球形拓扑下对budding和三凹形膜泡形状及其稳定性的研究
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摘要
由一个亲水的极性头部和两条疏水的非极性尾部组成的两亲分子在水溶液中可以自组织的形成双层,该双层在低浓度下由于疏水相互作用会形成闭合的膜泡。作为生物膜的一个简化模型,对类脂膜泡形状的研究将为我们理解复杂生物系统的行为提供有价值的信息。
为了解释膜泡的形状,德国液晶物理学家W.Helfrich根据向列相液晶与类脂膜的相似性,于1973年最先提出了膜的自发曲率(SC)弹性理论,并由此证明了膜的双凹盘形的存在;同时,考虑到膜泡双层间的面积差对膜泡能量的影响,1989年S.Svetian等人又提出了面积差(BC)模型,以后U.Seifert及苗玲等人又提出了面积差弹性(ADE)模型。
经过许多人多年的努力,这三种模型的相图在有些区域已经了解的很清楚了,但对有些区域还不了解。如在BC模型相图的左下角,胃形泡将转化到一个球形母泡和一个反转子球形用无限小细颈相连的极限情况佣L~(sto)表示),在该区域之外,稳定膜泡的形状是什么并不清楚。但在实验上,S.Svetina等人已经发现了很多有趣的形状,我们称之为内陷n-budding形状,它们并不能归到相图的已知区域。对此形状目前的理论研究是把它近似处理成用无穷小细颈连接起来的一个大的理想母泡球形和n个小的反转理想球形的简单情况,给出了约化体积等于0.85情况下的一些budding形状。在一般情况下这些膜泡的实际形状如何,这些形状是否稳定,它们在相图中处于什么区域,这些问题都是有待深入研究的。
病态红血球展现出的三凹形状引起了很多人的兴趣。Gerald Lim等人考虑了红血球的蛋白质骨架结构,用分子动力学模拟方法给出了三凹及其它一些红血球的反常结构。能否在只考虑膜泡的曲率能的情况下给出稳定的三凹形状成为人们关注的一个问题。
本文对内陷n-budding膜泡及红血球展现出的三凹形状进行了深入研究,我们得到的主要结果如下:
(1)对SC和BC模型的已有相图进行深入研究的基础上,对内陷n-budding形膜泡存在区间进行了探索,我们发现该形状位于曲线L~(sto)之下,这些区域在SC模型和BC模型两个相图中都是未知的。
(2)在实际计算中为了解决细颈部分所带来的困难,我们采用了两种计算能量的方法,stn及stp。stn可以保持颈部不至于断掉,而stp的收敛速度较快,并且对形状的稳定性进行分析时更为有效。我们计算方法的另一个特点是充分利用了BC模型和SC模型的关系及各自在计算上的特点。通过大量的计算,我们发现在SC模型下内陷2-budding膜泡是不稳定的,它通常总转化到胃形泡。而在BC模型下则可得到稳定的2-budding膜泡,这也提供了BC模型是更接近实际情况的模型的另一个例证。
(3)为了得到内陷2-budding膜泡,我们需要在BC模型下计算。为了提高计算速度,我们首先在SC模型下进行计算,然后转到BC模型下进行计算。我们发现在约化体积等于0.8,约化面积差等于0.485的情况下,稳定的形状是一个近似球形母泡和一个反转的长哑铃形通过细颈相连的形状。而对此形状以往的理论研究是把它近似处理成用无穷小细颈连接起来的一个大的理想球形和n个小的内陷理想球形的简单情况。
(4)对该形状进一步演化,我们发现内陷2-budding膜泡的budding过程是球形母泡和内陷于泡相连的细颈不断变细并分裂成一个球形和一个反转哑铃的形状,而不是通常文献中认为的直接分裂成一个球形母泡和两个反转子球形的过程。
(5)在约化体积等于0.8,约化面积差等于0.485的情况下,我们对3-budding形状也进行了研究,发现在该参数下,内陷3-budding形状也转化到内陷2-budding形状。为了得到内陷3-budding形状,必须在更小的约化面积差下进行计算。
(6)为了对比,我们研究了上下对称的双内陷泡的形状,U.Seifert等人曾研究过该形状,认为该形状是不稳定的,将转化到胃形泡。我们通过研究发现,上下对称性确实被打破,但只在约化面积差较大时转化到胃形泡;在约化面积差较小时,则转化到上下不对称的双内陷形状。
(7) Zhang yong等人宣称在SC模型下用Surface-Evolver给出了一个三凹膜泡的形状。我们用不同的初始形状重复了他们的工作,发现在他们所用的参数下用stn能量长时间演化确实可以得到一个三凹形状,但该形状只是一个鞍点,如转换到stp能量下进行演化,该形状很快转到胃形泡。由于很多在SC模型下不稳定的形状在BC模型下是稳定的,我们进一步用BC模型进行了研究,发现在BC模型下三凹形状也是不稳定的,它将转化到扁椭盘形。我们的研究说明只考虑双层膜的曲率能并不能给出三凹形状,蛋白质骨架对三凹形状的出现也是至关重要的。
Amphiphilic molecules composed of a polar hyfrophilic head and two non-polar hydrophobic tail chains may aggregate spontaneously into the bilayer membrane,which will form closed vesicles in low concentration because of the hydrophobic interaction.Study on the lipid vesicles as a simplified model of the biomembrane provide valuable information to understand the complex biological system.
