Hostname: page-component-5d59c44645-klj7v Total loading time: 0 Render date: 2024-03-02T21:00:38.192Z Has data issue: false hasContentIssue false

Modelling fluid deformable surfaces with an emphasis on biological interfaces

Published online by Cambridge University Press:  10 June 2019

Alejandro Torres-Sánchez
Affiliation:
LaCàN, Universitat Politècnica de Catalunya – BarcelonaTech, 08034 Barcelona, Spain
Daniel Millán
Affiliation:
LaCàN, Universitat Politècnica de Catalunya – BarcelonaTech, 08034 Barcelona, Spain CONICET and Facultad de Ciencias Aplicadas a la Industria, Universidad Nacional de Cuyo, 5600 San Rafael, Argentina
Marino Arroyo*
Affiliation:
LaCàN, Universitat Politècnica de Catalunya – BarcelonaTech, 08034 Barcelona, Spain Institute for Bioengineering of Catalonia, The Barcelona Institute of Science and Technology, 08028 Barcelona, Spain
*
Email address for correspondence: marino.arroyo@upc.edu

Abstract

Fluid deformable surfaces are ubiquitous in cell and tissue biology, including lipid bilayers, the actomyosin cortex or epithelial cell sheets. These interfaces exhibit a complex interplay between elasticity, low Reynolds number interfacial hydrodynamics, chemistry and geometry, and govern important biological processes such as cellular traffic, division, migration or tissue morphogenesis. To address the modelling challenges posed by this class of problems, in which interfacial phenomena tightly interact with the shape and dynamics of the surface, we develop a general continuum mechanics and computational framework for fluid deformable surfaces. The dual solid–fluid nature of fluid deformable surfaces challenges classical Lagrangian or Eulerian descriptions of deforming bodies. Here, we extend the notion of arbitrarily Lagrangian–Eulerian (ALE) formulations, well-established for bulk media, to deforming surfaces. To systematically develop models for fluid deformable surfaces, which consistently treat all couplings between fields and geometry, we follow a nonlinear Onsager formalism according to which the dynamics minimizes a Rayleighian functional where dissipation, power input and energy release rate compete. Finally, we propose new computational methods, which build on Onsager’s formalism and our ALE formulation, to deal with the resulting stiff system of higher-order partial differential equations. We apply our theoretical and computational methodology to classical models for lipid bilayers and the cell cortex. The methods developed here allow us to formulate/simulate these models in their full three-dimensional generality, accounting for finite curvatures and finite shape changes.

Type
JFM Papers
Copyright
© 2019 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

