Skip to main content Accessibility help

Modelling fluid deformable surfaces with an emphasis on biological interfaces

  • Alejandro Torres-Sánchez (a1), Daniel Millán (a1) (a2) and Marino Arroyo (a1) (a3)


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.


Corresponding author

Email address for correspondence:


Hide All
Aris, R. 1962 Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Prentice-Hall.
Arroyo, M. & DeSimone, A. 2009 Relaxation dynamics of fluid membranes. Phys. Rev. E 79 (3), 031915.
Arroyo, M., DeSimone, A. & Heltai, L.2010 The role of membrane viscosity in the dynamics of fluid membranes. arXiv:1007.4934.
Arroyo, M., Heltai, L., Millán, D. & DeSimone, A. 2012 Reverse engineering the euglenoid movement. Proc. Natl Acad. Sci. USA 109 (44), 1787417879.10.1073/pnas.1213977109
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.10.1007/978-3-319-56348-0_6
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), 32723277.10.1073/pnas.0408215102
Barrett, J. W., Garcke, H. & Nürnberg, R. 2008 On the parametric finite element approximation of evolving hypersurfaces in R3. J. Comput. Phys. 227 (9), 42814307.10.1016/
Barrett, J. W., Garcke, H. & Nürnberg, R. 2015 Numerical computations of the dynamics of fluidic membranes and vesicles. Phys. Rev. E 92 (5), 052704.10.1103/PhysRevE.92.052704
Barrett, J. W., Garcke, H. & Nürnberg, R. 2016 A stable numerical method for the dynamics of fluidic membranes. Numer. Math. 134 (4), 783822.10.1007/s00211-015-0787-5
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, 119135.10.1017/S002211208500341X
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, 483506.10.1146/annurev.physchem.012809.103450
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), 524529.10.1038/ncb3134
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.
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.
Bray, D. & White, J. G. 1988 Cortical flow in animal cells. Science 239 (4842), 883888.10.1126/science.3277283
Brezzi, F. & Fortin, M. 2012 Mixed and Hybrid Finite Element Methods. Springer Science & Business Media.
Brochard-Wyart, F. & de Gennes, P.-G. 2002 Adhesion induced by mobile binders: dynamics. Proc. Natl Acad. Sci. USA 99 (12), 78547859.10.1073/pnas.112221299
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, 188207.10.1016/j.cma.2014.10.044
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), 028102.10.1103/PhysRevLett.116.028102
Callan-Jones, A. C. & Voituriez, R. 2013 Active gel model of amoeboid cell motility. New J. Phys. 15 (2), 025022.10.1088/1367-2630/15/2/025022
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), 12481256.10.1016/j.bpj.2013.01.051
Capovilla, R. & Guven, J. 2002 Stresses in lipid membranes. J. Phys. A 35 (30), 62336247.10.1088/0305-4470/35/30/302
do Carmo, M. P. 1992 Riemannian Geometry, vol. 115. Birkhäuser.10.1007/978-1-4757-2201-7
do Carmo, M. P. 2016 Differential Geometry of Curves and Surfaces, 2nd edn. Dover.
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, 339351.10.1017/S0022112005006695
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), 689697.10.1038/ncb3525
Cirak, F. & Long, Q. 2011 Subdivision shells with exact boundary control and non-manifold geometry. Intl J. Numer. Meth. Engng 88, 897923.10.1002/nme.3206
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), 20392072.10.1002/(SICI)1097-0207(20000430)47:12<2039::AID-NME872>3.0.CO;2-1
Cirak, F. & Ortiz, M. 2001 Fully C1-conforming subdivision elements for finite deformation thin-shell analysis. Intl J. Numer. Meth. Engng 51 (7), 813833.10.1002/nme.182.abs
Clark, A. G., Dierkes, K. & Paluch, E. K. 2013 Monitoring actin cortex thickness in live cells. Biophys. J. 105 (3), 570580.10.1016/j.bpj.2013.05.057
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.
Doi, M. 2011 Onsager’s variational principle in soft matter. J. Phys.: Condens. Matter 23 (28), 284118.
Donea, J. & Huerta, A. 2003 Finite Element Methods for Flow Problems, reprint edn. John Wiley & Sons.10.1002/0470013826
Dörries, G. & Foltin, G. 1996 Energy dissipation of fluid membranes. Phys. Rev. E 53 (3), 25472550.
