On realizing shapes in the theory of RNA neutral networks

It is known (Reidys et al., 1997b. Bull. Math. Biol. 59(2), 339-397) that for any two secondary structures S,S′ there exists an RNA sequence compatible with both, and that this result does not extend to more than two secondary structures. Indeed, a simple formula for the number of RNA sequences compatible with secondary structures S,S′ plays a role in the algorithms of Flamm et al. (2001. RNA 7, 254-265) and of Abfalter et al. (2003. Proceedings of the German Conference on Bioinformatics, http://www.tbi.univie.ac.at/papers/ Abstracts/03-018.pdf) to design an RNA switch. Here we show that a natural extension of this problem is NP-complete. Unless P=NP, there is no polynomial time algorithm, which when given secondary structures S1,...,Sk, for k≥4, determines the least number of positions, such that after removal of all base pairs incident to these positions there exists an RNA nucleotide sequence compatible with the given secondary structures. We also consider a restricted version of this problem with a "fixed maximum" number of possible stars and show that it has a simple polynomial time solution.