Topological Methods for RNA Pseudoknots

1
Topological Methods for RNA Pseudoknots

• Nicole A. Larsen
• Georgia Institute of Technology
• Department of Mathematics

Math 4803 04/21/2008
2
Overview

• Introduction to Pseudoknots
• Topological Representation and Classification
• Thermodynamic Calculations
• Conclusions and Open Problems

3
Pseudoknots

• RNA secondary structures with crossing base
  pairs
• Prevalent in nature
• Telomerase
• Viruses such as Hepatitis C, SARS Coronavirus,
  and even several strains of HIV

Coronavirus
4
The Trouble with Pseudoknots

• Cannot be represented as a plane tree
• Current energy calculation methods do not hold
• About the only thing we can do is use recursive
  methods

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Representing Pseudoknots
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Topological Genus

• For a surface in 3-space g0 for a sphere, g1
  for a single-holed torus, g2 for a double-holed
  torus gn for an n-holed torus.
• The genus of an RNA structure is defined by Bon
  et al. to be the minimum g such that the disk
  diagram can be drawn on a surface of genus g with
  no crossing arcs.

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Calculating Genus

• Where P is the number of arcs in the diagram and
  L is the number of loops.

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Properties of Genus

• Pseudoknot-free structures have genus 0.
• Stacked base pairs do not contribute to genus.
• For concatenated structures, genus is the sum of
  the two substructures.
• For nested structures, genus is the sum of the
  two substructures.

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RNA Structures with Genus 1
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Classification Results

• There are 4 primitive pseudoknots of genus 1
• Pseudobase Contains 246 pseudoknots
• 238 were H-pseudoknots or nested H-pseudoknots
• Only 1 had genus gt1
• World Wide Protein Database (wwPDB)
• Even very long RNA structures (2000 bases) have
  low genus (lt18)
• Primitive pseudoknots have genus 1 or 2
• Expected genus for random RNA sequences
  length/4

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Classification Results

• (Left) Genus as a function of length of the RNA
  structure. (Right) A histogram of the genus of
  primitive RNA structures found in the wwPDB (Bon
  et al.)

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What good is it, anyway?

• Genus gives us a way to measure the complexity
  of a pseudoknot
• If we can determine a relationship between
  topological genus and energy then we can use a
  minimum free energy approach for prediction

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Thermodynamics and Quantum Matrix Field Theory

• RNA disk diagrams --------- Feynman diagrams

Feynman diagrams representing the Lamb shift
Nothing to do with RNA at all!
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Partition Function

• Thermodynamic partition function

where the sum ranges over all possible Feynman
diagrams D for a given RNA sequence and E(D) is
the energy of diagram D
where the sum ranges over all possible Feynman
diagrams D for a given RNA sequence and E(D) is
the energy of diagram D
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Results

• Vernizzi and Orland use a Monte Carlo method to
  generate RNA structures weighed by the partition
  function
• Where ? is a topological potential energy and g
  is genus. By adjusting ? you can allow RNA
  structures of any genus, or restrict to small
  genus structures. Useful for rapidly exploring
  energy regions to find minimum energy structures.
• When ? goes to infinity (PKF) results agree with
  mfold predictions.
• g/L 0.23 for random sequences

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Modeling with a Cubic Lattice

• Infinitely flexible polymer sequence
• Given by a self-avoiding random walk on a cubic
  lattice
• Each base lies on a vertex of the lattice
• Bases only bond with neighboring bases, modeled
  by spin vectors

where the sum ranges over all possible Feynman
diagrams D for a given RNA sequence and E(D) is
the energy of diagram D
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Results
where the sum ranges over all possible Feynman
diagrams D for a given RNA sequence and E(D) is
the energy of diagram D
Average genus per unit energy
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Results
Average genus per unit length for the low-energy
phase (left) and the high-energy phase
(right) ltg/Lgt 0.141 0.003 for low energy and
ltg/Lgt (585 8) x 10-6 for high energy
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Conclusions

• Topological genus provides a nice, relatively
  easy classification scheme for pseudoknots
• Thermodynamic predictions based on genus agree
  with observations and with predictions given by
  mfold
• Low-genus structures are more likely to be found
  in nature.

