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Modelling fluid deformable surfaces with an emphasis on biological interfaces

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

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.

Copyright

Corresponding author

Email address for correspondence: marino.arroyo@upc.edu

References

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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.
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.
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.
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.
Barrett, J. W., Garcke, H. & Nürnberg, R. 2015 Numerical computations of the dynamics of fluidic membranes and vesicles. Phys. Rev. E 92 (5), 052704.
Barrett, J. W., Garcke, H. & Nürnberg, R. 2016 A stable numerical method for the dynamics of fluidic membranes. Numer. Math. 134 (4), 783822.
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.
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.
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.
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.
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.
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.
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.
Callan-Jones, A. C. & Voituriez, R. 2013 Active gel model of amoeboid cell motility. New J. Phys. 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.
Capovilla, R. & Guven, J. 2002 Stresses in lipid membranes. J. Phys. A 35 (30), 62336247.
do Carmo, M. P. 1992 Riemannian Geometry, vol. 115. Birkhäuser.
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.
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.
Cirak, F. & Long, Q. 2011 Subdivision shells with exact boundary control and non-manifold geometry. Intl J. Numer. Meth. Engng 88, 897923.
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.
Cirak, F. & Ortiz, M. 2001 Fully C1-conforming subdivision elements for finite deformation thin-shell analysis. Intl J. Numer. Meth. Engng 51 (7), 813833.
Clark, A. G., Dierkes, K. & Paluch, E. K. 2013 Monitoring actin cortex thickness in live cells. Biophys. J. 105 (3), 570580.
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.
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.
Dziuk, G. & Elliott, C. M. 2013 Finite element methods for surface PDEs. Acta Numer. 22, 289396.
Elliott, C. M. & Styles, V. 2012 An ALE ESFEM for solving PDEs on evolving surfaces. Milan J. Math. 80 (2), 469501.
Evans, E. & Yeung, A. 1994 Hidden dynamics in rapid changes of bilayer shape. Chem. Phys. Lipids 73 (1), 3956.
Farutin, A. & Misbah, C. 2012 Rheology of vesicle suspensions under combined steady and oscillating shear flows. J. Fluid Mech. 700, 362381.
Feng, F. & Klug, W. S. 2006 Finite element modeling of lipid bilayer membranes. J. Comput. Phys. 220 (1), 394408.
Fischer, F. D., Svoboda, J. & Petryk, H. 2014 Thermodynamic extremal principles for irreversible processes in materials science. Acta Mater. 67, 120.
Fournier, J.-B. 2015 On the hydrodynamics of bilayer membranes. Intl J. Non-Linear Mech. 75, 6776.
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.
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.
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.
Gompper, G. & Kroll, D. M. 2004 Triangulated-surface models of fluctuating membranes. In Statistical Mechanics of Membranes and Surfaces, pp. 359426. World Scientific.
Gross, B. J. & Atzberger, P. J. 2018 Hydrodynamic flows on curved surfaces: spectral numerical methods for radial manifold shapes. J. Comput. Phys. 371, 663689.
Hamm, M. & Kozlov, M. M. 1998 Tilt model of inverted amphiphilic mesophases. Eur. Phys. J. B 6 (4), 519528.
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.
Helfrich, W. 1973 Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. 28c, 693703.
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.
Ho, J. S. & Baumgärtner, A. 1990 Simulations of fluid self-avoiding membranes. Europhys. Lett. 12 (4), 295.
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.
Jülicher, F. & Lipowsky, R. 1993 Domain-induced budding of vesicles. Phys. Rev. Lett. 70 (19), 29642967.
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.
Kantsler, V., Segre, E. & Steinberg, V. 2007 Vesicle dynamics in time-dependent elongation flow: wrinkling instability. Phys. Rev. Lett. 99 (17), 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.
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.
Koba, H., Liu, C. & Giga, Y. 2017 Energetic variational approaches for incompressible fluid systems on an evolving surface. Q. Appl. Maths 75 (2), 359389.
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.
Kroll, D. M. & Gompper, G. 1992 The conformation of fluid membranes: Monte Carlo simulations. Science 255 (5047), 968971.
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.
Lebon, G., Jou, D. & Casas-Vázquez, J. 2008 Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers. Springer.
Levayer, R. & Lecuit, T. 2012 Biomechanical regulation of contractility: spatial control and dynamics. Trends Cell Biol. 22 (2), 6181.
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.
Lew, A., Marsden, J. E., Ortiz, M. & West, M. 2004 Variational time integrators. Intl J. Numer. Meth. Engng 60 (1), 153212.
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.
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.
Lipowsky, R. 1991 The conformation of membranes. Nature 349 (6309), 475481.
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.
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.
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.
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.
