Hostname: page-component-7479d7b7d-767nl Total loading time: 0 Render date: 2024-07-11T13:46:21.548Z Has data issue: false hasContentIssue false
  • English
  • Français

Transport phenomena in fluid films with curvature elasticity

Published online by Cambridge University Press:  23 October 2020

Arijit Mahapatra Open the ORCID record for Arijit Mahapatra [Opens in a new window] ,
David Saintillan Open the ORCID record for David Saintillan [Opens in a new window]  and
Padmini Rangamani Open the ORCID record for Padmini Rangamani [Opens in a new window]
Arijit Mahapatra
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, CA92093, USA
David Saintillan
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, CA92093, USA
Padmini Rangamani*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, CA92093, USA
*
†Email address for correspondence: prangamani@ucsd.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Cellular membranes are elastic lipid bilayers that contain a variety of proteins, including ion channels, receptors and scaffolding proteins. These proteins are known to diffuse in the plane of the membrane and to influence the bending of the membrane. Experiments have shown that lipid flow in the plane of the membrane is closely coupled with the diffusion of proteins. Thus, there is a need for a comprehensive framework that accounts for the interplay between these processes. Here, we present a theory for the coupled in-plane viscous flow of lipids, diffusion of transmembrane proteins and elastic deformation of lipid bilayers. The proteins in the membrane are modelled such that they influence membrane bending by inducing a spontaneous curvature. We formulate the free energy of the membrane with a Helfrich-like curvature elastic energy density function modified to account for the chemical potential energy of proteins. We derive the conservation laws and equations of motion for this system. Finally, we present results from dimensional analysis and numerical simulations and demonstrate the effect of coupled transport processes in governing the dynamics of membrane bending and protein diffusion.

JFM classification

Biological Fluid Dynamics: Membranes Interfacial Flows (free surface): Thin films Biological Fluid Dynamics: Capsule/cell dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 905 , 25 December 2020 , A8
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K. & Walter, P. 1985 Molecular Biology of the Cell. Garland Science.Google Scholar
Alimohamadi, H. & Rangamani, P. 2018 Modeling membrane curvature generation due to membrane–protein interactions. Biomolecules 8, 120–147.CrossRefGoogle Scholar
Anjali, T. G. & Basavaraj, M. G. 2019 Shape-anisotropic colloids at interfaces. Langmuir 35, 3–20.CrossRefGoogle ScholarPubMed
Antonny, B. 2011 Mechanisms of membrane curvature sensing. Annu. Rev. Biochem. 80, 101–123.CrossRefGoogle ScholarPubMed
Aris, R. 1989 Vectors, Tensors and Basic Equation of Fluid Mechanics. Dover.Google Scholar
Arroyo, M. & DeSimone, A. 2009 Relaxation dynamics of fluid membranes. Phys. Rev. E 79, 031915.CrossRefGoogle ScholarPubMed
Arroyo, M., Walani, N., Torres-Sanchez, 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 The Role of Mechanics in the Study of Lipid Bilayers (ed. Steigmann, D. J.), pp. 1–53. Springer.Google Scholar
Bahmani, F., Christenson, J. & Rangamani, P. 2016 Analysis of lipid flow on minimal surfaces. Contin. Mech. Thermodyn. 28, 503–513.CrossRefGoogle Scholar
Bassereau, P., Jin, R., Baumgart, T., Deserno, M., Dimova, R., Frolov, V. A., Bashkirov, P. V., Grubmüller, H., Jahn, R. & Risselada, H. J. et al. 2018 The 2018 biomembrane curvature and remodeling roadmap. J. Phys. D: Appl. Phys. 51, 343001.CrossRefGoogle ScholarPubMed
Baumgart, T., Hess, S. T. & Webb, W. W. 2003 Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension. Nature 425, 821–824.CrossRefGoogle ScholarPubMed
Blazek, A. D., Paleo, B. J. & Weisleder, N. 2015 Plasma membrane repair: a central process for maintaining cellular homeostasis. Physiology 30, 438–448.CrossRefGoogle ScholarPubMed
