SC模型下新的开口膜泡形状的探究
|
摘要
类脂分子是一种双亲分子,它由一个亲水的极性头部和一条或两条疏水的非极性尾部(烃链)组成,类脂分子在水溶液中倾向于形成头部向外、烃链向内的双分子层,类脂双层在适当浓度会形成闭合的泡。实验上观测到形状丰富的闭合膜泡,我们把这种膜泡看作是在特定条件下的一个平衡态。
Canham首先提出膜的曲率弹性模型,后来Helfrich提出了自发曲率(SC)模型。考虑到膜的双层结构,人们又提出了双层耦合(BC)模型和面积差弹性(ADE)模型。这些模型的要点在于认为膜泡的构型并不是由其表面张力决定,而是由其表面弯曲能决定的。基于三个曲率模型,人们对闭合膜泡的形状已进行了大量的计算。
以往人们认为只有闭合的膜泡是稳定的,然而近期的研究发现一些有机化学试剂如Protein Talin(一种蛋白质)能够在类脂膜上打开稳定的孔洞。1998年,日本的A.Saitoh等人首次使用一定浓度的Talin分子将闭合膜泡打开,并且开口的大小随着Talin浓度而变,这一过程在一定条件下还是可逆的。
在理论上,墨西哥的R.Capovilla等人首先给出了在一定线张力下,开口膜泡的平衡方程及边界条件。涂展春和欧阳将外微分法用于处理曲面上的变分问题,导出了同样的结果。对开口膜泡,其欧拉-拉格朗日方程与闭合膜泡一样,但在边界上要加上两个独立的边界条件,其数值解法也要作相应的改变。
由于膜泡的形状方程为高阶非线性偏微分方程,目前只知道有限的特解。为了与实验结果比较,只能采用数值计算的方法。对膜泡形状的数值计算,分两种情况:
(1)对旋转对称性的形状,该方程可化为常微分方程进行数值求解。
(2)对非旋转对称性形状,因没有求解该偏微分方程的一般数值方法,通常采用直接极小化法。目前最常采用的的是在Evolver下编程计算。
对开口膜泡,目前还没有直接极小化的结果发表。对旋转对称性膜泡,T.Umeda等人在SC模型及ADE模型下数值计算了杯形及管型的平衡形状。相比于闭合膜泡的丰富形状,目前对开口膜泡形状的探索还很有限。
本文研究SC模型下闭合膜泡和开口膜泡的形状及其演化。旨在提供一种数值解闭合膜泡和开口膜泡的方法,结合Mathematica软件,对轴对称的膜泡形状我们得到的主要结果如下:
(1)通过与少数已知的解析结果如球形的对比,我们深入讨论了计算过程的误差问题;
(2)通过对计算方法的不断改进,找到了求解闭合膜泡的方法,定义为二维法,得到了丰富的闭合膜泡,长椭,扁椭,三角形,梨形,胃型等;
(3)理论推导出n-budding闭合形状对应的约化自发曲率c_0=2 n~(1/2):在此基础上得出n-budding开口膜泡可存在于c_0＜2 n~(1/2)处。理论推导出开口膜泡的三个边界之间的关系,得到只有两个是独立的;
(4)利用二维法,计算了开口膜泡形状并对开口膜泡的标度不变性进行验证;
(5)建立求解开口膜泡的三维法。对c_0=1时采用我们的方法对杯形开口膜泡形状及能量进行计算,与Y.Suezaki和T.Umeda的结果一致;
(6)运用三维法在c_0=2.3时,得到了在2-budding开口的新形状,给出了膜泡的形状以及能量与线张力系数的关系;
(7)运用三维法在c_0=3及c_0=3.5时,得到了3-budding开口形状。
Lipid molecule is a parent molecule, which consists of a hydrophilic polar head and one or two of the non-polar hydrophobic tail (hydrocarbon chain) , composed of lipid molecules in aqueous solution tend to form outside the head, hydrocarbon chains inside the bilayer, the lipid bilayer will be formed at an appropriate concentration of the foam closure. Experimentally observed shape of the closed-rich membrane vesicles, we have this film as a bubble under certain conditions an equilibrium state.
