(l=l_{0}\), the potential energy of the spring system can be calculated. \[E_{AB}=0.5k(l-l_{0})^{2} \tag{2}\]  
For a given conformation of the origami tile, \(l\) is set. The potential energy can be modulated by varying \(k\) and \(l_{0}\) (i.e., changing the spring) in each calculation from a state to the next. Overall, two million calculation steps were carried out, which took about two days of computation. After each calculation, the results were analyzed by visualizing the structures in cogli2 and by extracting the directional and positional data. The conformation with the minimum of global potential energy was chosen as the next state. The distance between two nucleotides from the cut of each linker staple was measured from the positional data to obtain a vector pointing from one terminal to the other. In the flattening process, external linear spring forces were introduced by two force planes in oxDNA (see oxDNA documents for details) and applied on all 13,360 nucleotides of the origami tile to force them to an imaginary center plane through a series of deformations. If the plane is aligned perpendicular to the \(x\)-axis, the loading of any single nucleotide will be as shown in Figure 2b.