# Mechanics and Physics of Lipid Bilayers

## Abstract

In this chapter we review recent work by the writer and coworkers on various aspects of the mechanics and physics of lipid bilayers. A framework for lipid bilayer surface, based on a dimension reduction procedure applied to three-dimensional liquid crystal theory, is reviewed in Sect. 1. This accommodates the non-standard effects of lipid distension and tilt. A special case of the general model in which tilt is suppressed but distension, and accompanying surface dilation, are permitted, is also derived. This is further specialized, in Sect. 2, to obtain a model of the classical type, due to Canham and Helfrich. Our approach facilitates understanding of the place of the classical theory, and its logical extensions, in a larger context. Section 3 provides a further development of the theory with surface dilation—reported here for the first time—to accommodate dissipative effects, including intra-membrane viscous flow and the diffusion of trans-membrane embedded proteins. This may be viewed as a theory of generalized capillarity, accounting for various higher order gradient effects of the Cahn–Hilliard type in the constitutive equations. A simpler variant of this model is described in Sect. 4, in which non-standard gradient effects are suppressed. This furnishes the simplest thermodynamically consistent extension of the classical theory to cover diffusion and viscosity. Finally, Sect. 5 is devoted to the electromechanical theory. This is limited to the simplest extension of the classical model to accommodate surface flexo-electricity and the coupling of surface shape with a polarization field. Restrictions on the latter, consistent with the three-dimensional electromechanical theory for liquid crystals, yield a relatively simple generalization of the classical theory appropriate for analyzing membrane response to a remote applied electric field.

## Keywords

Liquid Crystal Variational Derivative Liquid Crystal Molecule Director Gradient Surface Dilation## References

