K-Partite RNA Secondary Structures

Abstract

RNA secondary structure prediction is a fundamental problem in structural bioinformatics. The prediction problem is difficult because RNA secondary structures may contain pseudoknots formed by crossing base pairs. We introduce k-partite secondary structures as a simple classification of RNA secondary structures with pseudoknots. An RNA secondary structure is k-partite if it is the union of k pseudoknot-free sub-structures. Most known RNA secondary structures are either bipartite or tripartite. We show that there exists a constant number k such that any secondary structure can be modified into a k-partite secondary structure with approximately the same free energy. This offers a partial explanation of the prevalence of k-partite secondary structures with small k. We give a complete characterization of the computational complexities of recognizing k-partite secondary structures for all k ≥ 2, and show that this recognition problem is essentially the same as the k-colorability problem on circle graphs. We present two simple heuristics, iterated peeling and first-fit packing, for finding k-partite RNA secondary structures. For maximizing the number of base pair stackings, our iterated peeling heuristic achieves a constant approximation ratio of at most k for 2 ≤ k ≤ 5, and at most \(\frac6{1-(1-6/k)^k} \le \frac6{1-e^{-6}} < 6.01491\) for k ≥ 6. Experiment on sequences from PseudoBase shows that our first-fit packing heuristic outperforms the leading method HotKnots in predicting RNA secondary structures with pseudoknots. Source code, data set, and experimental results are available at http://www.cs.usu.edu/~mjiang/rna/kpartite/.