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Due to the discovery and study of opening-up lipid vesicles, the theoretical analysis and numerical calculation have aroused increasing interests of researchers. In the previous study, Suezaki and Umeda gave the opening-up vesicles near the spherical vesicles, such as the dish and cup shapes with one hole, and the tube and funnel shapes with two holes. These shapes are found at relatively low values of reduced, relaxed area difference a0. However, what are the stable shapes for high values of a0 is not known. Kang et al. found solutions of opening up dumbbell shapes with one hole. Whether or not there exist dumbbell shapes with two holes, and the phase transformation behavior between them remains unknown. The purpose of this paper is to explore a new kind of two-hole dumbbell shaped lipid vesicles and phase transformations between this kind of vesicle and previously found vesicles. Based on the area-difference-elasticity model, this paper tries to explore new solutions of the Euler-Lagrange equations of the opening-up membrane vesicles which meet the boundary conditions by using the relaxation method. A new branch of solution of dumbbell shapes with two holes is found. The phase transformations of closed dumbbell shapes and opening-up dumbbell shapes with one hole and two holes are studied in detail. To explore whether these shapes could be found in experiments, the energy of the cup, tube, and funnel shaped vesicles are also compared with the opening-up dumbbell shapes. It is found that at high values of a0, all the cup, tube, and funnel shapes will transform into closed spherical vesicles. So the energy of new opening-up dumbbell vesicles can be compared to that of closed spherical vesicles and closed dumbbell vesicles. It is found that the dumbbell shapes with one hole and two holes all have stable regions, implying that it is possible for these open dumbbells to be observed. Since the distance in the functional space is too far between the open dumbbell shapes and spherical vesicles, experimental test is needed to verify whether the dumbbell shapes with two holes will evolve continuously to the closed dumbbell shapes or to the closed spherical vesicles. It has been noticed that for relatively small values of a0, two holes vesicles may exhibit symmetrical tube shapes and asymmetric funnel shapes between which the phase transformation is continuous, because the funnel solutions bifurcate from the tube solutions. In order to check whether there exist asymmetric opening-up dumbbell shapes with two holes and the similar bifurcation behavior, a thorough search is made in the parameter space. So far no asymmetric dumbbell shape with two holes is found.
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Keywords:
- area-difference-elasticity model /
- relaxation method /
- opening-up vesicles /
- numerical calculations
[1] Lieber M R, Steck T L 1982 J. Biol. Chem. 257 11651
[2] Lieber M R, Steck T L 1982 J. Biol. Chem. 257 11660
[3] Saitoh A, Takiguchi K, Tanaka Y Hotani H 1998 Proc. Natl. Acad. Sci. USA 95 1026
[4] de Gennes P G, Tauppin C 1982 J. Phys. Chem. 86 2294
[5] Bar-Ziv R, Frisch T, Moses E 1995 Phys. Rev. Lett. 75 3481
[6] Zhelev D V, Needham D 1993 Biochim. Biophys. Acta 1147 89
[7] Capovilla R, Guven J, Santiago J A 2002 Phys. Rev. E 66 021607
[8] Tu Z C, Ouyang Z C 2003 Phys. Rev. E 68 061915
[9] Li S L, Zhang S G 2010 Acta Phys. Sin. 59 5202 (in Chinese) [李树玲, 张劭光 2010 物理学报 59 5202]
[10] Umeda T, Suezaki Y 2005 Phys. Rev. E 71 011913
[11] Kang W B, Zhang S G, Wang Y, Mu Y R, Huang C 2011 Sci. China: Phys. Mech. Astron. 54 2243
[12] Huang C, Zhang S G 2013 J. Shaanxi Normal Univ. (Natural Science Edition) 41 0031 (in Chinese) [黄聪, 张劭光 2013 陕西师范大学学报 (自然科学版) 41 0031]
[13] Helfrich W 1973 Z. Naturforsch. C 28 693
[14] Miao L, Seifert U, Wortis M, Dobereinert H G 1994 Phys. Rev. E 49 5389
[15] Ouyang Z C, Helfrich W 1989 Phys. Rev. A 39 5280
[16] Tu Z C 2010 J. Chem. Phys. 132 084111
[17] Press W H, Teukolsky S A, Vetterling S A, Flannery B P 1996 Numerical Recipes in Fortran (Second Edition) (U.K.: Cambridge University Press) pp1316-1320
[18] He G Y, Gao Y L 2002 Visual Fortran Commonly Used Numerical Algorithms (First Edition) (Beijing: Science Press) pp657-678 (in Chinese) [何光渝, 高永利 2002 Visual Fortran 常用数值算法 (第一版) (北京: 科学出版社) 第 657-678 页]
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[1] Lieber M R, Steck T L 1982 J. Biol. Chem. 257 11651
[2] Lieber M R, Steck T L 1982 J. Biol. Chem. 257 11660
[3] Saitoh A, Takiguchi K, Tanaka Y Hotani H 1998 Proc. Natl. Acad. Sci. USA 95 1026
[4] de Gennes P G, Tauppin C 1982 J. Phys. Chem. 86 2294
[5] Bar-Ziv R, Frisch T, Moses E 1995 Phys. Rev. Lett. 75 3481
[6] Zhelev D V, Needham D 1993 Biochim. Biophys. Acta 1147 89
[7] Capovilla R, Guven J, Santiago J A 2002 Phys. Rev. E 66 021607
[8] Tu Z C, Ouyang Z C 2003 Phys. Rev. E 68 061915
[9] Li S L, Zhang S G 2010 Acta Phys. Sin. 59 5202 (in Chinese) [李树玲, 张劭光 2010 物理学报 59 5202]
[10] Umeda T, Suezaki Y 2005 Phys. Rev. E 71 011913
[11] Kang W B, Zhang S G, Wang Y, Mu Y R, Huang C 2011 Sci. China: Phys. Mech. Astron. 54 2243
[12] Huang C, Zhang S G 2013 J. Shaanxi Normal Univ. (Natural Science Edition) 41 0031 (in Chinese) [黄聪, 张劭光 2013 陕西师范大学学报 (自然科学版) 41 0031]
[13] Helfrich W 1973 Z. Naturforsch. C 28 693
[14] Miao L, Seifert U, Wortis M, Dobereinert H G 1994 Phys. Rev. E 49 5389
[15] Ouyang Z C, Helfrich W 1989 Phys. Rev. A 39 5280
[16] Tu Z C 2010 J. Chem. Phys. 132 084111
[17] Press W H, Teukolsky S A, Vetterling S A, Flannery B P 1996 Numerical Recipes in Fortran (Second Edition) (U.K.: Cambridge University Press) pp1316-1320
[18] He G Y, Gao Y L 2002 Visual Fortran Commonly Used Numerical Algorithms (First Edition) (Beijing: Science Press) pp657-678 (in Chinese) [何光渝, 高永利 2002 Visual Fortran 常用数值算法 (第一版) (北京: 科学出版社) 第 657-678 页]
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