Directed Overlap-inclusion Graphs as Representations of Ciliate Genes

Ciliates form a large and old group of unicellular eukaryotes. One of their characteristics is that each ciliate contains two types of functionally different nuclei: the germline nuclei (micronuclei) and the somatic nuclei (the macronuclei), each having multiple copies. The genes are differently organized in the two types of nuclei: micronuclear genes are split into blocks (called MDSs), which are separated by noncoding blocks. The MDSs come in a shuffled order, some of them being also inverted. Each MDS M ends with a short sequence of nucleotides (called pointer) that has a second occurrence in the beginning of the MDS that should follow M in the orthodox order. Macronuclear genes have all the MDSs spliced together (or assembled) on their common pointers. During sexual reproduction, all macronuclei are destroyed and new macronuclei are formed starting from a copy of a micronucleus. During this process, micronuclear genes get transformed into macronuclear genes by having excised all noncoding blocks and assembling the MDSs in the orthodox order. The process is called gene assembly and has been subject to intense combinatorial and computational research in the last decade. We refer for details to [6], [1], and [23] and references therein. Several molecular models were considered for the gene assembly process, see [1]. Among them is the simple intramolecular model introduced in [8]. Unlike the other models, the simple intramolecular model postulates that gene assembly takes place as a result of local interactions, where only neighboring MDSs are able to interact with each other. The model was shown in [15] to predict correctly the assembly of all currently known gene patterns, see the database discussed in [5] for an up-to-date list. The simple model was modeled mathematically as a sorting of signed permutations in [16], and as a string rewriting system in [3, 4]. Both formal frameworks turned out to be limited in capturing the details of the local interactions postulated by the simple model and made it difficult to characterize, e.g., all gene patterns that can be assembled through simple operations. A similar difficulty in the case of the general intramolecular model was overcome by extending the model to signed overlap graphs, see [6]. In