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), 2846.10.1002/nme.5731
Dziuk, G. & Elliott, C. M. 2013 Finite element methods for surface PDEs. Acta Numer. 22, 289396.10.1017/S0962492913000056
Elliott, C. M. & Styles, V. 2012 An ALE ESFEM for solving PDEs on evolving surfaces. Milan J. Math. 80 (2), 469501.10.1007/s00032-012-0195-6
Evans, E. & Yeung, A. 1994 Hidden dynamics in rapid changes of bilayer shape. Chem. Phys. Lipids 73 (1), 3956.10.1016/0009-3084(94)90173-2
Farutin, A. & Misbah, C. 2012 Rheology of vesicle suspensions under combined steady and oscillating shear flows. J. Fluid Mech. 700, 362381.10.1017/jfm.2012.137
Feng, F. & Klug, W. S. 2006 Finite element modeling of lipid bilayer membranes. J. Comput. Phys. 220 (1), 394408.10.1016/
Fischer, F. D., Svoboda, J. & Petryk, H. 2014 Thermodynamic extremal principles for irreversible processes in materials science. Acta Mater. 67, 120.10.1016/j.actamat.2013.11.050
Fournier, J.-B. 2015 On the hydrodynamics of bilayer membranes. Intl J. Non-Linear Mech. 75, 6776.10.1016/j.ijnonlinmec.2015.02.006
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), 018102.10.1103/PhysRevLett.102.018102
Frenkel, D. & Smit, B. 2001 Understanding Molecular Simulation: From Algorithms to Applications, 2nd edn. Academic Press.
Fries, T.-P. 2018 Higher-order surface FEM for incompressible Navier–Stokes flows on manifolds. Intl J. Numer. Meth. Fluids 88 (2), 5578.10.1002/fld.4510
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), 757767.10.1091/mbc.e12-06-0485
Gompper, G. & Kroll, D. M. 2004 Triangulated-surface models of fluctuating membranes. In Statistical Mechanics of Membranes and Surfaces, pp. 359426. World Scientific.10.1142/9789812565518_0012
Gross, B. J. & Atzberger, P. J. 2018 Hydrodynamic flows on curved surfaces: spectral numerical methods for radial manifold shapes. J. Comput. Phys. 371, 663689.10.1016/
Hamm, M. & Kozlov, M. M. 1998 Tilt model of inverted amphiphilic mesophases. Eur. Phys. J. B 6 (4), 519528.10.1007/s100510050579
Hamm, M. & Kozlov, M. M. 2000 Elastic energy of tilt and bending of fluid membranes. Eur. Phys. J. E 3 (4), 323335.
Hansbo, P., Larson, M. G. & Larsson, K.2016 Analysis of finite element methods for vector laplacians on surfaces. arXiv:1610.06747.
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), 10411045.10.1016/j.bpj.2011.07.038
Helfrich, W. 1973 Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. 28c, 693703.10.1515/znc-1973-11-1209
Henle, M. L. & Levine, A. J. 2010 Hydrodynamics in curved membranes: the effect of geometry on particulate mobility. Phys. Rev. E 81 (1), 117.
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), 227253.10.1016/0021-9991(74)90051-5
Ho, J. S. & Baumgärtner, A. 1990 Simulations of fluid self-avoiding membranes. Europhys. Lett. 12 (4), 295.10.1209/0295-5075/12/4/002
Howard, J. 2001 Mechanics of Motor Proteins and the Cytoskeleton. Sinauer Associates, Publishers.
Hu, D., Zhang, P. & Weinan, E. 2007 Continuum theory of a moving membrane. Phys. Rev. E 75 (4), 111.