- A. Agrawal, D.J. Steigmann, Boundary-value problems in the theory of lipid membranes. Contin. Mech. Thermodyn.
**21**, 57–82 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - A. Agrawal, D.J. Steigmann, A model for surface diffusion of trans-membrane proteins on lipid bilayers. ZAMP
**62**, 549–563 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - R. Aris,
*Vectors, Tensors and the Basic Equations of Fluid Mechanics*(Dover, N.Y., 1989)zbMATHGoogle Scholar - M. Arroyo, A. DeSimone, Relaxation dynamics of fluid membranes. Phy. Rev.
**E79**(031915), 1–17 (2009)MathSciNetGoogle Scholar - E. Baesu, R.E. Rudd, J. Belak, M. McElfresh, Continuum modeling of cell membranes. Int. J. Non-linear Mech.
**39**, 369–377 (2004)CrossRefzbMATHGoogle Scholar - M. Barham, D.J. Steigmann, D. White, Magnetoelasticity of highly deformable thin films: theory and simulation. Int. J. Non-linear Mech.
**47**, 185–196 (2012)CrossRefGoogle Scholar - R. Bustamante, A. Dorfmann, R.W. Ogden, Nonlinear electroelastostatics: a variational framework. ZAMP
**60**, 154–177 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys.
**28**, 258–267 (1958)CrossRefGoogle Scholar - P. Canham, The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theor. Biol.
**26**, 61–81 (1970)CrossRefGoogle Scholar - R. Capovilla, J. Guven, Geometry of lipid vesicle adhesion. Phys. Rev. E
**66**(041604), 1–6 (2002)Google Scholar - P.G. Ciarlet,
*Mathematical Elasticity, Vol. 1: Three Dimensional Elasticity*(North-Holland, Amsterdam, 1993)Google Scholar - C.M. Dafermos, Disclinations in liquid crystals. Q. J. Mech. Appl. Math.
**23**, S49–S64 (1970). Supplement No. 1: Mechanics of Liquid CrystalsCrossRefGoogle Scholar - P.G. DeGennes, J. Prost,
*The Physics of Liquid Crystals*(Oxford University Press, Oxford, 1992)Google Scholar - L. Deseri, M.D. Piccioni, G. Zurlo, Derivation of a new free energy for biological membranes. Contin. Mech. Thermodyn.
**20**, 255–273 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - M. Deserno, M.M. Müller, J. Guven, Contact lines for fluid surface adhesion. Phys. Rev. E
**76**(011605), 1–10 (2007)Google Scholar - L. Dorfmann, R.W. Ogden,
*Nonlinear Theory of Electroelastic and Magnetoelastic Interactions*(Springer, N.Y., 2014)CrossRefzbMATHGoogle Scholar - A. Embar, J. Dolbow, E. Fried, Microdomain evolution on giant unilamellar vesicles. Biomech. Model. Mechanobiol.
**12**, 597–615 (2013)CrossRefGoogle Scholar - J.L. Ericksen, Conservation laws for liquid crystals. Trans. Soc. Rheol.
**5**, 23–34 (1961)MathSciNetCrossRefGoogle Scholar - J.L. Ericksen, Hydrostatic theory of liquid crystals. Arch. Ration. Mech. Anal.
**9**, 371–378 (1962)MathSciNetzbMATHGoogle Scholar - J.L. Ericksen, Equilibrium theory of liquid crystals, in
*Advances in Liquid Crystals*, ed. by G.H. Brown (Academic Press, N.Y., 1976), pp. 233–298Google Scholar - J.L. Ericksen, Theory of Cosserat surfaces and its application to shells, interfaces and cell membranes, in
*Proceeding of the International Symposium on Recent Developments in the Theory and Application of Generalized and Oriented Media*, ed. by P.G. Glockner, M. Epstein, D.J. Malcolm (Calgary, Alberta, 1979), pp. 27–39Google Scholar - E.A. Evans, R. Skalak,
*Mechanics and Thermodynamics of Biomembranes*(CRC Press, Boca Raton, 1980)Google Scholar - H. Frischleder, G. Peinel, Quantum-chemical and statistical calculations on phospholipids. Chem. Phys. Lipids
**30**, 121–158 (1982)CrossRefGoogle Scholar - L.T. Gao, X.-Q. Feng, Y.-J. Yin, H. Gao, An electromechanical liquid crystal model of vesicles. J. Mech. Phys. Solids
**56**, 2844–2862 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - G. Gioia, R.D. James, Micromagnetics of very thin films. Proc. R. Soc. Lond.
**A453**, 213 (1997)CrossRefGoogle Scholar - L.M. Graves, The Weierstrass condition for multiple integral variation problems. Duke Math. J.
**5**, 656–660 (1939)MathSciNetCrossRefzbMATHGoogle Scholar - W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch.
**28**, 693–703 (1973)Google Scholar - M.G. Hilgers, A.C. Pipkin, Energy-minimizing deformations of elastic sheets with bending stiffness. J. Elast.
**31**, 125–139 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - J.T. Jenkins, The equations of mechanical equilibrium of a model membrane. SIAM J. Appl. Math.
**32**(755–764), 00 (1977)MathSciNetGoogle Scholar - C.-I. Kim, D.J. Steigmann, Distension-induced gradient capillarity in lipid membranes. Contin. Mech. Thermodyn.
**27**, 609–621 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - A. Kovetz,
*Electromagnetic Theory*(Oxford University Press, Oxford, 2000)Google Scholar - K. Mandadapu, Private communication (2016)Google Scholar
