Skip to main content
SpringerLink
Log in
Book cover

Mathematical Approaches to Biomolecular Structure and Dynamics pp 39–58Cite as

Energy Functions for Knots: Beginning to Predict Physical Behavior

  • Chapter

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 82))

Abstract

Several definitions have been proposed for the “energy” of a knot. The intuitive goal is to define a number u(K) that somehow measures how “tangled” or “crumpled” a knot K is. Typically, one starts with the idea that a small piece of the knot somehow repels other pieces, and then adds up the contributions from all the pieces. From a purely mathematical standpoint, one may hope to define new knot-type invariants, e.g by considering the minimum of u(K) as K ranges over all the knots of a given knot-type. We also are motivated by the desire to understand and predict how knot-type affects the behavior of physically real knots, in particular DNA loops in gel electrophoresis or random knotting experiments. Despite the physical naiveté of recently studied knot energies, there now is enough laboratory data on relative gel velocity, along with computer calculations of idealized knot energies, to justify the assertion that knot energies can predict relative knot behavior in physical systems. The relationships between random knot frequencies and either gel velocities or knot energies is less clear at this time.

Partially supported by NSF DMS-9407132.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H.W. Benjamin, M.M. Matzuk, M.A. Krasnow, and N.R. Cozzarelli, Recombination site selection by Tn3 resolvase: topological tests of a tracking mechanism, Cell 40 (1985), 147–158.

    Article  CAS  Google Scholar 

  2. K. Brakke, Surface Evolver Manual (v. 1.92), Minnesota Geometry Center, Research Report GCG55, July 1993.

    Google Scholar 

  3. S. Bryson, M.H. Freedman, Z.X. He, and Z. Wang, Mobius invariance of knot energy, Bull. Amer. Math. Soc. 28 (1993), 99–103.

    Article  Google Scholar 

  4. G. Buck and J. Simon, Knots as dynamical systems, Topology Appl. 51 (1993), 229–246.

    Article  Google Scholar 

  5. G. Buck and J. Orloff, A simple energy function for knots, preprint 12/92 (to appear, Topology Appl.).

    Google Scholar 

  6. G. Buck, Random knots and energy: Elementary considerations, Journal of Knot Theory and its Ramifications 3 (1994), 355–364.

    Article  Google Scholar 

  7. G. Buck, The projection energy bounds crossing number, preprint 10/93.

    Google Scholar 

  8. G. Buck and J. Orloff, Computing canonical conformations for knots, Topology Appl. 51 (1993), 246–253.

    Google Scholar 

  9. N. Cozarelli, pers. comm. 7/94.

    Google Scholar 

  10. F.B. Dean, A. Stasiak, T. Koller, and N.R. Cozzarelli, Duplex DNA knots produced by escherichia coli topoisomerasel, Journal of Biological Chemistry 260 (1985), 4975–4983.

    CAS  Google Scholar 

  11. T. Deguchi and K. Tsurusaki, A statistical study of random knotting using the Vassiliev invariants, Journal of Knot Theory and its Ramifications 3 (1994), 321–353.

    Article  Google Scholar 

  12. P. Dröge and N.R. Cozzarelli, Topological structure of DNA knots and catenanes, Methods in Enzymology 212 (1992), 120–130.

    Article  Google Scholar 

  13. P. Dröge and N.R. Cozzarelli, Recombination of knotted substrates by Tn3 resolvase, Proc. Natl. Acad. Sci. USA 86 (1989), 6062–6066.

    Article  Google Scholar 

  14. C. Ernst and D.W. Sumners, The growth of the number of prime knots, Math. Proc. Camb. Phil. Soc. 102 (1987), 303–315.

    Article  Google Scholar 

  15. M. Freedman and Z.-X. He, Divergence free fields: Energy and asymptotic crossing number, Ann. Math. 133 (1991), 189–229.