Jankuhn, T., Olshanskii, M. A. & Reusken, A. 2018 Incompressible fluid problems on embedded surfaces: modeling and variational formulations. Interfaces Free Boundaries 20, 353378.10.4171/IFB/405
Jülicher, F. & Lipowsky, R. 1993 Domain-induced budding of vesicles. Phys. Rev. Lett. 70 (19), 29642967.10.1103/PhysRevLett.70.2964
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, 131146.10.1016/j.cma.2016.01.005
Kantsler, V., Segre, E. & Steinberg, V. 2007 Vesicle dynamics in time-dependent elongation flow: wrinkling instability. Phys. Rev. Lett. 99 (17), 178102.10.1103/PhysRevLett.99.178102
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), 49244933.10.1529/biophysj.108.136077
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), 879890.10.1016/j.bpj.2014.07.015
Koba, H., Liu, C. & Giga, Y. 2017 Energetic variational approaches for incompressible fluid systems on an evolving surface. Q. Appl. Maths 75 (2), 359389.10.1090/qam/1452
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, 7292.10.1038/ncomms8292
Kroll, D. M. & Gompper, G. 1992 The conformation of fluid membranes: Monte Carlo simulations. Science 255 (5047), 968971.10.1126/science.1546294
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, 271299.10.1016/
Lebon, G., Jou, D. & Casas-Vázquez, J. 2008 Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers. Springer.10.1007/978-3-540-74252-4
Levayer, R. & Lecuit, T. 2012 Biomechanical regulation of contractility: spatial control and dynamics. Trends Cell Biol. 22 (2), 6181.10.1016/j.tcb.2011.10.001
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), 038102.10.1103/PhysRevLett.93.038102
Lew, A., Marsden, J. E., Ortiz, M. & West, M. 2004 Variational time integrators. Intl J. Numer. Meth. Engng 60 (1), 153212.10.1002/nme.958
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, 334348.10.1016/j.jmps.2018.06.022
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), 15991603.10.1016/j.bpj.2015.02.007
Lipowsky, R. 1991 The conformation of membranes. Nature 349 (6309), 475481.10.1038/349475a0
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), 17221749.10.1016/j.cma.2005.05.049
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), 659672.10.1016/j.cell.2015.01.007
Loop, C.1987 Smooth subdivision surfaces based on triangles. PhD thesis, University of Utah.
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), 58165835.10.1016/
Marsden, J. E. & Hughes, T. J. R. 1994 Mathematical Foundations of Elasticity. Courier Corporation.
Martyushev, L. M. & Seleznev, V. D. 2006 Maximum entropy production principle in physics, chemistry and biology. Phys. Rep. 426 (1), 145.10.1016/j.physrep.2005.12.001
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.
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), 164503.10.1103/PhysRevLett.120.164503
Mielke, A. 2012 Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions. Discrete Continuous Dyn. Syst. – Ser. S 6 (2), 479499.10.3934/dcdss.2013.6.479
Mietke, A., Jülicher, F. & Sbalzarini, I. F. 2019 Self-organized shape dynamics of active surfaces. Proc. Natl Acad. Sci. USA 116 (1), 2934.10.1073/pnas.1810896115
Millán, D., Rosolen, A. & Arroyo, M. 2011 Thin shell analysis from scattered points with maximum-entropy approximants. Intl J. Numer. Meth. Engng 85 (6), 723751.10.1002/nme.2992
Morris, R. G. & Turner, M. S. 2015 Mobility measurements probe conformational changes in membrane proteins due to tension. Phys. Rev. Lett. 115 (19), 198101.10.1103/PhysRevLett.115.198101
Nelson, P., Powers, T. & Seifert, U. 1995 Dynamical theory of the pearling instability in cylindrical vesicles. Phys. Rev. Lett. 74 (17), 33843387.10.1103/PhysRevLett.74.3384
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), 147191.10.1007/s00332-017-9405-2
Nitschke, I., Voigt, A. & Wensch, J. 2012 A finite element approach to incompressible two-phase flow on manifolds. J. Fluid Mech. 708, 418438.10.1017/jfm.2012.317
Noguchi, H. & Gompper, G. 2005 Shape transitions of fluid vesicles and red blood cells in capillary flows. Proc. Natl Acad. Sci. USA 102 (40), 1415914164.10.1073/pnas.0504243102
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), 496502.10.1038/s41567-019-0425-8
Olshanskii, M., Quaini, A., Reusken, A. & Yushutin, V. 2018 A finite element method for the surface stokes problem. SIAM J. Sci. Comput. 40 (4), A2492A2518.10.1137/18M1166183
Ortiz, M. & Stainier, L. 1999 The variational formulation of viscoplastic constitutive updates. Comput. Meth. Appl. Mech. Engng 171 (3), 419444.10.1016/S0045-7825(98)00219-9
Paltridge, G. W. 1975 Global dynamics and climate – a system of minimum entropy exchange. Q. J. R. Meteorol. Soc. 101 (429), 475484.10.1002/qj.49710142906
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, 320336.10.1016/
Peletier, M.2014 Variational modelling: energies, gradient flows, and large deviations. Preprint, arXiv:1402.1990.
