# Mechanical Design of DNA Origami in the Classroom
## 3 Results
### Mechanical Modeling of the Compliant Hinge Joint Angles  
To guide the design of the CHJ structures and similar compliant hinge joints, we previously developed mechanical models that describe the structure deformation that results from the entropic force of the ssDNA linkers, causing bending of the joint [35; 47]. In our previous work, we modeled the large bending deflection of the six-helix layer using Euler-Bernoulli beam theory [48]. These models, however, require the use of differential equations. Here, we aimed to develop simpler models that are more readily relatable to basic undergraduate-level engineering mechanics. We present two models to quantify the CHJ deformation and compare them with experiment data [35] (see Supplemental Materials for angle measurement tutorial). Given the high bending stiffness of the two arm extensions of the CHJ, we assumed that they are rigid, maintaining a straight configuration, so the angle is fully described by the bending of the six-helix layer, which has a lower bending stiffness. Here, we modeled the six-helix layer as a beam whose cross-section contains six helices (each helix being modeled as a circular shape) connected in the appropriate single-layer honeycomb cross-section (Figure 5A). Bending of the six-helix layer is caused by the entropic force of the ssDNA linkers. As the six-helix layer bends, the end-to-end distance of the ssDNA linkers changes, thereby reducing the entropic force. In equilibrium, the entropic force generated by the ssDNA linkers in a particular configuration is balanced by the resistance force caused by the bending strain energy in the six-helix layer. To model the force from the ssDNA linker strands, we regard them as non-linear springs whose force-extension behavior is described by the Marko-Siggia approximation to the worm-like-chain (WLC) [49]:  
\[F_{WLC}(L_{c},\chi)=\frac{k_{B}T}{L_{p}}\left[\frac{1}{4\left(1-\frac{\chi}{L_ {c}}\right)^{2}}-\frac{1}{4}+\frac{x}{L_{c}}\right] \tag{1}\]  
where \(k_{B}\) is the Bolzman factor, \(T\) is the absolute temperature, \(L_{p}\) is the ssDNA persistence length, \(x\) is the end-to-end distance of the ssDNA linkers, and \(L_{c}\) is the ssDNA contour  
Figure 5: (**A**) The CHJ vertex is modeled as a symmetric bent beam. The extension force of the ssDNA linker balances the force required to bend the beam. (**B**) The symmetric bent beam is further simplified by PRBM or SDM. Both capture the deflection of the beam at the end, which defines the hinge angle. PRBM replaces the bent beam as the combination of a torsional spring and a solid beam. SDM applies a small-angle approximation, assuming small deformation (i.e., the beam is only under a pure bending moment and the buckling force effect is negligible). (**C**) Both models result in good agreement with experimental data for MF and LF. HF was not included in the modeling as the springs had 0 ssDNA length. The standard error from experiment data was less than 1\({}^{\circ}\), which is not shown in the bar plot (for a detailed geometrical description of the HF version, please refer to Supplementary Figure S6). length, which is a parameter that varies for different versions of the CHJ. The WLC is a model that is often used to describe the nonlinear force-extension behavior of biopolymers. All six ssDNA linkers that span the joint connecting the bottom layer of the 18-helix bundle arms apply a force. Due to the hexagonal lattice structure [2], the end-to-end distance of the three upper ssDNA springs of the bottom layer are not the same length as the three lower ssDNA springs in the bottom layer if the hinge angle is not 180\({}^{\circ}\). To accommodate this spatial difference, the upper three linkers have the additional length of ssDNA (Table 1), while the lower three helices have 10 extra dsDNA base pairs (\(\sim\)3.4 nm).