This paper considers cyclic DNA codes of arbitrary length over the ring $R=\F_2[u]/u^4-1$. A mapping is given between the elements of $R$ and the alphabet $\{A,C,G,T\}$ which allows the additive stem distance to be extended to this ring. Cyclic codes over $R$ are designed such that their images under the mapping are also cyclic or quasi-cyclic of index 2. The additive distance and hybridization energy are functions of the neighborhood energy.