×

A gauge-fixing procedure for spherical fluid membranes and application to computations. (English) Zbl 1506.76225

Summary: A distinguishing feature of lipid (bilayer) membranes is their in-plane fluidity caused by free-flowing lipid molecules on the membrane surface. In continuum models for lipid membranes (e.g., the Helfrich-Canham model), fluidity manifests as invariance of the free energy to change in parametrization of the reference surface; a property termed reparametrization invariance. Two different parametric equations of the surface, related through a reparametrization, have identical equilibrium and stability properties. They can therefore be considered equivalent representations of the same surface. Since there are infinitely many ways to parametrize a surface, there are infinitely many equivalent representations for the surface. This highly redundant representation for a surface poses significant challenges to computations. For example, in computational studies using finite element analysis, extreme mesh distortion and spurious zero-energy modes are reported [F. Feng and W. S. Klug, J. Comput. Phys. 220, No. 1, 394–408 (2006; Zbl 1102.92011); L. Ma and W. S. Klug, J. Comput. Phys. 227, No. 11, 5816–5835 (2008; Zbl 1168.74050)]. In this work, by viewing reparametrization invariance as a form of gauge symmetry, we propose a gauge-fixing procedure for the case of topologically spherical membranes. We show that this procedure breaks gauge symmetry and tames the extreme redundancy of the system. We also demonstrate that this procedure is suitable for efficient numerical computations. We obtain accurate equilibrium configurations for the Helfrich-Canham model while circumventing computational issues noted above.

MSC:

76Z05 Physiological flows
53Z30 Applications of differential geometry to engineering
74K35 Thin films

Software:

LBFGS-B; L-BFGS; L-BFGS-B
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dietrich, C.; Bagatolli, L.; Volovyk, Z.; Thompson, N.; Levi, M.; Jacobson, K.; Gratton, E., Lipid rafts reconstituted in model membranes, Biophys. J., 80, 3, 1417-1428 (2001)
[2] Das, S. L.; Jenkins, J. T., A higher-order boundary layer analysis for lipid vesicles with two fluid domains, J. Fluid Mech., 597, 429 (2008) · Zbl 1133.76061
[3] Peetla, C.; Stine, A.; Labhasetwar, V., Biophysical interactions with model lipid membranes: applications in drug discovery and drug delivery, Mol. Pharm., 6, 5, 1264-1276 (2009)
[4] Feng, F.; Klug, W. S., Finite element modeling of lipid bilayer membranes, J. Comput. Phys., 220, 1, 394-408 (2006) · Zbl 1102.92011
[5] Ma, L.; Klug, W. S., Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics, J. Comput. Phys., 227, 11, 5816-5835 (2008) · Zbl 1168.74050
[6] Lenaz, G., Lipid fluidity and membrane protein dynamics, Biosci. Rep., 7, 11, 823-837 (1987)
[7] Los, D. A.; Mironov, K. S.; Allakhverdiev, S. I., Regulatory role of membrane fluidity in gene expression and physiological functions, Photosynt. Res., 116, 2-3, 489-509 (2013)
[8] Ortiz, G. G.; Pacheco-Moisés, F. P.; Flores-Alvarado, L. J.; Macías-Islas, M. A.; Velázquez-Brizuela, I. E.; Ramírez-Anguiano, A. C.; Tórres-Sánchez, E. D.; Moráles-Sánchez, E. W.; Cruz-Ramos, J. A.; Ortiz-Velázquez, G. E., Alzheimer disease and metabolism: role of cholesterol and membrane fluidity (2013)
[9] Sameni, S.; Malacrida, L.; Tan, Z.; Digman, M. A., Alteration in fluidity of cell plasma membrane in huntington disease revealed by spectral phasor analysis, Sci. Rep., 8, 1, 1-10 (2018)
[10] Capovilla, R.; Guven, J.; Santiago, J., Deformations of the geometry of lipid vesicles, J. Phys. A: Math. Gen., 36, 23, 6281 (2003) · Zbl 1066.53125
[11] Canham, P. B., The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell, J. Theoret. Biol., 26, 1, 61-81 (1970)
[12] Helfrich, W., Elastic properties of lipid bilayers: theory and possible experiments, Z. Nat.forsch. C, 28, 11-12, 693-703 (1973)
[13] Elliott, C. M.; Stinner, B., Computation of two-phase biomembranes with phase dependent material parameters using surface finite elements, Commun. Comput. Phys., 13, 2, 325-336 (2013) · Zbl 1373.74089
[14] Sauer, R. A.; Duong, T. X.; Mandadapu, K. K.; Steigmann, D. J., A stabilized finite element formulation for liquid shells and its application to lipid bilayers, J. Comput. Phys., 330, 436-466 (2017) · Zbl 1378.74066
[15] Zhao, S.; Healey, T.; Li, Q., Direct computation of two-phase icosahedral equilibria of lipid bilayer vesicles, Comput. Methods Appl. Mech. Engrg., 314, 164-179 (2017) · Zbl 1439.74166
[16] Healey, T. J.; Dharmavaram, S., Symmetry-breaking global bifurcation in a surface continuum phase-field model for lipid bilayer vesicles, SIAM J. Math. Anal., 49, 2, 1027-1059 (2017) · Zbl 1391.35148
[17] Deserno, M., Fluid lipid membranes-a primer (2007), See http://www.cmu.edu/biolphys/deserno/pdf/membrane_theory.pdf
[18] Rangamani, P.; Agrawal, A.; Mandadapu, K. K.; Oster, G.; Steigmann, D. J., Interaction between surface shape and intra-surface viscous flow on lipid membranes, Biomech. Model. Mechanobiol., 12, 4, 833-845 (2013)
[19] Rangarajan, R.; Gao, H., A finite element method to compute three-dimensional equilibrium configurations of fluid membranes: Optimal parameterization, variational formulation and applications, J. Comput. Phys., 297, 266-294 (2015) · Zbl 1349.76255
[20] Torres-Sánchez, A.; Millán, D.; Arroyo, M., Modelling fluid deformable surfaces with an emphasis on biological interfaces, J. Fluid Mech., 872, 218-271 (2019) · Zbl 1430.76503
[21] Sahu, A.; Omar, Y. A.; Sauer, R. A.; Mandadapu, K. K., Arbitrary Lagrangian-Eulerian finite element method for curved and deforming surfaces: I. General theory and application to fluid interfaces, J. Comput. Phys., 407, Article 109253 pp. (2020) · Zbl 07504720
[22] Baumgart, T.; Hess, S. T.; Webb, W. W., Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension, Nature, 425, 6960, 821-824 (2003)
[23] Cirak, F.; Ortiz, M.; Schröder, P., Subdivision surfaces: a new paradigm for thin-shell finite-element analysis, Internat. J. Numer. Methods Engrg., 47, 12, 2039-2072 (2000) · Zbl 0983.74063
[24] Carmo, M. P.d., Riemannian Geometry (1992), Birkhäuser · Zbl 0752.53001
[25] Jenkins, J., Static equilibrium of configurations of a model red blood cell membrane, Biophys. J., 13, 926-939 (1973)
[26] Dharmavaram, S.; Healey, T. J., On the equivalence of local and global area-constraint formulations for lipid bilayer vesicles, Z. Angew. Math. Phys., 66, 5, 2843-2854 (2015) · Zbl 1360.37181
[27] Steigmann, D.; Baesu, E.; Rudd, R. E.; Belak, J.; McElfresh, M., On the variational theory of cell-membrane equilibria, Interfaces Free Bound., 5, 4, 357-366 (2003) · Zbl 1057.35077
[28] Schmid, R., Infinite dimentional Lie groups with applications to mathematical physics, J. Geom. Symmetry Phys., 1, 54-120 (2004) · Zbl 1063.22020
[29] Dharmavaram Muralidharan, S., Phase transitions in lipid bilayer membranes via bifurcation (2014)
[30] Jackson, J. D., From Lorenz to Coulomb and other explicit gauge transformations, Amer. J. Phys., 70, 9, 917-928 (2002)
[31] Jost, J., Harmonic Map between Surfaces (1980), Springer-Verlag
[32] Jost, J., Riemannian Geometry and Geometric Analysis (2008), Springer-Verlag · Zbl 1143.53001
[33] Eells, J.; LeMaire, L., A report on harmonic maps, Bull. Lond. Math. Soc., 10, 1-68 (1978) · Zbl 0401.58003
[34] Eells, J.; Sampson, J. H., Harmonic mappings of Riemannian manifolds, Am. J. Math., 86, 1, 109-160 (1964) · Zbl 0122.40102
[35] Jost, J.; Schoen, R., On the existence of harmonic diffeomorphisms between surfaces, Invent. Math., 66, 353-359 (1982) · Zbl 0488.58009
[36] Sideris, T. C., Global existence of harmonic maps in Minkowski space, Commun. Pure Appl. Math., 42, 1, 1-13 (1989) · Zbl 0685.58016
[37] Jost, J., Compact Riemann Surfaces (2002), Springer-Verlag · Zbl 1086.30038
[38] Gu, X.; Yau, S.-T., Computing conformal structures of surfaces, Commun. Inf. Syst., 2, 2, 121-146 (2002) · Zbl 1092.14514
[39] Gu, X.; Wang, Y.; Chan, T. F.; Thompson, P. M.; Yau, S.-T., Genus zero surface conformal mapping and application to brain sufrace mapping, IEEE Trans. Med. Imaging, 23, 7, 1-8 (2004)
[40] Arnold, D. N.; Rogness, J., Möbius transformations revealed, Notes AMS, 55, 10, 1226-1231 (2008) · Zbl 1200.30036
[41] Zhu, C.; Byrd, R. H.; Lu, P.; Nocedal, J., Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization, ACM Trans. Math. Softw., 23, 4, 550-560 (1997) · Zbl 0912.65057
[42] Bian, X.; Litvinov, S.; Koumoutsakos, P., Bending models of lipid bilayer membranes: Spontaneous curvature and area-difference elasticity, Comput. Methods Appl. Mech. Engrg., 359, Article 112758 pp. (2020) · Zbl 1441.74298
[43] Seifert, U.; Berndl, K.; Lipowsky, R., Shape transformations of vesicles: Phase diagram for spontaneous-curvature and bilayer-coupling models, Phys. Rev. A, 44, 2, 1182 (1991)
[44] Elliott, C. M.; Stinner, B., Modeling and computation of two phase geometric biomembranes using surface finite elements, J. Comput. Phys., 229, 18, 6585-6612 (2010) · Zbl 1425.74323
[45] Evans, L. C., Partial Differential Equations (1998), Amer. Math. Soc.: Amer. Math. Soc. Providence · Zbl 0902.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.