柱状流体膜的静力学和动力学性质研究
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摘要
自20世纪80年代以来，软物质或称软凝聚态物质的奇异特性便引起了广大物理、化学、和生物科学家的广泛关注。软物质物理代表了21世纪凝聚态物理发展的重要趋势，膜和泡则是重要的软物质。
双亲分子所构成的膜泡，其行为和性质类似于向列相液晶。Helfrich通过与液晶的类比得出了流体膜的弹性能量以及膜泡的形状方程。本文在简要介绍了Helfrich的弹性理论及膜方程的基础上，开创性地引入Taylor级数法，在轴对称膜的边界条件下，对Helfrich膜方程以Taylor级数法求解，统一地得到了目前已知的两个特解：常平均曲率曲面和红血球形状解。
本文还研究了柱状膜的静力学方程。由于开口的柱状膜的不稳定性，对柱状膜的研究只能采取近似方法。利用环拓扑结构近似，我们得到了无限长柱状膜的方程，但此方程没有给出任何有用的信息；利用球拓扑近似，无论直接变分还是间接变分，都给出了一组条件，规定了参数的一个取值范围，所以，球拓扑近似比环拓扑近似更实用一些。同时，通过我们的分析，发现Helfrich方程中积分常数C为零的情况仅适用于具有球拓扑结构的膜泡。
在光镊的作用下，柱状膜表现出令人惊讶的动力学行为，即珠链状结构的形成及其发展后期的水珠运动现象。对于此动力学行为，本文从膜的微观结构出发，建立了一个简单的模型，解释了珠链状膜的形成以及光镊熄灭后水珠的运动。并且定量计算了水珠的运动速度，与实验结果吻合，从而说明了本模型的合理性。
Since eighties of the 20th century, membranes and vesicles as main parts of soft matter or soft condensed matter have aroused many interests of scientists, such as physicists, chemists and biologists, because of their attractive characteristics. The development of soft matter physics indicates the progress of condensed matter physics in the 21st century.
Recognizing that the vesicle consisting of amphiphiles bilayer membrane is just like a nematic liquid crystal cell, Helfrich gave the elastic energy of the fluid membrane and the shape equation. In this paper, we give a brief introduction to the Helfrich elastic theory. To solve the Helfrich equation under the physical conditions of vesicles, a Taylor series method is introduced, which offers a unified method to reproduce the exact solution including the famous axisymmetrical constant-curvature surfaces and the biconcave shape solution.
The equilibrium condition for the cylindrical configuration in fluid membrane is re-examined in this paper, using the direct and the indirect variational method and taking account of possible influences of its topology. Since the pure cylindrical vesicles without ends are unstable even non-existence, approximations of torus and cylinder with spherical ends are utilized. The result from the former approximation means the condition of infinite long cylinder, which gives us no useful information. By spherically topological approximation, we get a set of conditions, all of which specify the range of parameters. Evidently, spherically topological approximation is more usable than the other. When our result is used to examine a debate related to the integral constant C, a clean conclusion is reached, that Helfrich shape equation with C=0 is only valid for spherical topology vesicles.
Applying laser tweezers to cylindrical vesicles of lipid bilayers induces an instability, which propagates down the vesicle leaving behind it a peristaltic state, which appears under the microscope as pearls on a string. We make up a simple model based on the microscopic mechanism of the membrane to investigate this phenomenon. The model proves reasonable after we quantify the velocity of the pearls drift slowly towards the laser trap.
双亲分子所构成的膜泡，其行为和性质类似于向列相液晶。Helfrich通过与液晶的类比得出了流体膜的弹性能量以及膜泡的形状方程。本文在简要介绍了Helfrich的弹性理论及膜方程的基础上，开创性地引入Taylor级数法，在轴对称膜的边界条件下，对Helfrich膜方程以Taylor级数法求解，统一地得到了目前已知的两个特解：常平均曲率曲面和红血球形状解。
本文还研究了柱状膜的静力学方程。由于开口的柱状膜的不稳定性，对柱状膜的研究只能采取近似方法。利用环拓扑结构近似，我们得到了无限长柱状膜的方程，但此方程没有给出任何有用的信息；利用球拓扑近似，无论直接变分还是间接变分，都给出了一组条件，规定了参数的一个取值范围，所以，球拓扑近似比环拓扑近似更实用一些。同时，通过我们的分析，发现Helfrich方程中积分常数C为零的情况仅适用于具有球拓扑结构的膜泡。
在光镊的作用下，柱状膜表现出令人惊讶的动力学行为，即珠链状结构的形成及其发展后期的水珠运动现象。对于此动力学行为，本文从膜的微观结构出发，建立了一个简单的模型，解释了珠链状膜的形成以及光镊熄灭后水珠的运动。并且定量计算了水珠的运动速度，与实验结果吻合，从而说明了本模型的合理性。
Since eighties of the 20th century, membranes and vesicles as main parts of soft matter or soft condensed matter have aroused many interests of scientists, such as physicists, chemists and biologists, because of their attractive characteristics. The development of soft matter physics indicates the progress of condensed matter physics in the 21st century.