    Article  Google Scholar 

  16. M. Freedman, X. He, [and Z. Wang], On the “energy” of knots and unknots, preprint 12/91 [12/92].

    Google Scholar 

  17. S. Fukuhara, Energy of a knot, Fete of Topology (Matsumoto et al., Eds.), Academic Press, New York, 1988, 443–451.

    Google Scholar 

  18. R. Kanaar, A. Klippel, E. Shekhtman, J.M. Dungan, R. Kahmann, and N.R. Cozzarelli, Processive recombination by the phage Mu Gin system: Implications fo rthe mechanisms of DNA strand exchange, DNA site alignment, and enhancer action, Cell 62 (1990), 353–366.

    Article  CAS  Google Scholar 

  19. D. Kim and R. Kusner, Torus knots extremizing the Möbius energy, Experimental Math. 2 (1993), 1–9.

    Google Scholar 

  20. R. Kusner and J. Sullivan, Möbius energies for knots and links, surfaces and submanifolds, preprint 4/94.

    Google Scholar 

  21. H.A. Lim and E.J. Janse van Rensburg, A numerical simulation of electrophoresis of knotted DNA, Supercomputer Computations Research Institute (Florida State Univ.), report #FSU-SCRI-91–163, to appear J. Modelling Sci. Corn-put., Oxford.

    Google Scholar 

  22. H.A. Lim, M.T. Carroll, and E.J. van Rensburg, Electrophoresis of knotted DNA in a regular and random electrophoretic medium, Biomedical Modeling and Simulation (J. Eisenfeld et al., Eds.), Elsevier Science Pub., New York, 1992, 213–223.

    Google Scholar 

  23. S. Lomonaco, The modern legacies of Thompson’s atomic vortex theory in classical electrodynamics, Amer. Math. Soc. Proc. in Appl. Math., to appear; also pers. comm., preprint 1994, and talks at several meetings.

    Google Scholar 

  24. J.P.J. Michels and F.W. Wiegel, On the topology of a polymer ring, Proc. Roy. Soc. A 403 (1986), 269–284.

    Article  CAS  Google Scholar 

  25. J.P.J. Michels and F.W. Wiegel, Phys. Let. 90A (1982), 381–384.

    Article  CAS  Google Scholar 

  26. K. Millett, Knotting of regular polygons in 3-space, Journal of Knot Theory and its Ramifications, 3 (1994), 263–278.

    Article  Google Scholar 

  27. J. Milnor, On the total curvature of knots, Ann. Math. 52 (1950), 248–257.

    Article  Google Scholar 

  28. K. Moffatt, The energy spectrum of knots and links, Nature 347 (Sept. 1990), 367–369.

    Article  Google Scholar 

  29. J. O’hara, Energy of a knot, Topology 30 (1991), 241–247.

    Article  Google Scholar 

  30. J. O’hara, Family of energy functionals of knots, Topology Appl. 48 (1992), 147–161.

    Article  Google Scholar 

  31. J. O’hara, Energy functionals of knots, Topology-Hawaii (K.H. Doverman, Ed.), (Proc. of 1991 conference), World Scientific, Singapore, 1992, 201–214.

    Google Scholar 

  32. J. O’hara, Energy functionals of knots II, Topology Appl. 56 (1994), 45–61.

    Article  Google Scholar 

  33. N. Pippenger, Knots in random walks, Disc. Appl. Math. 25 (1989), 273–278.

    Google Scholar 

  34. R. Randell, An elementary invariant of knots, Journal of Knot Theory and its Ramifications, 3 (1994), 279–286.

    Article  Google Scholar 

  35. E.J. Janse van Rensburg and S.G. Whittington, The knot probability in lattice polygons, J. Phys. A: Math. Gen. 23 (1990), 3573–3590.

    Article  Google Scholar 

  36. E.J. Janse van Rensburg and S.G. Whittington, The dimensions of knotted polygons, J. Phys. A: Math. Gen 24 (1991), 3935–3948.

