The Generation of Random Equilateral Polygons

Abstract

Freely jointed random equilateral polygons serve as a common model for polymer rings, reflecting their statistical properties under theta conditions. To generate equilateral polygons, researchers employ many procedures that have been proved, or at least are believed, to be random with respect to the natural measure on the space of polygonal knots. As a result, the random selection of equilateral polygons, as well as the statistical robustness of this selection, is of particular interest. In this research, we study the key features of four popular methods: the Polygonal Folding, the Crankshaft Rotation, the Hedgehog, and the Triangle Methods. In particular, we compare the implementation and efficacy of these procedures, especially in regards to the population distribution of polygons in the space of polygonal knots, the distribution of edge vectors, the local curvature, and the local torsion. In addition, we give a rigorous proof that the Crankshaft Rotation Method is ergodic.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Amzallag, A., Vaillant, C., Jacob, J., Unser, M., Bednar, J., Kahn, J.D., Dubochet, J., Stasiak, A., Maddocks, J.H.: 3d reconstruction and comparison of shapes of DNA minicircles observed by cryo-electron microscopy. Nucleic Acids Res. 34, e125-1-8 (2006)

    Article  Google Scholar 

  2. 2.

    Arsuaga, J., Vazquez, M., Trigueros, S., Sumners, D.W., Roca, J.: Knotting probability of DNA molecules confined in restricted volumes: DNA knotting in phage capsids. Proc. Natl. Acad. Sci. USA 99, 5373–5377 (2002)

    Article  ADS  Google Scholar 

  3. 3.

    Arsuaga, J., Vazquez, M., McGuirk, P., Trigueros, S., Sumners, D.W., Roca, J.: DNA knots reveal a chiral organization of DNA in phage capsids. Proc. Natl. Acad. Sci. USA 102(26), 9165–9169 (2005)

    Article  ADS  Google Scholar 

  4. 4.

    Calvo, J.A.: Geometric knot theory: the classification of spatial polygons with a small number of edges. Ph.D. thesis, University of California, Santa Barbara (1998)

  5. 5.

    Calvo, J.A., Millett, K.C.: Minimal edge piecewise linear knots. In: Ideal Knots, pp. 107–128. World Sci. Publishing, Singapore (1998)

    Google Scholar 

  6. 6.

    Calvo, J.A., Millett, K.C.: Minimal edge piecewise linear knots. In: Ideal Knots. Ser. Knots Everything, vol. 19, pp. 107–128. World Sci. Publ., River Edge (1998)

    Google Scholar 

  7. 7.

    Cerny, V.: A thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. J. Optim. Theory Appl. 45, 51–51 (1985)

    Article  MathSciNet  Google Scholar 

  8. 8.

    Chakravarti, I.M., Laha, R.G., Roy, J.: Handbook of Methods of Applied Statistics, vol. I. Wiley, New York (1967)

    Google Scholar 

  9. 9.

    de Gennes, P.G.: Collapse of a polymer chain in poor solvents. J. Phys. Lett. 36, 55–57 (1975)

    Article  ADS  Google Scholar 

  10. 10.

    de Gennes, P.G.: Scaling Concepts in Polymer Physics. Cornell University Press, Ithaca (1979)

    Google Scholar 

  11. 11.

    Deguchi, T., Shimamura, M.K.: Topological effects on the average size of random knots. In: Physical Knots: Knotting, Linking, and Folding Geometric Objects in ℝ3. Contemp. Math., vol. 304, pp. 93–114. Amer. Math. Soc., Providence (2002)

    Google Scholar 

  12. 12.

    Deguchi, T., Tsurusaki, K.: A statistical study of random knotting using the Vassiliev invariants. J. Knot Theory Ramif. 3(3), 321–353 (1994). Random knotting and linking (Vancouver, BC, 1993)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Deguchi, T., Tsurusaki, K.: Numerical application of knot invariants and universality of random knotting. In: Knot Theory, Warsaw, 1995. Banach Center Publ., vol. 42, pp. 77–85. Polish Acad. Sci., Warsaw (1998)

    Google Scholar 

  14. 14.

    des Cloizeaux, J.: Ring polymers in solution: Topological effects. J. Phys. Lett. 42, 433–436 (1981)

    Article  Google Scholar 

  15. 15.

    Deutsch, J.M.: Equilibrium size of large ring molecules. Phys. Rev. E 59(3), 2539–2541 (1999)

    Article  ADS  Google Scholar 

  16. 16.

    Diao, Y., Ernst, C., Janse van Rensburg, E.J.: In search of a good polygonal knot energy. J. Knot Theory Ramif. 6(5), 633–657 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. 17.