Recognizing that the vesicle consisting of amphiphiles bilayer membrane is just like a nematic liquid crystal cell, Helfrich gave the elastic energy of the fluid membrane and the shape equation. In this paper, we give a brief introduction to the Helfrich elastic theory. To solve the Helfrich equation under the physical conditions of vesicles, a Taylor series method is introduced, which offers a unified method to reproduce the exact solution including the famous axisymmetrical constant-curvature surfaces and the biconcave shape solution.
The equilibrium condition for the cylindrical configuration in fluid membrane is re-examined in this paper, using the direct and the indirect variational method and taking account of possible influences of its topology. Since the pure cylindrical vesicles without ends are unstable even non-existence, approximations of torus and cylinder with spherical ends are utilized. The result from the former approximation means the condition of infinite long cylinder, which gives us no useful information. By spherically topological approximation, we get a set of conditions, all of which specify the range of parameters. Evidently, spherically topological approximation is more usable than the other. When our result is used to examine a debate related to the integral constant C, a clean conclusion is reached, that Helfrich shape equation with C=0 is only valid for spherical topology vesicles.
Applying laser tweezers to cylindrical vesicles of lipid bilayers induces an instability, which propagates down the vesicle leaving behind it a peristaltic state, which appears under the microscope as pearls on a string. We make up a simple model based on the microscopic mechanism of the membrane to investigate this phenomenon. The model proves reasonable after we quantify the velocity of the pearls drift slowly towards the laser trap.
引文
[1] P.G. de Gennes,软物质与硬科学，湖南教育出版社，2000
[2] 欧阳钟灿，从液晶显示到液晶生物膜理论：软凝聚态物理在交叉学科发展中的创新机遇，物理，1999，28(1)
[3] P. M. Chaikin, T. C. Lubensky, Principles of condensed matter physics, New York, Cambridge university press, 1995
[4] 赵南明，周海梦，生物物理学，北京，高等教育出版社，2000
[5] W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments, Z. Natureforsch, 1973, C28:693-703
[6] H. J. Deuling, W. Helfrich, The curvature elasticity of fluid membranes: A catalogue of vesicle shape, Journal of Physics, 1976, 37:1335-1345
[7] Q. H. Liu, Z. Haijun, J. X. Liu et al., Spheres and prolate and oblate ellipsoids from an analytical solution of the spontaneous-curvature fluid-membrane model, Physical Review E, 1999, 60:3227-3233
[8] S. S. Xie, W. Z. Li, L. X. Qian et al., Equilibrium shape equations and possible shapes of carbon nanotubes, Physical Review B., 1996, 54(23): 16 436-16 439
[9] M. A. Peterson, Deformation energy of vesicles at fixed volume and surface area in the spherical limit, Physical Review A, 1989, 39(5): 2643-2645
[10] Zhang S. G., Ou-Yang Z. C., Periodic cylindrical surface solution for fluid bilayer membranes, Physical Review E, 1996, 53(4): 4206-4208
[11] H. Naito, M. Okuda, On-Yang Z. C., Polygonal shape transformation of a circular biconcave vesicle induced by osmotic pressure, 1996, Physical Review E, 54(3): 2816-2826
[12] V. Kumaran, Spontaneous formation of vesicles by weakly charged membranes, Journal of Chemical Physics, 1993, 99(7): 5490-5499
[13] U. Seifert, Configurations of fluid membranes and vesicles, Advanced in Physics, 1997, 46(1): 13-137
[14] T. Sekimura, H. HOtani, The morphogenesis of liposomes viewed from the aspect of bending energy, Journal of Theoretical Biology, 1991, 149:325-337
[15] H. J. Deuling, W. Helfrich, Red blood cell shapes as explained on the basis of curvature elasticity, biophysical Journal, 1976, 16:861-868
[16] P. B. Canham, the minimum energy of bending as a possible explanation of the bi-
concave shape of the human red blood cell, Journal of theoretical Biology, 1970,26: 61-81
[17] Ou-Yang Zhong-Can, Xie Yu-Zhang, Curvature elasticity modulus and local flicker effect of red blood cells, Physical Review A, 1990, 41(6) : 3381-3384
[18] L. Backman, J. B. Jonasson, P. Horstedt, Phosphoinositide metabolism and shape control in sheep red blood cells, Molecular Membrane Biology, 1998, 15: 27-32
[19] M. Rasia, A. Bollini, Red blood cell shape as a function of medium's ionic strenth and pH, Biochimica et Biophysica Acta, 1998, 1372: 198-204
[20] L. Miao, B. Fourcade, M. Rao et al. Equilibrium budding and vesiculation in the curvature model of lipid vesicles,Physical Review A, 1991, 43(12) : 6843-6856
[21] U. Seifert, K. Berndl, and R. Lipowsky, Shape transformations of vesicles: Phase diagram for spontaneous curvature and bilayer-coupling models, Physical Review A, 1991(2) , 44:1182-1202
[22] L Miao, U. Seifert, M. Wortis et al. Budding transitions of fluid-bilayer vesicles: The effect of area-difference elasticity, Physical Review E, 1994, 49(6) : 5389-5407
[23] X. Michalet , D. Bensimon, Vesicles of toroidal topology: Observed morphology and shape transformations, Journal of Physics, 1995, 5: 263-187
[24] W. Heinrich, B. Boztc, S. Svetina, Vesicle deformation by an axial load: from elongated shapes to tetherd vesicles, biophysical Journal, 1999, 76: 2056-2071
[25] O. Sandre, C. Menager, J. Prost et al., Shape transitions of giant liposomes induced by an anisotropic spontaneous curvaure, Physical Review E., 62: 3865-3870
[26] 欧阳钟灿，谢毓章，双亲分子流体膜的弹性与热涨落--生物膜的 Heifrich 理论，物理学进展, 1991 , 11 ( 1 ): 84-105
[27] Ou-Yang Zhong-Can, W. Heifrich, Instability and deformation of a spherical vesicle by pressure, Physical Review Letters, 1987, 59(21) : 2484-2488
[28] A. Fogden, S. T. Hyde, Continuous transformations of cubic minimal surfaces, The European Physical Journal B, 1999, 7: 91-104
[29] M. D. Lacasse, G. S. Grest, D. Levine, Deforamtion of small compressed droplets, Physical Review E, 1996, 54(5) : 5436-5446
[30] M. D. Lacasse, G. S. Grest, D. Levine et al., Model for the Elasticity of Compressed Emulsions, Physical Review Letters, 1996, 76(18) : 3448-3451
[31] I. Koltover, J. O. Raedler, T. Salditt et al., Phase behavior and interactions of the membrane-protein bacteriorhodopsin, Physical Review Letters, 1999, 82(15) : 3184-3187
[32] Xie Y. Z., Ou-Yang Z. C., Helfrich theory of biomembrane, Mordern Physics letters B, 1992, 6(15): 917-933
[33] G. Beblik, R. M. Servnss, W. Helfrich, Bilayer bending rigidty of some synthetic lecithins, Journal of Physics, 1985, 46:1773-1778
[34] 梅向明，黄敬之，微分几何(第二版)，北京，高等教育出版社，1988．
[35] J. J. Stoker, Differential Geometry, New York. London. Sydney. Toronto, Wiley-Interscience Press, 1986
[36] M. Daoud, C. E. Williams, Soft matter Physics, Verlag Berlin Heidlberg New York, Springer, 1995.