In 1973 German physicist W.Helfrich firstly proposed the spontaneous curvature model(SC) based on the similarities of the lipid membrane and the nematic phase liquid crystal,thus proved the existence of the biconcave shape.At the same time,taking into account impact of the energy from the area difference between the bilayers in 1989 S.Svetian et al advanced the bilayer couple model(BC) and then U.Seffet also proposed other area difference elasticity model(ADE) with Miao Ling.
After many years of effort by some people,we have a very clear understanding about phase diagrams of the three models,still don't clearly know some regions.It is showed the stomatocytes vesicles will transform the limit shape which have a spherical mother vesicle and a inverted spherical daughter vesicle with a infinitesimal neck,which is expressed with L~(sto) in the left corner of phase diagram of the BC model,but stable shapes are not clear beyond the region.However in experiments S.Svetina and others observed many interesting vesicle shapes named inside n-budding,the region known of the phase diagram don't show these shapes.The current theoretical research on n-budding shapes deal with the approximate shapes composed of one big ideal mother sphere linking n small inverted perfcet daughter sphere,then give some budding shapes with the reduced volume v=0.85.we need lucubrate such problems as what these actual budding shape are,whether these shape are stable,what region they are in.
The clinical triconcave shape of the morbid red blood cell caused a lot of people's interest,so considering the protein skeleton Gerald Lin et al obtained the triconcave and some other abnormal structure of red blood cell by molecular dynamics simulation method.It is a matter of concern whether the triconcave shape can be stable only taking account of the curvature energe.
We make an in-depth study on the inside n-budding vesicles and the triconcave of red blood cell, main results can be summarized as follows:
(1) We research the region in which the inside n-budding vesicles exist and find the existing region is under the curve of L~(sto) that is unknown region in the phase diagram of SC and BC model.
(2) In practice we calculate the energy with two methods of stn and stp in order to solve the difficulties from the thin neck.The algorithms of stn may keep the neck from repture,the stp may have faster convergence speed and is effective to analysis the stability of vesicle shapes.At the same time we make full use of the relationship between the SC and BC models,and their respective calculational characteristics.It is proved that inside two-budding vesicles are unstable in SC model and the shape will transform into stomatocytes.However two-budding vesicles are stable in BC model,so the example proves the BC model is closer to the actual situation.
(3) In order to gain inside 2-budding vesicle,we need to calculate in the BC model.For advancing calculation speed,firstly we calculate in SC model,secondly switch to BC model.We find that a nearly spherical mother vesicle linking inverted dumbbell by one infinitesimal neck with reduced volume v=0.8 and reduced area difference△a=0.485.In past,The shape is simpled the approximate situation under which the big mother and all the small daughter vesicles are ideal spheres in the past theoretical research.
(4) Through the further evolution of the shape,we find that the neck linking spherial mother vesicle and the inverted daughter vesicle constantly become thinner and thinner in the budding process,at last the shape split into one sphere and inverted dumbbell,rather than what the usual literature showed the shape directly split into a spherical mother vesicle and two inverted spherical daughter vesicles.
(5) We observe that inside three-budding vesicle will transform into inside two-budding vesicle during the study on three-budding vesicle with reduced volume v=0.8 and reduced area difference△a=0.485,so one ought to calculate in a smaller reduced area difference to obtain the inside three-budding shape.
(6) U.Seifert et al studied symmetric shape of a sphere enclosing above and below two smaller spheres,and thought the shape is unstable and it will transit stomatocyte discovering through research, it is broken from top to bottom symmetry,but only in more reduced area difference it transformed into stomatocyte.In littler reduced area difference it transformed symmetric shape of a sphere enclosing above and below two smaller spheres into asymmetric shape.
(7) Zhang yong et al claimed that triconcave is given with Surface-Evolver in SC model,we do their works again in different initial shapes,and find that triconcave is got with stn energy long evolveing in parameters that they used.However,the shape is only a saddle,if the shape is evolved in stp energy,it is faster switched to stomatocyte.Because many triconcave are stable shapes in BC modle but in SC modle,we study futher in BC modle,and find that triconcave is unstable switching to oblate ellipsoidal vesicle.The study show that triconcave is not unstable only given bilayer membranaceous curvature energy,the protein skeleton is vital for triconcave.
