On pseudoknot-bordered words and their properties

a r t i c l e i n f o a b s t r a c t We study a generalization of the classical notions of bordered and unbordered words, motivated by biomolecular computing. DNA strands can be viewed as finite strings over the alphabet {A, G, C, T}, and are used in biomolecular computing to encode information. Due to the fact that A is Watson–Crick complementary to T and G to C, DNA single strands that are Watson–Crick complementary can bind to each other or to themselves forming so-called secondary structures. Most of these secondary structures are undesirable for biomolecular computational purposes since the strands they involve cannot further interact with other strands. This paper studies pseudoknot-bordered words, a mathematical formalization of pseudoknot-like inter-and intra-molecular structures. In this context, pseudoknot-unbordered words model DNA or RNA strands that will be free of such secondary structures. We obtain several properties of pseudoknot-bordered and-unbordered words. We also address following problem: Given a pseudoknot-unbordered word u, does {u} + consist of pseudoknot-unbordered words only? We show that this is not generally true. We find that a sufficient condition for {u} + to consist of pseudoknot-unbordered words only is that u be not primitive. All of our results hold for arbitrary antimorphic involutions, of which the DNA Watson–Crick complementarity function is a particular case.