Elasticity and Hereditariness

  • Luca DeseriEmail author
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 577)


This chapter collects the lecture notes of the module “Elasticity and Hereditatiness of Lipid Bilayers” delivered at CISM in July 2016. Such material is based primarily on three papers coauthored by this lecturer, and which have been contributing to shed light on the mechanical behavior of lipid bilayers. In particular, the breakthrough from this research is that the underlying nonlinear elastic response of lipid bilayers is fully determined as long as the membrane energy is obtained. Bending and saddle splay rigidities are shown here to be directly obtainable from the membranal response, as well as the line tension, holding together domains in which lipids are in different phases. The power law hereditariness of lipid membranes strikingly shown through rheometric tests has been analyzed in this work through a suitable energetics obtained by the author and coworkers and penalizing small perturbations of ground configurations of such systems.


Lipid Membrane Total Potential Energy Line Tension Elastic Case Spontaneous Curvature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The author wishes to thank the organizer of this course, David Steigmann, for his invitation to contribute to this course. The other lecturers are also acknowledged for the nice and extended discussions that allowed for exchange of ideas on the topic of this course.

The author is extremely grateful to Giuseppe Zurlo (National University of Galway, Ireland), formerly his Ph.D. student, for the very extensive discussions and long-standing collaboration from his early days in 2002. His key contribution to this research has had huge impact in its assessment and development. Timothy J. Healey (Cornell University) and Roberto Paroni (University of Sassari, Italy) also gratefully acknowledged for the very extended discussions on the early stages of the 2008 work.

Grateful acknowledgements go to Massimiliano Zingales (University of Palermo), Kaushik Dayal (Carnegie Mellon University) as collaborators on key aspects related to the hereditary response of lipid bilayers. Massimiliano Fraldi (University of Napoli-Federico II) is also gratefully ackowledged for his illuminating remarks and insights on biological tissues and biomechanics, as well as Valentina Piccolo (University of Trento), a graduate student working with myself and other people on various topics, who also provided new perspectives on the applications of Fractional Analysis to lipid membranes and helped a lot to edit this work.

The author is grateful to the financial support provided by (i) the NSF Grant no.DMS-0635983 of the Center for Nonlinear Analysis, Carnegie Mellon University, (ii) for the direct financial support of the Dept. of Mechanical Engineering and Materials Science-MEMS of the University of Pittsburgh for, and also to (iii) the support of the EU Grant “INSTABILITIES” ERC-2013-ADG Instabilities and nonlocal multiscale modelling of materials held by Prof. Davide Bigoni from the University of Trento.


