Journal of Mathematical Chemistry

, Volume 54, Issue 2, pp 358–374 | Cite as

Surface tension and Laplace pressure in triangulated surface models for membranes without fixed boundary

  • Hiroshi Koibuchi
  • Andrey Shobukhov
  • Hideo Sekino
Original Paper


A Monte Carlo (MC) study is performed to evaluate the surface tension \(\gamma \) of spherical membranes that may be regarded as the models of the lipid layers. We use the canonical surface model defined on the self-avoiding triangulated lattices. The surface tension \(\gamma \) is calculated by keeping the total surface area A constant during the MC simulations. In the evaluation of \(\gamma \), we use A instead of the projected area \(A_p\), which is unknown due to the fluctuation of the spherical surface without boundary. The pressure difference \({\varDelta }p \) between the inner and the outer sides of the surface is also calculated by maintaining the enclosed volume constant. Using \({\varDelta }p \) and the Laplace formula, we obtain the tension, which is considered to be equal to the frame tension \(\tau \) conjugate to \(A_p\), and check whether or not \(\gamma \) is consistent with \(\tau \). We find reasonable consistency between \(\gamma \) and \(\tau \) in the region of sufficiently large bending rigidity \(\kappa \) or sufficiently large A / N. It is also found that \(\tau \) becomes constant in the limit of \(A{/}N\rightarrow \infty \) both in the tethered and fluid surfaces.


Surface tension Frame tension Membranes Laplace formula 



This work is supported in part by the Grant-in-Aid for Scientific Research (C) Number 26390138. We acknowledge the support of the Promotion of Joint Research 2014, Toyohashi University of Technology. We are grateful to K. Osari and S. Usui for the computer analyses.


