# Elucidating the Mechanical Energy for Cyclization of a DNA Origami Tile
## 3 Results
### Energy Driven Planarization  
If an origami tile is designed in a honeycomb lattice (as illustrated in Figure 3a), it will be flat in its intrinsic state. Then, the cyclization process of such a planar origami includes flexing only. For the twisted origami tile (in a square lattice), however, two deformation modes occur simultaneously during the cyclization, overcoming the initial twist and flexing the tile into a tube. It is difficult to separate the two modes. Therefore, we perform MD simulations to study a deformation that purely overcomes the twisting (i.e., flattening). This will also verify the conceptual two-step process in our previous study [27]. All of the 13,360 nucleotides were pulled to the imaginary center flat plane (dashed lines in Figure 5a) by applying the spring forces in a series of quasi-equilibrium deformations. Since the middle part of the origami tile is closer to flat compared with four corners, the imaginary center plane is placed through the middle. Then, strong forces are applied to the middle part in order to hold it in place, while the corners are subject to smaller flattening forces such that they will approach the center plane gradually and steadily. For example, the forces on dsDNA bundle number 12 through 21 are about 3 times stronger than those on the rest of the bundles, when transitioning from state 0 to state 1. If much stronger forces were applied at the corners, they would be compressed quickly and might be folded instead of moving toward the center plane. In states2, 3, and 6, for example, the upper corners hovered on top of the flat part due to thermal fluctuation. They were allowed to deform toward the center plane due to small forces applied instead of folding locally. Moreover, much more energy would have been consumed, thus we could not determine the minimum energy for flattening. Figure 5a shows the gradual deformations of the tile from the twisted curvature to the flat conformation. We observe that the two upper corners along with the top few dsDNA bundles were not in the center plane until the last state. It is worth noting that a step to find the starting conformation was not needed in the flattening, because the origami tile does not have two directions to deform--it deforms only toward the center plane. Similar to the cyclization process, the structure was deformed in a multistep process for flattening. The external forces and structural deformation were halted when the origami became flat enough. To develop a criterion, we defined the flatness as the sum of the  
Figure 5: Flattening process of a DNA origami tile in MD simulations. (**a**) The origami tile flattens from the initial state (state 0) and undergoes gradual planarization until the final state (state 8). The flattening is induced by pulling all the nucleotides to the imaginary flat plane at the center (dashed turquoise line) with hypothetical springs. In the final state, the origami tile is planar. (**b**) External potential landscape during the flattening process. The external work was calculated in a similar approach as in the cyclization process. (**c**) Flatness during the flattening. This is a sum of the distance squared of each nucleotide to the imaginary flat plane. Given the radius of a dsDNA bundle (-1.1 nm), the flatness of a planar tile is -2 \(\times\) 10\({}^{4}\) nm\({}^{2}\). During the deformation, the flatness starts from \(\sim\)6.5 \(\times\) 10\({}^{5}\) nm\({}^{2}\) and ends at -23 \(\times\) 10\({}^{4}\) nm\({}^{2}\). distance squared of each nucleotide to the imaginary flat plane. For example, 0 flatness means perfectly flat. Since the origami tile has a non-zero thickness, its flatness is always greater than 0. The radius of regular, right-handed B-DNA is 1.1 nm. The flatness is \(\sim\)2 \(\times\) 10\({}^{4}\) nm\({}^{2}\) if we consider a reference state where all the nucleotides are 1.1 nm from the imaginary center plane. When the flatness of the origami tile is below 2.5 \(\times\) 10\({}^{4}\) nm\({}^{2}\), the simulations were halted.