Skip to main content
SpringerLink
Log in

The Study of Complex Shapes of Fluid Membranes, the Helfrich Functional and New Applications

  • Conference paper
  • First Online:
Mathematical Analysis, Probability and Applications – Plenary Lectures (ISAAC 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 177))

  • 1025 Accesses

Abstract

The theoretical study of complex configurations of fluid membranes is reported on the basis of the Helfrich functional. Series of analytical results on the governing equations of closed lipid vesicles and open lipid vesicles with holes are surveyed. The concepts of stress tensor and moment tensor in fluid membranes are investigated from two different viewpoints: the balance of forces (moments) and the generalized variational principle of free energy. Several examples on new applications of the Helfrich functional in understanding the growth mechanism of some mesoscopic structures are illustrated.

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

Access this chapter

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 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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

Institutional subscriptions

References

  1. Plateau, J.: Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires. Gauthier-Villars, Paris (1873)

    Google Scholar 

  2. Young, T.: An essay on the cohesion of fluids. Philos. Trans. R. Soc. Lond. 95, 65–87 (1805)

    Article  Google Scholar 

  3. Laplace, P.: Traité de Mécanique Céleste. Gauthier-Villars, Paris (1839)

    Google Scholar 

  4. Alexandrov, A.: Uniqueness theorems for surfaces in the large. Amer. Math. Soc. transl. 21, 341–416 (1962)

    Article  MathSciNet  Google Scholar 

  5. Poisson, S.: Traité de Mécanique. Bachelier, Paris (1833)

    Google Scholar 

  6. Willmore, T.: Total Curvature in Riemannian Geometry. Wiley, New York (1982)

    MATH  Google Scholar 

  7. Marques, F., Neves, A.: The Willmore conjecture. Jahresber. Dtsch. Math-Ver. 116, 201–222 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Willmore, T.: Note on embedded surfaces. An. Ştiinţ. Univ. ‘Al.I. Cuza’ Iaşi, Mat. (N.S.) B 11, 493–496 (1965)

    Google Scholar 

  9. Marques, F., Neves, A.: Min-max theory and the Willmore conjecture. Ann. Math. 179, 683–782 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Canham, P.: The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theor. Biol. 26, 61–81 (1970)

    Article  Google Scholar 

  11. Singer, S., Nicolson, G.: The fluid mosaic model of cell membranes. Science 175, 720–731 (1972)

    Article  Google Scholar 

  12. Helfrich, W.: Elastic properties of lipid bilayers-theory and possible experiments. Z. Naturforsch. C 28, 693–703 (1973)

    Google Scholar 

  13. Deuling, H., Helfrich, W.: Red blood cell shapes as explained on the basis of curvature elasticity. Biophys. J. 16, 861–868 (1976)

    Article  Google Scholar 

  14. Lipowsky, R.: The conformation of membranes. Nature 349, 475–481 (1991)

    Article  Google Scholar 

  15. Ou-Yang, Z., Liu, J., Xie, Y.: Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases. World Scientific, Singapore (1999)

    Book  MATH  Google Scholar 

  16. Seifert, U.: Configurations of fluid membranes and vesicles. Adv. Phys. 46, 13–137 (1997)

    Article  Google Scholar 

  17. Chern, S., Chen, W.: Lecture on Differential Geometry. Beijing University Press, Beijing (1983)

    Google Scholar 

  18. Tu, Z., Ou-Yang, Z.: Lipid membranes with free edges. Phys. Rev. E 68, 061915 (2003)

    Article  Google Scholar 

  19. Tu, Z., Ou-Yang, Z.: A geometric theory on the elasticity of bio-membranes. J. Phys. A Math. Gen. 37, 11407–11429 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tu, Z., Ou-Yang, Z.: Elastic theory of low-dimensional continua and its applications in bio- and nano-structures. J. Comput. Theor. Nanosci. 5, 422–448 (2008)

    Article  Google Scholar 

  21. Westenholz, C.: Differential Forms in Mathematical Physics. North-Holland, Amsterdam (1981)

    MATH  Google Scholar 

  22. Ou-Yang, Z., Helfrich, W.: Instability and deformation of a spherical vesicle by pressure. Phys. Rev. Lett. 59, 2486–2488 (1987)

    Article  Google Scholar 

  23. Ou-Yang, Z., Helfrich, W.: Bending energy of vesicle membranes: general expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. Phys. Rev. A 39, 5280–5288 (1989)