    Article  Google Scholar 

  37. E.J.J. van Rensburg and S.D. Promislaw, Minimal knots in the cubic lattice, preprint 12/93.

    Google Scholar 

  38. At the July 1994 IMA conference, after the talk which included the data and computationally estimated slope of 1.647, several colleagues (including E.J.J. Van Rensburg, A. Stasiak, and J. Sullivan) suggested that the limiting slope was, in fact, (math), and this has been verified.

    Google Scholar 

  39. V.Y. Rybekov, N.R. Cozzarelli, and A.V. Vologodskii, Probability of DNA knotting and the effective diameter of the DNA double helix, Proc. Natl. Acad. Sci. USA 90 (1993), 5307–5311.

    Article  Google Scholar 

  40. S.Y. Shaw and J.C. Wang, Science 260 (1993), 533.

    Article  CAS  Google Scholar 

  41. S.Y. Shaw and J.C. Wang, DNA knot formation is aqueous solutions, Journal of Knot Theory and its Ramifications, 3 (1994), 287–298.

    Article  Google Scholar 

  42. S. Spengler, A. Stasiak, and N.R. Cozzarelli, The stereostructure of knots and catenanes produced by phage λ integrative recombination: implications for mechanism and DNA structure, Cell 42 (1985), 325–334.

    Article  CAS  Google Scholar 

  43. J.K. Simon, Energy functions for polygonal knots, Journal of Knot Theory and its Ramifications, 3 (1994), 299–320.

    Article  Google Scholar 

  44. In a conversation during the July 1994 IMA conference, Andrzej Stasiak suggested the appealing term in virtuo to describe computer experiments.

    Google Scholar 

  45. De Witt Sumners and S.G. Whittington, Knots in self-avoiding walks, J. Phys. A: Math. Gen. 21 (1988), 1689–1694.

    Article  Google Scholar 

  46. M.C. Tesi, E.J. Janse van Rensburg, E. Orlandini, and S.G. Whittington, Knot probability for lattice polygons in confined geometries, J. Phys. A: Math. Gen. 27 (1994), 347–360.

    Article  Google Scholar 

  47. M.C. Tesi, E.J. Janse van Rensburg, E. Orlandini, D.W. Sumners, and S.G. Whittington, Knotting and supercoiling in circular DNA: A model incorporating the effect of added sait, Phys. Rev. E: 49 (1994), 868–872.

    Article  CAS  Google Scholar 

  48. A.V. Vologodski, A.V. Lukashin, M.D. Frank-Kaminetskii, and A.V. Ahshelevich, The knot probability in statistical mechanics of polymer chains, Sov. Phys. JETP 39 (1974), 1059–1063.

    Google Scholar 

  49. S. Wasserman and N.R. Cozzarelli, Supercoiled DNA-directed knotting by T4 topoisomerase, Journal of Biological Chemistry 266 (1991), 20567–20573.

    CAS  Google Scholar 

  50. S. Whittington, Topology of polymers, in New Scientific Applications of Geometry and Topology (D.W. Sumners, Ed.), Amer. Math. Soc. PSAM 45 (1992), 73–95.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA

    Jonathan Simon

Authors
  1. Jonathan Simon
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Computer Science Department, Boston University, 111 Cummington Street, Boston, MA, 02215, USA

    Jill P. Mesirov

  2. Beckman Institute, Theoretical Biophysics Group, University of Illinois, 405 N. Mathews Avenue, Urbana, IL, 61801, USA

    Klaus Schulten

  3. Department of Mathematics, Florida State University, Tallahassee, FL, 32306-3027, USA

    De Witt Sumners

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag New York, Inc.

About this chapter

Cite this chapter

Simon, J. (1996). Energy Functions for Knots: Beginning to Predict Physical Behavior. In: Mesirov, J.P., Schulten, K., Sumners, D.W. (eds) Mathematical Approaches to Biomolecular Structure and Dynamics. The IMA Volumes in Mathematics and its Applications, vol 82. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4066-2_4

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-4066-2_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-94838-6

  • Online ISBN: 978-1-4612-4066-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation