European Biophysics Journal

, Volume 22, Issue 2, pp 97–103 | Cite as

The existence of non-axisymmetric bilayer vesicle shapes predicted by the bilayer couple model

  • Vera Kralj-Iglič
  • Saša Svetina
  • Boštjan Žekš


The existence of non-axisymmetric shapes with minimal bending energy is proved by means of a mathematical model. A parametric model is used; the shapes considered have an elliptical top view whilst their front view contour is described using Cassim ovals. Taking into account the bilayer couple model, the minimization of the membrane bending energy is performed at a constant membrane area A, a constant enclosed volume V and a constant difference between the two membrane leaflet areas ΔA. It is shown that for certain sets of A, V and ΔA the non-axisymmetric shapes calculated with the use of the parametric model have lower energy than the corresponding axisymmetric shapes obtained by the exact solution of the general variational problem. As an exact solution of the general variational problem for non-axisymmetric shapes would yield even lower energy, this indicates the existence of non-axisymmetric shapes with minimal bending energy in a region of the V/4A phase diagram.

Key words

Bilayer couple Membrane Non-axisymmetric shapes Phospholipid vesicles 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Canham PB (1970) The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J Theor Biol 26:61–81Google Scholar
  2. Deuling HL, Helfrich W (1976) The curvature elasticity of fluid membranes: A catalogue of vesicle shapes. J Phys (Paris) 37:1335–1345Google Scholar
  3. Evans E (1974) Bending resistance and chemically induced moments in membrane bilayers. Biophys J 14:923–931Google Scholar
  4. Heinrich V, Brumen M, Heinrich R, Svetina S, Žekš B (1992) Nearly spherical vesicle shapes calculated by use of spherical harmonics: axisymmetric and non-axisymmetric shapes and their stability. J Phys 11 France 2:1081–1108Google Scholar
  5. Hotani H (1984) Transformation pathways of liposomes. J Mol Biol 178:113–120Google Scholar
  6. Khodadad JK, Weinstein RS (1983) The band 3 - rich membrane of llama erythrocytes - studies on cell shape and the organization of membrane proteins. J Membr Biol 72:161–171Google Scholar
  7. Lipowsky R (1991) The conformation of biomembranes. Nature 349: 475–481Google Scholar
  8. Palek J (1987) Hereditary elliptocytosis, spherocytosis and related disorders: consequences of a deficiency or a mutation of membrane skeletal proteins. Blood Rev 1:147–168Google Scholar
  9. Peterson MA (1985) An instability of the red blood cell shape. J Appl Phys 57:1739–1742Google Scholar
  10. Peterson MA (1989) Deformation energy of vesicles at fixed volume and surface area in the spherical limit. Phys Rev A 39:2643–2645Google Scholar
  11. Seifert U, Berndl K, Lipowsky R (1991) Shape transformations of vesicles: phase diagram for spontaneous — curvature and bilayer-coupling models. Phys Rev 44:1182–1202Google Scholar
  12. Sekimura T, Hotani H (1991) The morphogenesis of liposomes viewed from the aspect of bending energy. J Theor Biol 149: 325–337Google Scholar
  13. Sheetz MP, Singer SJ (1974) Biological membranes as bilayer couples. A mechanism of drug-erythrocyte interactions. Proc Natl Acad Sci, USA 71:4457–4461Google Scholar
  14. Svetina S, Žekš B (1989) Membrane bending energy and shape determination of phospholipid vesicles and red blood cells. Eur Biophys J 17:101–111Google Scholar
  15. Svetina S, Ottova-Leitmannova A, Glaser R (1982) Membrane bending energy in relation to bilayer couples concept of red blood cell shape transformations. J Theor Biol 94:13–23Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Vera Kralj-Iglič
    • 1
  • Saša Svetina
    • 2
  • Boštjan Žekš
    • 1
    • 2
  1. 1.Medical FacultyInstitute of BiophysicsLjubljanaSlovenia
  2. 2.J. Stefan InstituteLjubljanaSlovenia

Personalised recommendations