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Lipid membrane is a continuous barrier between cell and organelle, providing relatively separate room for the vital biological reaction to take place and guarantee substance, energy and information exchange between cells and organelles. Helfrich proposed a spontaneous curvature model to describe the free energy of lipid bilayer. This article reviews the equations describing the equilibrium morphologies of closed lipid membranes and lipid membranes with free edge based on the spontaneous model, and some analytic solutions are provided as well. The practicality of proving linking condition for splitting vesicle is also discussed.
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Keywords:
- biomembrane /
- spontaneous curvature model /
- morphology
[1] Singer S J, Nicolson G L 1972 Science 175 720
[2] Fung Y C, Tong P 1968 Biophys. J. 8 175
[3] Pinder D N 1972 J. Theor. Biol. 34 407
[4] Lopez L, Duck I M, Hunt W A 1968 Biophys. J. 8 1228
[5] Greer M A, Baker R F 1970 Proceedings of the 7th International Congress of Electron Microscopy Grenoble, France
[6] Murphy J R 1965 J. Lab. Clin. Med. 65 756
[7] Seeman P, Cheng D, Lies G H 1973 J. Cell Biol. 56 519
[8] Canham P B 1970 J. Theor. Biol. 26 61
[9] Helfrich W 1973 Z. Naturforsch. C 28 693
[10] Ouyang Z C, Helfrich W 1987 Phys. Rev. Lett. 59 2486
[11] Ouyang Z C, Helfrich W 1989 Phys. Rev. A 39 5280
[12] Ouyang Z C 1990 Phys. Rev. A 41 4517
[13] Mutz M, Bensimon D 1991 Phys. Rev. A 43 4525
[14] Rudolph A S, Ratna B R, Kahn B 1991 Nature 3 52
[15] Lin Z 1994 Langmuir 10 1008
[16] Willmore T J, 1965 Al. I. Cuza Iasi Sect. I a Mat. An. Sti. Univ. 11B 493
[17] Marques F C, Neves A 2012 Annals of Mathematics 179 83
[18] Ouyang Z C, Liu J X, Xie Y Z 1999 Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases (Singapore: World Scientific)
[19] Hu J G, Ouyang Z C 1993 Phys. Rev. E 47 461
[20] Zheng W M, Liu J X 1993 Phys. Rev. E 48 2856
[21] Naito H, Okuda M, Ouyang Z C 1993 Phys. Rev. E 48 2304
[22] Saitoh A, Takiguchi K, Tanaka Y, Hotani H 1998 Proc. Nat. Acad. Sci. 95 1026
[23] Capovilla R, Guven J, Santiago J 2002 Phys. Rev. E 66 021607
[24] Tu Z C, Ouyang Z C 2003 Phys. Rev. E 68 061915
[25] Tu Z C, Ouyang Z C 2004 J. Phys. A: Math. Gen. 37 11407
[26] Tu Z C, Ouyang Z C 2008 J. Comput. Theor. Nanosci. 5 422
[27] Tu Z C 2010 J. Chem. Phys. 132 084111
[28] Tu Z C 2011 J. Geom. Symmetry Phys. 24 45
[29] Tu Z C, Ouyang Z C 2014 Adv. Colloid Interface Sci. 208 66
[30] Tu Z C 2013 Chin. Phys. B 22 028701
[31] Seifert U, Berndl K, Lipowsky R 1991 Phys. Rev. A 44 1182
[32] Fourcade B, Miao L, Rao M, Mortis M, Zia R K P 1994 Phys. Rev. E 49 5276
[33] Jlicher F, Lipowsky R 1996 Phys. Rev. E 53 2670
[34] Seifert U 1997 Adv. Phys. 46 13
[35] Xie Y Z, Liu J X, Ouyang Z C 2003 Elastic Theory of Surfaces of Biomembrane-Vesicle (Shanghai: Shanghai Scientific Technical Publishers) (in Chinese) [谢毓章, 刘寄星, 欧阳钟灿 2003 生物膜泡曲面弹性理论 (上海: 上海科学技术出版社)]
[36] Du Q, Liu C, Wang X 2006 J. Comput. Phys. 212 757
[37] Du Q, Liu C, Ryham R, Wang X 2005 Commun. Pure Appl. Anal. 4 537
[38] Yan J, Liu Q H, Liu J X, Ouyang Z C 1998 Phys. Rev. E 58 4730
[39] Su Y C, Chen J Z 2015 Soft Matter 11 4054
[40] Zhang P W, Shi A C 2015 Chin. Phys. B 24 128707
[41] Zhou X H, Zhang S G 2006 Acta Phys. Sin. 55 5568 (in Chinese) [周晓华, 张劭光 2006 55 5568]