Aris, R. 1962 Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Prentice-Hall.Google Scholar
Arroyo, M. & DeSimone, A. 2009 Relaxation dynamics of fluid membranes. Phys. Rev. E 79 (3), 031915.Google Scholar
Arroyo, M., DeSimone, A. & Heltai, L.2010 The role of membrane viscosity in the dynamics of fluid membranes. arXiv:1007.4934.Google Scholar
Arroyo, M., Heltai, L., Millán, D. & DeSimone, A. 2012 Reverse engineering the euglenoid movement. Proc. Natl Acad. Sci. USA 109 (44), 1787417879.Google Scholar
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 The Role of Mechanics in the Study of Lipid Bilayers (ed. Steigmann, D. J.), pp. 287332. Springer.Google Scholar
Bacia, K., Schwille, P. & Kurzchalia, T. 2005 Sterol structure determines the separation of phases and the curvature of the liquid-ordered phase in model membranes. Proc. Natl Acad. Sci. USA 102 (9), 32723277.Google Scholar
Barrett, J. W., Garcke, H. & Nürnberg, R. 2008 On the parametric finite element approximation of evolving hypersurfaces in R3. J. Comput. Phys. 227 (9), 42814307.Google Scholar
Barrett, J. W., Garcke, H. & Nürnberg, R. 2015 Numerical computations of the dynamics of fluidic membranes and vesicles. Phys. Rev. E 92 (5), 052704.Google Scholar
Barrett, J. W., Garcke, H. & Nürnberg, R. 2016 A stable numerical method for the dynamics of fluidic membranes. Numer. Math. 134 (4), 783822.Google Scholar
Barthes-Biesel, D. & Sgaier, H. 1985 Role of membrane viscosity in the orientation and deformation of a spherical capsule suspended in shear flow. J. Fluid Mech. 160, 119135.Google Scholar
Baumgart, T., Capraro, B. R., Zhu, C. & Das, S. L. 2011 Thermodynamics and mechanics of membrane curvature generation and sensing by proteins and lipids. Annu. Rev. Phys. Chem. 62, 483506.Google Scholar
Bergert, M., Erzberger, A., Desai, R. A., Aspalter, I. M., Oates, A. C., Charras, G., Salbreux, G. & Paluch, E. K. 2015 Force transmission during adhesion-independent migration. Nat. Cell Biol. 17 (4), 524529.Google Scholar
Biermann, H., Levin, A. & Zorin, D. 2000 Piecewise smooth subdivision surfaces with normal control. In Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, pp. 113120. ACM Press/Addison-Wesley.Google Scholar
Biria, A., Maleki, M. & Fried, E. 2013 Chapter one – continuum theory for the edge of an open lipid bilayer. In Advances in Applied Mechanics (ed. Bordas, S. P. A.), vol. 46, pp. 168. Elsevier.Google Scholar
Bray, D. & White, J. G. 1988 Cortical flow in animal cells. Science 239 (4842), 883888.Google Scholar
Brezzi, F. & Fortin, M. 2012 Mixed and Hybrid Finite Element Methods. Springer Science & Business Media.Google Scholar
Brochard-Wyart, F. & de Gennes, P.-G. 2002 Adhesion induced by mobile binders: dynamics. Proc. Natl Acad. Sci. USA 99 (12), 78547859.Google Scholar
Burman, E., Hansbo, P. & Larson, M. G. 2015 A stabilized cut finite element method for partial differential equations on surfaces: the Laplace–Beltrami operator. Comput. Meth. Appl. Mech. Engng 285, 188207.Google Scholar
Callan-Jones, A. C., Ruprecht, V., Wieser, S., Heisenberg, C. P. & Voituriez, R. 2016 Cortical flow-driven shapes of nonadherent cells. Phys. Rev. Lett. 116 (2), 028102.Google Scholar
Callan-Jones, A. C. & Voituriez, R. 2013 Active gel model of amoeboid cell motility. New J. Phys. 15 (2), 025022.Google Scholar
Campillo, C., Sens, P., Köster, D., Pontani, L.-L., Lévy, D., Bassereau, P., Nassoy, P. & Sykes, C. 2013 Unexpected membrane dynamics unveiled by membrane nanotube extrusion. Biophys. J. 104 (6), 12481256.Google Scholar
Capovilla, R. & Guven, J. 2002 Stresses in lipid membranes. J. Phys. A 35 (30), 62336247.Google Scholar
do Carmo, M. P. 1992 Riemannian Geometry, vol. 115. Birkhäuser.Google Scholar
do Carmo, M. P. 2016 Differential Geometry of Curves and Surfaces, 2nd edn. Dover.Google Scholar
Cermelli, P., Fried, E. & Gurtin, M. E. 2005 Transport relations for surface integrals arising in the formulation of balance laws for evolving fluid interfaces. J. Fluid Mech. 544, 339351.Google Scholar
Chugh, P., Clark, A. G., Smith, M. B., Cassani, D. A. D., Dierkes, K., Ragab, A., Roux, P. P., Charras, G., Salbreux, G. & Paluch, E. K. 2017 Actin cortex architecture regulates cell surface tension. Nat. Cell Biol. 19 (6), 689697.Google Scholar
Cirak, F. & Long, Q. 2011 Subdivision shells with exact boundary control and non-manifold geometry. Intl J. Numer. Meth. Engng 88, 897923.Google Scholar
Cirak, F. & Ortiz, M. 2000 Schroder (2000) Subdivision surfaces: a new paradigm for thin shell finite-element analysis. Intl J. Numer. Meth. Engng 47 (12), 20392072.Google Scholar
Cirak, F. & Ortiz, M. 2001 Fully C1-conforming subdivision elements for finite deformation thin-shell analysis. Intl J. Numer. Meth. Engng 51 (7), 813833.Google Scholar
Clark, A. G., Dierkes, K. & Paluch, E. K. 2013 Monitoring actin cortex thickness in live cells. Biophys. J. 105 (3), 570580.Google Scholar
Dimova, R., Aranda, S., Bezlyepkina, N., Nikolov, V., Riske, K. A. & Lipowsky, R. 2006 A practical guide to giant vesicles. Probing the membrane nanoregime via optical microscopy. J. Phys.: Condens. Matter 18 (28), S1151S1176.Google Scholar
Doi, M. 2011 Onsager’s variational principle in soft matter. J. Phys.: Condens. Matter 23 (28), 284118.Google Scholar
Donea, J. & Huerta, A. 2003 Finite Element Methods for Flow Problems, reprint edn. John Wiley & Sons.Google Scholar
Dörries, G. & Foltin, G. 1996 Energy dissipation of fluid membranes. Phys. Rev. E 53 (3), 25472550.Google Scholar
Dortdivanlioglu, B., Krischok, A., Beirão da Veiga, L. & Linder, C. 2018 Mixed isogeometric analysis of strongly coupled diffusion in porous materials: mixed IGA of strongly coupled diffusion in porous materials. Intl J. Numer. Meth. Engng 114 (1), 2846.Google Scholar
Dziuk, G. & Elliott, C. M. 2013 Finite element methods for surface PDEs. Acta Numer. 22, 289396.Google Scholar
Elliott, C. M. & Styles, V. 2012 An ALE ESFEM for solving PDEs on evolving surfaces. Milan J. Math. 80 (2), 469501.Google Scholar
Evans, E. & Yeung, A. 1994 Hidden dynamics in rapid changes of bilayer shape. Chem. Phys. Lipids 73 (1), 3956.Google Scholar
Farutin, A. & Misbah, C. 2012 Rheology of vesicle suspensions under combined steady and oscillating shear flows. J. Fluid Mech. 700, 362381.Google Scholar
Feng, F. & Klug, W. S. 2006 Finite element modeling of lipid bilayer membranes. J. Comput. Phys. 220 (1), 394408.Google Scholar
Fischer, F. D., Svoboda, J. & Petryk, H. 2014 Thermodynamic extremal principles for irreversible processes in materials science. Acta Mater. 67, 120.Google Scholar
Fournier, J.-B. 2015 On the hydrodynamics of bilayer membranes. Intl J. Non-Linear Mech. 75, 6776.Google Scholar
Fournier, J.-B., Khalifat, N., Puff, N. & Angelova, M. I. 2009 Chemically triggered ejection of membrane tubules controlled by intermonolayer friction. Phys. Rev. Lett. 102 (1), 018102.Google Scholar
Frenkel, D. & Smit, B. 2001 Understanding Molecular Simulation: From Algorithms to Applications, 2nd edn. Academic Press.Google Scholar
Fries, T.-P. 2018 Higher-order surface FEM for incompressible Navier–Stokes flows on manifolds. Intl J. Numer. Meth. Fluids 88 (2), 5578.Google Scholar
Fritzsche, M., Lewalle, A., Duke, T., Kruse, K. & Charras, G. 2013 Analysis of turnover dynamics of the submembranous actin cortex. Mol. Biol. Cell 24 (6), 757767.Google Scholar