    Diao, Y., Ernst, C.: The average crossing number of Gaussian random walks and polygons. In: Physical and Numerical Models in Knot Theory. Series on Knots and Everything, vol. 36, pp. 275–292. World Sci. Publ., River Edge (2005)

    Google Scholar 

  18. 18.

    Diao, Y., Dobay, A., Kusner, R.B., Millett, K.C., Stasiak, A.: The average crossing number of equilateral random polygons. J. Phys. A 36(46), 11,561–11,574 (2003)

    Article  MathSciNet  Google Scholar 

  19. 19.

    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall Inc., Englewood Cliffs (1976). Translated from the Portuguese

    Google Scholar 

  20. 20.

    Dobay, A., Sottas, P., Dubochet, J., Stasiak, A.: Predicting optimal lengths of random knots. Lett. Math. Phys. 55(3), 239–247 (2001). Topological and geometrical methods (Dijon, 2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. 21.

    Dobay, A., Dubochet, J., Millett, K.C., Sottas, P., Stasiak, A.: Scaling behavior of random knots. Proc. Natl. Acad. Sci. USA 100(10), 5611–5615 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. 22.

    Drube, E., Alim, K., Witz, G., Dietler, G., Frey, E.: Excluded volume effects on semiflexible ring polymers. Nano Lett. 10, 1445–1449 (2010)

    Article  ADS  Google Scholar 

  23. 23.

    Ercolini, E., Valle, E., Adamcik, J., Witz, G., Metzler, R., De Los Rios, P., Roca, J., Dietler, G.: Fractal dimension and localization of DNA knots. Phys. Rev. Lett. 98 058102 (2007)

    Article  ADS  Google Scholar 

  24. 24.

    Ewing, B., Millett, K.C.: Computational algorithms and the complexity of link polynomials. In: Progress in Knot Theory and Related Topics, pp. 51–68. Hermann, Paris (1997)

    Google Scholar 

  25. 25.

    Ferrenberg, A.M., Swendsen, R.H.: Application of the Monte Carlo method to the lattice-gas model. Phys. Rev. Lett. 61, 2635–2638 (1988)

    Article  ADS  Google Scholar 

  26. 26.

    Forsén, S. (ed.): Nobel Lectures, Chemistry 1971–1980. World Scientific, Singapore (1993)

    Google Scholar 

  27. 27.

    Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K.C., Ocneanu, A.: A new polynomial invariant of knots and links. Bull., New Ser., Am. Math. Soc. 12(2), 239–246 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  28. 28.

    Geyer, C.J.: Markov chain Monte Carlo maximum likelihood. In: Computing Science and Statistics: Proc. 23rd Symp. on the Interface, pp. 156–163 (1991)

    Google Scholar 

  29. 29.

    Grosberg, A.Y.: Critical exponents for random knots. Phys. Rev. Lett. 85(18), 3858–3861 (2000)

    Article  ADS  Google Scholar 

  30. 30.

    Grosberg, A.Y.: Total curvature and total torsion of a freely jointed circular polymer with n≫1 segments. Macromolecules 41(12), 4524–4527 (2008)

    Article  ADS  Google Scholar 

  31. 31.

    Haahr, M.: True random number service. http://www.random.org. Cited Nov 2007

  32. 32.

    Hoste, J.: The enumeration and classification of knots and links. In: Handbook of Knot Theory, pp. 209–232. Elsevier, Amsterdam (2005)

    Google Scholar 

  33. 33.

    Hoste, J., Thistlethwaite, M., Weeks, J.: The first 1,701,936 knots. Math. Intell. 20(4), 33–48 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  34. 34.

    Kapovich, M., Millson, J.J.: The symplectic geometry of polygons in Euclidean space. J. Differ. Geom. 44, 479–513 (1996)

    MATH  MathSciNet  Google Scholar 

  35. 35.

    Kirpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  36. 36.

    Klenin, K.V., Vologodskii, A.V., Anshelevich, V.V., Dykhne, A.M., Frank-Kamenetskii, M.D.: Effect of excluded volume on topological properties of circular DNA. J. Biomol. Struct. Dyn. 5, 1173–1185 (1988)

    Google Scholar 

  37. 37.

    Lal, M.: Monte Carlo computer simulation of chain molecules. I. Mol. Phys. 17, 57–64 (1969)

    Article  ADS  Google Scholar 

  38. 38.

    Larsen, R.J., Marx, M.L.: An Introduction to Mathematical Statistics and Its Applications, 3rd edn. Prentice Hall, Upper Saddle River (2001)

    Google Scholar 

  39. 39.