Canham first put forward membrane curvature elastic model, and later Helfrich made spontaneous curvature (SC) model. Taking into account the double-membrane structure, it also raised the double-coupled (BC) model and the area difference elasticity (ADE) model. The main points of these models is that the structure of membrane vesicles is not determined by their surface tension, but its curved surface. Based on three curvature models, the shape of the bubble membrane closure has been a great deal of calculation.
People in the past believed that only the closure of the membrane vesicles is stable, but recent studies have found a number of organic reagents such as Protein Talin (a protein) can open the lipid membrane of the holes which can be stable. In 1998, such as Japan's A.Saitoh first used a certain concentration of the elements Talin to open the closed membrane vesicles, and the size of the opening varies with the concentration of Talin, a process that under certain conditions is reversible.
In theory, R. Capovilla from Mexico's first gave the opening film of the bubble equilibrium equations and boundary conditions in a certain line tension. Mr Tu zhan -chun and Ou-Yang used differentiation to deal with outside surface of the variational problem, derived the same result. Membrane vesicles of openings, the Euler- Lagrange equation is the same to the closed membrane vesicles, but the border would like to add two independent boundary conditions, so the numerical method should make the corresponding changes.
Due to the shape of the bubble membrane equations is higher-order nonlinear partial differential equations, people are only aware of the limited special solution. In order to compare with experimental results we must use the numerical calculation. Vesicle shape on the numerical calculation, points to two situations:
(1) the shape of rotational symmetry, the equation can be translated into the numerical solution of ordinary differential equations.
(2) non-rotational symmetry of shape, because there is no solution of partial differential equations of the general numerical method, commonly used method of direct minimization. At present, the most commonly used is calculated under the Evolver.
Membrane vesicles of openings, there is no direct minimization of the results. Rotational symmetry of the vesicle, T. Umeda and others in the SC model and the ADE model has made numerical calculation of the cup and the shape of the balance tube. compared to the wealth of closed membrane vesicles, the current to explore the shape of the opening-up vesicles is still very limited.
In this paper, under the SC model we study the closed and openings membrane vesicles and their evolution. Numerical solution is designed to solue a closed-membrane vesicles and vesicle openings approach, combined with Mathematica software, for axisymmetric bubble shape of the membrane we get the main results are as follows:
(1)With a small number of known analytical comparison of the results, such as spherical, we have discussed in depth the issue of calculation errors;
(2)Through continuous improvement methodology, we find the solution method of closed membrane vesicles, defined as two-dimensional method, the closure membrane vesicles has been enriched, oblate,prolate,sanye, pear-shaped, stomach, etc;
(3) theory derived the corresponding spontaneous curvature c_0 = 2n~(1/2) from the shape of n-budding closed vesicles . On this basis of n-budding vesicle openings may exist in the department of c_0 < 2n~(1/2) .Theory derived the relationship of the opening three border membrane vesicles and received only two are independent;
(4) Use the two-dimensional method for calculating the shape of the openings and validate its scale invariance;
(5) Finding the method of three-dimensional method vesicle for the openings. On the method used in ours,we give the cup-shaped openings on the vesicle shape and energy calculation, which is the same to T. Umeda and Y. Suezaki's results;
(6) When use the three-dimensional method in c_0 = 2.3, the 2-budding in the opening of the new shape is obtained and given the shape of the bubble membrane as well as the energy and the relationship between line tension coefficient;
(7)When use the three-dimensional method in c_0 = 3 and c_0 =3.5 we have the shape of 3-budding openings.
Canham首先提出膜的曲率弹性模型,后来Helfrich提出了自发曲率(SC)模型。考虑到膜的双层结构,人们又提出了双层耦合(BC)模型和面积差弹性(ADE)模型。这些模型的要点在于认为膜泡的构型并不是由其表面张力决定,而是由其表面弯曲能决定的。基于三个曲率模型,人们对闭合膜泡的形状已进行了大量的计算。
以往人们认为只有闭合的膜泡是稳定的,然而近期的研究发现一些有机化学试剂如Protein Talin(一种蛋白质)能够在类脂膜上打开稳定的孔洞。1998年,日本的A.Saitoh等人首次使用一定浓度的Talin分子将闭合膜泡打开,并且开口的大小随着Talin浓度而变,这一过程在一定条件下还是可逆的。
在理论上,墨西哥的R.Capovilla等人首先给出了在一定线张力下,开口膜泡的平衡方程及边界条件。涂展春和欧阳将外微分法用于处理曲面上的变分问题,导出了同样的结果。对开口膜泡,其欧拉-拉格朗日方程与闭合膜泡一样,但在边界上要加上两个独立的边界条件,其数值解法也要作相应的改变。
由于膜泡的形状方程为高阶非线性偏微分方程,目前只知道有限的特解。为了与实验结果比较,只能采用数值计算的方法。对膜泡形状的数值计算,分两种情况:
(1)对旋转对称性的形状,该方程可化为常微分方程进行数值求解。
(2)对非旋转对称性形状,因没有求解该偏微分方程的一般数值方法,通常采用直接极小化法。目前最常采用的的是在Evolver下编程计算。
对开口膜泡,目前还没有直接极小化的结果发表。对旋转对称性膜泡,T.Umeda等人在SC模型及ADE模型下数值计算了杯形及管型的平衡形状。相比于闭合膜泡的丰富形状,目前对开口膜泡形状的探索还很有限。
本文研究SC模型下闭合膜泡和开口膜泡的形状及其演化。旨在提供一种数值解闭合膜泡和开口膜泡的方法,结合Mathematica软件,对轴对称的膜泡形状我们得到的主要结果如下:
(1)通过与少数已知的解析结果如球形的对比,我们深入讨论了计算过程的误差问题;
(2)通过对计算方法的不断改进,找到了求解闭合膜泡的方法,定义为二维法,得到了丰富的闭合膜泡,长椭,扁椭,三角形,梨形,胃型等;
(3)理论推导出n-budding闭合形状对应的约化自发曲率c_0=2 n~(1/2):在此基础上得出n-budding开口膜泡可存在于c_0＜2 n~(1/2)处。理论推导出开口膜泡的三个边界之间的关系,得到只有两个是独立的;
(4)利用二维法,计算了开口膜泡形状并对开口膜泡的标度不变性进行验证;
(5)建立求解开口膜泡的三维法。对c_0=1时采用我们的方法对杯形开口膜泡形状及能量进行计算,与Y.Suezaki和T.Umeda的结果一致;
(6)运用三维法在c_0=2.3时,得到了在2-budding开口的新形状,给出了膜泡的形状以及能量与线张力系数的关系;
(7)运用三维法在c_0=3及c_0=3.5时,得到了3-budding开口形状。
Lipid molecule is a parent molecule, which consists of a hydrophilic polar head and one or two of the non-polar hydrophobic tail (hydrocarbon chain) , composed of lipid molecules in aqueous solution tend to form outside the head, hydrocarbon chains inside the bilayer, the lipid bilayer will be formed at an appropriate concentration of the foam closure. Experimentally observed shape of the closed-rich membrane vesicles, we have this film as a bubble under certain conditions an equilibrium state.
Canham first put forward membrane curvature elastic model, and later Helfrich made spontaneous curvature (SC) model. Taking into account the double-membrane structure, it also raised the double-coupled (BC) model and the area difference elasticity (ADE) model. The main points of these models is that the structure of membrane vesicles is not determined by their surface tension, but its curved surface. Based on three curvature models, the shape of the bubble membrane closure has been a great deal of calculation.
People in the past believed that only the closure of the membrane vesicles is stable, but recent studies have found a number of organic reagents such as Protein Talin (a protein) can open the lipid membrane of the holes which can be stable. In 1998, such as Japan's A.Saitoh first used a certain concentration of the elements Talin to open the closed membrane vesicles, and the size of the opening varies with the concentration of Talin, a process that under certain conditions is reversible.
In theory, R. Capovilla from Mexico's first gave the opening film of the bubble equilibrium equations and boundary conditions in a certain line tension. Mr Tu zhan -chun and Ou-Yang used differentiation to deal with outside surface of the variational problem, derived the same result. Membrane vesicles of openings, the Euler- Lagrange equation is the same to the closed membrane vesicles, but the border would like to add two independent boundary conditions, so the numerical method should make the corresponding changes.