Gompper, G. & Kroll, D. M. 2004 Triangulated-surface models of fluctuating membranes. In Statistical Mechanics of Membranes and Surfaces, pp. 359426. World Scientific.Google Scholar
Gross, B. J. & Atzberger, P. J. 2018 Hydrodynamic flows on curved surfaces: spectral numerical methods for radial manifold shapes. J. Comput. Phys. 371, 663689.Google Scholar
Hamm, M. & Kozlov, M. M. 1998 Tilt model of inverted amphiphilic mesophases. Eur. Phys. J. B 6 (4), 519528.Google Scholar
Hamm, M. & Kozlov, M. M. 2000 Elastic energy of tilt and bending of fluid membranes. Eur. Phys. J. E 3 (4), 323335.Google Scholar
Hansbo, P., Larson, M. G. & Larsson, K.2016 Analysis of finite element methods for vector laplacians on surfaces. arXiv:1610.06747.Google Scholar
Hawkins, R. J., Poincloux, R., Bénichou, O., Piel, M., Chavrier, P. & Voituriez, R. 2011 Spontaneous contractility-mediated cortical flow generates cell migration in three-dimensional environments. Biophys. J. 101 (5), 10411045.Google Scholar
Helfrich, W. 1973 Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. 28c, 693703.Google Scholar
Henle, M. L. & Levine, A. J. 2010 Hydrodynamics in curved membranes: the effect of geometry on particulate mobility. Phys. Rev. E 81 (1), 117.Google Scholar
Hirt, C. W., Amsden, A. A. & Cook, J. L. 1974 An arbitrary Lagrangian–Eulerian computing method for all flow speeds. J. Comput. Phys. 14 (3), 227253.Google Scholar
Ho, J. S. & Baumgärtner, A. 1990 Simulations of fluid self-avoiding membranes. Europhys. Lett. 12 (4), 295.Google Scholar
Howard, J. 2001 Mechanics of Motor Proteins and the Cytoskeleton. Sinauer Associates, Publishers.Google Scholar
Hu, D., Zhang, P. & Weinan, E. 2007 Continuum theory of a moving membrane. Phys. Rev. E 75 (4), 111.Google Scholar
Jankuhn, T., Olshanskii, M. A. & Reusken, A. 2018 Incompressible fluid problems on embedded surfaces: modeling and variational formulations. Interfaces Free Boundaries 20, 353378.Google Scholar
Jülicher, F. & Lipowsky, R. 1993 Domain-induced budding of vesicles. Phys. Rev. Lett. 70 (19), 29642967.Google Scholar
Jüttler, B., Mantzaflaris, A., Perl, R. & Rumpf, M. 2016 On numerical integration in isogeometric subdivision methods for PDEs on surfaces. Comput. Meth. Appl. Mech. Engng 302, 131146.Google Scholar
Kantsler, V., Segre, E. & Steinberg, V. 2007 Vesicle dynamics in time-dependent elongation flow: wrinkling instability. Phys. Rev. Lett. 99 (17), 178102.Google Scholar
Khalifat, N., Puff, N., Bonneau, S., Fournier, J.-B. & Angelova, M. I. 2008 Membrane deformation under local pH gradient: mimicking mitochondrial cristae dynamics. Biophys. J. 95 (10), 49244933.Google Scholar
Khalifat, N., Rahimi, M., Bitbol, A.-F., Seigneuret, M., Fournier, J.-B., Puff, N., Arroyo, M. & Angelova, M. I. 2014 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), 879890.Google Scholar
Koba, H., Liu, C. & Giga, Y. 2017 Energetic variational approaches for incompressible fluid systems on an evolving surface. Q. Appl. Maths 75 (2), 359389.Google Scholar
Kosmalska, A. J., Casares, L., Elosegui-Artola, A., Thottacherry, J. J., Moreno-Vicente, R., González-Tarragó, V., del Pozo, M. Á., Mayor, S., Arroyo, M., Navajas, D. et al. 2015 Physical principles of membrane remodelling during cell mechanoadaptation. Nat. Commun. 6, 7292.Google Scholar
Kroll, D. M. & Gompper, G. 1992 The conformation of fluid membranes: Monte Carlo simulations. Science 255 (5047), 968971.Google Scholar
Laadhari, A., Saramito, P., Misbah, C. & Székely, G. 2017 Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and Newton methods. J. Comput. Phys. 343, 271299.Google Scholar
Lebon, G., Jou, D. & Casas-Vázquez, J. 2008 Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers. Springer.Google Scholar
Levayer, R. & Lecuit, T. 2012 Biomechanical regulation of contractility: spatial control and dynamics. Trends Cell Biol. 22 (2), 6181.Google Scholar
Levine, A. J., Liverpool, T. B. & MacKintosh, F. C. 2004 Dynamics of rigid and flexible extended bodies in viscous films and membranes. Phys. Rev. Lett. 93 (3), 038102.Google Scholar
Lew, A., Marsden, J. E., Ortiz, M. & West, M. 2004 Variational time integrators. Intl J. Numer. Meth. Engng 60 (1), 153212.Google Scholar
Li, B., Millán, D., Torres-Sánchez, A., Roman, B. & Arroyo, M. 2018 A variational model of fracture for tearing brittle thin sheets. J. Mech. Phys. Solids 119, 334348.Google Scholar
Lieber, A. D., Schweitzer, Y., Kozlov, M. M. & Keren, K. 2015 Front-to-rear membrane tension gradient in rapidly moving cells. Biophys. J. 108 (7), 15991603.Google Scholar
Lipowsky, R. 1991 The conformation of membranes. Nature 349 (6309), 475481.Google Scholar
Liu, W. K., Liu, Y., Farrell, D., Zhang, L., Wang, X. S., Fukui, Y., Patankar, N., Zhang, Y., Bajaj, C., Lee, J., Hong, J., Chen, X. & Hsu, H. 2006 Immersed finite element method and its applications to biological systems. Comput. Meth. Appl. Mech. Engng 195 (13–16), 17221749.Google Scholar
Liu, Y.-J., Le Berre, M., Lautenschlaeger, F., Maiuri, P., Callan-Jones, A., Heuzé, M., Takaki, T., Voituriez, R. & Piel, M. 2015 Confinement and low adhesion induce fast amoeboid migration of slow mesenchymal cells. Cell 160 (4), 659672.Google Scholar
Loop, C.1987 Smooth subdivision surfaces based on triangles. PhD thesis, University of Utah.Google Scholar
Ma, L. & Klug, W. S. 2008 Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics. J. Comput. Phys. 227 (11), 58165835.Google Scholar
Marsden, J. E. & Hughes, T. J. R. 1994 Mathematical Foundations of Elasticity. Courier Corporation.Google Scholar
Martyushev, L. M. & Seleznev, V. D. 2006 Maximum entropy production principle in physics, chemistry and biology. Phys. Rep. 426 (1), 145.Google Scholar
Miao, L., Seifert, U., Wortis, M. & Döbereiner, H. G. 1994 Budding transitions of fluid-bilayer vesicles: the effect of area-difference elasticity. Phys. Rev. E 49 (6), 53895407.Google Scholar
Mickelin, O., Słomka, J., Burns, K. J., Lecoanet, D., Vasil, G. M., Faria, L. M. & Dunkel, J. 2018 Anomalous chained turbulence in actively driven flows on spheres. Phys. Rev. Lett. 120 (16), 164503.Google Scholar
Mielke, A. 2012 Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions. Discrete Continuous Dyn. Syst. – Ser. S 6 (2), 479499.Google Scholar
Mietke, A., Jülicher, F. & Sbalzarini, I. F. 2019 Self-organized shape dynamics of active surfaces. Proc. Natl Acad. Sci. USA 116 (1), 2934.Google Scholar
Millán, D., Rosolen, A. & Arroyo, M. 2011 Thin shell analysis from scattered points with maximum-entropy approximants. Intl J. Numer. Meth. Engng 85 (6), 723751.Google Scholar
Morris, R. G. & Turner, M. S. 2015 Mobility measurements probe conformational changes in membrane proteins due to tension. Phys. Rev. Lett. 115 (19), 198101.Google Scholar
Nelson, P., Powers, T. & Seifert, U. 1995 Dynamical theory of the pearling instability in cylindrical vesicles. Phys. Rev. Lett. 74 (17), 33843387.Google Scholar
Nestler, M., Nitschke, I., Praetorius, S. & Voigt, A. 2018 Orientational order on surfaces: the coupling of topology, geometry, and dynamics. J. Nonlinear Sci. 28 (1), 147191.Google Scholar
Nitschke, I., Voigt, A. & Wensch, J. 2012 A finite element approach to incompressible two-phase flow on manifolds. J. Fluid Mech. 708, 418438.Google Scholar
Noguchi, H. & Gompper, G. 2005 Shape transitions of fluid vesicles and red blood cells in capillary flows. Proc. Natl Acad. Sci. USA 102 (40), 1415914164.Google Scholar