- R.B. Meyer, Piezoelectric effects in liquid crystals. Phys. Rev. Lett.
**22**, 918–921 (1969)CrossRefGoogle Scholar - P. Mohammidi, L.P. Liu, P. Sharma, A theory of flexo-electric membranes and effective properties of heterogeneous membranes. ASME J. Appl. Mech.
**81**, 011007-1–011007-11 (2014)Google Scholar - M.M. Müller, M. Deserno, J. Guven, Interface-mediated interactions between particles: A geometrical approach. Phys. Rev.
**E72**(061407), 1–17 (2005a)MathSciNetGoogle Scholar - M.M. Müller, M. Deserno, J. Guven, Geometry of surface-mediated interactions. Europhys. Lett.
**69**, 482–488 (2005b)CrossRefGoogle Scholar - M.M. Müller, M. Deserno, J. Guven, Balancing torques in membrane-mediated interactions: Exact results and numerical illustrations. Phys. Rev.
**E76**(011921), 1–16 (2007)Google Scholar - P.M. Naghdi, Theory of Shells and Plates, in
*Handbuch der Physik*, vol. VIa/2, ed. by C. Truesdell (Springer, Berlin, 1972)Google Scholar - Z.-C. Ou-Yang, J.-X. Liu, Y.-Z. Xie,
*Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases*(World Scientific, Singapore, 1999)CrossRefzbMATHGoogle Scholar - A.G. Petrov,
*The Lyotropic State of Matter*(Gordon and Breach, Amsterdam, 1999)Google Scholar - P. Rangamani, A. Agrawal, K. Mandadapu, G. Oster, D.J. Steigmann, Interaction between surface shape and intra-surface viscous flow on lipid membranes. Biomech. Model. Mechanobiol.
**12**, 833–845 (2013)CrossRefGoogle Scholar - P. Rangamani, A. Benjamani, A. Agrawal, B. Smit, D.J. Steigmann, G. Oster, Small scale membrane mechanics. Biomech. Model. Mechanobiol.
**13**, 697–711 (2014)CrossRefGoogle Scholar - P. Rangamani, D.J. Steigmann, Variable tilt on lipid membranes. Proc. R. Soc. Lond. A
**470**(20140463) (2014). https://doi.org/10.1098/rspa.2014.0463 - L.E. Scriven, Dynamics of a fluid interface. Chem. Eng. Sci.
**12**, 98–108 (1960)CrossRefGoogle Scholar - J. Seelig, \(^{31}\)P nuclear magnetic resonance and the head group structure of phospholipids in membranes. Biochim. Biophys. Acta
**515**, 105–140 (1978)CrossRefGoogle Scholar - I.S. Sokolnikoff,
*Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua*(Wiley, N.Y., 1964)zbMATHGoogle Scholar - D.J. Steigmann, A note on pressure potentials. J. Elast.
**26**, 87–93 (1991)MathSciNetCrossRefGoogle Scholar - D.J. Steigmann, On the relationship between the Cosserat and Kirchhoff-Love theories of elastic shells. Math. Mech. Solids
**4**, 275–88 (1999a)MathSciNetCrossRefzbMATHGoogle Scholar - D.J. Steigmann, Fluid films with curvature elasticity. Arch. Ration. Mech. Anal.
**150**, 127–52 (1999b)MathSciNetCrossRefzbMATHGoogle Scholar - D.J. Steigmann, Applications of polyconvexity and strong ellipticity to nonlinear elasticity and elastic plate theory, in
*CISM Course on Applications of Poly-, Quasi-, and Rank-One Convexity in Applied Mechanics*, vol. 516, ed. by J. Schröder, P. Neff (Springer, Wien and New York, 2010), pp. 265–299CrossRefGoogle Scholar - D.J. Steigmann, A model for lipid membranes with tilt and distension based on three-dimensional liquid crystal theory. Int. J. Non-linear Mech.
**56**, 61–70 (2013)CrossRefGoogle Scholar - D.J. Steigmann, A. Agrawal, Electromechanics of polarized lipid bilayers. Math. Mech. Complex Syst.
**4–1**, 31–54 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - D.J. Steigmann, E. Baesu, R.E. Rudd, M. McElfresh, J. Belak, On the variational theory of cell-membrane equilibria. Interfaces Free Bound.
**5**, 357–366 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - R.A. Toupin, The elastic dielectric. J. Ration. Mech. Anal.
**5**, 849–915 (1956)MathSciNetzbMATHGoogle Scholar - C. Truesdell,
*A First Course in Rational Continuum Mechanics*(Academic Press, N.Y., 1991)zbMATHGoogle Scholar - E.G. Virga,
*Variational Theories for Liquid Crystals*(Chapman and Hall, London, 1994)CrossRefzbMATHGoogle Scholar - D.T. Warshaviak, M.J. Muellner, M. Chachisvilis, Effect of membrane tension on the electric field and dipole potential of lipid bilayer membrane. Biochim. Biophys. Acta
**2608–2617**, 2011 (1808)Google Scholar - Q.-S. Zheng, Two-dimensional tensor function representation for all kinds of material symmetry. Proc. R. Soc. Lond. A
**443**, 127–138 (1993)MathSciNetCrossRefzbMATHGoogle Scholar