Bulletin of Mathematical Biology

, Volume 61, Issue 3, pp 437–467 | Cite as

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

  • Christian Haslinger
  • Peter F. StadlerEmail author


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.


Secondary Structure Contact Structure Neutral Network Outerplanar Graph Diagram Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Society for Mathematical Biology 1999

Authors and Affiliations

  1. 1.Institut für Theoretische ChemieUniversität WienWienAustria
  2. 2.The Sante Fe InstituteSante FeUSA

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