    Article  Google Scholar 

  24. Hu, J., Ou-Yang, Z.: Shape equations of the axisymmetric vesicles. Phys. Rev. E 47, 461–467 (1993)

    Article  Google Scholar 

  25. Zheng, W., Liu, J.: Helfrich shape equation for axisymmetric vesicles as a first integral. Phys. Rev. E 48, 2856–2860 (1993)

    Article  Google Scholar 

  26. Naito, H., Okuda, M., Ou-Yang, Z.: New solutions to the helfrich variation problem for the shapes of lipid bilayer vesicles: beyond delaunay’s surfaces. Phys. Rev. Lett. 74, 4345–4348 (1995)

    Article  Google Scholar 

  27. Mladenov, I.: New solutions of the shape equation. Eur. Phys. J. B 29, 327–330 (2002)

    Article  Google Scholar 

  28. Ou-Yang, Z.: Anchor ring-vesicle membranes. Phys. Rev. A 41, 4517–4520 (1990)

    Article  Google Scholar 

  29. Ou-Yang, Z.: Selection of toroidal shape of partially polymerized membranes. Phys. Rev. E 47, 747–749 (1993)

    Article  Google Scholar 

  30. Castro-Villarreal, P., Guven, J.: Inverted catenoid as a fluid membrane with two points pulled together. Phys. Rev. E 76, 011922 (2007)

    Article  Google Scholar 

  31. Zhang, S., Ou-Yang, Z.: Periodic cylindrical surface solution for fluid bilayer membranes. Phys. Rev. E 53, 4206–4208 (1996)

    Article  Google Scholar 

  32. Vassilev, V., Djondjorov, P., Mladenov, I.: Cylindrical equilibrium shapes of fluid membranes. J. Phys. A Math. Theor. 41, 435201 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhou, X.: Periodic-cylinder vesicle with minimal energy. Chin. Phys. B 19, 058702 (2010)

    Article  Google Scholar 

  34. Naito, H., Okuda, M., Ou-Yang, Z.: Counterexample to some shape equations for axisymmetric vesicles. Phys. Rev. E 48, 2304–2307 (1993)

    Article  Google Scholar 

  35. Naito, H., Okuda, M., Ou-Yang, Z.: Polygonal shape transformation of a circular biconcave vesicle induced by osmotic pressure. Phys. Rev. E 54, 2816–2826 (1996)

    Article  Google Scholar 

  36. Mutz, M., Bensimon, D.: Observation of toroidal vesicles. Phys. Rev. A 43, 4525–4527 (1991)

    Article  Google Scholar 

  37. Seifert, U.: Vesicles of toroidal topology. Phys. Rev. Lett. 66, 2404–2407 (1991)

    Article  Google Scholar 

  38. Fourcade, B., Mutz, M., Bensimon, D.: Experimental and theoretical study of toroidal vesicles. Phys. Rev. Lett. 68, 2551–2554 (1992)

    Article  Google Scholar 

  39. Evans, E., Fung, Y.: Improved measurements of the erythrocyte geometry. Microvasc. Res. 4, 335–347 (1972)

    Article  Google Scholar 

  40. Saitoh, A., Takiguchi, K., Tanaka, Y., Hotani, H.: Opening-up of liposomal membranes by Talin. Proc. Natl. Acad. Sci. 95, 1026–1031 (1998)

    Article  Google Scholar 

  41. Capovilla, R., Guven, J., Santiago, J.: Lipid membranes with an edge. Phys. Rev. E 66, 021607 (2002)

    Article  Google Scholar 

  42. Tu, Z.: Compatibility between shape equation and boundary conditions of lipid membranes with free edges. J. Chem. Phys. 132, 084111 (2010)

    Article  Google Scholar 

  43. Umeda, T., Suezaki, Y., Takiguchi, K., Hotani, H.: Theoretical analysis of opening-up vesicles with single and two holes. Phys. Rev. E 71, 011913 (2005)

    Article  Google Scholar 

  44. Wang, X., Du, Q.: Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches. J. Math. Biol. 56, 347–371 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  45. Tu, Z.: Geometry of membranes. J. Geom. Symmetry Phys. 24, 45–75 (2011)

    MathSciNet  MATH  Google Scholar 

  46. Tu, Z.: Challenges in theoretical investigations of configurations of lipid membranes. Chin. Phys. B 22, 028701 (2013)

    Article  Google Scholar 

  47. Tu, Z., Ou-Yang, Z.: Recent theoretical advances in elasticity of membranes following Helfrich’s spontaneous curvature model. Adv. Colloid Interface Sci. 208, 66–75 (2014)

    Article  Google Scholar 

  48. Koch, E., Fischer, W.: Flat points of minimal balance surfaces. Acta Cryst. A 46, 33–40 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  49. Giomi, L., Mahadevan, L.: Minimal surfaces bounded by elastic lines. Proc. R. Soc. A 468, 1851–1864 (2012)

