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) [1]. 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.

The simple intramolecular model for gene assembly in ciliates consists of three molecular operations based on local DNA manipulations. It was shown to predict correctly the assembly of all currently known ciliate gene patterns. 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. A different formalization in terms of overlap-inclusion graphs, recently introduced by Brijder and Hoogeboom, proved well-suited to describe two of the three operations of the model and their combinatorial properties. We introduce in this paper an extension of the framework of Brijder and Hoogeboom in terms of directed overlap-inclusion graphs where more of the linear structure of the ciliate genes is described. We investigate a number of combinatorial properties of these graphs, including a necessary property in terms of forbidden induced subgraphs.