    Le Bret, M.: Monte Carlo computation of the supercoiling energy. The sedimentation constant, and the radius of gyration of unknotted and knotted circular DNA. Biopolymers 19(3), 619–637 (1980)

    Article  Google Scholar 

  40. 40.

    Madras, N., Slade, G.: The Self-Avoiding Walk. Birkhäuser, Boston (1996)

    Google Scholar 

  41. 41.

    Madras, N., Sokal, A.D.: The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk. J. Stat. Phys. 50, 109–186 (1988)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  42. 42.

    Marenduzzo, D., Orlandini, E., Stasiak, A., Sumners, D.W., Tubiana, L., Micheletti, C.: Application of the Monte Carlo method to the lattice-gas model. Proc. Natl. Acad. Sci. USA 106, 22269–22274 (2009)

    Article  ADS  Google Scholar 

  43. 43.

    Metropolis, N.C., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953)

    Article  ADS  Google Scholar 

  44. 44.

    Micheletti, C., Marenduzzo, D., Orlandini, E., Sumners, D.W.: Application of the Monte Carlo method to the lattice-gas model. J. Chem. Phys. 124, 64903-1-10 (2006)

    Google Scholar 

  45. 45.

    Micheletti, C., Marenduzzo, D., Orlandini, E., Sumners, D.W.: Application of the Monte Carlo method to the lattice-gas model. Biophys. J. 95, 3591–3599 (2008)

    Article  ADS  Google Scholar 

  46. 46.

    Millett, K.C.: Knotting of regular polygons in 3-space. J. Knot Theory Ramif. 3(3), 263–278 (1994). Random knotting and linking (Vancouver, BC, 1993)

    Article  MATH  MathSciNet  Google Scholar 

  47. 47.

    Millett, K.C.: Monte Carlo explorations of polygonal knot spaces. In: Knots in Hellas ’98 (Delphi). Ser. Knots Everything, vol. 24, pp. 306–334. World Sci. Publ., River Edge (2000)

    Google Scholar 

  48. 48.

    Millett, K.C.: An investigation of equilateral knot spaces and ideal physical knot configurations. In: Physical Knots: Knotting, Linking, and Folding Geometric Objects in ℝ3. Contemp. Math., vol. 304, pp. 77–91. Amer. Math. Soc., Providence (2002)

    Google Scholar 

  49. 49.

    Millett, K.C., Rawdon, E.J.: Energy ropelength, and other physical aspects of equilateral knots. J. Comput. Phys. 186(2), 426–456 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  50. 50.

    Millett, K.C., Dobay, A., Stasiak, A.: Linear random knots and their scaling behavior. Macromolecules 38(2), 601–606 (2005)

    Article  ADS  Google Scholar 

  51. 51.

    Millett, K.C., Piatek, M., Rawdon, E.J.: Polygonal knot space near ropelength-minimized knots. J. Knot Theory Ramif. 17(5), 601–631 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  52. 52.

    Millett, K.C., Rawdon, E.J., Tran, V.T., Stasiak, A.: Symmetry-breaking in cumulative measures of shapes of polymer models. J. Chem. Phys. 133(15), 154113 (2010). Also cross-listed in the Virtual Journal of Biological Physics Research in the November 1, 2010 issue (volume 20, issue 9)

    Article  ADS  Google Scholar 

  53. 53.

    Moore, N.T., Grosberg, A.Y.: Limits of analogy between self-avoidance and topology-driven swelling of polymer loops. Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 72(6), 061803-1-10 (2005)

    Article  Google Scholar 

  54. 54.

    Moore, N.T., Grosberg, A.Y.: The abundance of unknots in various models of polymer loops. J. Phys. A, Math. Gen. 39, 9081–9092 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  55. 55.

    Moore, N.T., Lua, R.C., Grosberg, A.Y.: Topologically driven swelling of a polymer loop. Proc. Natl. Acad. Sci. USA 101(37), 13431–13435 (2004)

    Article  ADS  Google Scholar 

  56. 56.

    Moore, N.T., Lua, R.C., Grosberg, A.Y.: Under-knotted and over-knotted polymers: 1. Unrestricted loops. In: Calvo, J.A., Millett, K.C., Rawdon, E.J., Stasiak, A. (eds.) Physical and Numerical Models in Knot Theory. Ser. Knots Everything, vol. 36, pp. 363–384. World Sci. Publishing, Singapore (2005)

    Google Scholar 

  57. 57.

