Watson–Crick pairing, the Heisenberg group and Milnor invariants

  • Siddhartha GadgilEmail author


We study the secondary structure of RNA determined by Watson–Crick pairing without pseudo-knots using Milnor invariants of links. We focus on the first non-trivial invariant, which we call the Heisenberg invariant. The Heisenberg invariant, which is an integer, can be interpreted in terms of the Heisenberg group as well as in terms of lattice paths. We show that the Heisenberg invariant gives a lower bound on the number of unpaired bases in an RNA secondary structure. We also show that the Heisenberg invariant can predict allosteric structures for RNA. Namely, if the Heisenberg invariant is large, then there are widely separated local maxima (i.e., allosteric structures) for the number of Watson–Crick pairs found.


RNA secondary structure Stem-loop Free groups Milnor invariants Lower central series 

Mathematics Subject Classification (2000)

Primary: 57N10 Secondary: 53A10 


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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