[37] A. W(?)rger, Bending elasticity of surfactant films: the role of the hydrophobic tails, Physical Review Letters, 2000, 85(2): 337-340
[38] Yan Jie, Liu Quanhui, Liu Jixing et al., Numerical observation of nonaxisymmetric vesicles in fluid membranes, Physical Review E, 1998, 58(4): 4730-4736
[39] Hu Jian-Guo, Ou-Yang Zhong-Can, Shape equations of the axisymmetric vesicles, Physical Review E, 1993, 47(1): 461-467
[40] W. M. Zheng, J. X. Liu, Helfrich shape equation for axisymmetric vesicles as a first integral, Physical Review E, 1993, 48(4): 2856-2860
[41] Quan-Hui Liu, Jie Yan, Ou-Yang Zhong-Can, Apex shift of a circular biconcave vesicle induced by osmotic pressure, Physics letters A, 1999, 260:162-167
[42] B.N.斯米尔诺夫著，宋正译，高等数学教程(第二版)，北京，人民教育出版社，1964
[43] H. Naito, M. Okuda, Ou-Yang Zhong-Can, New solutions to the Helfrich variation proglem for the shapes of lipid bilayer vesicles: beyond Delaunay's surfaces, Physical Review Letter, 1995, 74(21): 4345-4348
[44] Hu B., Liu Q. H., and Zhang H., Tayler series method solving the nonlinear equation for axisymmetrical fluid membrane shape, The Third Joint Meeting of Chinese Physicist Worldwide, Hongkong, Hongkong Town Press, 1999:113
[45] Ou-Yang Zhong-Can, W. Helfrich, Bending energy of vesicle membranes: general expressions for the first, second, and third variation of the shape energy and applicaions to spheres and cylinders, Physical Review A, 1989, 39(10): 5280-5288
[46] W. M. Zheng, J. X. Liu, Helfrich shape equation for axisymmetric vesicles as a first integral, Physical Review E, 1993, 48:2858-2860
[47] F. J(?)licher and U. Seifert, Shape equations for axisymmetric vesicles: a clarification, Physical Review E., 1994, 49(5): 4728-4731
[48] S. Chateb, S. Rica, Spontaneous curvature-induced pearling instability, Physical Review E, 1998, 58(6) : 7733-7737
[49] R. Podgornik, S. Svetina, B. Zeks, Parametriztion invariance and shape equations of elastic axisymmetric vesicles, Physical Review E, 1995, 51: 544-547
[50] B. Boztc, S. Svetina, B. Zeks, Theoretical analysis of the formation of membrane microtubes on axially strained vesicles, Physical Review E, 1997, 55: 5834-5842
[51] T. Umeda, H. Nakajima, H. hotani, Theoretical analysis of shape transformations of liposomes caused by microtubule assembly, Journal of the Physical Society of Japan, 1998, 67(2) : 682-688
[52] W. M. Zheng, J. X. Liu, Relation between shape equations for axisymmetric vesicles, Communication of Theoretical Physics, 1994, 21(3) : 369-380
[53] Evans E., W. Rawicz, Entropy-driven tension and bending elasticity in condensed-fluid membranes, Physical Review Letters, 1990, 64: 2094-2097
[54] R. Bar-Ziv, E. Moses, and P. Nelson, Dynamic Excitation in Membranes Induced by Optical Tweezers, Biophysical Journal, 1998, 75, 294-320．
[55] Bar-Ziv R., T. Rrisch, and E. Moses, Entropic expulsion in vesicles, Physical Review Letters, 1995, 75: 3481-3484
[56] Bar-Ziv R., A. Meller, T. Tlusty, et al., Localized dynamic light scattering: probing dynamics at the nanoscale, Physical Review Letters, 1997, 78: 154-156
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[58] Bar-Ziv R., E. Moses, Instability and "pearling" states produced in tubular membranes by competition of curvature and tension, Physical Review Letters, 1994, 73: 1392-1395
[59] Bar-Ziv R., T. Tlusty and E. Moses, Critical dynamics in the pearling instability of membranes, Physical Review Letters, 1997, 79: 1158-1161
[60] R. E. Goldstein, P. Nelson, T. Power, Front propagation in the pearling instability of tubular vesicles, Journal of Physics Ⅱ France 1996, 6: 767-796．
[61] J. L. Goveas, S. T. Milner and W. B. Russel, Late Stages of the "Pearling11 Instability in Lipid Bilayers, Journal of Physics, 1997, 7, 1185-1204．
[62] 李承祖，赵凤章，电动力学教程，长沙，国防科技大学出版社， 1994．
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[66] P. D. Olmsted and F. C. Macintosh, Instability and Front Propagation in Laser-Tweezed Lipid Bilayer Tubules, Journal of Physics, 1997, 7, 139-156.
[67] J.I.朗道，E.M.栗弗席茨著，孔祥言，徐燕侯，庄礼贤译，流体力学(第一版)，北京，高等教育出版社，1983．
[68] H. Lamb, Hydrodynamics(Sixth Edition), Cambridge, University Press, 1932.