Noselli, G., Bean, A., Arroyo, M. & DeSimone, A. 2019 Swimming Euglena respond to confinement with a behavioural change enabling effective crawling. Nat. Phys. 15 (5), 496502.Google Scholar
Olshanskii, M., Quaini, A., Reusken, A. & Yushutin, V. 2018 A finite element method for the surface stokes problem. SIAM J. Sci. Comput. 40 (4), A2492A2518.Google Scholar
Ortiz, M. & Stainier, L. 1999 The variational formulation of viscoplastic constitutive updates. Comput. Meth. Appl. Mech. Engng 171 (3), 419444.Google Scholar
Paltridge, G. W. 1975 Global dynamics and climate – a system of minimum entropy exchange. Q. J. R. Meteorol. Soc. 101 (429), 475484.Google Scholar
Peco, C., Rosolen, A. & Arroyo, M. 2013 An adaptive meshfree method for phase-field models of biomembranes. Part II: a Lagrangian approach for membranes in viscous fluids. J. Comput. Phys. 249, 320336.Google Scholar
Peletier, M.2014 Variational modelling: energies, gradient flows, and large deviations. Preprint, arXiv:1402.1990.Google Scholar
Peng, Z., Li, X., Pivkin, I. V., Dao, M., Karniadakis, G. E. & Suresh, S. 2013 Lipid bilayer and cytoskeletal interactions in a red blood cell. Proc. Natl Acad. Sci. USA 110 (33), 1335613361.Google Scholar
Piegl, L. & Tiller, W. 2012 The NURBS Book. Springer Science & Business Media.Google Scholar
Poincloux, R., Collin, O., Lizárraga, F., Romao, M., Debray, M., Piel, M. & Chavrier, P. 2011 Contractility of the cell rear drives invasion of breast tumor cells in 3D Matrigel. Proc. Natl Acad. Sci. USA 108 (5), 19431948.Google Scholar
Prost, J., Jülicher, F. & Joanny, J.-F. 2015 Active gel physics. Nat. Phys. 11 (2), 111.Google Scholar
Rahimi, M.2013 Shape dynamics and lipid hydrodynamics of bilayer membranes: modeling, simulation and experiments. PhD thesis, Universitat Politècnica de Catalunya – BarcelonaTech.Google Scholar
Rahimi, M. & Arroyo, M. 2012 Shape dynamics, lipid hydrodynamics, and the complex viscoelasticity of bilayer membranes. Phys. Rev. E 86 (1), 011932.Google Scholar
Rahimi, M., DeSimone, A. & Arroyo, M. 2013 Curved fluid membranes behave laterally as effective viscoelastic media. Soft Matt. 9 (46), 1103311045.Google Scholar
Rangarajan, R. & Gao, H. 2015 A finite element method to compute three-dimensional equilibrium configurations of fluid membranes: optimal parameterization, variational formulation and applications. J. Comput. Phys. 297, 266294.Google Scholar
Rangarajan, R., Kabaria, H. & Lew, A. 2019 An algorithm for triangulating smooth three-dimensional domains immersed in universal meshes: meshing domains immersed in universal meshes. Intl J. Numer. Meth. Engng 117 (1), 84117.Google Scholar
Reuther, S. & Voigt, A. 2015 The interplay of curvature and vortices in flow on curved surfaces. Multiscale Model. Simul. 13 (2), 632643.Google Scholar
Reuther, S. & Voigt, A. 2016 Incompressible two-phase flows with an inextensible Newtonian fluid interface. J. Comput. Phys. 322, 850858.Google Scholar
Reuther, S. & Voigt, A. 2018a Erratum: the interplay of curvature and vortices in flow on curved surfaces. Multiscale Model. Simul. 16 (3), 14481453.Google Scholar
Reuther, S. & Voigt, A. 2018b Solving the incompressible surface Navier–Stokes equation by surface finite elements. Phys. Fluids 30 (1), 012107.Google Scholar
Reymann, A.-C., Staniscia, F., Erzberger, A., Salbreux, G. & Grill, S. W. 2016 Cortical flow aligns actin filaments to form a furrow. Elife 5, 125.Google Scholar
Rodrigues, D. S., Ausas, R. F., Mut, F. & Buscaglia, G. C. 2015 A semi-implicit finite element method for viscous lipid membranes. J. Comput. Phys. 298, 565584.Google Scholar
Roux, A., Cappello, G., Cartaud, J., Prost, J., Goud, B. & Bassereau, P. 2002 A minimal system allowing tubulation with molecular motors pulling on giant liposomes. Proc. Natl Acad. Sci. USA 99 (8), 53945399.Google Scholar