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Open Questions

• Create an algorithm for predicting secondary
  structures that may have pseudoknots
• Pillsbury, Orland, and Zee steepest-descent
  method that takes O(L6) just to calculate
  partition function, much less optimal structures!
• Experimental measurement and cataloging of
  low-genus structures
• How does genus depend on temperature?
• Can genus be used to predict asymptotic behavior
  of very long sequences?
• Incorporation of higher-order considerations such
  as entropy

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References

• Key Sources
• Bon, Michael, Graziano Vernizzi, Henri Orland,
  A. Zee. Topological Classification of RNA
  Structures. ArXiv Quantitative Biology e-prints
  (2006) arXivq-bio/0607032v1.
• Orland, Henri, A. Zee. RNA Folding and Large N
  Matrix Theory. Nucl.Phys. B620 (2002) 456-476.
• Vernizzi, Graziano, and Henri Orland. Large-N
  Random Matrices for RNA Folding. Acta Physica
  Polonica B 36(2005) 2821-2827.
• Vernizzi, Graziano, Paulo Ribeca, Henri Orland,
  A. Zee. Topology of Pseudoknotted Homopolymers.
  Physical Review E 73(2006).
• Mathematics Sources (found in MathSciNet)
• Karp, Richard M. Mathematical Challenges from
  Genomics and Molecular Biology. Notices of the
  AMS 49(2002) 544-553.
• Pillsbury, M., J. A. Taylor, H. Orland, A. Zee.
  An Algorithm for RNA Pseudoknots. ArXiv
  Condensed Matter e-prints (2005)
  arXivcond-mat/0310505.
• Rivas, Elena, and Sean R. Eddy. A Dynamic
  Programming Algorithm for RNA Structure
  Prediction Including Pseudoknots. Journal of
  Molecular Biology, Vol. 285 No 5 (5 February
  1999), pp 2053-2068.
• Vernizzi, Graziano, Henri Orland, A. Zee.
  Enumeration of RNA Structures by Matrix Models.
  Phys Rev Lett. 94(2006).
• Zee, A. Random Matrix Theory and RNA Folding.
  Acta Physica Polonica B 36(2005) 2829-2836.

• Biology Sources (found in PubMed)
• Brierley, Ian, Simon Pennell, and Robert J. C.
  Gilbert. Viral RNA Pseudoknots Versatile Motifs
  in Gene Expression and Replication. Nature
  Reviews Microbiology 5(2007) 598-610.
• Chen, Jiunn-Liang, and Carol W. Greider.
  Functional Analysis of the Pseudoknot Structure
  in Human Telomerase RNA. Proceedings of the
  National Academy of Sciences 102(2005)
  8080-8085.
• Maugh, Thomas H. RNA Viruses The Age of
  Innocence Ends. Science, New Series, Vol. 183,
  No. 4130. (Mar. 22, 1974), pp. 1181-1185.
• Tu, Chialing, Tzy-Hwa Tzeng, and Jeremy A.
  Bruenn. Ribosomal Movement Impeded at a
  Pseudoknot Required for Frameshifting.
  Proceedings of the National Academy of Sciences
  of the United States of America, Vol. 89, No. 18.
  (Sep. 15, 1992), pp. 8636-8640.
• Other Sources
• Rong, Yongwu. Feynman diagrams, RNA folding, and
  the transition polynomial. IMA Annual Program
  Year Workshop RNA in Biology, Bioengineering and
  Nanotechnology. October 29-November 2, 2007.
• Staple DW, Butcher SE (2005) Pseudoknots RNA
  Structures with Diverse Functions. PLoS Biol
  3(6) (2005), e213 doi10.1371/journal.pbio.0030213
  .

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THE END