[2] 欧阳钟灿，从液晶显示到液晶生物膜理论：软凝聚态物理在交叉学科发展中的创新机遇，物理，1999，28(1)
[3] P. M. Chaikin, T. C. Lubensky, Principles of condensed matter physics, New York, Cambridge university press, 1995
[4] 赵南明，周海梦，生物物理学，北京，高等教育出版社，2000
[5] W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments, Z. Natureforsch, 1973, C28:693-703
[6] H. J. Deuling, W. Helfrich, The curvature elasticity of fluid membranes: A catalogue of vesicle shape, Journal of Physics, 1976, 37:1335-1345
[7] Q. H. Liu, Z. Haijun, J. X. Liu et al., Spheres and prolate and oblate ellipsoids from an analytical solution of the spontaneous-curvature fluid-membrane model, Physical Review E, 1999, 60:3227-3233
[8] S. S. Xie, W. Z. Li, L. X. Qian et al., Equilibrium shape equations and possible shapes of carbon nanotubes, Physical Review B., 1996, 54(23): 16 436-16 439
[9] M. A. Peterson, Deformation energy of vesicles at fixed volume and surface area in the spherical limit, Physical Review A, 1989, 39(5): 2643-2645
[10] Zhang S. G., Ou-Yang Z. C., Periodic cylindrical surface solution for fluid bilayer membranes, Physical Review E, 1996, 53(4): 4206-4208
[11] H. Naito, M. Okuda, On-Yang Z. C., Polygonal shape transformation of a circular biconcave vesicle induced by osmotic pressure, 1996, Physical Review E, 54(3): 2816-2826
[12] V. Kumaran, Spontaneous formation of vesicles by weakly charged membranes, Journal of Chemical Physics, 1993, 99(7): 5490-5499
[13] U. Seifert, Configurations of fluid membranes and vesicles, Advanced in Physics, 1997, 46(1): 13-137
[14] T. Sekimura, H. HOtani, The morphogenesis of liposomes viewed from the aspect of bending energy, Journal of Theoretical Biology, 1991, 149:325-337
[15] H. J. Deuling, W. Helfrich, Red blood cell shapes as explained on the basis of curvature elasticity, biophysical Journal, 1976, 16:861-868
[16] P. B. Canham, the minimum energy of bending as a possible explanation of the bi-
concave shape of the human red blood cell, Journal of theoretical Biology, 1970,26: 61-81
[17] Ou-Yang Zhong-Can, Xie Yu-Zhang, Curvature elasticity modulus and local flicker effect of red blood cells, Physical Review A, 1990, 41(6) : 3381-3384
[18] L. Backman, J. B. Jonasson, P. Horstedt, Phosphoinositide metabolism and shape control in sheep red blood cells, Molecular Membrane Biology, 1998, 15: 27-32
[19] M. Rasia, A. Bollini, Red blood cell shape as a function of medium's ionic strenth and pH, Biochimica et Biophysica Acta, 1998, 1372: 198-204
[20] L. Miao, B. Fourcade, M. Rao et al. Equilibrium budding and vesiculation in the curvature model of lipid vesicles,Physical Review A, 1991, 43(12) : 6843-6856
[21] U. Seifert, K. Berndl, and R. Lipowsky, Shape transformations of vesicles: Phase diagram for spontaneous curvature and bilayer-coupling models, Physical Review A, 1991(2) , 44:1182-1202
[22] L Miao, U. Seifert, M. Wortis et al. Budding transitions of fluid-bilayer vesicles: The effect of area-difference elasticity, Physical Review E, 1994, 49(6) : 5389-5407
[23] X. Michalet , D. Bensimon, Vesicles of toroidal topology: Observed morphology and shape transformations, Journal of Physics, 1995, 5: 263-187
[24] W. Heinrich, B. Boztc, S. Svetina, Vesicle deformation by an axial load: from elongated shapes to tetherd vesicles, biophysical Journal, 1999, 76: 2056-2071
[25] O. Sandre, C. Menager, J. Prost et al., Shape transitions of giant liposomes induced by an anisotropic spontaneous curvaure, Physical Review E., 62: 3865-3870
[26] 欧阳钟灿，谢毓章，双亲分子流体膜的弹性与热涨落--生物膜的 Heifrich 理论，物理学进展, 1991 , 11 ( 1 ): 84-105
[27] Ou-Yang Zhong-Can, W. Heifrich, Instability and deformation of a spherical vesicle by pressure, Physical Review Letters, 1987, 59(21) : 2484-2488
[28] A. Fogden, S. T. Hyde, Continuous transformations of cubic minimal surfaces, The European Physical Journal B, 1999, 7: 91-104
[29] M. D. Lacasse, G. S. Grest, D. Levine, Deforamtion of small compressed droplets, Physical Review E, 1996, 54(5) : 5436-5446
[30] M. D. Lacasse, G. S. Grest, D. Levine et al., Model for the Elasticity of Compressed Emulsions, Physical Review Letters, 1996, 76(18) : 3448-3451
[31] I. Koltover, J. O. Raedler, T. Salditt et al., Phase behavior and interactions of the membrane-protein bacteriorhodopsin, Physical Review Letters, 1999, 82(15) : 3184-3187
[32] Xie Y. Z., Ou-Yang Z. C., Helfrich theory of biomembrane, Mordern Physics letters B, 1992, 6(15): 917-933
[33] G. Beblik, R. M. Servnss, W. Helfrich, Bilayer bending rigidty of some synthetic lecithins, Journal of Physics, 1985, 46:1773-1778
[34] 梅向明，黄敬之，微分几何(第二版)，北京，高等教育出版社，1988．
[35] J. J. Stoker, Differential Geometry, New York. London. Sydney. Toronto, Wiley-Interscience Press, 1986
[36] M. Daoud, C. E. Williams, Soft matter Physics, Verlag Berlin Heidlberg New York, Springer, 1995.
