Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-17T16:03:52.931Z Has data issue: false hasContentIssue false
  • English
  • Français

Fast Simulation of Lipid Vesicle Deformation Using Spherical Harmonic Approximation

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Physiological, cellular and medical topics Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  05 December 2016

Michael Mikucki  and
Yongcheng Zhou
Michael Mikucki*
Affiliation:
Department of Applied Mathematics & Statistics, Colorado School of Mines, Golden, Colorado, 80401, USA
Yongcheng Zhou*
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, Colorado, 80523, USA
*
*Corresponding author. Email addresses:[email protected] (M. Mikucki), [email protected] (Y. Zhou)
*Corresponding author. Email addresses:[email protected] (M. Mikucki), [email protected] (Y. Zhou)
Get access

Abstract

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.

Keywords

Lipid bilayercurvature energysurface harmonicsfast algorithm

MSC classification

Secondary: 35Q92: PDEs in connection with biology and other natural sciences 92C40: Biochemistry, molecular biology 65M70: Spectral, collocation and related methods
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 1 , January 2017 , pp. 40 - 64
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bagchi, Prosenjit. Mesoscale simulation of blood flow in small vessels. Biophys. J., 92(6):1858–1877, 2007.CrossRefGoogle ScholarPubMed
[2] Bahrami, Amir Houshang and Jalali, Mir Abbas. Vesicle deformations by clusters of transmembrane proteins. J. Chem. Phys., 134:085106, 2011.CrossRefGoogle ScholarPubMed
[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):61–81, 1970.Google Scholar
[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.CrossRefGoogle Scholar
[5] Cohen, Fredric S., Eisenberg, Robert, and Ryham, Rolf J.. A dynamic model of open vesicles in fluids. Commun. Math. Sci., 10:1273–1285, 2012.CrossRefGoogle Scholar
[6] Das, Sovan and Du, Qiang. Adhesion of vesicles to curved substrates. Phys. Rev. E, 77:011907, Jan 2008.CrossRefGoogle ScholarPubMed
[7] Du, Qiang, Liu, Chun, Ryham, Rolf, and Wang, Xiaoqiang. A phase field formulation of the Willmore problem. Nonlinearity, 18:1249, 2005.Google Scholar
[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):450–468, 2004.Google Scholar
[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):757–777, 2006.Google Scholar
[10] Eggleton, C. D. and Popel, A. S.. Large deformation of red blood cell ghosts in a simple shear flow. Phys. Fluids, 10(8):1834–1845, 1998.Google Scholar
[11] Evans, E.A.. Bending resistance and chemically induced moments in membrane bilayers. Biophys. J., 14(12):923–931, 1974.Google Scholar
[12] Evans, Evan and Fung, Yuan-Cheng. Improved measurements of the erythrocyte geometry. Microvasc. Res., 4(4):335–347, 1972.Google Scholar
[13] Farsad, Khashayar and De Camilli, Pietro. Mechanisms of membrane deformation. Curr. Opin. Cell Biol., 15(4):372–381, 2003.Google Scholar
[14] Feng, Feng and Klug, William S.. Finite element modeling of lipid bilayer membranes. J. Comput. Phys., 220(1):394–408, 2006.Google Scholar
[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., 28:1198–1223, 2012.Google ScholarPubMed
[16] Heinrich, Volkmar, Bozic, Bojan, Svetina, Sasa, and Zeks, Bostjan. Vesicle deformation by an axial load: From elongated shapes to tethered vesicles. Biophys. J., 76:2056–2071, 1999.Google Scholar
[17] Helfrich, W. et al. Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. C, 28(11):693–703, 1973.Google Scholar
[18] Kamm, Roger D.. cellular fluid mechanics. Annu. Rev. Fluid Mech., 34(1):211–232, 2002.CrossRefGoogle ScholarPubMed
[19] Keiner, Jens and Potts, Daniel. Fast evaluation of quadrature formulae on the sphere. Math. Comp., 77:397–419, 2008.CrossRefGoogle Scholar
[20] Khairy, Khaled and Howard, Jonathon. Minimum-energy vesicle and cell shapes calculated using spherical harmonics parameterization. Soft Matter, 7:2138–2143, 2011.CrossRefGoogle Scholar
[21] Kunis, Stefan and Potts, Daniel. Fast spherical Fourier algorithms. J. Comput. Appl. Math., 161(1):75–98, 2003.Google Scholar
[22] Li, Shuwang, Lowengrub, John, and Voigt, Axel. Locomotion, wrinkling, and budding of a multicomponent vesicle in viscous fluids. Commun. Math. Sci., 10:645–670, 2012.CrossRefGoogle Scholar
[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):5816–5835, 2008.CrossRefGoogle Scholar
[24] Mikucki, M. and Zhou, Y.. Electrostatic forces on charged surfaces of bilayer lipid membranes. SIAM J. Appl.Math., 74(1):1–21, 2014.CrossRefGoogle Scholar
[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.Google Scholar
[26] Powers, Thomas R.. Mechanics of lipid bilayer membranes. In Yip, Sidney, editor, Handbook of Materials Modeling, pages 2631–2643. Springer Netherlands, 2005.CrossRefGoogle Scholar
[27] Rokhlin, Vladimir and Tygert, Mark. Fast Algorithms for Spherical Harmonic Expansions. SIAM J. Sci. Comput., 27(6):1903–28, 2006.CrossRefGoogle Scholar
[28] Seifert, Udo. Configurations of fluid membranes and vesicles. Adv. Phys., 46(1):13–137, 1997.CrossRefGoogle Scholar
[29] Seifert, Udo, Berndl, Karin, and Lipowsky, Reinhard. Shape transformations of vesicles: Phase diagram for spontaneous-curvature and bilayer-coupling models. Phys. Rev. A, 44:1182–1202, Jul 1991.Google Scholar
[30] Sidi, Avram. Application of class Sm variable transformations to numerical integration over surfaces of spheres. J. Comput. Appl. Math., 184(2):475–492, 2005.CrossRefGoogle Scholar
[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):119–144, 2010.Google Scholar
[32] Sokolnikoff, I.S.. Tensor analysis: theory and applications to geometry and mechanics of continua. Applied mathematics series. Wiley, 1964.Google Scholar
[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):3357–3363, 2005.Google Scholar
[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., 230:375–393, 2011.Google Scholar
[35] Wang, Xiaoqiang and Du, Qiang. Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches. J. Math. Biol., 56:347–371, 2008.CrossRefGoogle ScholarPubMed
[36] Wei, Guo-Wei. Differential geometry based multiscale models. Bulletin of Mathematical Biology, 72(6):1562–1622, 2010.CrossRefGoogle ScholarPubMed
[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):414–446, 2007.Google Scholar
[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:912–936, 2014.Google Scholar
[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):5897–5909, 2012.CrossRefGoogle Scholar
[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):5610–5634, 2011.Google Scholar
[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):364–394, 2006.Google Scholar
[42] Zhong-can, Ou-Yang and Helfrich, W.. Instability and deformation of a spherical vesicle by pressure. Phys. Rev. Lett., 59:2486–2488, 1987.CrossRefGoogle ScholarPubMed
[43] Zhou, Y. C., Lu, B., and Gorfe, A. A.. Continuum electromechanical modeling of protein-membrane interactions. Phys. Rev. E, 82(4):041923, 2010.CrossRefGoogle ScholarPubMed