Journal of Mathematical Biology

, Volume 68, Issue 3, pp 647–665 | Cite as

Microphysical derivation of the Canham–Helfrich free-energy density

  • Brian Seguin
  • Eliot FriedEmail author


The Canham–Helfrich free-energy density for a lipid bilayer has drawn considerable attention. Aside from the mean and Gaussian curvatures, this free-energy density involves a spontaneous mean-curvature that encompasses information regarding the preferred, natural shape of the lipid bilayer. We use a straightforward microphysical argument to derive the Canham–Helfrich free-energy density. Our derivation (1) provides a justification for the common assertion that spontaneous curvature originates primarily from asymmetry between the leaflets comprising a bilayer and (2) furnishes expressions for the splay and saddle-splay moduli in terms of derivatives of the underlying potential.


Lipid bilayer Biomembrane Canham–Helfrich energy 

Mathematics Subject Classification




We thank Mohsen Maleki for very fruitful discussions.


  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 26:61–81CrossRefGoogle Scholar
  2. Cosserat E, Cosserat F (1909) Théorie des Corps Deformables. Herman et fils, ParisGoogle Scholar
  3. Döbereiner HG, Selchow O, Lipowsky R (1999) Spontaneous curvature of fluid vesicles induced by trans-bilayer sugar asymmetry. Eur Biophys J 28:174–178CrossRefGoogle Scholar
  4. Föppl A (1907) Vorlesungen über technische Mechanik, Bd. 5, Die wichtigsten Lehren der höheren Elastizitätstheorie. Teubner, LeipzigGoogle Scholar
  5. Germain S (1821) Recherches sur la Théorie des Surfaces Élastique. Huzard-Courcier, ParisGoogle Scholar
  6. Gurtin ME, Fried E, Anand L (2010) The Mechanics and Thermodynamics of Continua. Cambridge University Press, New YorkCrossRefGoogle Scholar
  7. Helfrich W (1973) Elastic properties of lipid bilayers: Theory and possible experiments. Zeitschrift für Naturforschung 28c:693–703Google Scholar
  8. Keller JB, Merchant GJ (1991) Flexural rigidity of a liquid surface. J Stat Phys 63:1039–1051CrossRefMathSciNetGoogle Scholar
  9. Lasic DD (1988) The mechanism of liposome formation. A review. Biochem J 256:1–11Google Scholar
  10. Lipowsky R (1990) Shape fluctuations and critical phenomena. In: van Beijeren H (ed) Fundamental Problems in Statistical Mechanics VII. North-Holland, Amsterdam, pp 139–170Google Scholar
  11. Ljunggren S, Eriksson JC (1985) Comments on the origin of the curvature elasticity of vesicle bilayers. J Colloid Interface Sci 107:138–145Google Scholar
  12. Luisi PL, Walade P (2000) Giant Vesicles. Wiley, ChichesterGoogle Scholar
  13. Maïer W, Saupe A (1958) Eine einfache molekulare Theorie des nematischen Kristallinflüssigen Zustands. Zeitschrift für Naturforschg 13a:564–566Google Scholar
  14. Maleki M, Seguin B, Fried E (2012) Kinematics, material symmetry, and energy densities for lipid bilayers with spontaneous curvature. Biomech Model Mechanobiol. doi: 10.1007/s10237-012-0459-7
  15. McMahon HT, Gallop JL (2005) Membrane curvature and mechanisms of dynamic cell membrane remodelling. Nature 438:590–596CrossRefGoogle Scholar
  16. Paunov VN, Sandler SI, Kaler EW (2000) A simple molecular model for the spontaneous curvature and the bending constants of nonionic surfactant monolayers at the oil/water interface. Langmuir 16:8917–8925CrossRefGoogle Scholar
  17. Poisson SD (1812) Mémoire sur les surfaces élastiques. Mémoire de Classe des Sciences Mathématiques et Physiques de I’Institut de France 2nd pt, pp 167–225Google Scholar
  18. Sackmman E (1994) The seventh Datta Lecture. Membrane bending energy concept of vesicle- and cell-shapes and shape-transitions. FEBS Lett 346:3–16CrossRefGoogle Scholar
  19. Safran SA (1994) Statistical Thermodynamics of Surfaces, Interfaces, and Membranes. Addison-Wesley, ReadingGoogle Scholar
  20. Seguin B, Fried E (2012) Statistical foundations of liquid-crystal theory I. Discrete systems of rod-like molecules. Arch Ration Mech Anal 206:1039–1072CrossRefzbMATHMathSciNetGoogle Scholar
  21. Seifert U (1997) Configurations of fluid membranes and vesicles. Adv Phys 46:13–137CrossRefGoogle Scholar
  22. Steigmann DJ (1999) Fluid films with curvature elasticity. Arch Ration Mech Anal 150:127–152CrossRefzbMATHMathSciNetGoogle Scholar
  23. von Kámán Th (1910) Festigkeitsprobleme im Maschinenbau. In: Klein F, Müller C (eds) Encyklopädia der mathematischen Wissenschaften IV/4. Teubner, Berlin, pp 311–385Google Scholar
  24. Winterhalter M, Helfrich W (1988) Effect of surface charge on the curvature elasticity of membranes. J Phys Chem 92:6865–6867Google Scholar
  25. Yuan H (2010) A solvent-free coarse-grained model for biological and biomimetic fluid membranes. Ph. D. Thesis, Pennsylvania State UniversityGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMontrealCanada
  2. 2.Department of Mechanical EngineeringMontrealCanada

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