The denaturation transition of circular DNA is studied within a Poland-Scheraga type approach, generalized to account for the fact that the total linking number (LK), which measures the number of windings of one strand around the other, is conserved. In the model the LK conservation is maintained by invoking both overtwisting and writhing (supercoiling) mechanisms. This generalizes previous studies which considered each mechanism separately. The phase diagram of the model is analyzed as a function of the temperature and the elastic constant $kappa$ associated with the overtwisting energy for any given loop entropy exponent, $c$. As is the case where the two mechanisms apply separately, the model exhibits no denaturation transition for $c le 2$. For $c>2$ and $kappa=0$ we find that the model exhibits a first order transition. The transition becomes of higher order for any $kappa>0$. We also calculate the contribution of the two mechanisms separately in maintaining the conservation of the linking number and find that it is weakly dependent on the loop exponent $c$.
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