# Elucidating the Mechanical Energy for Cyclization of a DNA Origami Tile
## 3 Results
### Mechanical Energy Induced Cyclization  
Coarse-grained MD simulations were performed to analyze the detailed cyclization process. We realized a series of quasi-equilibrium deformations by defining multiple states and calculating potential energies from one state to the next. First, an equilibrium structure was set as the initial state (state 0). Later states were named from 1 to 10, as shown in Figure 4a. The structure was subject under hypothetical linear spring forces in each process from one state to the next. From state 0 to state 1, the forces were added between neighboring dsDNA bundles to create the starting conformation of the cyclization (state 1, see Supplementary Material Figure S1). This step was needed since the tile may fold in a random direction toward either side upon external forces, thus the cyclization may not be achieved. For state 1 to 2, 2 to 3, and all the rest until the final state, the forces were added between 13 pairs of cut sites on the linker staples (shown as dots in Figure 1). By pulling the linkers, the upper and lower edges will be brought together, cyclizing the origami tile. The total number of states were identified from the potential energy calculations. Figure 4b shows that each state has two (low and high) potential energies, except for states 0 and 10 (the initial and final states). The low energy is the ending potential energy of the calculation from the previous to the current state, while the high energy is the starting potential energy for the transition from the current state to the next. For example, the potential energies involved in the transition from state 2 to 3 are 7.5 and 5.86 kcal/mol, respectively. That is, the origami in state 2 was subject to a set of external forces with 7.5 kcal/mol, and the minimum potential energy of 5.86 kcal/mol was found after 2 million calculations. The difference (1.64 kcal/mol) is the mechanical energy into the DNA origami tile. If the forces were kept constant for further deformation, the upper and lower boundaries would not  
Figure 4: Deformation process of origami tile cyclization in MD simulations. (**a**) The DNA tile deforms from state 0 (initial) to state 10 (final). In the final state, the tile is cyclized into a tube. The cyclization is induced by adding hypothetical linear spring forces in the MD simulations. From state 0 to state 1, the forces were added between neighboring dsDNA bundles (illustrated by the red arrows in state 0) to create the starting conformation for the cyclization (state 1). From state 1 to 2, from 2 to 3, and all the rest until the final state, the forces were added between 13 pairs of cut sites on the linker staples (on the upper and lower edges), as illustrated by the red arrows in state 1. (**b**) Potential energy landscape associated with the applied spring forces for the cyclization. By extracting the initial and final potential energies (i.e., the potential energy of the springs) in a process from one state to the next, the external work is calculated. (**c**) Distance profile during the cyclization. This is the averaged distance between the cut sites of the linkers. The distance starts from \(\sim\)84 nm (32 rows of dsDNA bundles or the height of the origami tile) and ends below 11 nm, indicating a cyclized conformation in the final state. become closer. Therefore, a new stronger set of spring forces was needed (with higher force constant \(k\) and lower equilibrium length of the spring \(l_{0}\)). For the transition from state 3 to 4, multiple potential energies were tried and 10 kcal/mol was found to further deform the origami without excess energy. Calculations for other states were carried out similarly, by selecting the conformation at the potential minimum in the previous simulation as the starting point of the next computation. The simulations were halted when the upper and lower boundaries were close enough for the linkers to seal them together. Since each linker is 32 nt, the base-paired length is estimated as roughly 11 nm.