    Article  MathSciNet  Google Scholar 

  50. Capovilla, R., Guven, J.: Stresses in lipid membranes. J. Phys. A: Math. Gen. 35, 6233–6247 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  51. Müller, M., Deserno, M., Guven, J.: Interface-mediated interactions between particles: a geometrical approach. Phys. Rev. E 72, 061407 (2005)

    Article  MathSciNet  Google Scholar 

  52. Müller, M., Deserno, M., Guven, J.: Balancing torques in membrane-mediated interactions: exact results and numerical illustrations. Phys. Rev. E 76, 011921 (2007)

    Article  Google Scholar 

  53. Deserno, M.: Fluid lipid membranes: from differential geometry to curvature stresses. Chem. Phys. Lipids 185, 11–45 (2015)

    Article  Google Scholar 

  54. Yang, P., Tu, Z.: General neck condition for the limit shape of budding vesicles. arXiv:1508.02151

  55. Jülicher, F., Lipowsky, R.: Shape transformations of vesicles with intramembrane domains. Phys. Rev. E 53, 2670–2683 (1996)

    Article  Google Scholar 

  56. Du, Q., Guven, J., Tu, Z., Vázquez-Montejo, P.: Fluid membranes bounded by semi-flexible polymers (in preparation)

    Google Scholar 

  57. Naito, H., Okuda, M., Ou-Yang, Z.: Equilibrium shapes of smectic-A phase grown from isotropic phase. Phys. Rev. Lett. 70, 2912–2915 (1993)

    Article  Google Scholar 

  58. Naito, H., Okuda, M., Ou-Yang, Z.: Preferred equilibrium structures of a smectic-A phase grown from an isotropic phase: origin of focal conic domains. Phys. Rev. E 52, 2095–2098 (1995)

    Article  Google Scholar 

  59. Ou-Yang, Z., Su, Z., Wang, C.: Coil formation in multishell carbon nanotubes: competition between curvature elasticity and interlayer adhesion. Phys. Rev. Lett. 78, 4055–4058 (1997)

    Article  Google Scholar 

  60. Yan, X., Cui, Y., He, Q., Wang, K., Li, J., Mu, W., Wang, B., Ou-Yang, Z.: Reversible transitions between peptide nanotubes and vesicle-like structures including theoretical modeling studies. Chem. Eur. J. 14, 5974–5980 (2008)

    Article  Google Scholar 

  61. Friedel, G.: Les états mésomorphes de la matiére. Ann. Phys. 18, 273–474 (1922)

    Google Scholar 

  62. Bragg, W.: Liquid crystals. Nature 133, 445–456 (1934)

    Article  Google Scholar 

  63. Langer, J., Singer, D.: The total squared curvature of closed curves. J. Differ. Geom. 20, 1–22 (1984)

    MathSciNet  MATH  Google Scholar 

  64. Zhang, X., Zhang, X., Bernaerts, D., Vantendeloo, G., Amelinckx, S., Vanlanduyt, J., Ivanov, V., Nagy, J., Lambin, P., Lucas, A.: The texture of catalytically grown coil-shaped carbon nanotubules. Europhys. Lett. 27, 141–146 (1994)

    Article  Google Scholar 

  65. Sabitov, I.: Some integral formulas for compact surfaces. TWMS J. Pure Appl. Math. 1, 123–131 (2010)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the financial support from the National Natural Science Foundation of China (Grant Nos. 11274046 and 10704009).

Author information

Authors and Affiliations

  1. Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100080, China

    Zhong-Can Ou-Yang

  2. Department of Physics, Beijing Normal University, Beijing, 100875, China

    Zhan-Chun Tu

Authors
  1. Zhong-Can Ou-Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhan-Chun Tu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhan-Chun Tu .

Editor information

Editors and Affiliations

  1. Faculty of Science and Technology, University of Macau, Taipa, Macao

    Tao Qian

  2. Department of Mathematics, University of Torino, Torino, Italy

    Luigi G. Rodino

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Ou-Yang, ZC., Tu, ZC. (2016). The Study of Complex Shapes of Fluid Membranes, the Helfrich Functional and New Applications. In: Qian, T., Rodino, L. (eds) Mathematical Analysis, Probability and Applications – Plenary Lectures. ISAAC 2015. Springer Proceedings in Mathematics & Statistics, vol 177. Springer, Cham. https://doi.org/10.1007/978-3-319-41945-9_4

Download citation

Publish with us

Policies and ethics

Search

Navigation