Due to the shape of the bubble membrane equations is higher-order nonlinear partial differential equations, people are only aware of the limited special solution. In order to compare with experimental results we must use the numerical calculation. Vesicle shape on the numerical calculation, points to two situations:
(1) the shape of rotational symmetry, the equation can be translated into the numerical solution of ordinary differential equations.
(2) non-rotational symmetry of shape, because there is no solution of partial differential equations of the general numerical method, commonly used method of direct minimization. At present, the most commonly used is calculated under the Evolver.
Membrane vesicles of openings, there is no direct minimization of the results. Rotational symmetry of the vesicle, T. Umeda and others in the SC model and the ADE model has made numerical calculation of the cup and the shape of the balance tube. compared to the wealth of closed membrane vesicles, the current to explore the shape of the opening-up vesicles is still very limited.
In this paper, under the SC model we study the closed and openings membrane vesicles and their evolution. Numerical solution is designed to solue a closed-membrane vesicles and vesicle openings approach, combined with Mathematica software, for axisymmetric bubble shape of the membrane we get the main results are as follows:
(1)With a small number of known analytical comparison of the results, such as spherical, we have discussed in depth the issue of calculation errors;
(2)Through continuous improvement methodology, we find the solution method of closed membrane vesicles, defined as two-dimensional method, the closure membrane vesicles has been enriched, oblate,prolate,sanye, pear-shaped, stomach, etc;
(3) theory derived the corresponding spontaneous curvature c_0 = 2n~(1/2) from the shape of n-budding closed vesicles . On this basis of n-budding vesicle openings may exist in the department of c_0 < 2n~(1/2) .Theory derived the relationship of the opening three border membrane vesicles and received only two are independent;
(4) Use the two-dimensional method for calculating the shape of the openings and validate its scale invariance;
(5) Finding the method of three-dimensional method vesicle for the openings. On the method used in ours,we give the cup-shaped openings on the vesicle shape and energy calculation, which is the same to T. Umeda and Y. Suezaki's results;
(6) When use the three-dimensional method in c_0 = 2.3, the 2-budding in the opening of the new shape is obtained and given the shape of the bubble membrane as well as the energy and the relationship between line tension coefficient;
(7)When use the three-dimensional method in c_0 = 3 and c_0 =3.5 we have the shape of 3-budding openings.
引文
[1]S.J.Singer,G.L.Nicolson.The fluid mosaic model of the structure of cell membranes[J].Science,1972,175:720-731.
[2]彼得J 柯林斯.液晶:自然界的奇妙物相[M].阮丽真 译.上海:科技教育出版社,2002:232.
[3]杨福愉.生物膜[M].北京:科学出版社,2005:405-406.
[4]谢毓章,刘寄星,欧阳钟灿生物膜泡曲面弹性理论[M].上海:上海科学技术出版社,2003:12.
[5]Berndl K,Lipowsky R,Sackmann E,Seifert U.Shape transformations of giant vesicles:extreme sensitivity to bilayer asymmetry[J].Europhysics Letters,1990,13:659-664.
[6]Canham P The minimum energy of bending as a possible explanation of the Biconca-ve shap ofthe human red blood cell[J].Journal of Theoretical Biology,1970,26:61-81.
[7]Helfrich W.Elastic properties of lipid bilayers:theory and possible experiments[J].Z.Naturforsch,1973,28c:693-703.
[8]Deuling H J,Helfrich W.The curvature elasticity of fluid membranes:A catalogue of vesicle shapes.J Phys(Paris) 1976,37:1335-1345
[9]Miao L,Seifert U,Wortis M,et al.Budding transitions of Fluid bilayer vesicles:The effect of area-difference elasticity[J].Physical Review E,1994,49:5389-5407
[10]Svetina S,Zeks B.Membrane bending energy and shap determination of phosphor-lipid vesicles and red blood cells[J].European Biophysics Journal,1989,17:101-111.
[11]Seifert U,Bemdl K,Lipowsky R.Shepe transformations of vesicles:Phase diagram for spontaneous-curvature and bilayer-coupling models[J].Physical Review A,1991,44:1182-1202.
[12]谢毓章,刘寄星,欧阳钟灿.生物膜泡曲面弹性理论[M].上海:上海科学技术出版社,2003.