Ruprecht, V., Wieser, S., Callan-Jones, A., Smutny, M., Morita, H., Sako, K., Barone, V., Ritsch-Marte, M., Sixt, M., Voituriez, R. & Heisenberg, C.-P. 2015 Cortical contractility triggers a stochastic switch to fast amoeboid cell motility. Cell 160 (4), 673685.Google Scholar
Rustom, A., Saffrich, R., Markovic, I., Walther, P. & Gerdes, H.-H. 2004 Nanotubular highways for intercellular organelle transport. Science 303 (5660), 10071010.Google Scholar
Saffman, P. G. & Delbrück, M. 1975 Brownian motion in biological membranes. Proc. Natl Acad. Sci. USA 72 (8), 31113113.Google Scholar
Saha, A., Nishikawa, M., Behrndt, M., Heisenberg, C.-P., Jülicher, F. & Grill, S. W. 2016 Determining physical properties of the cell cortex. Biophys. J. 110 (6), 14211429.Google Scholar
Sahu, A., Sauer, R. A. & Mandadapu, K. K. 2017 Irreversible thermodynamics of curved lipid membranes. Phys. Rev. E 96 (4), 042409.Google Scholar
Salac, D. & Miksis, M. 2011 A level set projection model of lipid vesicles in general flows. J. Comput. Phys. 230 (22), 81928215.Google Scholar
Salbreux, G., Charras, G. & Paluch, E. 2012 Actin cortex mechanics and cellular morphogenesis. Trends Cell Biol. 22 (10), 536545.Google Scholar
Salbreux, G. & Jülicher, F. 2017 Mechanics of active surfaces. Phys. Rev. E 96 (3), 032404.Google Scholar
Salbreux, G., Prost, J. & Joanny, J. F. 2009 Hydrodynamics of cellular cortical flows and the formation of contractile rings. Phys. Rev. Lett. 103 (5), 058102.Google Scholar
Santos-Oliván, D., Torres-Sánchez, A., Vilanova, G. & Arroyo, M.2019 A macroelement approach for inextensible flows with subdivision finite elements. (in preparation).Google Scholar
Sauer, R. A., Duong, T. X., Mandadapu, K. & Steigmann, D. 2017 A stabilized finite element formulation for liquid shells and its application to lipid bilayers. J. Comput. Phys. 330, 119.Google Scholar
Scriven, L. E. 1960 Dynamics of a fluid interface Equation of motion for Newtonian surface fluids. Chem. Engng Sci. 12 (2), 98108.Google Scholar
Secomb, T. W. & Skalak, R. 1982 Surface flow of viscoelastic membranes in viscous fluids. Q. J. Mech. Appl. Maths 35 (2), 233247.Google Scholar
Seifert, U. 1997 Configurations of fluid membranes and vesicles. Adv. Phys. 46 (1), 13137.Google Scholar
Seifert, U. & Langer, S. A. 1993 Viscous modes of fluid bilayer membranes. Europhys. Lett. 23 (1), 7176.Google Scholar
Sens, P., Johannes, L. & Bassereau, P. 2008 Biophysical approaches to protein-induced membrane deformations in trafficking. Curr. Opin. Cell Biol. 20 (4), 476482.Google Scholar
Shen, Z., Fischer, T. M., Farutin, A., Vlahovska, P. M., Harting, J. & Misbah, C. 2018 Blood crystal: emergent order of red blood cells under wall-confined shear flow. Phys. Rev. Lett. 120 (26), 268102.Google Scholar
Shibata, Y., Hu, J., Kozlov, M. M. & Rapoport, T. A. 2009 Mechanisms shaping the membranes of cellular organelles. Annu. Rev. Cell Dev. Biol. 25, 329354.Google Scholar
Sigurdsson, J. K. & Atzberger, P. J. 2016 Hydrodynamic coupling of particle inclusions embedded in curved lipid bilayer membranes. Soft Matt. 12 (32), 66856707.Google Scholar
Skalak, R. 1970 Extensions of extremum principles for slow viscous flows. J. Fluid Mech. 42 (3), 527548.Google Scholar
Sprong, H., van der Sluijs, P. & van Meer, G. 2001 How proteins move lipids and lipids move proteins. Nat. Rev. Mol. Cell Biol. 2 (7), 504513.Google Scholar
Stam, J. 1999 Evaluation of Loop subdivision surfaces. In SIGGRAPH’99 Course Notes. Los Angeles, CA.Google Scholar
Staneva, G., Angelova, M. I. & Koumanov, K. 2004 Phospholipase A2 promotes raft budding and fission from giant liposomes. Chem. Phys. Lipids 129 (1), 5362.Google Scholar
