The First and Second Variation of the Total Energy of Closed Duplex DNA   in Planar Case
  DNA's shape mostly lies on its total energy $F$. Its corresponding equilibrium shape equations can be got by classical variation method: letting the first energy variation $\delta^{(1)}F=0$. Here we not only provide the first variation $\delta^{(1)}F$ but also give the second variation $\delta^{(2)}F$ in planar case. Moreover, the general shape equations of DNA are obtained and a mistake in [Zhang, \emph{et al}. {\it Phys. Rev. E} {\bf 70} 051902 (2004)] is pointed out. 
Xiaohua Zhou