[42] Zhou X H, Zhang S G, Yang J Q, Qu X M, Liu Y S, Wang S G 2007 Acta Phys. Sin. 56 6137 (in Chinese) [周晓华, 张劭光, 杨继庆, 屈学民, 刘渊声, 王斯刚 2007 56 6137]
[43] Xia B K, Li J F, Li W H, Zhang H D, Qiu F 2013 Acta Phys. Sin. 62 248701 (in Chinese) [夏彬凯, 李剑锋, 李卫华, 张红东, 邱枫 2013 62 248701]
[44] Yang K, Ma Y Q 2010 Nat. Nanotechnol. 5 579
[45] Ding H M, Ma Y Q 2012 Biomaterials 33 5798
[46] Ding H M, Ma Y Q 2015 Small 11 1055
-
[1] Singer S J, Nicolson G L 1972 Science 175 720
[2] Fung Y C, Tong P 1968 Biophys. J. 8 175
[3] Pinder D N 1972 J. Theor. Biol. 34 407
[4] Lopez L, Duck I M, Hunt W A 1968 Biophys. J. 8 1228
[5] Greer M A, Baker R F 1970 Proceedings of the 7th International Congress of Electron Microscopy Grenoble, France
[6] Murphy J R 1965 J. Lab. Clin. Med. 65 756
[7] Seeman P, Cheng D, Lies G H 1973 J. Cell Biol. 56 519
[8] Canham P B 1970 J. Theor. Biol. 26 61
[9] Helfrich W 1973 Z. Naturforsch. C 28 693
[10] Ouyang Z C, Helfrich W 1987 Phys. Rev. Lett. 59 2486
[11] Ouyang Z C, Helfrich W 1989 Phys. Rev. A 39 5280
[12] Ouyang Z C 1990 Phys. Rev. A 41 4517
[13] Mutz M, Bensimon D 1991 Phys. Rev. A 43 4525
[14] Rudolph A S, Ratna B R, Kahn B 1991 Nature 3 52
[15] Lin Z 1994 Langmuir 10 1008
[16] Willmore T J, 1965 Al. I. Cuza Iasi Sect. I a Mat. An. Sti. Univ. 11B 493
[17] Marques F C, Neves A 2012 Annals of Mathematics 179 83
[18] Ouyang Z C, Liu J X, Xie Y Z 1999 Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases (Singapore: World Scientific)
[19] Hu J G, Ouyang Z C 1993 Phys. Rev. E 47 461
[20] Zheng W M, Liu J X 1993 Phys. Rev. E 48 2856
[21] Naito H, Okuda M, Ouyang Z C 1993 Phys. Rev. E 48 2304
[22] Saitoh A, Takiguchi K, Tanaka Y, Hotani H 1998 Proc. Nat. Acad. Sci. 95 1026
[23] Capovilla R, Guven J, Santiago J 2002 Phys. Rev. E 66 021607
[24] Tu Z C, Ouyang Z C 2003 Phys. Rev. E 68 061915
[25] Tu Z C, Ouyang Z C 2004 J. Phys. A: Math. Gen. 37 11407
[26] Tu Z C, Ouyang Z C 2008 J. Comput. Theor. Nanosci. 5 422
[27] Tu Z C 2010 J. Chem. Phys. 132 084111
[28] Tu Z C 2011 J. Geom. Symmetry Phys. 24 45
[29] Tu Z C, Ouyang Z C 2014 Adv. Colloid Interface Sci. 208 66
[30] Tu Z C 2013 Chin. Phys. B 22 028701
[31] Seifert U, Berndl K, Lipowsky R 1991 Phys. Rev. A 44 1182
[32] Fourcade B, Miao L, Rao M, Mortis M, Zia R K P 1994 Phys. Rev. E 49 5276
[33] Jlicher F, Lipowsky R 1996 Phys. Rev. E 53 2670
[34] Seifert U 1997 Adv. Phys. 46 13
[35] Xie Y Z, Liu J X, Ouyang Z C 2003 Elastic Theory of Surfaces of Biomembrane-Vesicle (Shanghai: Shanghai Scientific Technical Publishers) (in Chinese) [谢毓章, 刘寄星, 欧阳钟灿 2003 生物膜泡曲面弹性理论 (上海: 上海科学技术出版社)]
[36] Du Q, Liu C, Wang X 2006 J. Comput. Phys. 212 757
[37] Du Q, Liu C, Ryham R, Wang X 2005 Commun. Pure Appl. Anal. 4 537
[38] Yan J, Liu Q H, Liu J X, Ouyang Z C 1998 Phys. Rev. E 58 4730
[39] Su Y C, Chen J Z 2015 Soft Matter 11 4054
[40] Zhang P W, Shi A C 2015 Chin. Phys. B 24 128707
[41] Zhou X H, Zhang S G 2006 Acta Phys. Sin. 55 5568 (in Chinese) [周晓华, 张劭光 2006 55 5568]
[42] Zhou X H, Zhang S G, Yang J Q, Qu X M, Liu Y S, Wang S G 2007 Acta Phys. Sin. 56 6137 (in Chinese) [周晓华, 张劭光, 杨继庆, 屈学民, 刘渊声, 王斯刚 2007 56 6137]
[43] Xia B K, Li J F, Li W H, Zhang H D, Qiu F 2013 Acta Phys. Sin. 62 248701 (in Chinese) [夏彬凯, 李剑锋, 李卫华, 张红东, 邱枫 2013 62 248701]
[44] Yang K, Ma Y Q 2010 Nat. Nanotechnol. 5 579
[45] Ding H M, Ma Y Q 2012 Biomaterials 33 5798
[46] Ding H M, Ma Y Q 2015 Small 11 1055
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