Staykova, M., Arroyo, M., Rahimi, M. & Stone, H. A. 2013 Confined bilayers passively regulate shape and stress. Phys. Rev. Lett. 110 (2), 028101.Google Scholar
Steigmann, D. J. 1999 Fluid films with curvature elasticity. Arch. Rat. Mech. Anal. 150 (2), 127152.Google Scholar
Stone, H. A. & Ajdari, A. 1998 Hydrodynamics of particles embedded in a flat surfactant layer overlying a subphase of finite depth. J. Fluid Mech. 369, 151173.Google Scholar
Torres-Sánchez, A.2017 A theoretical and computational study of the mechanics of biomembranes at multiple scales. PhD thesis, Universitat Politècnica de Catalunya.Google Scholar
Torres-Sánchez, A., Santos-Oliván, D. & Arroyo, M.2019 Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations. arXiv:1904.06390.Google Scholar
Tsafrir, I., Caspi, Y., Guedeau-Boudeville, M.-A., Arzi, T. & Stavans, J. 2003 Budding and tubulation in highly oblate vesicles by anchored amphiphilic molecules. Phys. Rev. Lett. 91 (13), 138102.Google Scholar
Tu, Z. C. & Ou-Yang, Z. C. 2004 A geometric theory on the elasticity of bio-membranes. J. Phys. A 37 (47), 1140711429.Google Scholar
Turlier, H., Audoly, B., Prost, J. & Joanny, J.-F. 2014 Furrow constriction in animal cell cytokinesis. Biophys. J. 106 (1), 114123.Google Scholar
Veerapaneni, S. K., Rahimian, A., Biros, G. & Zorin, D. 2011 A fast algorithm for simulating vesicle flows in three dimensions. J. Comput. Phys. 230 (14), 56105634.Google Scholar
Willmore, T. J. 1996 Riemannian Geometry. Oxford University Press.Google Scholar
Woodhouse, F. G. & Goldstein, R. E. 2012 Shear-driven circulation patterns in lipid membrane vesicles. J. Fluid Mech. 705, 165175.Google Scholar
Wu, J.-Z., Yang, Y.-T., Luo, Y.-B. & Pozrikidis, C. 2005 Fluid kinematics on a deformable surface. J. Fluid Mech. 541, 371381.Google Scholar
Yavari, A., Ozakin, A. & Sadik, S. 2016 Nonlinear elasticity in a deforming ambient space. J. Nonlinear Sci. 26 (6), 16511692.Google Scholar
Zhang, K. & Arroyo, M. 2014 Understanding and strain-engineering wrinkle networks in supported graphene through simulations. J. Mech. Phys. Solids 72, 6174.Google Scholar
Zhou, Y. & Yan, D. 2005 Real-time membrane fission of giant polymer vesicles. Angew. Chem. Intl Ed. Engl. 44 (21), 32233226.Google Scholar
Ziegler, H. 1958 An attempt to generalize Onsager’s principle, and its significance for rheological problems. Z. Angew. Math. Phys. 9 (5–6), 748763.Google Scholar
Ziegler, H. & Wehrli, C. 1987 The derivation of constitutive relations from the free energy and the dissipation function. In Advances in Applied Mechanics (ed. Wu, T. Y. & Hutchinson, J. W.), vol. 25, pp. 183238. Elsevier.Google Scholar

Torres-Sánchez supplementary movie 1

Remeshing during the relaxation dynamics of an inextensible monolayer with bending energy.

Download Torres-Sánchez supplementary movie 1(Video)
Video 6 MB

Torres-Sánchez supplementary movie 2

Relaxation dynamics of a density disturbance of 25% in the outer monolayer of a lipid bilayer modelled with the Seifert-Langer model. The density disturbance drives flows and also shape changes before relaxing to an equilibrium.

Download Torres-Sánchez supplementary movie 2(Video)
Video 467 KB

Torres-Sánchez supplementary movie 3

Self-polarization of a compressed and non-adherent cell leading to cell migration. An initial density disturbance and a sufficiently large contractile activity lead to cortical flows and shape changes that result in the self-polarization of the cell, with a steady state in which a continuous flow from the front to the rear of the cell is sustained. Friction with the confining plates leads to cell migration.

Download Torres-Sánchez supplementary movie 3(Video)
Video 1 MB