# The Free-Energy Landscape of a Mechanically Bistable DNA Origami
## 3 Results
### Free-Energy Decomposition
By keeping track of the distance order parameters of each joint as well as \(R\) and \(\theta\) in the simulations of the complete origami, we can transform the free-energy landscapes of the individual joints to be functions of \(\theta\) (Figure S1). We next check if the free energy of the complete origami where the motion of the joints is coupled can be represented as the sum of the contributions from the individual joints. The comparison in Figure 5 shows that this is a very good approximation. Having established this, we now use these landscapes to understand why oxDNA predicts the closed state S1 to be unstable. The contribution to the free energy of the complete origami associated with joint 1 shows two minima at \(\theta\approx 20^{\circ}\) and at \(\theta\approx 90^{\circ}\), and a maximum at around \(\theta\approx 45^{\circ}\) (Figure 6). These positions generally agree with the angles for the S1, U and S2 states observed experimentally [24]. Furthermore, this component of the free-energy landscape is very similar to that predicted by the mechanical model used in that paper, albeit with a slightly larger barrier that is probably due to the somewhat high values of persistence lengths that oxDNA predicts for DNA origami [29]. These results indicate that the designed mechanism for stress accumulation due to bending of the compliant block can be captured by oxDNA. Figure 4: Close-ups of each joint for the configurations illustrated in Figure 3. Specifically, these are for (**a**) Joint 1 (**b**) Joint 2 (**c**) Joint 3 (**d**) Joint 4 (part 1) (**e**) Joint 4 (part 2, view from outside) (**f**) Joint 4 (part 2, view from inside). (The outside and inside of joint 4 are defined relative to the quadrilateral formed by the four blocks.) Single-stranded sections of the scaffold that link the blocks are coloured grey. (**d–f**) clearly show the difference in steric repulsion between the helices near the joint for open and closed configurations of joint 4. The free-energy contribution of joint 2 is a broad minimum centred around the preferred opening angle of the joint (Figure S1), and so adding this contribution does not significantly alter the general shape of the landscape. It slightly lowers the barrier and penalizes configurations with more extreme values of \(\theta\), particularly those with \(\theta>90^{\circ}\), because of the steric repulsions between the crank and frame blocks at these joint angles. Further including the contribution from joint 3 results in an appreciable change in the shape of the landscape. Most noticeably, the minimum corresponding to S1 becomes a flat region with a higher free energy. The contribution of joint 3 has a more significant effect than joint 2 because the structure of the origami necessitates smaller opening angles at joint 3 than joint 2. It is these small opening angles, and their associated free-energy cost, that particularly lead to the destabilization of S1. Finally, including the contribution from joint 4 completely removes the second minimum, because the inward bending of this joint that is associated with the closed configuration leads to a significantly larger free-energy cost. Figure 5: Comparison of the free-energy landscape and the sum of the contributions from the joints. Figure 6: The Free-energy contributions of each joint to the overall free-energy landscape.