DOI QR코드

DOI QR Code

Mechanics of lipid membranes subjected to boundary excitations and an elliptic substrate interactions

  • Kim, Chun IL
  • Received : 2015.12.21
  • Accepted : 2016.06.28
  • Published : 2016.07.25

Abstract

We present relatively simple derivations of the Helfrich energy potential that has been widely adopted in the analysis of lipid membranes without detailed explanations. Through the energy variation methods (within the limit of Helfrich energy potential), we obtained series of analytical solutions in the case when the lipid membranes are excited through their edges. These affordable solutions can be readily applied in the related membrane experiments. In particular, it is shown that, in case of an elliptic cross section of a rigid substrate differing slightly from a circle and subjected to the incremental deformations, exact analytical expressions describing deformed configurations of lipid membranes can be obtained without the extensive use of Mathieu's function.

Keywords

lipid membranes;bilayers;shape equation;substrate-membrane interaction;elliptical contact domain;analytic solution

References

  1. Agrawal, A. and Steigmann, D.J. (2009), "Modeling protein-mediated morphology in biomembranes", Biomech. Model. Mechanobio., 8(5), 371-379. https://doi.org/10.1007/s10237-008-0143-0
  2. Agrawal, A. and Steigmann, D.J. (2009), "Boundary-value problems in the theory of lipid membranes", Continuum Mech. Therm., 21(1), 57-82. https://doi.org/10.1007/s00161-009-0102-8
  3. Belay, T., Kim, C.I. and Schiavone, P. (2016), "Analytical solution of lipid membrane morphology subjected to boundary forces on the edges of rectangular membranes", Continuum Mech. Therm., 28(1-2), 305-315. https://doi.org/10.1007/s00161-015-0426-5
  4. Belay, T., Kim, C.I. and Schiavone, P. (2016), "Interaction-induced morphological transitions of lipid membranes in contact with an elliptical cross section of a rigid substrate", J. Appl. Mech., 83(1), 011001.
  5. Helfrich, W. (1973), "Elastic properties of lipid bilayers: theory and possible experiments", Zeitschrift fur Naturforschung, 28(11), 693-703.
  6. Hianik, T. and Passechnik, V.I. (1995), "Bilayer lipid membranes. Structure and mechanical properties", Springer Science & Business Media.
  7. Kim, C.I. and Steigmann, D.J. (2015), "Distension-induced gradient capillarity in lipid membranes", Continuum Mech. Therm., 27(4-5), 609-621. https://doi.org/10.1007/s00161-014-0333-1
  8. Kalandiya, A. I. (1975), "Mathematical methods of two-dimensional elasticity", MIR, Moscow.
  9. Muskhelishvili, N.I. (1953), "Some basic problems of the mathematical theory of elasticity", Noordhoff, Groningen, The Netherlands.
  10. Mindlin, R.D. and Tiersten, H.F. (1962), "Effects of couple-stresses in linear elasticity", Arch. Ration. Mech. Anal., 11(1), 415-448. https://doi.org/10.1007/BF00253946
  11. Zhong-Can, O.Y., Ji-Xing, L. and Yu-Zhang, X. (1999), Geometric methods in the elastic theory of membranes in liquid crystal phases, 2, World Scientific.
  12. Rosso, R. and Virga, E.G. (1999), "Adhesive borders of lipid membranes", Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 455(1992), 4145-4168, The Royal Society. https://doi.org/10.1098/rspa.1999.0495
  13. Sokolnikoff, I.S. (1951), Tesnsor analysis: Theory and applications, John Wiley & Sons, Inc, New York.
  14. Steigmann, D.J. (1999), "Fluid films with curvature elasticity", Arch. Ration. Mech. Anal., 150, 127-152. https://doi.org/10.1007/s002050050183
  15. Steigmann, D.J., Baesu, E., Rudd, R.E., Belak, J. and McElfresh, M. (2003), "On the variational theory of cell-membrane equilibria", Interfaces Free Bound., 5, 357-366.
  16. Steigmann, D.J. (2013), "A model for lipid membranes with tilt and distension based on three-dimensional liquid crystal theory", Int .J. Non-Linear Mech., 56, 61-70. https://doi.org/10.1016/j.ijnonlinmec.2013.02.006
  17. Truesdell, C. and Toupin, R.A. (1960), "The classical field theories", Flugge, S (ed.) Handbuch der Physik, 3(1), 226-902.

Cited by

  1. A discussion on the mechanics of lipid membranes: Lagrange multipliers and a singular substrate vol.68, pp.4, 2017, https://doi.org/10.1007/s00033-017-0825-5

Acknowledgement

Supported by : Natural Sciences and Engineering Research Council of Canada