Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For the reader who is not familiar with this standard terminology, we recall that the right-handed Fourier transform of a given relaxation function represents the “complex modulus” of a viscoelastic material; its real part is the “storage modulus”, while its imaginary part is its “loss modulus”.
References
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)
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)
E. Baesu, R.E. Rudd, J. Belak, M. McElfresh, Continuum modeling of cell membranes. Int. J. Non-Linear Mech. 39(3), 369–377 (2004)
T. Baumgart, W.W. Webb, S.T. Hess, Imaging coexisting domains in biomembrane models coupling curvature and line tension. Nature 423, 821–824 (2003)
H. Bermúdez, D.A. Hammer, D.E. Discher, Effect of bilayer thickness on membrane bending rigidity. Langmuir 20, 540–543 (2004)
S. Breuer, E. Onat, On the determination of free energies in linear viscoelastic solids. ZAMP 15, 184–191 (1964)
J.W. Cahn, J.E. Hilliard, Free energy of a non-uniform system i - interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)
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)
M. Caputo, Elasticità e Dissipazione (Zanichelli, Bologna, 1969)
R. Choksi, M. Morandotti, M. Veneroni, Global Minimizers for Axisymmetric Multiphase Membranes, arXiv preprint (2012), arXiv:1204.6673
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)
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)
G. Del Piero, L. Deseri, On the analytic expression of the free energy in linear viscoelasticity. J. Elast. 43, 247–278 (1996)
G. Del Piero, L. Deseri, On the concepts of state and free energy in linear viscoelasticity. Arch. Ration. Mech. Anal. 138, 1–35 (1997)
L. Deseri, G. Zurlo, The stretching elasticity of biomembranes determines their line tension and bending rigidity. Biomech. Model. Mechanobiol. 12, 1233–1242 (2013)
L. Deseri, G. Gentili, M.J. Golden, An expression for the minimal free energy in linear viscoelasticity. J. Elast. 54, 141–185 (1999)
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)
L. Deseri, M. Piccioni, G. Zurlo, Derivation of a new free energy for biological membranes. Contin. Mech. Term 20(5), 255–273 (2008)
L. Deseri, M. Di Paola, M. Zingales, Free energy and states of fractional-order hereditariness. Int. J. Solids Struct. 51, 3156–3167 (2014)
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)
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)
E.A. Evans, Bending resistance and chemically induced moments in membrane bilayers. Biophys. J. 14, 923–931 (1974)
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)
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)
Y.C. Fung, P. Tong, Theory of sphering of red blood cells. Biophys. J. 8, 175–198 (1968)
R.E. Goldstein, S. Leibler, Model for lamellar phases of interacting lipid membranes. Phys. Rev. Let. 61(19), 2213–2216 (1988)
R.E. Goldstein, S. Leibler, Structural phase transitions of interacting membranes. Phys. Rev. A. 40(2) (1989)
M. Hamm, M.M. Kozlov, Elastic energy of tilt and bending of fluid membranes. Eur. Phys. J. E 3, 323–335 (2000)
C.W. Harland, M.J. Bradley, R. Parthasarathy, Phospholipid bilayers are viscoelastic. PNAS 107(45), 19146–19150 (2010)
T.J. Healey, R. Paroni, L. Deseri, Material gamma-limits for biological in-plane fluid plates. (2017)
W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch [C], 28(11), 693–703 (1973)
M. Hu, J.J. Briguglio, M. Deserno, Determining the gaussian curvature modulus of lipid membranes in simulations. Biophys. J. 102, 1403–1410 (2012)
F. Jahnig, Critical effects from lipid-protein interaction in membranes. Biophys. J. 36, 329–345 (1981)
F. Jahnig, What is the surface tension of a lipid bilayer membrane? Biophys. J. 71, 1348–1349 (1996)
J.B. Keller, G.J. Merchant, Flexural rigidity of a liquid surface. J. Stat. Phys. 63(5–6), 1039–1051 (1991)
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)
W.T. Koiter, On the nonlinear theory of thin elastic shells. Proc. K. Ned. Akad. Wet. B 69, 1–54 (1966)
S. Komura, H. Shirotori, P.D. Olmsted, D. Andelman. Lateral phase separation in mixtures of lipids and cholesterol. Europhys. Lett. 67(2) (2004)
R. Lipowsky, E. Sackmann (eds.), Handbook of Biological Physics-Structure and Dynamics of Membranes, vol. 1 (Elsevier Science B.V, Amsterdam, 1995)
R.L. Magin, Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl. 59(5), 1586–1593 (2010)
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)
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)
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)
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)
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)
I. Podlubny, Fractional Differential Equation (Academic, New York, 1998)
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)
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)
S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications (Gordon & Breach Science Publishers, London, 1987)
G.W. Scott-Blair, Psychoreology: links between the past and the present. J. Texture Stud. 5, 3–12 (1974)
S. Semrau, T. Idema, L. Holtzer, T. Schmict, C. Storm, Accurate determination of elastic parameters for multicomponent membranes. PRL 100(088101) (2008)
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)
D.J. Steigmann, Fluid films with curvature elasticity. Arch. Ration. Mech. Anal. 150, 127–152 (1999)
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)
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)
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)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 CISM International Centre for Mechanical Sciences
About this chapter
Cite this chapter
Deseri, L. (2018). Elasticity and Hereditariness. In: Steigmann, D. (eds) The Role of Mechanics in the Study of Lipid Bilayers. CISM International Centre for Mechanical Sciences, vol 577. Springer, Cham. https://doi.org/10.1007/978-3-319-56348-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-56348-0_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-56347-3
Online ISBN: 978-3-319-56348-0
eBook Packages: EngineeringEngineering (R0)