Campelo, F. & Hernández-Machado, A. 2007 Model for curvature-driven pearling instability in membranes. Phys. Rev. Lett. 99, 088101.CrossRefGoogle ScholarPubMed
Canham, P. B. 1970 The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theor. Biol. 26, 61–81.CrossRefGoogle ScholarPubMed
Chabanon, M. & Rangamani, P. 2018 Gaussian curvature directs the distribution of spontaneous curvature on bilayer membrane necks. Soft Matt. 12, 2281–2294.CrossRefGoogle Scholar
Chabanon, M. & Rangamani, P. 2019 Geometric coupling of helicoidal ramps and curvature-inducing proteins in organelle membranes. J. R. Soc. Interface 16, 20190354.CrossRefGoogle ScholarPubMed
Dannhauser, P. N. & Ungewickell, E. J. 2012 Reconstitution of clathrin-coated bud and vesicle formation with minimal components. Nat. Cell Biol. 14, 634–639.CrossRefGoogle ScholarPubMed
Do, C. 1976 Differential Geometry of Curves and Surfaces. Prentice Hall.Google Scholar
Elliott, C. M., Graser, C., Hobbs, G., Kornhuber, R. & Wolf, M. A. 2016 A variational approach to particles in lipid membranes. Arch. Rat. Mech. Anal. 222, 1011–1075.CrossRefGoogle Scholar
Elliott, C. M. & Stinner, B. 2013 Computation of two-phase biomembranes with phase dependent material parameters using surface finite elements. Commun. Comput. Phys. 13, 325–360.CrossRefGoogle Scholar
Gera, P. & Salac, D. 2017 Cahn–Hilliard on surfaces: a numerical study. Appl. Math. Lett. 73, 56–61.CrossRefGoogle Scholar
Glasmaster, K., Larsson, C., Hook, F. & Kasemo, B. 2002 Protein adsorption on supported phospholipid bilayer. J. Colloid Interface Sci. 246, 40–47.CrossRefGoogle Scholar
Glowinski, R., Pan, T.-W., Juarez, V. L. H. & Dean, E. 2005 Numerical methods for the simulation of incompressible viscous flow: an introduction. In Multidisciplinary Methods for Analysis Optimization and Control of Complex Systems (ed. Capasso, V. & Périaux, J.), pp. 49–175. Springer.CrossRefGoogle Scholar
Gov, N. S. 2018 Guided by curvature: shaping cells by coupling curved membrane proteins and cytoskeletal forces. Phil. Trans. R. Soc. Lond. B 373, 20170115.CrossRefGoogle ScholarPubMed
Gozdz, W. T. 2011 Shape transformation of lipid vesicle induced by diffusing macromolecules. J. Chem. Phys. 134, 371–379.CrossRefGoogle ScholarPubMed
Góźdź, W. T. & Gompper, G. 1998 Composition-driven shape transformations of membranes of complex topology. Phys. Rev. Lett. 80, 4213–4216.CrossRefGoogle Scholar
Gruenberg, J. 2001 The endocytic pathway: A mosaic of domains. Nat. Rev. Mol. Cell Biol. 2, 721–730.CrossRefGoogle ScholarPubMed
Harayama, T. & Riezman, H. 2018 Understanding the diversity of membrane lipid composition. Nat. Rev. Mol. Cell Biol. 19, 281–296.CrossRefGoogle ScholarPubMed
Hassinger, J. E., Oster, G., Drubin, D. G. & Rangamani, P. 2017 Design principles for robust vesiculation in clathrin-mediated endocytosis. Proc. Natl Acad. Sci. USA 114, 1118–1127.CrossRefGoogle ScholarPubMed
Haucke, V. & Kozlov, M. M. 2018 Membrane remodeling in clathrin-mediated endocytosis. J. Cell Sci. 131, jcs216812.CrossRefGoogle ScholarPubMed
Helfrich, W. 1973 Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. C 5, 693–703.CrossRefGoogle Scholar
Horner, A., Antonenko, Y. N. & Pohl, P. 2009 Coupled diffusion of peripherally bound peptides along the outer and inner membrane leaflets. Biophys. J. 96, 2689–2695.CrossRefGoogle ScholarPubMed
Hurley, J. H., Boura, E., Carlson, L.-A. & Różycki, B. 2010 Membrane budding. Cell 143, 875–887.CrossRefGoogle ScholarPubMed
Iglič, A., Babnik, B., Gimsa, U. & Kralj-Iglič, V. 2005 On the role of membrane anisotropy in the beading transition of undulated tubular membrane structures. J. Phys. A 40, 8527–8536.CrossRefGoogle Scholar
Jahn, R., Lang, T. & Südhof, T. C. 2003 Membrane fusion. Cell 112, 519–533.CrossRefGoogle ScholarPubMed
Jenkins, J. T. 1977 The equation of mechanical equilibrium of a model membrane. SIAM J. Appl. Maths 4, 693–703.Google Scholar
