Formalization of Side-Aware DNA Origami Words and Their Rewriting System, and Equivalent Classes

DNA origami is a powerful technique for constructing nanoscale structures by folding a single-stranded DNA scaffold with short staple strands. While traditional models assume staples bind to a fixed side of the scaffold, we introduce a side-aware DNA origami framework that incorporates the directional binding of staples to either the left or right side. The graphical representation of DNA origami is described using rectangular basic modules of scaffolds and staples, which we refer to as symbols in side-aware DNA origami words. We further define the concatenation of these symbols to represent side-aware DNA origami words. A set of rewriting rules is introduced to define equivalent words that correspond to the same graphical structure. Finally, we compute the number of possible structures by determining the equivalence classes of these words.