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A conformal variational approach for helices in nature

J. Math. Phys. 50, 103529 (2009); doi:10.1063/1.3236683

Published 26 October 2009

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Manuel Barros1 and Angel Ferrández2
1Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
2Departamento de Matemáticas, Universidad de Murcia Campus de Espinardo, 30100 Murcia, Spain
We propose a two step variational principle to describe helical structures in nature. The first one is governed by an energy action which is a linear function in both curvature and torsion allowing to describe nonclosed structures including elliptical, spherical, and conical helices. These appear as rhumb lines in right cylinders constructed over plane curves. The model is completed with a conformal alternative which, in particular, gives a description of closed structures. The energy action is linear in the curvatures when computed in a conformal spherical metric. Now, helices appear as making a constant angle with a Villarceau flow and so they are loxodromes in surfaces which are stereographic projections of Hopf tubes, in particular, anchor rings, revolution tori, and Dupin cyclides. The model satisfies the requirements of simplicity and beauty as reflected in the three main principles that head its construction: least action, topological, and quantization. According to the latter, the main entities and quantities associated with the model should not be multiplied unnecessarily but they are quantized. In this sense, a quantization principle, a la Dirac, is obtained for closed structures and also for the critical levels of energy. ©2009 American Institute of Physics
History: Received 23 July 2009; accepted 31 August 2009; published 26 October 2009
Permalink: http://link.aip.org/link/?JMAPAQ/50/103529/1
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KEYWORDS and PACS

Keywords
PACS
  • 03.65.-w
    Quantum mechanics
  • 02.30.Xx
    Calculus of variations
  • 02.40.-k
    Geometry, differential geometry, and topology
  • YEAR: 2009

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PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
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REFERENCES (31)
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  1. Arroyo, J., Barros, M., and Garay, O. J., “Models of relativistic particle with curvature and torsion revisited,” Gen. Relativ. Gravit. 36, 1441–1451 (2004).
  2. Balaeff, A., Mahadevan, L., and Schoulten, K., “Elastic rod model of a DNA loop in the Lac Operon,” Phys. Rev. Lett. 83, 4900–4903 (1999).
  3. Barros, M., “General helices and a theorem of Lancret,” Proc. Am. Math. Soc. 125, 1503–1509 (1997).
  4. Besse, A., Einstein Manifolds (Springer, New York, 1987).
  5. Borman, S. and Washington, C., “Tying up loose ends: New examples and applications of circular and knotted peptides and proteins are turning up,” Chem. Eng. News 82, 40–42 (2004).
  6. Cahill, K., e-print arXiv:q-bio.BM/0502043.
  7. Calugareanu, G., “Sur les classes d'isotopie des noeuds tridimensionnels et leurs invariants,” Czech. Math. J. 11, 588–625 (1961).
  8. Feoli, A., Nesterenko, V. V., and Scarpetta, G., “Functionals linear in curvature and statistics of helical proteins,” Nucl. Phys. B 705, 577–592 (2005).
  9. da Fonseca, A. L. and Malta, C. P., e-print arXiv:physics/0507105.
  10. da Fonseca, A. L., Malta, C. P., and Aguiar, M. A. M., “Resonant helical deformations in nonhomogeneous filaments,” Physica A 352, 547–557 (2005).
  11. da Fonseca, A. L., Malta, C. P., and Galvão, D. S., e-print arXiv:cond-mat/0507400).
  12. da Fonseca, A. L., Malta, C. P., and Galvão, D. S., “Elastic properties of nanowires,” J. Appl. Phys. 99, 094310 (2006).
  13. Goldstein, R. E., Goriely, A., Huber, G., and Wolgemuth, C. W., “Bistable helices,” Phys. Rev. Lett. 84, 1631–1634 (2000).
  14. Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. (CRC, Boca Raton, FL, 1998).
  15. Greub, W., Halperin, S., and Vanstone, R., Connections, Curvature and Cohomology (Academic, New York, 1972)
    16. Connections, Curvature and Cohomology (Academic, New York, 1973)
    Connections, Curvature and Cohomology (Academic, New York, 1976).
  16. Ivey, T. A. and Singer, D. A., “Knot types, homotopies and stability of closed elastic rods,” Proc. London Math. Soc. 79, 429–450 (1999).
  17. Langer, J. and Singer, D. A., “Lagrangian aspects of the Kirchhoff elastic rod,” SIAM Rev. 38, 605–618 (1996).
  18. Liu, Z. and Qin, L., “Electron diffraction from elliptical nanotubes,” Chem. Phys. Lett. 406, 106–110 (2005).
  19. Thamwattana, N., McCoy, J. A., and Hill, J. M., “Energy density functions for protein structures,” Q. J. Mech. Appl. Math. 61(3), 431–451 (2008).
  20. McCoy, J. A., “Helices for mathematical modelling of proteins, nucleid acids and polymers,” J. Math. Anal. Appl. 347, 255–265 (2008).
  21. Mendelson, N. H., Sarlls, J. E., Wolgemuth, C. W., and Goldstein, R. E., “Chiral self-propulsion of growing bacterial macrofibers on solid surfaces,” Phys. Rev. Lett. 84, 1627–1630 (2000).
  22. Morozov, V. F., Mamasakhlisov, E. S., Grigoryan, A. V., Badasyan, A. V., Hayryan, S., and Hu, C. -K., “Helix-coil transition in closed circular DNA,” Physica A 348, 327–338 (2005).
  23. O'Neill, B., Semi-Riemannian Geometry (Academic, New York, 1983).
  24. Pinkall, U., “Hopf tori in [openface S]3,” Invent. Math. 81, 379–386 (1985).
  25. Reichert, J. and Sühnel, J., “The IMB Jena Image Library of Biological Molecules:2002 updates,” Nucleic Acids Res. 30, 253–254 (2002).
  26. Reichert, M. and Stark, H., “Synchronization of rotating helices by hydrodynamic interactions,” Eur. Phys. J. E 17, 493–500 (2005).
  27. Urbantke, H. K., “The Hopf fibration, seven times in physics,” J. Geom. Phys. 46, 125–150 (2003).
  28. von Seggern, D. H., CRC Standard Curves and Surfaces (CRC, Boca Raton, FL, 1993).
  29. Weiner, J. L., “How helical can a closed, twisted space curve be?,” Am. Math. Monthly 107, 327–333 (2000).
  30. White, J., “Self-linking and the Gauss integral in higher dimensions,” Am. J. Math. 91, 693–728 (1969).
  31. Wu, Z. and Yung, E. K. N., “Axial mode elliptical cross-section helical antenna,” Microwave Opt. Technol. Lett. 48, 2080–2083 (2006).

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