    Moore, N.T., Lua, R.C., Grosberg, A.Y.: Under-knotted and over-knotted polymers: 2. Compact self-avoiding loops. In: Calvo, J.A., Millett, K.C., Rawdon, E.J., Stasiak, A. (eds.) Physical and Numerical Models in Knot Theory. Ser. Knots Everything, vol. 36, pp. 385–398. World Sci. Publishing, Singapore (2005)

    Google Scholar 

  58. 58.

    Orlandini, E., Whittington, S.: Statistical topology of closed curves: Some applications in polymer physics. Rev. Mod. Phys. 79, 611–642 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  59. 59.

    Orlandini, E., Tesi, M.C., Janse van Rensburg, E.J., Whittington, S.G.: Asymptotics of knotted lattice polygons. J. Phys. A 31(28), 5953–5967 (1998)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  60. 60.

    Plunkett, P., Piatek, M., Dobay, A., Kern, J.C., Millett, K.C., Stasiak, A., Rawdon, E.J.: Total curvature and total torsion of knotted polymers. Macromolecules 40(10), 3860–3867 (2007)

    Article  ADS  Google Scholar 

  61. 61.

    Rawdon, E.J., Scharein, R.G.: Upper bounds for equilateral stick numbers. In: Physical Knots: Knotting, Linking, and Folding Geometric Objects in ℝ3, Las Vegas, NV, 2001. Contemp. Math., vol. 304, pp. 55–75. Amer. Math. Soc., Providence (2002)

    Google Scholar 

  62. 62.

    Rawdon, E.J., Dobay, A., Kern, J.C., Millett, K.C., Piatek, M., Plunkett, P., Stasiak, A.: Scaling behavior and equilibrium lengths of knotted polymers. Macromolecules 41(12), 4444–4451 (2008)

    Article  ADS  Google Scholar 

  63. 63.

    Rawdon, E.J., Kern, J.C., Piatek, M., Plunkett, P., Millett, K.C., Stasiak, A.: The effect of knotting on the shape of polymers. Macromolecules 41, 8281–8287 (2008)

    Article  ADS  Google Scholar 

  64. 64.

    Rivetti, C., Walker, C., Bustamante, C.: Polymer chain statistics and conformational analysis of DNA molecules with bends or sections of different flexibility. J. Mol. Biol. 280, 058102 (1988)

    Google Scholar 

  65. 65.

    Salzburg, Z.W., Jacobson, J.D., Fickett, W., Wood, W.W.: Application of the Monte Carlo method to the lattice-gas model. J. Chem. Phys. 30, 65 (1959)

    Article  ADS  Google Scholar 

  66. 66.

    Shimamura, M.K., Deguchi, T.: Anomalous finite-size effects for the mean-squared gyration radius of Gaussian random knots. J. Phys. A 35(18), 241–246 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  67. 67.

    Shimamura, M.K., Deguchi, T.: Finite-size and asymptotic behaviors of the gyration radius of knotted cylindrical self-avoiding polygons. Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 65(5), 051802 (2002)

    Article  ADS  Google Scholar 

  68. 68.

    Tesi, M.C., Janse Van Rensburg, E.J., Orlandini, E., Whittington, S.G.: Monte Carlo study of the interacting self-avoiding walk model in three dimensions. J. Stat. Phys. 82, 155–181 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  69. 69.

    Toussaint, G.: The Erdös-Nagy theorem and its ramifications. Comput. Geom. 31, 219–236 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  70. 70.

    Valle, E., Favre, M., De Los Rios, A., Rosa, A., Dietler, G.: Scaling exponents and probability distributions of DNA end-to-end distance. Phys. Rev. Lett. 95, 158105 (2006)

    Article  ADS  Google Scholar 

  71. 71.

    Vanderzande, C.: Lattice Models of Polymers. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  72. 72.

    Vologodskii, A.V., Anshelevich, V.V., Lukashin, A.V., Frank-Kamenetskii, M.D.: Statistical mechanics of supercoils and the torsional stiffness of the DNA double helix. Nature 280, 294–298 (1979)

    Article  ADS  Google Scholar 

  73. 73.

    Wolfram Research, Inc.: Date and time functions. In: Wolfram Mathematica 7 Documentation. http://reference.wolfram.com/mathematica/tutorial/DateAndTimeFunctions.html. Cited Jan. 2010

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jorge Alberto Calvo.

Electronic Supplementary Material

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Alvarado, S., Calvo, J.A. & Millett, K.C. The Generation of Random Equilateral Polygons. J Stat Phys 143, 102–138 (2011). https://doi.org/10.1007/s10955-011-0164-4

Download citation

Keywords

  • Knot space
  • Probability of knotting
  • Monte Carlo method
  • Pivot method
  • Polygonal folding
  • Crankshaft rotations
  • Hedgehog method
  • Triangle method