Abstract
Lipid bilayers are unique soft materials operating in general in the low Reynolds limit. While their shape is predominantly dominated by curvature elasticity as in a solid shell, their in-plane behavior is that of a largely inextensible viscous fluid. Furthermore, lipid membranes are extremely responsive to chemical stimuli. Because in their biological context they are continuously brought out-of-equilibrium mechanically or chemically, it is important to understand their dynamics. Here, we introduce Onsager’s variational principle as a general and transparent modeling tool for lipid bilayer dynamics. We introduce this principle with elementary examples, and then use it to study the sorption of curved proteins on lipid membranes.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
S. Aimon, A. Callan-Jones, A. Berthaud, M. Pinot, G.E.S. Toombes, P. Bassereau, Membrane shape modulates transmembrane protein distribution. Dev. Cell 28(2), 212–218 (2014)
B. Antonny, Mechanisms of membrane curvature sensing. Annu. Rev. Biochem. 80, 101–123 (2011)
M. Arroyo, A. DeSimone, Relaxation dynamics of fluid membranes. Phy. Rev. E 79, 031915 (2009)
M. Arroyo, A. DeSimone, L. Heltai, The role of membrane viscosity in the dynamics of fluid membranes. arXiv 2007, 1–21 (2010)
M. Arroyo, L. Heltai, D. Millán, A. DeSimone, Reverse engineering the euglenoid movement. Proc. Natl. Acad. Sci. U. S. A. 44, 17874–17879 (2012)
J.W. Barrett, H. Garcke, R. Nürnberg, A Stable Numerical Method for the Dynamics of Fluidic Membranes, vol. 134 (Springer, Berlin, 2016)
M. Breidenich, R.R. Netz, R. Lipowsky, The shape of polymer-decorated membranes. 49, 431–437 (2000)
A. Callan-Jones, M. Durand, J.-B. Fournier, Hydrodynamics of bilayer membranes with diffusing transmembrane proteins. Soft Matter 12(6), 1791–1800 (2016)
R. Capovilla, J. Guven, Stresses in lipid membranes. J. Phy. A: Math. Gen. 35(30), 6233–6247 (2002)
M. Doi, Onsager’s variational principle in soft matter. J. Phys.: Condens. Matter 23(28), 284118 (2011)
C.M. Elliott, B. Stinner, Computation of two-phase biomembranes with phase dependent material parameters using surface finite elements. Commun. Comput. Phys. 13(2), 325–360 (2013)
M.B. Elowitz, M.G. Surette, P.E. Wolf, J.B. Stock, S. Leibler, Protein mobility in the cytoplasm of Escherichia coli. J. Bacteriol. 181(1), 197–203 (1999)
A. Embar, J. Dolbow, E. Fried, Microdomain evolution on giant unilamellar vesicles. Biomech. Model. Mechanobiol. 12(3), 597–615 (2013)
E. Evans, A. Yeung, Hidden dynamics in rapid changes of bilayer shape. Chem. Phy. Lipids 73(1–2), 39–56 (1994)
F. Feng, W.S. Klug, Finite element modeling of lipid bilayer membranes. J. Comput. Phys. 220(1), 394–408 (2006)
P.J. Flory, Thermodynamics of high polymer solutions. J. Chem. Phys. 10(1), 51–61 (1942)
J.-B. Fournier, N. Khalifat, N. Puff, M.I. Angelova, Chemically triggered ejection of membrane tubules controlled by intermonolayer friction. Phys. Rev. Lett. 102(1), 018102 (2009)
J.B. Fournier, On the hydrodynamics of bilayer membranes. Int. J. Non-Linear Mech. 75, 67–76 (2015)
H. Goldstein, Classical Mechanics, World student series (Addison-Wesley, Reading, 1980)
W.T. Góźdź, Shape transformation of lipid vesicles induced by diffusing macromolecules. J. Chem. Phys. 134(2) (2011)
J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media, vol. 1 (Springer Science & Business Media, 2012)
M. Heinrich, A. Tian, C. Esposito, T. Baumgart, Dynamic sorting of lipids and proteins in membrane tubes with a moving phase boundary. Proc. Natl. Acad. Sci. U. S. A. 107(16), 7208–7213 (2010a)
M.C. Heinrich, B.R. Capraro, A. Tian, J.M. Isas, R. Langen, T. Baumgart, Quantifying membrane curvature generation of drosophila amphiphysin N-BAR domains. J. Phys. Chem. Lett. 1(23), 3401–3406 (2010b)
W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. 28c, 693–703 (1973)
M.L. Huggins, Solutions of long chain compounds. J. Chem. Phys. 9(5), 440–440 (1941)
R. Jordan, D. Kinderlehrer, F. Otto, The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)
F. Jülicher, R. Lipowsky, Domain-induced budding of vesicles. Phys. Rev. Lett. 70(19), 2964–2967 (1993)
N. Khalifat, N. Puff, S. Bonneau, J.-B. Fournier, M.I. Angelova, Membrane deformation under local pH gradient: mimicking mitochondrial cristae dynamics. Biophys. J. 95(10), 4924–4933 (2008)
N. Khalifat, M. Rahimi, A.-F. Bitbol, M. Seigneuret, J.-B. Fournier, N. Puff, M. Arroyo, M.I. Angelova, Interplay of packing and flip-flop in local bilayer deformation. How phosphatidylglycerol could rescue mitochondrial function in a cardiolipin-deficient yeast mutant. Biophys. J. 107(4), 879–890 (2014)
A.J. Kosmalska, L. Casares, A. Elosegui-Artola, J.J. Thottacherry, R. Moreno-Vicente, V. González-Tarragó, M.Á. del Pozo, S. Mayor, M. Arroyo, D. Navajas, X. Trepat, N.C. Gauthier, P. Roca-Cusachs, Physical principles of membrane remodelling during cell mechanoadaptation. Nat. Commun. 6, 7292 (2015)
L.D. Landau, E.M. Lifshitz, Fluid Mechanics: Landau and Lifshitz: Course of Theoretical Physics, vol. 6 (Elsevier, Amsterdam, 2013)
A. Lew, J.E. Marsden, M. Ortiz, M. West, Variational time integrators. Int. J. Numer. Meth. Eng. 60(1), 153–212 (2004)
R. Lipowsky, The conformation of membranes. Nature 349(6309), 475–481 (1991)
R. Lipowsky, Spontaneous tubulation of membranes and vesicles reveals membrane tension generated by spontaneous curvature. Faraday Discuss. 161, 305–331 (2013)
J. Liu, Y. Sun, D.G. Drubin, G.F. Oster, The mechanochemistry of endocytosis. PLoS Biol. 7(9), e1000204 (2009)
R.I. Masel, Principles of Adsorption and Reaction on Solid Surfaces, vol. 3 (Wiley, New York, 1996)
H.T. McMahon, J.L. Gallop, Membrane curvature and mechanisms of dynamic cell membrane remodelling. Nature 438(7068), 590–596 (2005)
A. Mielke, A gradient structure for reactiondiffusion systems and for energy-drift-diffusion systems. Nonlinearity 24(4), 1329–1346 (2011a). doi:10.1088/0951-7715/24/4/016. ISSN 0951-7715
A. Mielke, On thermodynamically consistent models and gradient structures for thermoplasticity. GAMM Mitt. 34(1), 51–58 (2011b)
A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions. Discrete Continuous Dyn. Syst. - Ser. S 6(2), 479–499 (2012)
L. Onsager, Irreversible processes. Phys. Rev. 37, 237–241 (1931a)
L. Onsager, Reciprocal relations in irreversible processes I. Phys. Rev. 37(4), 405–426 (1931b)
M. Ortiz, E.A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47(2), 397–462 (1999)
H.C. Öttinger, Beyond Equilibrium Thermodynamics (Wiley, New York, 2005)
F. Otto, The geometry of dissipative evolution equations: The porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)
W. Pauli, C.P. Enz, Thermodynamics and the kinetic theory of gases, vol. 3 (Courier Corporation, 2000)
C. Peco, A. Rosolen, M. Arroyo, An adaptive meshfree method for phase-field models of biomembranes. Part II: A Lagrangian approach for membranes in viscous fluids. J. Comput. Phys. 249, 320–336 (2013)
M.A. Peletier, Variational modelling: energies, gradient flows, and large deviations, February 2014
C. Prévost, H. Zhao, J. Manzi, E. Lemichez, P. Lappalainen, A. Callan-Jones, P. Bassereau, IRSp53 senses negative membrane curvature and phase separates along membrane tubules. Nat. Commun. 6, 8529 (2015)
I. Prigogine, Introduction to Thermodynamics of Irreversible Processes (Interscience Publishers, 1967)
M. Rahimi, M. Arroyo, Shape dynamics, lipid hydrodynamics, and the complex viscoelasticity of bilayer membranes. Phys. Rev. E 86(1), 011932 (2012)
M. Rahimi, A. DeSimone, M. Arroyo, Curved fluid membranes behave laterally as effective viscoelastic media. Soft Matter 9(46), 11033 (2013)
S. Ramadurai, A. Holt, V. Krasnikov, G. van den Bogaart, J.A. Killian, B. Poolman, Lateral diffusion of membrane proteins. J. Am. Chem. Soc. 13135, 12650–12656 (2009)
P. Rangamani, A. Agrawal, K.K. Mandadapu, G. Oster, D.J. Steigmann, Interaction between surface shape and intra-surface viscous flow on lipid membranes. Biomech. Model. Mechanobiol. 12(4), 833–845 (2013)
R. Rangarajan, H. Gao, A finite element method to compute three-dimensional equilibrium configurations of fluid membranes: Optimal parameterization, variational formulation and applications. J. Comput. Phys. 297, 266–294 (2015)
Rayleigh, Proc. Math. Soc. London 363, 357 (1873)
D.S. Rodrigues, R.F. Ausas, F. Mut, G.C. Buscaglia, Numerical modeling of tether formation in viscous. XXXII, 19–22 (2013)
A. Rustom, R. Saffrich, I. Markovic, P. Walther, H.-H. Gerdes, Nanotubular highways for intercellular organelle transport. Sci. (New York, N.Y.) 303(5660), 1007–1010 (2004)
P.G. Saffman, M. Delbruck, Brownian motion in biological membranes. Proc. Natl. Acad. Sci. U. S. A. 72(8), 3111–3113 (1975)
R.A. Sauer, T.X. Duong, K. Mandadapu, D. Steigmann, A stabilized finite element formulation for liquid shells and its application to lipid bilayers. J. Comput. Phys. 330, 1–19 (2017)
U. Seifert, S.A. Langer, Viscous modes of fluid bilayer membranes. Europhys. Lett. (EPL) 23(1), 71–76 (1993)
U. Seifert, Configurations of fluid membranes and vesicles. Adv. Phys. (1997). July 2011
P. Sens, L. Johannes, P. Bassereau, Biophysical approaches to protein-induced membrane deformations in trafficking. Curr. Opin. Cell Biol. 20(4), 476–482 (2008)
Z. Shi, T. Baumgart, Membrane tension and peripheral protein density mediate membrane shape transitions. Nat. Commun. 6, 5974 (2015). May 2014
Y. Shibata, H. Junjie, M.M. Kozlov, T.A. Rapoport, Mechanisms shaping the membranes of cellular organelles. Annu. Rev. Cell Dev. Biol. 25, 329–354 (2009)
P. Singh, P. Mahata, T. Baumgart, S.L. Das, Curvature sorting of proteins on a cylindrical lipid membrane tether connected to a reservoir. Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys. 85(5), 1–10 (2012)
B. Sinha, D. Köster, R. Ruez, P. Gonnord, M. Bastiani, D. Abankwa, R.V. Stan, G. Butler-Browne, B. Vedie, L. Johannes, N. Morone, R.G. Parton, G. Raposo, P. Sens, C. Lamaze, P. Nassoy, Cells respond to mechanical stress by rapid disassembly of caveolae. Cell 144(3), 402–413 (2011)
B. Sorre, A. Callan-Jones, J.-B. Manneville, P. Nassoy, J.-F. Joanny, J. Prost, B. Goud, P. Bassereau, Curvature-driven lipid sorting needs proximity to a demixing point and is aided by proteins. Proc. Natl. Acad. Sci. U. S. A. 106(14), 5622–5626 (2009)
B. Sorre, A. Callan-Jones, J. Manzi, B. Goud, J. Prost, P. Bassereau, A. Roux, Nature of curvature coupling of amphiphysin with membranes depends on its bound density. Proc. Natl. Acad. Sci. U. S. A. 109(1), 173–178 (2012)
H. Sprong, P. van der Sluijs, G. van Meer, How proteins move lipids and lipids move proteins. Nat. Rev. Mol. Cell Biol. 2(7), 504–513 (2001)
J.C. Stachowiak, E.M. Schmid, C.J. Ryan, H.S. Ann, D.Y. Sasaki, M.B. Sherman, P.L. Geissler, D.A. Fletcher, C.C. Hayden, Membrane bending by proteinprotein crowding. Nat. Cell Biol. 14(9), 944–949 (2012)
M. Staykova, M. Arroyo, M. Rahimi, H.A. Stone, Confined bilayers passively regulate shape and stress. Phys. Rev. Lett. 110, 028101 (2013)
D.J. Steigmann, On the relationship between the Cosserat and Kirchhoff-Love theories of elastic shells. Math. Mech. Solids 4, 275–288 (1999)
M. Terasaki, T. Shemesh, N. Kasthuri, R.W. Klemm, R. Schalek, K.J. Hayworth, A.R. Hand, M. Yankova, G. Huber, J.W. Lichtman, T.A. Rapoport, M.M. Kozlov, Stacked endoplasmic reticulum sheets are connected by helicoidal membrane motifs. Cell 154(2), 285–296 (2013)
Z.C. Tu, Z.C. Ou-Yang, A geometric theory on the elasticity of bio-membranes. J. Phys. A: Math. Gen. 37(47), 11407–11429 (2004)
N. Yamaguchi, T. Mizutani, K. Kawabata, H. Haga, Leader cells regulate collective cell migration via Rac activation in the downstream signaling of integrin \(\beta \)1 and PI3K. Sci. Rep. 5, 7656 (2015)
C. Zhu, S.L. Das, T. Baumgart, Nonlinear sorting, curvature generation, and crowding of endophilin N-BAR on tubular membranes. Biophys. J. 102(8), 1837–1845 (2012)
J. Zimmerberg, M.M. Kozlov, How proteins produce cellular membrane curvature. Nat. Rev. Mol. Cell Biol. 7(1), 9–19 (2006)
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
Arroyo, M., Walani, N., Torres-Sánchez, A., Kaurin, D. (2018). Onsager’s Variational Principle in Soft Matter: Introduction and Application to the Dynamics of Adsorption of Proteins onto Fluid Membranes. 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_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-56348-0_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-56347-3
Online ISBN: 978-3-319-56348-0
eBook Packages: EngineeringEngineering (R0)