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
Single Hairpin Pseudoknot

5. Representing Pseudoknots

6. Topological Genus
For a surface in 3-space: g=0 for a sphere, g=1 for a single-holed torus, g=2 for a double-holed torus… g=n 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.

7. Calculating Genus
Where P is the number of arcs in the diagram and L is the number of loops.

8. 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.

9. RNA Structures with Genus 1

10. 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 >1
World Wide Protein Database (wwPDB)
Even very long RNA structures (~2000 bases) have low genus (<18)
Primitive pseudoknots have genus 1 or 2
Expected genus for random RNA sequences ~ length/4

11. 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.)

12. 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

13. Thermodynamics and Quantum Matrix Field Theory
RNA disk diagrams Feynman diagrams
Feynman diagrams representing the Lamb shift – Nothing to do with RNA at all!

14. 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

15. 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

16. 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

17. Results Average genus per unit energy
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

18. Results
Average genus per unit length for the low-energy phase (left) and the high-energy phase (right)
<g/L> = ± for low energy and <g/L> = (585 ± 8) x 10-6 for high energy

19. 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.

20. 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

21. References Key Sources Mathematics Sources (found in MathSciNet)
Bon, Michael, Graziano Vernizzi, Henri Orland, & A. Zee. “Topological Classification of RNA Structures.” ArXiv Quantitative Biology e-prints (2006): arXiv:q-bio/ v1.
Orland, Henri, & A. Zee. “RNA Folding and Large N Matrix Theory.” Nucl.Phys. B620 (2002):
Vernizzi, Graziano, and Henri Orland. “Large-N Random Matrices for RNA Folding.” Acta Physica Polonica B 36(2005):
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):
Pillsbury, M., J. A. Taylor, H. Orland, & A. Zee. “An Algorithm for RNA Pseudoknots.” ArXiv Condensed Matter e-prints (2005): arXiv:cond-mat/
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
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):
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):
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):
Maugh, Thomas H. “RNA Viruses: The Age of Innocence Ends.” Science, New Series, Vol. 183, No (Mar. 22, 1974), pp
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
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 doi: /journal.pbio

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