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), 1335613361.10.1073/pnas.1311827110
Piegl, L. & Tiller, W. 2012 The NURBS Book. Springer Science & Business Media.
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), 19431948.10.1073/pnas.1010396108
Prost, J., Jülicher, F. & Joanny, J.-F. 2015 Active gel physics. Nat. Phys. 11 (2), 111.10.1038/nphys3224
Rahimi, M.2013 Shape dynamics and lipid hydrodynamics of bilayer membranes: modeling, simulation and experiments. PhD thesis, Universitat Politècnica de Catalunya – BarcelonaTech.
Rahimi, M. & Arroyo, M. 2012 Shape dynamics, lipid hydrodynamics, and the complex viscoelasticity of bilayer membranes. Phys. Rev. E 86 (1), 011932.
Rahimi, M., DeSimone, A. & Arroyo, M. 2013 Curved fluid membranes behave laterally as effective viscoelastic media. Soft Matt. 9 (46), 1103311045.10.1039/c3sm51748a
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, 266294.10.1016/
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), 84117.10.1002/nme.5949
Reuther, S. & Voigt, A. 2015 The interplay of curvature and vortices in flow on curved surfaces. Multiscale Model. Simul. 13 (2), 632643.10.1137/140971798
Reuther, S. & Voigt, A. 2016 Incompressible two-phase flows with an inextensible Newtonian fluid interface. J. Comput. Phys. 322, 850858.10.1016/
Reuther, S. & Voigt, A. 2018a Erratum: the interplay of curvature and vortices in flow on curved surfaces. Multiscale Model. Simul. 16 (3), 14481453.10.1137/18M1176464
Reuther, S. & Voigt, A. 2018b Solving the incompressible surface Navier–Stokes equation by surface finite elements. Phys. Fluids 30 (1), 012107.10.1063/1.5005142
Reymann, A.-C., Staniscia, F., Erzberger, A., Salbreux, G. & Grill, S. W. 2016 Cortical flow aligns actin filaments to form a furrow. Elife 5, 125.10.7554/eLife.17807
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, 565584.10.1016/
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), 53945399.10.1073/pnas.082107299
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), 673685.10.1016/j.cell.2015.01.008
Rustom, A., Saffrich, R., Markovic, I., Walther, P. & Gerdes, H.-H. 2004 Nanotubular highways for intercellular organelle transport. Science 303 (5660), 10071010.10.1126/science.1093133
Saffman, P. G. & Delbrück, M. 1975 Brownian motion in biological membranes. Proc. Natl Acad. Sci. USA 72 (8), 31113113.10.1073/pnas.72.8.3111
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), 14211429.10.1016/j.bpj.2016.02.013
Sahu, A., Sauer, R. A. & Mandadapu, K. K. 2017 Irreversible thermodynamics of curved lipid membranes. Phys. Rev. E 96 (4), 042409.
Salac, D. & Miksis, M. 2011 A level set projection model of lipid vesicles in general flows. J. Comput. Phys. 230 (22), 81928215.10.1016/
Salbreux, G., Charras, G. & Paluch, E. 2012 Actin cortex mechanics and cellular morphogenesis. Trends Cell Biol. 22 (10), 536545.10.1016/j.tcb.2012.07.001
Salbreux, G. & Jülicher, F. 2017 Mechanics of active surfaces. Phys. Rev. E 96 (3), 032404.
Salbreux, G., Prost, J. & Joanny, J. F. 2009 Hydrodynamics of cellular cortical flows and the formation of contractile rings. Phys. Rev. Lett. 103 (5), 058102.10.1103/PhysRevLett.103.058102
Santos-Oliván, D., Torres-Sánchez, A., Vilanova, G. & Arroyo, M.2019 A macroelement approach for inextensible flows with subdivision finite elements. (in preparation).
