# The Free-Energy Landscape of a Mechanically Bistable DNA Origami
## 3 Results
### Free-Energy Decomposition  
To better understand the form of the oxDNA free-energy landscape, and in particular why there is no minimum corresponding to state S1, we attempted to compute the contributions to the free-energy landscape from the four joints. In the complete origami the motion of the four joints is coupled due to the overall structure of the origami. Therefore, to isolate the behaviour of the individual joints we performed a series of simulations with one or more blocks of the origami removed (see the configurations in Figure 3). Similar to the previous section, we then defined distance order parameters between the centres of mass of groups of nucleotides to probe the motion of a given joint. The definitions of the order parameters and their respective free-energy landscapes are shown in Figure 3.  
Joint 1 is designed to be rigid and, as expected, the free-energy minimum is when the frame and compliant blocks are perpendicular and unbent. The free-energy cost of changing the distance order parameter comes mainly from the bending of the compliant block instead of a change in the angle at the joint. This degree of freedom is the intended mode of stress accumulation in the original design that leads to disfavouring of the intermediate state. Joints 2 and 3 are both designed to act as hinges and have a similar local structure. However, as their free-energy landscapes show, the hinges are not completely flexible. They have preferred opening angles at around 55\({}^{\circ}\) and 70\({}^{\circ}\), respectively, and there is a non-negligible free-energy cost when the angles deviate from the preferred values. In particular, there are significant steric repulsions between the helices in the two blocks when the opening angle goes beyond 90\({}^{\circ}\) or approaches zero. Figure 2: (**a**) A configuration of the origami illustrating the definition of the order parameter \(R\) as the distance between the centres of mass of the two groups of nucleotides coloured in light blue. The value of \(\theta\) was tracked in the simulations, and is defined as the angle between the two vectors coloured in white. The vectors pass through the centres of mass of the two groups of nucleotides coloured in light red and light blue in the crank block and the frame block, respectively. (**b**) The free-energy landscape of the origami as a function of \(\theta\) at \(T=300\) K.  
Joint 4 was designed to be more flexible as the joint needs to bend both inward and outward to allow both the open and closed states to be easily accessed. The landscape for the joint has been computed in two sections because the distance order parameter on its own cannot distinguish whether the interior angle of the joint is obtuse (as in the open state) or reflex (as in the closed state). We find that the joint prefers to be roughly straight with an increasing free-energy penalty for more bent configurations; we note that we do not observe any coaxial stacking between the helix ends either side of the joint even when it adopts a straight configuration. (Note that the sharp rise in the free energy at large values of the distance order parameter is due to the stretching of the straight joint and is not relevant to the behaviour of the complete origami.) Particularly noteworthy is the asymmetry between bending the junction outwards and inwards, with a much higher cost  
Figure 3: The free-energy landscapes associated with each individual joint. The configurations illustrate the subsystems used to compute these landscapes and the definitions of the distance order parameters. The values of these order parameters in the depicted configurations are 86 nm, 45 nm, 51 nm, 78 nm and 70 nm, respectively. The landscape associated with joint 4 has been computed in two parts corresponding to whether the joint is pointing outwards (1) or inwards (2). Only when joint 4 is straight do these sub-landscapes sample equivalent configurations. for the inward bending due to the resulting steric repulsions between the helices near to the joint (Figure 4).