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.
Mielke, A. 2012 Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions. Discrete Continuous Dyn. Syst. – Ser. S 6 (2), 479499.
Mietke, A., Jülicher, F. & Sbalzarini, I. F. 2019 Self-organized shape dynamics of active surfaces. Proc. Natl Acad. Sci. USA 116 (1), 2934.
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.
Morris, R. G. & Turner, M. S. 2015 Mobility measurements probe conformational changes in membrane proteins due to tension. Phys. Rev. Lett. 115 (19), 198101.
Nelson, P., Powers, T. & Seifert, U. 1995 Dynamical theory of the pearling instability in cylindrical vesicles. Phys. Rev. Lett. 74 (17), 33843387.
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.
Nitschke, I., Voigt, A. & Wensch, J. 2012 A finite element approach to incompressible two-phase flow on manifolds. J. Fluid Mech. 708, 418438.
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.
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.
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.
Ortiz, M. & Stainier, L. 1999 The variational formulation of viscoplastic constitutive updates. Comput. Meth. Appl. Mech. Engng 171 (3), 419444.
Paltridge, G. W. 1975 Global dynamics and climate – a system of minimum entropy exchange. Q. J. R. Meteorol. Soc. 101 (429), 475484.
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.
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.
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.
Prost, J., Jülicher, F. & Joanny, J.-F. 2015 Active gel physics. Nat. Phys. 11 (2), 111.
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.
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.
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.
Reuther, S. & Voigt, A. 2015 The interplay of curvature and vortices in flow on curved surfaces. Multiscale Model. Simul. 13 (2), 632643.
Reuther, S. & Voigt, A. 2016 Incompressible two-phase flows with an inextensible Newtonian fluid interface. J. Comput. Phys. 322, 850858.
Reuther, S. & Voigt, A. 2018a Erratum: the interplay of curvature and vortices in flow on curved surfaces. Multiscale Model. Simul. 16 (3), 14481453.
Reuther, S. & Voigt, A. 2018b Solving the incompressible surface Navier–Stokes equation by surface finite elements. Phys. Fluids 30 (1), 012107.
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.
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.
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.
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.
Rustom, A., Saffrich, R., Markovic, I., Walther, P. & Gerdes, H.-H. 2004 Nanotubular highways for intercellular organelle transport. Science 303 (5660), 10071010.
Saffman, P. G. & Delbrück, M. 1975 Brownian motion in biological membranes. Proc. Natl Acad. Sci. USA 72 (8), 31113113.
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.
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.
Salbreux, G., Charras, G. & Paluch, E. 2012 Actin cortex mechanics and cellular morphogenesis. Trends Cell Biol. 22 (10), 536545.
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.
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.
Scriven, L. E. 1960 Dynamics of a fluid interface Equation of motion for Newtonian surface fluids. Chem. Engng Sci. 12 (2), 98108.
Secomb, T. W. & Skalak, R. 1982 Surface flow of viscoelastic membranes in viscous fluids. Q. J. Mech. Appl. Maths 35 (2), 233247.
Seifert, U. 1997 Configurations of fluid membranes and vesicles. Adv. Phys. 46 (1), 13137.
Seifert, U. & Langer, S. A. 1993 Viscous modes of fluid bilayer membranes. Europhys. Lett. 23 (1), 7176.
Sens, P., Johannes, L. & Bassereau, P. 2008 Biophysical approaches to protein-induced membrane deformations in trafficking. Curr. Opin. Cell Biol. 20 (4), 476482.
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.
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.
Sigurdsson, J. K. & Atzberger, P. J. 2016 Hydrodynamic coupling of particle inclusions embedded in curved lipid bilayer membranes. Soft Matt. 12 (32), 66856707.
Skalak, R. 1970 Extensions of extremum principles for slow viscous flows. J. Fluid Mech. 42 (3), 527548.
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.
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.
Staykova, M., Arroyo, M., Rahimi, M. & Stone, H. A. 2013 Confined bilayers passively regulate shape and stress. Phys. Rev. Lett. 110 (2), 028101.
Steigmann, D. J. 1999 Fluid films with curvature elasticity. Arch. Rat. Mech. Anal. 150 (2), 127152.
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.
Tu, Z. C. & Ou-Yang, Z. C. 2004 A geometric theory on the elasticity of bio-membranes. J. Phys. A 37 (47), 1140711429.
Turlier, H., Audoly, B., Prost, J. & Joanny, J.-F. 2014 Furrow constriction in animal cell cytokinesis. Biophys. J. 106 (1), 114123.
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.
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.
Wu, J.-Z., Yang, Y.-T., Luo, Y.-B. & Pozrikidis, C. 2005 Fluid kinematics on a deformable surface. J. Fluid Mech. 541, 371381.
Yavari, A., Ozakin, A. & Sadik, S. 2016 Nonlinear elasticity in a deforming ambient space. J. Nonlinear Sci. 26 (6), 16511692.
Zhang, K. & Arroyo, M. 2014 Understanding and strain-engineering wrinkle networks in supported graphene through simulations. J. Mech. Phys. Solids 72, 6174.
Zhou, Y. & Yan, D. 2005 Real-time membrane fission of giant polymer vesicles. Angew. Chem. Intl Ed. Engl. 44 (21), 32233226.
Ziegler, H. 1958 An attempt to generalize Onsager’s principle, and its significance for rheological problems. Z. Angew. Math. Phys. 9 (5–6), 748763.
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.
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