[13]Y.Suezaki,T.Umeda.The Opening-up of lipid bilayer vesicles bu guest molecules:The adsorption isotherm and numerical calculations[J].Journal of Optoelectronics and Advanced Materials,2005,Vol 7,No 1:25-34.
[14]H.Naito,M.Okuda,and Z.C.Ou-Yang.Counterexample to some shape equations for axisymmetric vesicles[J].Phys.Rev.E,1993,48:2304-2307.
[15]Z.C.Ou-Yang.Anchor ring-vesicle membranes[J].Phys.Rev.A,1990,41:4517-4520.
[16]Frank F C.On the theory of liquid crystals.Discuss Faraday Soc,1958,25:19-28
[17]W.Helfrich.Elastic properties of lipid bilayers:theory and possible experiments [J].Z Naturforsch.1973,28c:693-703.
[18]Ou-Yang Zhong-can and W.Helfrich.Instability and Deformation of a Spherical Vesicle by Pressure[J].Phys.Rev.Lett.,1987,59:2486-2488
[19]S.G.Zhang and Z.C.Ou-Yang.Periodic cylindrical surface solution for fluid bilayer membranes[J].Phys.Rev.E,1996,53:4206-4208.
[20]H.Naito,M.Okuda,and Z.C.Ou-Yang.New Solutions to the Helfrich Variation Problem for the Shapes of Lipid Bilayer Vesicles:Beyond Delaunay's Surfaces[J].Phys.Rev.Lett.,1995,74:4345-4348.
[21]Svetina S,Zeks B.Membrane bending energy and shape determination of phospholipid vesicles and red blood cells[J].,Eur Biophys J.,1989,17(2):101-111.
[22]Seifert U,Bemdl K,Lipowsky R.Shape transformations of vesicles:Phase diagram for spontaneous- curvature and bilayer-coupling models.Phys.Rev.A,1991,44(2):1182-1202.
[23]Jaric M,Seifert U,Wintz W,et al.Vesicular instabilities:The prolate-to-oblate transition and other shap instabilities of fluid bilayer membranes[J].Physical Review E,1995,52,(6):66236-634.
[24]周建军.闭合与开口生物膜泡的研究[D].北京:中科院理论研究所,2002.
[25]涂展春.生物膜弹性的几何理论及碳纳米管的若干物理性质[D].中国科学院2004,1-85
[26]Tamiki Umeda,Yukio Suezaki,et al.Theoretical analysis of opening-up vesicles with single and two holes[J].Physical Review E,2005,71,011913.
[27]J(u|¨)licher F,Seifert U.Shape equations for axisymmetricvesicles:Aclarification[J].Phys.Rew.E,1994,49(5):4728-4731.
[28]胡开福.用“打靶法”对轴对称膜泡形状的研究[D].西安:陕西师范大学,2008.
[29]Naito H,Okuda M,Ou-Yang Z C.Polygonal shape transformation of a circular biconcave vesicle induced by osmotic pressure[J].Physical Review E,1996,54(6):2816-2826.
[30]Y.Suezaki, Tamiki Umeda.The opening-up of lipid bilayer vesciles by guest molecules:The adsorption isotherm and numericsl cslculations[J].Journal of Optoelectronics and Advanced Material,2005,Vol.7,N0.1:25-34.
[2]彼得J 柯林斯.液晶:自然界的奇妙物相[M].阮丽真 译.上海:科技教育出版社,2002:232.
[3]杨福愉.生物膜[M].北京:科学出版社,2005:405-406.
[4]谢毓章,刘寄星,欧阳钟灿生物膜泡曲面弹性理论[M].上海:上海科学技术出版社,2003:12.
[5]Berndl K,Lipowsky R,Sackmann E,Seifert U.Shape transformations of giant vesicles:extreme sensitivity to bilayer asymmetry[J].Europhysics Letters,1990,13:659-664.
[6]Canham P The minimum energy of bending as a possible explanation of the Biconca-ve shap ofthe human red blood cell[J].Journal of Theoretical Biology,1970,26:61-81.
[7]Helfrich W.Elastic properties of lipid bilayers:theory and possible experiments[J].Z.Naturforsch,1973,28c:693-703.
