Stability analysis of kinked DNA in $\mathcal{F}(K,\tau)$ model

We phenomenologically analyze short DNA rings' stability by discussing the second variation of its elastic free energy. Through expanding the perturbation functions as Fourier series, we obtain DNA rings' stability condition in a general case. By reviewing the relationship between the Kirchhoff model and the worm-like road chain (WLRC) model, we insert a spontaneous curvature term which can partly reflect the twist angle's contribution to free energy in the WLRC model and name this extended model the EWLRC model. By choosing suitable spontaneous curvature, stability analysis in this model provide us with some useful results which are consistent with the experimental observations.