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. 1999;61(3):437–467. doi: 10.1006/bulm.1998.0085

RNA structures with pseudo-knots: Graph-theoretical, combinatorial, and statistical properties

Christian Haslinger 1, Peter F Stadler 1,2,✉
PMCID: PMC7197269  PMID: 17883226

Abstract

The secondary structures of nucleic acids form a particularly important class of contact structures. Many important RNA molecules, however, contain pseudo-knots, a structural feature that is excluded explicitly from the conventional definition of secondary structures. We propose here a generalization of secondary structures incorporating ‘non-nested’ pseudo-knots, which we call bi-secondary structures, and discuss measures for the complexity of more general contact structures based on their graph-theoretical properties. Bi-secondary structures are planar trivalent graphs that are characterized by special embedding properties. We derive exact upper bounds on their number (as a function of the chain length n) implying that there are fewer different structures than sequences. Computational results show that the number of bi-secondary structures grows approximately like 2.35n. Numerical studies based on kinetic folding and a simple extension of the standard energy model show that the global features of the sequence-structure map of RNA do not change when pseudo-knots are introduced into the secondary structure picture. We find a large fraction of neutral mutations and, in particular, networks of sequences that fold into the same shape. These neutral networks percolate through the entire sequence space.

Keywords: Secondary Structure, Contact Structure, Neutral Network, Outerplanar Graph, Diagram Graph

Contributor Information

Christian Haslinger, Email: grisu@tbi.univie.ac.at.

Peter F. Stadler, Email: studla@tbi.univie.ac.at

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