  1. S.A. Akimov, P.I. Kuzmin, J. Zimmerberg, An elastic theory for line tension at a boundary separating two lipid monolayer regions of different thickness. J. Electroanal. Chem. 564, 13–18 (2004)CrossRefGoogle Scholar
  2. G. Alberti, An approach via \(\Gamma -\)convergence, in Calculus of Variations and Partial Differential Equations, Topics on Geometrical Evolution Problems and Degree Theory, ed. by L. Ambrosio, N. Dancer (Springer, Berlin, 2000)Google Scholar
  3. E. Baesu, R.E. Rudd, J. Belak, M. McElfresh, Continuum modeling of cell membranes. Int. J. Non-Linear Mech. 39(3), 369–377 (2004)CrossRefzbMATHGoogle Scholar
  4. T. Baumgart, W.W. Webb, S.T. Hess, Imaging coexisting domains in biomembrane models coupling curvature and line tension. Nature 423, 821–824 (2003)CrossRefGoogle Scholar
  5. H. Bermúdez, D.A. Hammer, D.E. Discher, Effect of bilayer thickness on membrane bending rigidity. Langmuir 20, 540–543 (2004)CrossRefGoogle Scholar
  6. S. Breuer, E. Onat, On the determination of free energies in linear viscoelastic solids. ZAMP 15, 184–191 (1964)CrossRefzbMATHGoogle Scholar
  7. J.W. Cahn, J.E. Hilliard, Free energy of a non-uniform system i - interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)CrossRefGoogle Scholar
  8. P.B. Canham, The minimum energy as possible explanation of the biconcave shape of the human red blood cell. J. Theor. Biol. 26(1), 61–81 (1970)CrossRefGoogle Scholar
  9. M. Caputo, Elasticità e Dissipazione (Zanichelli, Bologna, 1969)Google Scholar
  10. R. Choksi, M. Morandotti, M. Veneroni, Global Minimizers for Axisymmetric Multiphase Membranes, arXiv preprint (2012), arXiv:1204.6673
  11. B.D. Coleman, D.C. Newman, On the rheology of cold drawing. i. elastic materials. J. Polym. Sci.: Part B: Polym. Phys. 26, 1801–1822 (1988)CrossRefGoogle Scholar
  12. D. Craiem, R.L. Magin, Fractional order models of viscoelasticity as an alternative in the analysis of red blood cell (rbc) membrane mechanics. Phys. Biol. 7(1), 13001 (2010)CrossRefGoogle Scholar
  13. G. Del Piero, L. Deseri, On the analytic expression of the free energy in linear viscoelasticity. J. Elast. 43, 247–278 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. G. Del Piero, L. Deseri, On the concepts of state and free energy in linear viscoelasticity. Arch. Ration. Mech. Anal. 138, 1–35 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. L. Deseri, G. Zurlo, The stretching elasticity of biomembranes determines their line tension and bending rigidity. Biomech. Model. Mechanobiol. 12, 1233–1242 (2013)CrossRefGoogle Scholar
  16. L. Deseri, G. Gentili, M.J. Golden, An expression for the minimal free energy in linear viscoelasticity. J. Elast. 54, 141–185 (1999)CrossRefzbMATHGoogle Scholar
  17. L. Deseri, M.J. Golden, M. Fabrizio, The concept of a minimal state in viscoelasticity: new free energies and applications to pdes. Arch. Ration. Mech. Anal. 181, 43–96 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. L. Deseri, M. Piccioni, G. Zurlo, Derivation of a new free energy for biological membranes. Contin. Mech. Term 20(5), 255–273 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. L. Deseri, M. Di Paola, M. Zingales, Free energy and states of fractional-order hereditariness. Int. J. Solids Struct. 51, 3156–3167 (2014)CrossRefGoogle Scholar
  20. L. Deseri, P. Pollaci, M. Zingales, K. Dayal, Fractional hereditariness of lipid membranes: Instabilities and linearized evolution. J. Mech. Behav. Biomed. Mater. 58, 11–27 (2016)CrossRefGoogle Scholar
  21. G. Espinosa, I. López-Montero, F. Monroy, D. Langevin, Shear rheology of lipid monolayers and insights on membrane fluidity. PNAS 108(15), 6008–6013 (2011)CrossRefGoogle Scholar
  22. E.A. Evans, Bending resistance and chemically induced moments in membrane bilayers. Biophys. J. 14, 923–931 (1974)CrossRefGoogle Scholar
  23. M.S. Falkovitz, M. Seul, H.L. Frisch, H.M. McConnell, Theory of periodic structures in lipid bilayer membranes. Proc. Natl. Acad. Sci. USA 79, 3918–3921 (1982)CrossRefGoogle Scholar
  24. Y.C. Fung, Theoretical considerations of the elasticity of red blood cells and small blood vessels. Proc. Fed. Am. Soc. Exp. Biol. 25(6), 1761–1772 (1966)Google Scholar
  25. Y.C. Fung, P. Tong, Theory of sphering of red blood cells. Biophys. J. 8, 175–198 (1968)CrossRefGoogle Scholar
  26. R.E. Goldstein, S. Leibler, Model for lamellar phases of interacting lipid membranes. Phys. Rev. Let. 61(19), 2213–2216 (1988)CrossRefGoogle Scholar
  27. R.E. Goldstein, S. Leibler, Structural phase transitions of interacting membranes. Phys. Rev. A. 40(2) (1989)Google Scholar
  28. M. Hamm, M.M. Kozlov, Elastic energy of tilt and bending of fluid membranes. Eur. Phys. J. E 3, 323–335 (2000)CrossRefGoogle Scholar
  29. C.W. Harland, M.J. Bradley, R. Parthasarathy, Phospholipid bilayers are viscoelastic. PNAS 107(45), 19146–19150 (2010)CrossRefGoogle Scholar
  30. T.J. Healey, R. Paroni, L. Deseri, Material gamma-limits for biological in-plane fluid plates. (2017)Google Scholar