[8]Deuling H J,Helfrich W.The curvature elasticity of fluid membranes:A catalogue of vesicle shapes.J Phys(Paris) 1976,37:1335-1345
[9]Miao L,Seifert U,Wortis M,et al.Budding transitions of Fluid bilayer vesicles:The effect of area-difference elasticity[J].Physical Review E,1994,49:5389-5407
[10]Svetina S,Zeks B.Membrane bending energy and shap determination of phosphor-lipid vesicles and red blood cells[J].European Biophysics Journal,1989,17:101-111.
[11]Seifert U,Bemdl K,Lipowsky R.Shepe transformations of vesicles:Phase diagram for spontaneous-curvature and bilayer-coupling models[J].Physical Review A,1991,44:1182-1202.
[12]谢毓章,刘寄星,欧阳钟灿.生物膜泡曲面弹性理论[M].上海:上海科学技术出版社,2003.
[13]Y.Suezaki,T.Umeda.The Opening-up of lipid bilayer vesicles bu guest molecules:The adsorption isotherm and numerical calculations[J].Journal of Optoelectronics and Advanced Materials,2005,Vol 7,No 1:25-34.
[14]H.Naito,M.Okuda,and Z.C.Ou-Yang.Counterexample to some shape equations for axisymmetric vesicles[J].Phys.Rev.E,1993,48:2304-2307.
[15]Z.C.Ou-Yang.Anchor ring-vesicle membranes[J].Phys.Rev.A,1990,41:4517-4520.
[16]Frank F C.On the theory of liquid crystals.Discuss Faraday Soc,1958,25:19-28
[17]W.Helfrich.Elastic properties of lipid bilayers:theory and possible experiments [J].Z Naturforsch.1973,28c:693-703.
[18]Ou-Yang Zhong-can and W.Helfrich.Instability and Deformation of a Spherical Vesicle by Pressure[J].Phys.Rev.Lett.,1987,59:2486-2488
[19]S.G.Zhang and Z.C.Ou-Yang.Periodic cylindrical surface solution for fluid bilayer membranes[J].Phys.Rev.E,1996,53:4206-4208.
[20]H.Naito,M.Okuda,and Z.C.Ou-Yang.New Solutions to the Helfrich Variation Problem for the Shapes of Lipid Bilayer Vesicles:Beyond Delaunay's Surfaces[J].Phys.Rev.Lett.,1995,74:4345-4348.
[21]Svetina S,Zeks B.Membrane bending energy and shape determination of phospholipid vesicles and red blood cells[J].,Eur Biophys J.,1989,17(2):101-111.
[22]Seifert U,Bemdl K,Lipowsky R.Shape transformations of vesicles:Phase diagram for spontaneous- curvature and bilayer-coupling models.Phys.Rev.A,1991,44(2):1182-1202.
[23]Jaric M,Seifert U,Wintz W,et al.Vesicular instabilities:The prolate-to-oblate transition and other shap instabilities of fluid bilayer membranes[J].Physical Review E,1995,52,(6):66236-634.
[24]周建军.闭合与开口生物膜泡的研究[D].北京:中科院理论研究所,2002.
[25]涂展春.生物膜弹性的几何理论及碳纳米管的若干物理性质[D].中国科学院2004,1-85
[26]Tamiki Umeda,Yukio Suezaki,et al.Theoretical analysis of opening-up vesicles with single and two holes[J].Physical Review E,2005,71,011913.
[27]J(u|¨)licher F,Seifert U.Shape equations for axisymmetricvesicles:Aclarification[J].Phys.Rew.E,1994,49(5):4728-4731.
[28]胡开福.用“打靶法”对轴对称膜泡形状的研究[D].西安:陕西师范大学,2008.
[29]Naito H,Okuda M,Ou-Yang Z C.Polygonal shape transformation of a circular biconcave vesicle induced by osmotic pressure[J].Physical Review E,1996,54(6):2816-2826.
[30]Y.Suezaki, Tamiki Umeda.The opening-up of lipid bilayer vesciles by guest molecules:The adsorption isotherm and numericsl cslculations[J].Journal of Optoelectronics and Advanced Material,2005,Vol.7,N0.1:25-34.