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, 119.10.1016/
Scriven, L. E. 1960 Dynamics of a fluid interface Equation of motion for Newtonian surface fluids. Chem. Engng Sci. 12 (2), 98108.10.1016/0009-2509(60)87003-0
Secomb, T. W. & Skalak, R. 1982 Surface flow of viscoelastic membranes in viscous fluids. Q. J. Mech. Appl. Maths 35 (2), 233247.10.1093/qjmam/35.2.233
Seifert, U. 1997 Configurations of fluid membranes and vesicles. Adv. Phys. 46 (1), 13137.10.1080/00018739700101488
Seifert, U. & Langer, S. A. 1993 Viscous modes of fluid bilayer membranes. Europhys. Lett. 23 (1), 7176.10.1209/0295-5075/23/1/012
Sens, P., Johannes, L. & Bassereau, P. 2008 Biophysical approaches to protein-induced membrane deformations in trafficking. Curr. Opin. Cell Biol. 20 (4), 476482.10.1016/
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), 268102.10.1103/PhysRevLett.120.268102
Shibata, Y., Hu, J., Kozlov, M. M. & Rapoport, T. A. 2009 Mechanisms shaping the membranes of cellular organelles. Annu. Rev. Cell Dev. Biol. 25, 329354.10.1146/annurev.cellbio.042308.113324
Sigurdsson, J. K. & Atzberger, P. J. 2016 Hydrodynamic coupling of particle inclusions embedded in curved lipid bilayer membranes. Soft Matt. 12 (32), 66856707.10.1039/C6SM00194G
Skalak, R. 1970 Extensions of extremum principles for slow viscous flows. J. Fluid Mech. 42 (3), 527548.10.1017/S0022112070001465
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), 504513.10.1038/35080071
Stam, J. 1999 Evaluation of Loop subdivision surfaces. In SIGGRAPH’99 Course Notes. Los Angeles, CA.
Staneva, G., Angelova, M. I. & Koumanov, K. 2004 Phospholipase A2 promotes raft budding and fission from giant liposomes. Chem. Phys. Lipids 129 (1), 5362.10.1016/j.chemphyslip.2003.11.005
Staykova, M., Arroyo, M., Rahimi, M. & Stone, H. A. 2013 Confined bilayers passively regulate shape and stress. Phys. Rev. Lett. 110 (2), 028101.10.1103/PhysRevLett.110.028101
Steigmann, D. J. 1999 Fluid films with curvature elasticity. Arch. Rat. Mech. Anal. 150 (2), 127152.10.1007/s002050050183
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.
Torres-Sánchez, A.2017 A theoretical and computational study of the mechanics of biomembranes at multiple scales. PhD thesis, Universitat Politècnica de Catalunya.
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.
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), 138102.10.1103/PhysRevLett.91.138102
Tu, Z. C. & Ou-Yang, Z. C. 2004 A geometric theory on the elasticity of bio-membranes. J. Phys. A 37 (47), 1140711429.10.1088/0305-4470/37/47/010
Turlier, H., Audoly, B., Prost, J. & Joanny, J.-F. 2014 Furrow constriction in animal cell cytokinesis. Biophys. J. 106 (1), 114123.10.1016/j.bpj.2013.11.014
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), 56105634.10.1016/
Willmore, T. J. 1996 Riemannian Geometry. Oxford University Press.
Woodhouse, F. G. & Goldstein, R. E. 2012 Shear-driven circulation patterns in lipid membrane vesicles. J. Fluid Mech. 705, 165175.10.1017/jfm.2012.118
Wu, J.-Z., Yang, Y.-T., Luo, Y.-B. & Pozrikidis, C. 2005 Fluid kinematics on a deformable surface. J. Fluid Mech. 541, 371381.10.1017/S0022112005005963
Yavari, A., Ozakin, A. & Sadik, S. 2016 Nonlinear elasticity in a deforming ambient space. J. Nonlinear Sci. 26 (6), 16511692.10.1007/s00332-016-9315-8
Zhang, K. & Arroyo, M. 2014 Understanding and strain-engineering wrinkle networks in supported graphene through simulations. J. Mech. Phys. Solids 72, 6174.10.1016/j.jmps.2014.07.012
Zhou, Y. & Yan, D. 2005 Real-time membrane fission of giant polymer vesicles. Angew. Chem. Intl Ed. Engl. 44 (21), 32233226.10.1002/anie.200462622
Ziegler, H. 1958 An attempt to generalize Onsager’s principle, and its significance for rheological problems. Z. Angew. Math. Phys. 9 (5–6), 748763.10.1007/BF02424793
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.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed