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Fast Simulation of Lipid Vesicle Deformation Using Spherical Harmonic Approximation

  • Michael Mikucki (a1) and Yongcheng Zhou (a2)

Lipid vesicles appear ubiquitously in biological systems. Understanding how the mechanical and intermolecular interactions deform vesicle membranes is a fundamental question in biophysics. In this article we develop a fast algorithm to compute the surface configurations of lipid vesicles by introducing surface harmonic functions to approximate themembrane surface. This parameterization allows an analytical computation of the membrane curvature energy and its gradient for the efficient minimization of the curvature energy using a nonlinear conjugate gradient method. Our approach drastically reduces the degrees of freedom for approximating the membrane surfaces compared to the previously developed finite element and finite difference methods. Vesicle deformations with a reduced volume larger than 0.65 can be well approximated by using as small as 49 surface harmonic functions. The method thus has a great potential to reduce the computational expense of tracking multiple vesicles which deform for their interaction with external fields.

Corresponding author
*Corresponding author. Email addresses: (M. Mikucki), (Y. Zhou)
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[1] Bagchi Prosenjit. Mesoscale simulation of blood flow in small vessels. Biophys. J., 92(6):18581877, 2007.
[2] Bahrami Amir Houshang and Jalali Mir Abbas. Vesicle deformations by clusters of transmembrane proteins. J. Chem. Phys., 134:085106, 2011.
[3] Canham P.B.. The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theor. Biol., 26(1):6181, 1970.
[4] Capovilla R., Guven J., and Santiago J. A.. Deformations of the geometry of lipid vesicles. J. Phys. A – Math. Gen., 36(23):6281, 2003.
[5] Cohen Fredric S., Eisenberg Robert, and Ryham Rolf J.. A dynamic model of open vesicles in fluids. Commun. Math. Sci., 10:12731285, 2012.
[6] Das Sovan and Du Qiang. Adhesion of vesicles to curved substrates. Phys. Rev. E, 77:011907, Jan 2008.
[7] Du Qiang, Liu Chun, Ryham Rolf, and Wang Xiaoqiang. A phase field formulation of the Willmore problem. Nonlinearity, 18:1249, 2005.
[8] Du Qiang, Liu Chun, and Wang Xiaoqiang. A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys., 198(2):450468, 2004.
[9] Du Qiang, Liu Chun, and Wang Xiaoqiang. Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Comput. Phys., 212(2):757777, 2006.
[10] Eggleton C. D. and Popel A. S.. Large deformation of red blood cell ghosts in a simple shear flow. Phys. Fluids, 10(8):18341845, 1998.
[11] Evans E.A.. Bending resistance and chemically induced moments in membrane bilayers. Biophys. J., 14(12):923931, 1974.
[12] Evans Evan and Fung Yuan-Cheng. Improved measurements of the erythrocyte geometry. Microvasc. Res., 4(4):335347, 1972.
[13] Farsad Khashayar and De Camilli Pietro. Mechanisms of membrane deformation. Curr. Opin. Cell Biol., 15(4):372381, 2003.
[14] Feng Feng and Klug William S.. Finite element modeling of lipid bilayer membranes. J. Comput. Phys., 220(1):394408, 2006.
[15] Feng Xin, Xia Kelin, Tong Yiying, and Wei Guo-Wei. Geometric modeling of subcellular structures, organelles, and multiprotein complexes. Internat. J. Numer. Methods Eng., 28:11981223, 2012.
[16] Heinrich Volkmar, Bozic Bojan, Svetina Sasa, and Zeks Bostjan. Vesicle deformation by an axial load: From elongated shapes to tethered vesicles. Biophys. J., 76:20562071, 1999.
[17] Helfrich W. et al. Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. C, 28(11):693703, 1973.
[18] Kamm Roger D.. cellular fluid mechanics. Annu. Rev. Fluid Mech., 34(1):211232, 2002.
[19] Keiner Jens and Potts Daniel. Fast evaluation of quadrature formulae on the sphere. Math. Comp., 77:397419, 2008.
[20] Khairy Khaled and Howard Jonathon. Minimum-energy vesicle and cell shapes calculated using spherical harmonics parameterization. Soft Matter, 7:21382143, 2011.
[21] Kunis Stefan and Potts Daniel. Fast spherical Fourier algorithms. J. Comput. Appl. Math., 161(1):7598, 2003.
[22] Li Shuwang, Lowengrub John, and Voigt Axel. Locomotion, wrinkling, and budding of a multicomponent vesicle in viscous fluids. Commun. Math. Sci., 10:645670, 2012.
[23] Ma L. and Klug W. S.. Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics. J. Comput. Phys., 227(11):58165835, 2008.
[24] Mikucki M. and Zhou Y.. Electrostatic forces on charged surfaces of bilayer lipid membranes. SIAM J. Appl.Math., 74(1):121, 2014.
[25] Nocedal Jorge and Wright Stephen J.. Numerical optimization. Springer series in operations research and financial engineering. Springer, New York, NY, 2. ed. edition, 2006.
[26] Powers Thomas R.. Mechanics of lipid bilayer membranes. In Yip Sidney, editor, Handbook of Materials Modeling, pages 26312643. Springer Netherlands, 2005.
[27] Rokhlin Vladimir and Tygert Mark. Fast Algorithms for Spherical Harmonic Expansions. SIAM J. Sci. Comput., 27(6):1903–28, 2006.
[28] Seifert Udo. Configurations of fluid membranes and vesicles. Adv. Phys., 46(1):13137, 1997.
[29] Seifert Udo, Berndl Karin, and Lipowsky Reinhard. Shape transformations of vesicles: Phase diagram for spontaneous-curvature and bilayer-coupling models. Phys. Rev. A, 44:11821202, Jul 1991.
[30] Sidi Avram. Application of class Sm variable transformations to numerical integration over surfaces of spheres. J. Comput. Appl. Math., 184(2):475492, 2005.
[31] Sohn Jin Sun, Tseng Yu-Hau, Li Shuwang, Voigt Axel, and Lowengrub John S.. Dynamics of multicomponent vesicles in a viscous fluid. J. Comput. Phys., 229(1):119144, 2010.
[32] Sokolnikoff I.S.. Tensor analysis: theory and applications to geometry and mechanics of continua. Applied mathematics series. Wiley, 1964.
[33] Solon Jerome, Gareil Olivier, Bassereau Patricia, and Gaudin Yves. Membrane deformations induced by the matrix protein of vesicular stomatitis virus in aminimal system. J. Gen. Virol., 86(12):33573363, 2005.
[34] Teigen Knut Erik, Song Peng, Lowengrub John, and Voigt Axel. A diffuse-interface method for two-phase flows with soluble surfactants. J. Comput. Phys., 230:375393, 2011.
[35] Wang Xiaoqiang and Du Qiang. Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches. J. Math. Biol., 56:347371, 2008.
[36] Wei Guo-Wei. Differential geometry based multiscale models. Bulletin of Mathematical Biology, 72(6):15621622, 2010.
[37] Wise Steven, Kim Junseok, and Lowengrub John. Solving the regularized, strongly anisotropic CahnHilliard equation by an adaptive nonlinear multigrid method. J. Comput. Phys., 226(1):414446, 2007.
[38] Xia Kelin, Feng Xin, Chen Zhan, Tong Yiying, and Wei Guo-Wei. Multiscale geometric modeling of macromolecules I: Cartesian representation. J. Comput. Phys., 257, Part A:912936, 2014.
[39] Xu Jian-Jun, Yang Yin, and Lowengrub John. A level-set continuum method for two-phase flows with insoluble surfactant. J. Comput. Phys., 231(17):58975909, 2012.
[40] Veerapaneni Shravan K., Rahimian Abtin, Biros George and Zorin Denis. A fast algorithm for simulating vesicle flows in three dimensions. J. Comput. Phys., 230(14):56105634, 2011.
[41] Yang Xiaofeng, James Ashley J., Lowengrub John, Zheng Xiaoming, and Cristini Vittorio. An adaptive coupled level-set/volume-of-fluid interface capturing method for unstructured triangular grids. J. Comput. Phys., 217(2):364394, 2006.
[42] Zhong-can Ou-Yang and Helfrich W.. Instability and deformation of a spherical vesicle by pressure. Phys. Rev. Lett., 59:24862488, 1987.
[43] Zhou Y. C., Lu B., and Gorfe A. A.. Continuum electromechanical modeling of protein-membrane interactions. Phys. Rev. E, 82(4):041923, 2010.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
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