  1. 1.
    H.L. Scott, Lipid–cholesterol interactions: Monte Carlo simulations and theory. Biophys. J. 59, 445–455 (1991)CrossRefGoogle Scholar
  2. 2.
    R.W. Pastor, Molecular dynamics and Monte Carlo simulations of lipid bilayers. Curr. Opin. Struct. Biol. 4, 486–492 (1994)CrossRefGoogle Scholar
  3. 3.
    K.M. Merz, Molecular dynamics simulations of lipid bilayers. Curr. Opin. Struct. Biol. 7, 511–517 (1997)CrossRefGoogle Scholar
  4. 4.
    D.J. Tobias, K. Tu, M.L. Klein, Atomic scale molecular dynamics simulations of lipid membranes. Curr. Opin. Struct. Biol. 2, 15–26 (1997)Google Scholar
  5. 5.
    J.F. Nagle, S. Tristram-Nagle, Structure of lipid bilayers. Curr. Opin. Struct. Biol. 10, 474–480 (2000)CrossRefGoogle Scholar
  6. 6.
    R.M. Venable, B.R. Brooks, R.W. Pastor, Molecular dynamics simulations of gel phase lipid bilayers in constant pressure and constant surface area ensembles. J. Chem. Phys. 112, 4822–4832 (2000)CrossRefGoogle Scholar
  7. 7.
    S.-W. Chiu, M.M. Clark, S. Subramaniam, H.L. Scott, E. Jakobsson, Incorporation of surface tension into molecular dynamics simulations of an interface: a fluid phase lipid bilayer membrane. Biophys. J. 69, 1230–1245 (1995)CrossRefGoogle Scholar
  8. 8.
    S.-W. Chiu, M. Clark, E. Jakobsson, S. Subramaniam, H.L. Scott, Application of a combined Monte Carlo and molecular dynamics method to the simulation of a dipalmitoyl phosphatidylcholine lipid bilayer. J. Comput. Chem. 20, 1153–1164 (1999)CrossRefGoogle Scholar
  9. 9.
    S.E. Feller, R.W. Pastor, On simulating lipid bilayers with an applied surface tension: periodic boundary conditions and undulations. Biophys. J. 71, 1350–1355 (1996)CrossRefGoogle Scholar
  10. 10.
    E. Lindahl, O. Edholm, Spatial and energetic-entropic decomposition of surface tension in lipid bilayers from molecular dynamics simulations. J. Chem. Phys. 113, 3882–3893 (2000)CrossRefGoogle Scholar
  11. 11.
    S.J. Marrink, A.E. Mark, Effect of undulations on surface tension in simulated bilayers. J. Phys. Chem. B 105, 6122–6127 (2001)CrossRefGoogle Scholar
  12. 12.
    E. Lindahl, O. Edholm, Mesoscopic undulations and thickness fluctuations in lipid bilayers from molecular dynamics simulations. Biophys. J. 79, 426–433 (2000)CrossRefGoogle Scholar
  13. 13.
    W. Cai, T.C. Lubensky, P. Nelson, T. Powers, Measure factors, tension, and correlations of fluid membranes. J. Phys. II Fr. 4, 931 (1994)Google Scholar
  14. 14.
    J. Ambjörn, B. Durhuus, T. Jonsson, Scaling of the string tension in a new class of regularized string theories. Phys. Rev. Lett. 58, 2619–2622 (1987)CrossRefGoogle Scholar
  15. 15.
    J. Ambjörn, A. Irbäck, J. Jurkiewicz, B. Petersson, The theory of dynamical random surfaces with extrinsic curvature. Nucl. Phys. B 393(3), 571–600 (1993)CrossRefGoogle Scholar
  16. 16.
    J.F. Wheater, Random surfaces: from polymer membranes to strings. J. Phys. A Math. Gen. 27, 3323–3353 (1994)CrossRefGoogle Scholar
  17. 17.
    H.-G. Döbereiner, G. Gompper, C.K. Haluska, D.M. Kroll, P.G. Petrov, K.A. Riske, Advanced flicker spectroscopy of fluid membranes. Phys. Rev. Lett. 91, 048301(1–4) (2003)CrossRefGoogle Scholar
  18. 18.
    J.-B. Fournier, C. Barbetta, Direct calculation from the stress tensor of the lateral surface tension of fluctuating fluid membranes. Phys. Rev. Lett. 100, 078103(1–4) (2008)CrossRefGoogle Scholar
  19. 19.
    A. Imparato, Surface tension in bilayer membranes with fixed projected area. J. Chem. Phys. 124, 154714(1–9) (2006)CrossRefGoogle Scholar
  20. 20.
    J. Pécréaux, H.-G. Döbereiner, J. Prost, J.-F. Joanny, P. Bassereau, Refined contour analysis of giant unilamellar vesicles. Euro. Phys. J. E 13, 277–290 (2004)CrossRefGoogle Scholar
  21. 21.
    W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch 28c, 693–703 (1973)Google Scholar
  22. 22.
    A.M. Polyakov, Fine structure of strings. Nucl. Phys. B 268, 406–412 (1986)CrossRefGoogle Scholar
  23. 23.
    D. Nelson, The statistical mechanics of membranes and interfaces, in Statistical Mechanics of Membranes and Surfaces, 2nd edn., ed. by D. Nelson, T. Piran, S. Weinberg (World Scientific, Singapore, 2004), pp. 1–17CrossRefGoogle Scholar
  24. 24.
    F. David, S. Leibler, Vanishing tension of fluctuating membranes. J. Phys. II Fr. 1, 959–976 (1991)Google Scholar
  25. 25.
    R.A. Foty, G. Forgacs, C.M. Pfleger, M.S. Steinberg, Liquid properties of embryonic tissues: measurement of interfacial tensions. Phys. Rev. Lett. 72, 2298–2301 (1994)CrossRefGoogle Scholar
  26. 26.
    R.A. Foty, C.M. Pfleger, G. Forgacs, M.S. Steinberg, Surface tensions of embryonic tissues predict their mutual envelopment behavior. Development 122, 1611–1620 (1996)Google Scholar
  27. 27.
    H. Koibuchi, A. Shobukhov, Surface tension, pressure difference and Laplace formula for membranes. In Proceedings of International Conference on Mathematical Modeling in Physical Sciences 2014, Journal of Physics: Conference Series, vol. 574 (IOP Publishing, Madrid Spain, 2015) p. 012101(1–5)Google Scholar
  28. 28.
    M. Doi, F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, Oxford, 1986)Google Scholar
  29. 29.
    M. Bowick, A. Cacciuto, G. Thorleifsson, A. Travesset, Universality classes of self-avoiding fixed connectivity membranes. Euro. Phys. J. E 5, 149–160 (2001)CrossRefGoogle Scholar
  30. 30.
    G. Gompper, D.M. Kroll, Phase diagram and scaling behavior of fluid vesicles. Phys. Rev. E 51, 514–525 (1995)CrossRefGoogle Scholar
  31. 31.
    F. David, A model of random surfaces with non-trivial critical behavior. Nucl. Phys. B 257(FS14), 543–576 (1985)CrossRefGoogle Scholar
  32. 32.
    H. Koibuchi, A. Shobukhov, Branched-polymer to inflated transition of self-avoiding fluid surfaces. Phys. A 410, 54–65 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Hiroshi Koibuchi
    • 1
  • Andrey Shobukhov
    • 2
  • Hideo Sekino
    • 3
  1. 1.Department of Mechanical and Systems Engineering, National Institute of TechnologyIbaraki CollegeHitachinakaJapan
  2. 2.Faculty of Computational Mathematics and CyberneticsLomonosov Moscow State UniversityMoscowRussia
  3. 3.Computer Science and EngineeringsToyohashi University of TechnologyToyohashiJapan

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