[37] A. W(?)rger, Bending elasticity of surfactant films: the role of the hydrophobic tails, Physical Review Letters, 2000, 85(2): 337-340
[38] Yan Jie, Liu Quanhui, Liu Jixing et al., Numerical observation of nonaxisymmetric vesicles in fluid membranes, Physical Review E, 1998, 58(4): 4730-4736
[39] Hu Jian-Guo, Ou-Yang Zhong-Can, Shape equations of the axisymmetric vesicles, Physical Review E, 1993, 47(1): 461-467
[40] W. M. Zheng, J. X. Liu, Helfrich shape equation for axisymmetric vesicles as a first integral, Physical Review E, 1993, 48(4): 2856-2860
[41] Quan-Hui Liu, Jie Yan, Ou-Yang Zhong-Can, Apex shift of a circular biconcave vesicle induced by osmotic pressure, Physics letters A, 1999, 260:162-167
[42] B.N.斯米尔诺夫著，宋正译，高等数学教程(第二版)，北京，人民教育出版社，1964
[43] H. Naito, M. Okuda, Ou-Yang Zhong-Can, New solutions to the Helfrich variation proglem for the shapes of lipid bilayer vesicles: beyond Delaunay's surfaces, Physical Review Letter, 1995, 74(21): 4345-4348
[44] Hu B., Liu Q. H., and Zhang H., Tayler series method solving the nonlinear equation for axisymmetrical fluid membrane shape, The Third Joint Meeting of Chinese Physicist Worldwide, Hongkong, Hongkong Town Press, 1999:113
[45] Ou-Yang Zhong-Can, W. Helfrich, Bending energy of vesicle membranes: general expressions for the first, second, and third variation of the shape energy and applicaions to spheres and cylinders, Physical Review A, 1989, 39(10): 5280-5288
[46] W. M. Zheng, J. X. Liu, Helfrich shape equation for axisymmetric vesicles as a first integral, Physical Review E, 1993, 48:2858-2860
[47] F. J(?)licher and U. Seifert, Shape equations for axisymmetric vesicles: a clarification, Physical Review E., 1994, 49(5): 4728-4731
[48] S. Chateb, S. Rica, Spontaneous curvature-induced pearling instability, Physical Review E, 1998, 58(6) : 7733-7737
[49] R. Podgornik, S. Svetina, B. Zeks, Parametriztion invariance and shape equations of elastic axisymmetric vesicles, Physical Review E, 1995, 51: 544-547
[50] B. Boztc, S. Svetina, B. Zeks, Theoretical analysis of the formation of membrane microtubes on axially strained vesicles, Physical Review E, 1997, 55: 5834-5842
[51] T. Umeda, H. Nakajima, H. hotani, Theoretical analysis of shape transformations of liposomes caused by microtubule assembly, Journal of the Physical Society of Japan, 1998, 67(2) : 682-688
[52] W. M. Zheng, J. X. Liu, Relation between shape equations for axisymmetric vesicles, Communication of Theoretical Physics, 1994, 21(3) : 369-380
[53] Evans E., W. Rawicz, Entropy-driven tension and bending elasticity in condensed-fluid membranes, Physical Review Letters, 1990, 64: 2094-2097
[54] R. Bar-Ziv, E. Moses, and P. Nelson, Dynamic Excitation in Membranes Induced by Optical Tweezers, Biophysical Journal, 1998, 75, 294-320．
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