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Computation of Two-Phase Biomembranes with Phase DependentMaterial Parameters Using Surface Finite Elements

Published online by Cambridge University Press:  03 June 2015

Charles M. Elliott  and
Björn Stinner
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Abstract

The shapes of vesicles formed by lipid bilayers with phase separation are governed by a bending energy with phase dependent material parameters together with a line energy associated with the phase interfaces. We present a numerical method to approximate solutions to the Euler-Lagrange equations featuring triangulated surfaces, isoparametric quadratic surface finite elements and the phase field approach for the phase separation. Furthermore, the method involves an iterative solution scheme that is based on a relaxation dynamics coupling a geometric evolution equation for the membrane surface with a surface Allen-Cahn equation. Remeshing and grid adaptivity are discussed, and in various simulations the influence of several physical parameters is investigated.

Keywords

49Q1065M6074S0574G1574L1582B2692C10Biomembranesurface finite elementsrelaxation dynamics
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 2 , February 2013 , pp. 325 - 360
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Andelman, D., Kawakatsu, T. and Kawasaki, K., Equilibrium shape of two-component unil-amellar membranes and vesicles, Europhys. Lett., 19 (1992), 57–62.Google Scholar
[2]Barrett, J. W., Garcke, H. and Nürnberg, R., On the parametric finite element approximation of evolving hypersurfaces in R3, J. Comput. Phys., 227 (2008), 4281–4307.CrossRefGoogle Scholar
[3]Barrett, J. W., Garcke, H. and Nürnberg, R., Parametric approximation of Willmore flow and related geometric evolution equations, SIAM J. Sci. Comput., 31 (2008), 225–253.Google Scholar
[4]Barrett, J. W., Garcke, H. and Nürnberg, R., Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies, Interf. Free Bound., 12 (2010), 187–234.Google Scholar
[5]Baumgart, T., Das, S., Webb, W. and Jenkins, J., Membrane elasticity in giant vesicles with fluid phase coexistence, Biophys. J., 89 (2005), 1067–1084.CrossRefGoogle ScholarPubMed
[6]Baumgart, T., Hess, S. and Webb, W., Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension, Nature, 425 (2003), 821–824.CrossRefGoogle ScholarPubMed
[7]Blowey, J. F. and Elliott, C. M., The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy part ii: Numerical analysis, Euro. J. Appl. Math., 3 (1993), 147–179.CrossRefGoogle Scholar
[8]Brenner, S. and Scott, L., The Mathematical Theory of Finite Element Methods, No. 15 in Texts in Applied Mathematics, Springer, Third ed., 2007.Google Scholar
[9]Campelo, F., Allain, J.-M. and Amar, M. B., Periodic lipidic membrane tubes, Europhys. Lett., 77 (2007), 38006.CrossRefGoogle Scholar
[10]Canham, P., The minimum energy of bending as a possible explanation of the biconcave shape of the red blood cell, J. Theor. Biol., 26 (1970), 61–81.Google Scholar
[11]Chen, L.-Q., Phase-field models for microstructure evolution, Ann. Rev. Mat. Res., 32 (2002), 113–140.CrossRefGoogle Scholar
[12]Das, S., Jenkins, J. and Baumgart, T., Neck geometry and shape transitions in vesicles with co-existing fluid phases: role of gaussian curvature stiffness vs. spontaneous curvature, Europhys. Lett., 86 (2009), 48003–1-6.CrossRefGoogle Scholar
[13]Davies, T. A., Algorithm 832: Umfpack, an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30 (2004), 196–199.Google Scholar
[14]Deckelnick, K., Dziuk, G. and Elliott, C. M., Computation of geometric partial differential equations and mean curvature flow, Acta Numer., 14 (2005), 139–232.Google Scholar
[15]Demlow, A., Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces, SIAM J. Numer. Anal., 47 (2009), 805–827.Google Scholar
[16]Du, Q., Liu, C. and Wang, X., A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, J. Comput. Phys., 198 (2004), 450–468.CrossRefGoogle Scholar
[17]Du, Q., Liu, C. and Wang, X., Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions, J. Comput. Phys., 212 (2006), 757–777.Google Scholar
[18]Dziuk, G., Finite elements for the Beltrami operator on arbitrary surfaces, in Partial Differential Equations and Calculus of Variations, Hildebrandt, S. and Leis, R., eds., Vol. 1357 of Lecture Notes in Mathematics, Springer, 1988, 142–155.Google Scholar
[19]Dziuk, G., An algorithm for evolutionary surfaces, Numerische Mathematik, 58 (1991), 603–611.Google Scholar
[20]Dziuk, G., Computational parametric Willmore flow, Numerische Mathematik, 111 (2008), 55–80.Google Scholar
[21]Dziuk, G. and Clarenz, U., Numerical methods for conformally parametrized surfaces, CPDw04-Interphase 2003: Numerical Methods for Free Boundary Problems, workshop at the Isaac Newton Institute, http://www.newton.cam.ac.uk/webseminars/pg+ws/2003/cpd /cpdw04/0415/dziuk/, (2003).Google Scholar
[22]Dziuk, G. and Elliott, C. M., Finite elements on evolving surfaces, IMA J. Numer. Anal., 25 (2007), 385–407.Google Scholar
[23]Dziuk, G. and Elliott, C. M., Surface finite elements for parabolic equations, J. Comput. Math., 25 (2007), 385–407.Google Scholar
[24]Elliott, C. and Stinner, B., Modeling and computation of two phase geometric biomembranes using surface finite elements, J. Comput. Phys., 229 (2010), 6585–6612.CrossRefGoogle Scholar
[25]Elliott, C. and Stinner, B., A surface phase field model for two-phase biological membranes, SIAM J. Appl. Math., 70 (2010), 2904–2928.Google Scholar
[26]Elliott, C. M. and Stuart, A. M., The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30 (1993), 1622–1663.Google Scholar
[27]Evans, E., Bending resistance and chemically induced moments in membrane bilayers, Biophys. J., 14 (1974), 923–931.Google Scholar
[28]Feng, F. and Klug, W., Finite element modeling of liquid bilayer membranes, J. Comput. Phys., 220 (2006), 394–408.Google Scholar
[29]Garcke, H., Nestler, B., Stinner, B. and Wendler, F., Allen-Cahn systems with volume constraints, Math. Mod. Meth. Appl. Sci., 18 (2008), 1347–1381.Google Scholar
[30]Heine, C.-J., Computations of Form and Stability of Rotating Drops with Finite Elements, PhD thesis, Faculty for Mathematics, Informatics and Natural Sciences, University of Aachen, 2003.Google Scholar
[31]Heine, C.-J., Isoparametric finite element approximation of curvature on hypersurfaces, Fakultät für Mathematik und Physik, University of Freiburg, Preprint, 26 (2004).Google Scholar
[32]Helfrich, W., Elastic properties of lipid bilayers: theory and possible experiments, Z. Natur-forschung, C28 (1973), 693–703.Google Scholar
[33]Helmers, M., Snapping elastic curves as a one-dimensional analogue of two-component lipid bilayers, to appear in Math. Mod. Meth. Appl. Sci., DOI: 10.1142/S0218202511005234, (2011).CrossRefGoogle Scholar
[34]Jülicher, F. and Lipowsky, R., Domain-induced budding of vesicles, Phys. Rev. Lett., 70 (1993), 2964–2967.CrossRefGoogle ScholarPubMed
[35]Jülicher, F. and Lipowsky, R., Shape transformations of vesicles with intramembrane domains, Phys. Rev. E, 53 (1996), 2670–2683.Google Scholar
[36]Kawakatsu, T., Andelman, D., Kawasaki, K. and Taniguchi, T., Phase transitions and shapes of two component membranes and vesicles I: strong segregation limit, J. Phys. II (France), 3 (1993), 971–997.Google Scholar
[37]Lowengrub, J. S., Rätz, A. and Voigt, A., Phase-field modeling of the dynamics of multicomponent vesicles: spinodal decomposition, coarsening, budding and fission, Phys. Rev. E, 79 (2009), 031926–1-13.CrossRefGoogle ScholarPubMed
[38]Ma, L. and Klug, W., Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics, J. Comput. Phys., 227 (2008), 5816–5835.Google Scholar
[39]Schmidt, A. and Siebert, K. G., Design of adaptive finite element software: the finite element toolbox ALBERTA, Vol. 42 of Lecture Notes in Computational Science and Engineering, Springer, 2005.Google Scholar
[40]Seifert, U., Configurations of fluid membranes and vesicles, Adv. Phys., 46 (1997), 1–137.Google Scholar
[41]Taniguchi, T., Shape deformations and phase separation dynamics of two-component vesicles, Phys. Rev. Lett., 76 (1996), 4444–4447.Google Scholar
[42]Taniguchi, T., Kawasaki, K., Andelman, D. and Kawakatsu, T., Phase transitions and shapes of two component membranes and vesicles II: weak segregation limit, J. Phys. II (France), 4 (1994), 1333–1362.CrossRefGoogle Scholar
[43]Veatch, S. L. and Keller, S. L., Separation of liquid phases in giant vesicles of ternary mixtures of phospholipids and cholesterol, Biophys. J., 85 (2003), 3074–3083.Google Scholar
[44]Wang, X. and Du, Q., Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches, J. Math. Biol., 56 (2008), 347–371.Google Scholar