Jülicher, F. & Lipowsky, R. 1993 Domain-induced budding of vesicles. Phys. Rev. Lett. 70, 2964–2967.CrossRefGoogle ScholarPubMed
Kahya, N., Scherfeld, D., Bacia, K. & Schwille, P. 2004 Lipid domain formation and dynamics in giant unilamellar vesicles explored by fluorescence correlation spectroscopy. J. Struct. Biol. 147, 77–89.CrossRefGoogle ScholarPubMed
Kozlovsky, Y. & Kozlov, M. M. 2003 Membrane fission: model for intermediate structures. Biophys. J. 85, 85–96.CrossRefGoogle ScholarPubMed
Kralj-Iglič, V., Hagerstand, H., Veranic, P., Jezernik, K., Babnik, B., Gauger, D. R. & Iglič, A. 2005 Amphiphile-induced tubular budding of the bilayer membrane. Eur. Biophys. J. 34, 1066–1070.CrossRefGoogle ScholarPubMed
Kreyszig, E. 1968 Advanced Engineering Mathematics. Wiley.Google Scholar
Kumar, P. B. S., Gompper, G. & Lipowsky, R. 2001 Budding dynamics of multicomponent membranes. Phys. Rev. Lett. 86, 3911.CrossRefGoogle Scholar
Leibler, S. & Andelman, D. 1987 Ordered and curved meso-structures in membranes and amphiphilic films. J. Phys. France 48, 2013–2018.CrossRefGoogle Scholar
Lipowsky, R. 1991 The conformation of membranes. Nature 349, 475–481.CrossRefGoogle ScholarPubMed
Lipowsky, R. 2013 Spontaneous tubulation of membranes and vesicles reveals membrane tension generated by spontaneous curvature. Faraday Discuss. 161, 305–331.CrossRefGoogle ScholarPubMed
Liu, I. B., Sharifi-Mood, N. & Stebe, K. J. 2018 Capillary assembly of colloids: interactions on planar and curved interfaces. Annu. Rev. Condens. Matter Phys. 9, 283–305.CrossRefGoogle Scholar
Lowengrub, J. S., Ratz, A. & Voigt, A. 2009 Phase-field modeling of the dynamics of multicomponent vesicles: spinodal decomposition, coarsening, budding, and fission. Phys. Rev. E 79, 031926.CrossRefGoogle ScholarPubMed
McMahon, H. T. & Gallop, J. L. 2005 Membrane curvature and mechanisms of dynamic cell membrane remodeling. Nature 438, 590–596.CrossRefGoogle Scholar
Mietke, A., Julicher, F. & Sbalzarini, I. F. 2019 Self-organized shape dynamics of active surfaces. Proc. Natl Acad. Sci. USA 116, 29–34.CrossRefGoogle ScholarPubMed
Mikucki, M. & Zhou, Y. C. 2017 Curvature-driven molecular flow on membrane surface. SIAM J. Appl. Math. 77, 1587–1605.CrossRefGoogle ScholarPubMed
Mujherjee, S. & Maxfield, E. R. 2000 Role of membrane organization and membrane domains in endocytic lipid trafficking. Traffic 1, 203–211.CrossRefGoogle Scholar
Nitschke, I, Voigt, A & Wensch, J 2012 A finite element approach to incompressible two-phase flow on manifolds. J. Fluid Mech. 708, 418–438.CrossRefGoogle Scholar
Noguchi, H. & Gompper, G. 2004 Fluid vesicles with viscous membranes in shear flow. Phys. Rev. Lett. 93, 258102.CrossRefGoogle ScholarPubMed
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Rahimi, M. & Arroyo, M. 2012 Shape dynamics, lipid hydrodynamics, and complex viscoelasdticity of bilayer membranes. Phys. Rev. E 86, 011932.CrossRefGoogle ScholarPubMed
Ramaswamy, S., Toner, J. & Prost, J. 2005 Nonequilibrium fluctuations, travelling waves, and instabilities in active membranes. Phys. Rev. Lett. 84, 3494–3497.CrossRefGoogle Scholar
Rangamani, P., Agrawal, A., Mandadapu, K., Oster, G. & Steigmann, D. 2013 Interaction between surface shape and intra-surface viscous flow on lipid membranes. Biomech. Model. Mechanobiol. 12, 833–845.CrossRefGoogle ScholarPubMed
Rangamani, P., Behzadan, A. & Holst, M. 2020 Local sensitivity analysis of the ‘membrane shape equation’ derived from Helfrich energy. arXiv:2005.12550.CrossRefGoogle Scholar
Rangamani, P., Mandadapu, K. & Oster, G. 2014 Protein induced membrane curvature alters local membrane tension. Biophys. J. 107, 751–762.CrossRefGoogle ScholarPubMed
Reynwar, B. J., Illya, G., Harmandaris, V. A., Müller, M. M., Kremer, K. & Deserno, M. 2007 Aggregation and vesiculation of membrane proteins by curvature-mediated interactions. Nature 447, 461–464.CrossRefGoogle ScholarPubMed
Sackmann, E., Duwe, H. P. & Engelhardt, H. 1986 Membrane bending elasticity and its role for shape fluctuations and shape transformations of cells and vesicles. Faraday Discuss. Chem. Soc. 81, 281–290.CrossRefGoogle Scholar
Sahu, A., Omar, Y. A. D., Sauer, R. A. & Mandadapu, K. K. 2020 Arbitrary Lagrangian–Eulerian finite element method for curved and deforming surfaces: I. General theory and application to fluid interfaces. J. Comput. Phys. 407, 109253.CrossRefGoogle Scholar
Sahu, A., Sauer, R. A. & Mandadapu, K. K. 2017 Irreversible thermodynamics of curved lipid membranes. Phys. Rev. E 96, 042409.CrossRefGoogle ScholarPubMed
Salbreux, G. & Jülicher, F. 2017 Mechanics of active surfaces. Phys. Rev. E 96, 032404.CrossRefGoogle ScholarPubMed
Sauer, R. A., Duong, T. X., Mandadapu, K. K. & Steigmann, D. J. 2017 A stabilized finite element formulation for liquid shells and its application to lipid bilayers. J. Comput. Phys. 330, 436–466.CrossRefGoogle Scholar
Scriven, L. E. 1960 Dynamics of a fluid surface. Chem. Engng Sci. 12, 98–108.CrossRefGoogle Scholar
Seifert, U. 1993 Curvature-induced lateral phase segregation in two-component vesicles. Phys. Rev. Lett. 70, 1335–1338.CrossRefGoogle ScholarPubMed
Seifert, U. & Langer, S. A. 1993 Viscous modes of fluid bilayer membranes. Europhys. Lett. 23, 65–72.CrossRefGoogle Scholar
Singer, S. 1974 The molecular organization of membranes. Annu. Rev. Biochem. 43, 805–833.CrossRefGoogle ScholarPubMed
Snead, W. T., Hayden, C. C., Gadok, A. K., Zhao, C., Lafer, E. M., Rangamani, P. & Stachowiak, J. C. 2017 Membrane fission by protein crowding. Proc. Natl Acad. Sci. 114, 3258–3267.CrossRefGoogle ScholarPubMed
Sokolnikoff, I. S. 1951 Tensor Analysis: Theory and Applications. Wiley.Google Scholar
Stachowiak, J. C., Hayden, C. C. & Saski, D. Y. 2010 Steric confinement of proteins on lipid membranes can drive curvature and tubulation. Proc. Natl Acad. Sci. 107, 7781–7786.CrossRefGoogle ScholarPubMed
Steigmann, D. & Agrawal, A. 2011 A model for surface diffusion of trans-membrane protein on lipid bilayers. Z. Angew. Math. Phys. 62, 449–563.Google Scholar
Steigmann, D., Baesu, E., Rudd, R. E., Belak, J. & McEleresh, M. 2003 On the variational theory of cell-membrane equilibria. Interface. Free Bound. 5, 357–366.CrossRefGoogle Scholar
Steigmann, D. J. 1999 Fluid films with curvature elasticity. Arch. Rat. Mech. Anal. 150, 257–272.CrossRefGoogle Scholar
Steigmann, D. J. 2018 Mechanics and physics of lipid bilayers. In The Role of Mechanics in the Study of Lipid Bilayers (ed. Steigmann, D. J.), pp. 1–61. Springer.CrossRefGoogle Scholar
Torres-Sanchez, A., Millan, D. & Arroyo, M. 2019 Modelling fluid deformable surfaces with an emphasis on biological interfaces. J. Fluid Mech. 872, 218–271.CrossRefGoogle Scholar
Tozzi, C., Walani, N. & Arroyo, M. 2019 Out-of-equilibrium mechanochemistry and self-organization of fluid membranes interacting with curved proteins. New J. Phys. 21, 093004.CrossRefGoogle Scholar
Tran-Son-Tay, R., Sutera, S. P. & Rao, P. R. 1984 Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion. Biophys. J. 46, 1335–1338.CrossRefGoogle ScholarPubMed
Vasan, R., Rudraraju, S., Akamatsu, M., Garikipati, K. & Rangamani, P. 2020 A mechanical model reveals that non-axisymmetric buckling lowers the energy barrier associated with membrane neck constriction. Soft Matt. 16, 784–797.CrossRefGoogle ScholarPubMed
Veksler, A. & Gov, N. S. 2007 Phase transitions of the coupled membrane-cytoskeleton modify cellular shape. Biophys. J. 93, 3798–3810.CrossRefGoogle ScholarPubMed
Zhdanov, V. P. & Kasemo, B. 2010 Adsorption of proteins on a lipid bilayer. Eur. Biophys. J. 39, 1477–1482.CrossRefGoogle ScholarPubMed
Zhong-Can, O.-Y. & Helfrich, W. 1989 Bending energy of vesicle membranes: general expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. Phys. Rev. A 39, 5280–5288.CrossRefGoogle ScholarPubMed
Zimmerberg, J. & Kozlov, M. M. 2006 How proteins produce cellular membrane curvature. Nat. Rev. Mol. Cell Biol. 7, 9–19.CrossRefGoogle ScholarPubMed