Watson-Crick Powers of a Word

Abstract

In this paper we define and investigate the binary word operation of strong-$\theta$-catenation (denoted by $\otimes$) where $\theta$ is an antimorphic involution modelling the Watson-Crick complementarity of DNA single strands. When iteratively applied to a word u, this operation generates all the strong-$\theta$-powers of u (defined as any word in $\{u, \theta (u)\}^+$), which amount to all the Watson-Crick powers of u when $\theta = \theta _{DNA}$ (the antimorphic involution on the DNA alphabet $\varDelta = \{A, C, G, T\}$ that maps A to T and C to G). In turn, the Watson-Crick powers of u represent DNA strands usually undesirable in DNA computing, since they attach to themselves via intramolecular Watson-Crick complementarity that binds u to $\theta _{DNA}(u)$, and thus become unavailable for other computational interactions. We find necessary and sufficient conditions for two words u and v to commute with respect to the operation of strong-$\theta$-catenation. We also define the concept of $\otimes$-primitive root pair of a word, and prove that it always exists and is unique.

This work was partially supported by Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant R2824A01 to L.K.