Concatenation, Splicing and Insertion-Deletion Operations on Cyclic DNA Molecules

Splicing systems (H-systems) and insertion-deletion systems are well-studied DNA computing paradigms inspired by biological operations observed in nature. They are modelled in a formal, information-theoretic manner. In these systems, splicing, insertion and deletion operations are used. In this paper, we study these operations on circular words, which naturally model cyclic DNA molecules, focusing on the relationship between algebraic operations and biologically inspired string transformations. In particular, we formally compare concatenation with circular splicing and insertion. While the splicing operation has already been defined and used for circular words in the literature, for insertion-deletion systems, e.g., circular insertion was defined to insert a linearized variant of a circular string (cyclic molecule) into a linear (ordinary) string (or linear molecule that can be represented by a string). In this paper, we provide a formal definition of insertion operators acting on circular words. We prove that if we use circular splicing and insertion in a somewhat context-independent way (using the empty word in the formal rules), these computational operations coincide with the one-wheel concatenation operation. Our results support the theoretical understanding of these operations and their biological interpretations, with implications for DNA computing and synthetic biology.