The simple intramolecular model for gene assembly in ciliates consists of three molecular operations, simple ld, simple hi and simple dlad. Mathematical models in terms of signed permutations and signed strings proved limited in capturing some of the combinatorial details of the simple gene assembly process. Brijder and Hoogeboom introduced a new model in terms of overlap-inclusion graphs which could describe two of the three operations of the model and their combinatorial properties. To capture the third operation, we extended their framework to directed overlap-inclusion (DOI) graphs in Azimi et al. (2011). In this paper we introduce DOI graph-based rewriting rules that capture all three operations of the simple gene assembly model and prove that they are equivalent to the string-based formalization of the model.