为了解释膜泡的形状,德国液晶物理学家W.Helfrich根据向列相液晶与类脂膜的相似性,于1973年最先提出了膜的自发曲率(SC)弹性理论,并由此证明了膜的双凹盘形的存在;同时,考虑到膜泡双层间的面积差对膜泡能量的影响,1989年S.Svetian等人又提出了面积差(BC)模型,以后U.Seifert及苗玲等人又提出了面积差弹性(ADE)模型。
经过许多人多年的努力,这三种模型的相图在有些区域已经了解的很清楚了,但对有些区域还不了解。如在BC模型相图的左下角,胃形泡将转化到一个球形母泡和一个反转子球形用无限小细颈相连的极限情况佣L~(sto)表示),在该区域之外,稳定膜泡的形状是什么并不清楚。但在实验上,S.Svetina等人已经发现了很多有趣的形状,我们称之为内陷n-budding形状,它们并不能归到相图的已知区域。对此形状目前的理论研究是把它近似处理成用无穷小细颈连接起来的一个大的理想母泡球形和n个小的反转理想球形的简单情况,给出了约化体积等于0.85情况下的一些budding形状。在一般情况下这些膜泡的实际形状如何,这些形状是否稳定,它们在相图中处于什么区域,这些问题都是有待深入研究的。
病态红血球展现出的三凹形状引起了很多人的兴趣。Gerald Lim等人考虑了红血球的蛋白质骨架结构,用分子动力学模拟方法给出了三凹及其它一些红血球的反常结构。能否在只考虑膜泡的曲率能的情况下给出稳定的三凹形状成为人们关注的一个问题。
本文对内陷n-budding膜泡及红血球展现出的三凹形状进行了深入研究,我们得到的主要结果如下:
(1)对SC和BC模型的已有相图进行深入研究的基础上,对内陷n-budding形膜泡存在区间进行了探索,我们发现该形状位于曲线L~(sto)之下,这些区域在SC模型和BC模型两个相图中都是未知的。
(2)在实际计算中为了解决细颈部分所带来的困难,我们采用了两种计算能量的方法,stn及stp。stn可以保持颈部不至于断掉,而stp的收敛速度较快,并且对形状的稳定性进行分析时更为有效。我们计算方法的另一个特点是充分利用了BC模型和SC模型的关系及各自在计算上的特点。通过大量的计算,我们发现在SC模型下内陷2-budding膜泡是不稳定的,它通常总转化到胃形泡。而在BC模型下则可得到稳定的2-budding膜泡,这也提供了BC模型是更接近实际情况的模型的另一个例证。
(3)为了得到内陷2-budding膜泡,我们需要在BC模型下计算。为了提高计算速度,我们首先在SC模型下进行计算,然后转到BC模型下进行计算。我们发现在约化体积等于0.8,约化面积差等于0.485的情况下,稳定的形状是一个近似球形母泡和一个反转的长哑铃形通过细颈相连的形状。而对此形状以往的理论研究是把它近似处理成用无穷小细颈连接起来的一个大的理想球形和n个小的内陷理想球形的简单情况。
(4)对该形状进一步演化,我们发现内陷2-budding膜泡的budding过程是球形母泡和内陷于泡相连的细颈不断变细并分裂成一个球形和一个反转哑铃的形状,而不是通常文献中认为的直接分裂成一个球形母泡和两个反转子球形的过程。
(5)在约化体积等于0.8,约化面积差等于0.485的情况下,我们对3-budding形状也进行了研究,发现在该参数下,内陷3-budding形状也转化到内陷2-budding形状。为了得到内陷3-budding形状,必须在更小的约化面积差下进行计算。
(6)为了对比,我们研究了上下对称的双内陷泡的形状,U.Seifert等人曾研究过该形状,认为该形状是不稳定的,将转化到胃形泡。我们通过研究发现,上下对称性确实被打破,但只在约化面积差较大时转化到胃形泡;在约化面积差较小时,则转化到上下不对称的双内陷形状。
(7) Zhang yong等人宣称在SC模型下用Surface-Evolver给出了一个三凹膜泡的形状。我们用不同的初始形状重复了他们的工作,发现在他们所用的参数下用stn能量长时间演化确实可以得到一个三凹形状,但该形状只是一个鞍点,如转换到stp能量下进行演化,该形状很快转到胃形泡。由于很多在SC模型下不稳定的形状在BC模型下是稳定的,我们进一步用BC模型进行了研究,发现在BC模型下三凹形状也是不稳定的,它将转化到扁椭盘形。我们的研究说明只考虑双层膜的曲率能并不能给出三凹形状,蛋白质骨架对三凹形状的出现也是至关重要的。
Amphiphilic molecules composed of a polar hyfrophilic head and two non-polar hydrophobic tail chains may aggregate spontaneously into the bilayer membrane,which will form closed vesicles in low concentration because of the hydrophobic interaction.Study on the lipid vesicles as a simplified model of the biomembrane provide valuable information to understand the complex biological system.
