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The numerical and the analytical studies of the opening-up vesicles have become hot topics since the experimental observation by A. Saitoh et al. This paper deals with how to obtain the analytical solution of an opening-up shape from the Ouyang biconcave analytical solution. We find only two of the three boundary conditions for the vesicle rims to be independent. The second boundary condition can be satisfied by the Gaussian bending modulus kg=-2, then we obtain the geometric equation for the rims of an opening-up vesicle from the first boundary condition. By analyzing the Ouyang analytic solutions for a closed circular biconcave vesicle and periodic noduloidlike vesicle, we obtain three kinds of shapes with tube topology, which are the convex tube, the toruslike tube, and the catenoidlike tube.
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Keywords:
- opening-up vesicles /
- boundary conditions /
- analytical solution
[1] Bassis M 1973 Living Blood Cells and Their Ultrastructure (Berlin: Springer) p10
[2] Saitoh A, Takiguchi K, Tanaka Y, Hotani H 1998 Proc. Natl. Acad. Sci. USA 95 1026
[3] Lieber M R, Steck T L 1982 J. Biol. Chem. 257 11651
[4] Lieber M R, Steck T L 1982 J. Biol. Chem. 257 11660
[5] Capovilla R, Guven J, Santiago J A 2002 Phys. Rev. E 66 021607
[6] Tu Z C, Ouyang Z C 2003 Phys. Rev. E 68 061915
[7] Helfrich W 1973 Z. Naturforsch. C 28 693
[8] Umeda T, Suezaki Y 2005 Phys. Rev. E 71 011913
[9] Naito H, Okuda M, Ouyang Z C 1993 Phys. Rev. E 48 2304
[10] Naito H, Okuda M, Ouyang Z C 1995 Phys. Rev. Lett. 74 4345
[11] Liu Q H, Zhou H J, Liu J X, Ouyang Z C 1999 Phys. Rev. E 60 3227
[12] Ouyang Z C, Helfrich W 1987 Phys. Rev. Lett. 59 2486
[13] Ouyang Z C, Helfrich W 1988 Phys. Rev. A 39 5280
[14] Hu J G, Ouyang Z C 1993 Phys. Rev. E 47 461
[15] Seifert U, Berndl K, Lipowsky R 1991 Phys. Rev. A 44 1182
[16] Miao L, Seifert U, Wortis M, Dbereiner H G 1994 Phys. Rev. E 49 5389
[17] Zhou X H, Zhang S G 2006 Acta Phys. Sin. 55 5568 (in Chinese) [周晓华、张劭光 2006 物理学报 55 5568]
[18] 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 物理学报 56 6137]
[19] Helfrich W 1974 Phys. Lett. A 50 115
[20] Lipowsky R 1993 Biophys. J. 64 1133
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[1] Bassis M 1973 Living Blood Cells and Their Ultrastructure (Berlin: Springer) p10
[2] Saitoh A, Takiguchi K, Tanaka Y, Hotani H 1998 Proc. Natl. Acad. Sci. USA 95 1026
[3] Lieber M R, Steck T L 1982 J. Biol. Chem. 257 11651
[4] Lieber M R, Steck T L 1982 J. Biol. Chem. 257 11660
[5] Capovilla R, Guven J, Santiago J A 2002 Phys. Rev. E 66 021607
[6] Tu Z C, Ouyang Z C 2003 Phys. Rev. E 68 061915
[7] Helfrich W 1973 Z. Naturforsch. C 28 693
[8] Umeda T, Suezaki Y 2005 Phys. Rev. E 71 011913
[9] Naito H, Okuda M, Ouyang Z C 1993 Phys. Rev. E 48 2304
[10] Naito H, Okuda M, Ouyang Z C 1995 Phys. Rev. Lett. 74 4345
[11] Liu Q H, Zhou H J, Liu J X, Ouyang Z C 1999 Phys. Rev. E 60 3227
[12] Ouyang Z C, Helfrich W 1987 Phys. Rev. Lett. 59 2486
[13] Ouyang Z C, Helfrich W 1988 Phys. Rev. A 39 5280
[14] Hu J G, Ouyang Z C 1993 Phys. Rev. E 47 461
[15] Seifert U, Berndl K, Lipowsky R 1991 Phys. Rev. A 44 1182
[16] Miao L, Seifert U, Wortis M, Dbereiner H G 1994 Phys. Rev. E 49 5389
[17] Zhou X H, Zhang S G 2006 Acta Phys. Sin. 55 5568 (in Chinese) [周晓华、张劭光 2006 物理学报 55 5568]
[18] 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 物理学报 56 6137]
[19] Helfrich W 1974 Phys. Lett. A 50 115
[20] Lipowsky R 1993 Biophys. J. 64 1133
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