  31. W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch [C], 28(11), 693–703 (1973)Google Scholar
  32. M. Hu, J.J. Briguglio, M. Deserno, Determining the gaussian curvature modulus of lipid membranes in simulations. Biophys. J. 102, 1403–1410 (2012)CrossRefGoogle Scholar
  33. F. Jahnig, Critical effects from lipid-protein interaction in membranes. Biophys. J. 36, 329–345 (1981)CrossRefGoogle Scholar
  34. F. Jahnig, What is the surface tension of a lipid bilayer membrane? Biophys. J. 71, 1348–1349 (1996)CrossRefGoogle Scholar
  35. J.B. Keller, G.J. Merchant, Flexural rigidity of a liquid surface. J. Stat. Phys. 63(5–6), 1039–1051 (1991)Google Scholar
  36. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)zbMATHGoogle Scholar
  37. W.T. Koiter, On the nonlinear theory of thin elastic shells. Proc. K. Ned. Akad. Wet. B 69, 1–54 (1966)MathSciNetGoogle Scholar
  38. S. Komura, H. Shirotori, P.D. Olmsted, D. Andelman. Lateral phase separation in mixtures of lipids and cholesterol. Europhys. Lett. 67(2) (2004)Google Scholar
  39. R. Lipowsky, E. Sackmann (eds.), Handbook of Biological Physics-Structure and Dynamics of Membranes, vol. 1 (Elsevier Science B.V, Amsterdam, 1995)zbMATHGoogle Scholar
  40. R.L. Magin, Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl. 59(5), 1586–1593 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  41. M. Maleki, B. Seguin, E. Fried, Kinematics, material symmetry, and energy densities for lipid bilayers with spontaneous curvature. Biomech. Model. Mechanobiol. 12(5), 997–1017 (2013)CrossRefGoogle Scholar
  42. D. Norouzi, M.M. Müller, M. Deserno, How to determine local elastic properties of lipid bilayer membranes from atomic-force-microscope measurements: a theoretical analysis. Phys. Rev. E, 74 (2006)Google Scholar
  43. J.C. Owicki, H.M. McConnell, Theory of protein-lipid and protein-protein interactions in bilayer membranes. Proc. Natl. Acad. Sci. USA 76, 4750–4754 (1979)CrossRefGoogle Scholar
  44. J.C. Owicki, M.W. Springgate, H.M. McConnell, Theoretical study of protein-lipid interactions in bilayer membranes. Proc. Natl. Acad. Sci. USA 75, 1616–1619 (1978)CrossRefGoogle Scholar
  45. J. Pan, S. Tristram-Nagle, J.F. Nagle, Effect of cholesterol on structural and mechanical properties of membranes depends on lipid chain saturation. Phys. Rev. E: Stat. Nonlinear 80(021931) (2009)Google Scholar
  46. I. Podlubny, Fractional Differential Equation (Academic, New York, 1998)zbMATHGoogle Scholar
  47. W. Rawicz, K.C. Olbrich, T. McIntosh, D. Needham, E. Evans, Effect of chain length and unsaturation on elasticity of lipid bilayers. Biophys. J. 79, 328–339 (2000)CrossRefGoogle Scholar
  48. A.S. Reddy, D. Toledo Warshaviak, M. Chachisvilis, Effect of membrane tension on the physical properties of dopc lipid bilayer membrane. Bioch. Biophys. Acta 1818, 2271–2281 (2012)Google Scholar
  49. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications (Gordon & Breach Science Publishers, London, 1987)zbMATHGoogle Scholar
  50. G.W. Scott-Blair, Psychoreology: links between the past and the present. J. Texture Stud. 5, 3–12 (1974)CrossRefGoogle Scholar
  51. S. Semrau, T. Idema, L. Holtzer, T. Schmict, C. Storm, Accurate determination of elastic parameters for multicomponent membranes. PRL 100(088101) (2008)Google Scholar
  52. D.P. Siegel, M.M. Kozlov, The gaussian curvature elastic modulus of n-monomethylated dioleoylphosphatidylethanolamine: Relevance to membrane fusion and lipid phase behavior. Biophys. J. 87, 366–374 (2004)CrossRefGoogle Scholar
  53. D.J. Steigmann, Fluid films with curvature elasticity. Arch. Ration. Mech. Anal. 150, 127–152 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  54. M. Trejo, M. Ben, Amar. Effective line tension and contact angles between membrane domains in biphasic vesicles. Eur. Phys. J. E 34(8), 2–14 (2011)Google Scholar
  55. S.L. Veatch, V.I. Polozov, K. Gawrisch, S.L. Keller, Liquid domains in vescicles investigated by nmr and fluorescence microscopy. Biophys. J. 86, 2910–2922 (2004)CrossRefGoogle Scholar
  56. G. Zurlo. Material and geometric phase transitions in biological membranes. Dissertation for the Fulfillment of the Doctorate of Philosophy in Structural Engineering, University of Pisa, etd-11142006-173408 (2006)Google Scholar

Copyright information

© CISM International Centre for Mechanical Sciences 2018

Authors and Affiliations

  1. 1.Department of Mechanical, Aerospace and Civil Engineering-MACEBrunel University LondonUxbridgeUK
  2. 2.Department of Mechanical Engineering and Materials Science-MEMSUniversity of PittsburghPittsburghUSA
  3. 3.Department of Mechanical, Civil and Environmental Engineering-DICAMUniversity of TrentoTrentoItaly
  4. 4.Department of Mechanical EngineeringCarnegie Mellon UniversityPittsburghUSA
  5. 5.Department of NanomedicineThe Methodist Hospital Research InstituteHoustonUSA

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