In 1973 German physicist W.Helfrich firstly proposed the spontaneous curvature model(SC) based on the similarities of the lipid membrane and the nematic phase liquid crystal,thus proved the existence of the biconcave shape.At the same time,taking into account impact of the energy from the area difference between the bilayers in 1989 S.Svetian et al advanced the bilayer couple model(BC) and then U.Seffet also proposed other area difference elasticity model(ADE) with Miao Ling.
After many years of effort by some people,we have a very clear understanding about phase diagrams of the three models,still don't clearly know some regions.It is showed the stomatocytes vesicles will transform the limit shape which have a spherical mother vesicle and a inverted spherical daughter vesicle with a infinitesimal neck,which is expressed with L~(sto) in the left corner of phase diagram of the BC model,but stable shapes are not clear beyond the region.However in experiments S.Svetina and others observed many interesting vesicle shapes named inside n-budding,the region known of the phase diagram don't show these shapes.The current theoretical research on n-budding shapes deal with the approximate shapes composed of one big ideal mother sphere linking n small inverted perfcet daughter sphere,then give some budding shapes with the reduced volume v=0.85.we need lucubrate such problems as what these actual budding shape are,whether these shape are stable,what region they are in.
The clinical triconcave shape of the morbid red blood cell caused a lot of people's interest,so considering the protein skeleton Gerald Lin et al obtained the triconcave and some other abnormal structure of red blood cell by molecular dynamics simulation method.It is a matter of concern whether the triconcave shape can be stable only taking account of the curvature energe.
We make an in-depth study on the inside n-budding vesicles and the triconcave of red blood cell, main results can be summarized as follows:
(1) We research the region in which the inside n-budding vesicles exist and find the existing region is under the curve of L~(sto) that is unknown region in the phase diagram of SC and BC model.
(2) In practice we calculate the energy with two methods of stn and stp in order to solve the difficulties from the thin neck.The algorithms of stn may keep the neck from repture,the stp may have faster convergence speed and is effective to analysis the stability of vesicle shapes.At the same time we make full use of the relationship between the SC and BC models,and their respective calculational characteristics.It is proved that inside two-budding vesicles are unstable in SC model and the shape will transform into stomatocytes.However two-budding vesicles are stable in BC model,so the example proves the BC model is closer to the actual situation.
(3) In order to gain inside 2-budding vesicle,we need to calculate in the BC model.For advancing calculation speed,firstly we calculate in SC model,secondly switch to BC model.We find that a nearly spherical mother vesicle linking inverted dumbbell by one infinitesimal neck with reduced volume v=0.8 and reduced area difference△a=0.485.In past,The shape is simpled the approximate situation under which the big mother and all the small daughter vesicles are ideal spheres in the past theoretical research.
(4) Through the further evolution of the shape,we find that the neck linking spherial mother vesicle and the inverted daughter vesicle constantly become thinner and thinner in the budding process,at last the shape split into one sphere and inverted dumbbell,rather than what the usual literature showed the shape directly split into a spherical mother vesicle and two inverted spherical daughter vesicles.
(5) We observe that inside three-budding vesicle will transform into inside two-budding vesicle during the study on three-budding vesicle with reduced volume v=0.8 and reduced area difference△a=0.485,so one ought to calculate in a smaller reduced area difference to obtain the inside three-budding shape.
(6) U.Seifert et al studied symmetric shape of a sphere enclosing above and below two smaller spheres,and thought the shape is unstable and it will transit stomatocyte discovering through research, it is broken from top to bottom symmetry,but only in more reduced area difference it transformed into stomatocyte.In littler reduced area difference it transformed symmetric shape of a sphere enclosing above and below two smaller spheres into asymmetric shape.
(7) Zhang yong et al claimed that triconcave is given with Surface-Evolver in SC model,we do their works again in different initial shapes,and find that triconcave is got with stn energy long evolveing in parameters that they used.However,the shape is only a saddle,if the shape is evolved in stp energy,it is faster switched to stomatocyte.Because many triconcave are stable shapes in BC modle but in SC modle,we study futher in BC modle,and find that triconcave is unstable switching to oblate ellipsoidal vesicle.The study show that triconcave is not unstable only given bilayer membranaceous curvature energy,the protein skeleton is vital for triconcave.
引文
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