# Elucidating the Mechanical Energy for Cyclization of a DNA Origami Tile
## 4 Discussion
### Free Energy Change in Cyclization  
The cyclization process was studied experimentally and theoretically in our previous report [27], as well as by MD simulations in this work. As discussed above, the cyclization was theorized into two independent processes of flattening and flexing by introducing a hypothetical intermediate planar conformation. The energy for cyclization equals the summation of energy changes for flattening and flexing:  
\[E_{cyclization}=E_{flattening}+E_{flexing} \tag{7}\]  
Figure 6 summarizes the cyclization and the conceptual two-step process. The energies extracted from all three approaches are in remarkable agreement. Note that it was impossible to determine the energies needed for flattening or flexing in experiments. The mechanical analysis and MD computation yielded \(E_{flattening}\) of approximately 25.6 and 26.3 kcal/mol, respectively. It is worth noting that flexing was not performed in MD simulations, because the origami could twist again once subjected under flexing forces. For any given initial and final states, various deformation pathways may be proposed to calculate the related energies. However, whether the process is realistic or not may not be revealed by the theory alone, even if it is a free energy driven reaction. Therefore, performing simulations along with the theory will be beneficial. Since direct numerical studies on flexing are not practical, the flexing energy may be obtained from Equation 7 using the cyclization and flattening energies, which is about  
Figure 6: Overall analysis of the cyclization process. The cyclization of a DNA origami tile with a twist may be conceptualized as a two-step (flattening and flexing) process by introducing a hypothetical intermediate planar conformation. Blue, purple, and red arrows indicate the experiment, theory, and simulation, respectively. Solid arrows denote the direct results, while dashed arrows mark the indirect results from summation or subtraction of direct results. Experimentally determined free energy change for cyclization is approximately 32.4 kcal/mol. The theoretical value for the cyclization energy is obtained as \(\sim\)32.3 kcal/mol by adding energies for flattening and flexing steps. The MD simulations yield a cyclization energy of about 33.1 kcal/mol, consistent with the experimental and theoretical values. 6.8 kcal/mol. This value also agrees well with the theoretical value of \(\sim\)6.7 kcal/mol. The combined experiment, theory, and computation all agree well and verify the effectiveness of the simulation approach in this work. In summary, coarse-grained MD simulations were performed on the oxDNA platform to provide critical insights on the cyclization of a DNA origami tile with initial twisting. This work advances our understanding of fundamental mechanics of DNA origami in general and should be useful for energy driven processes including adaptive reconfiguration, mechanical (or other) energy absorption mechanisms, and the interactions with other biomolecules. For example, the energy consumption determined by numerical simulations may be used to design the length of linkers and the C-G ratio for reconfiguration of an origami structure. Mobile nanodevices such as DNA walkers could also be designed and optimized by the computation of energy landscape [59; 60; 61; 62]. The following are available online at [[https://www.mdpi.com/2076-3](https://www.mdpi.com/2076-3)]([https://www.mdpi.com/2076-3](https://www.mdpi.com/2076-3)) 417/11/5/2357/s1, Figure S1: Starting conformation for cyclization, Figure S2: Potential energy landscape for the cyclization of the DNA origami tile, Figure S3: Potential energy landscape for the flattening of the DNA origami tile. Conceptualization, R.L., H.C. and J.H.C. ; methodology, R.L. ; software, R.L. ; validation, R.L. and H.L. ; formal analysis, R.L. ; investigation, R.L. ; resources, R.L. ; data curation, R.L. ; writing--original draft preparation, R.L. ; writing--review and editing, R.L. and J.H.C. ; visualization, R.L. ; supervision, J.H.C. ; project administration, J.H.C.