In addition there are other lipids and proteins in the membrane, the latter typically in the form of isolated rafts.
The simplest component of a membrane is the lipid bilayer which has a thickness that is much smaller than the length scale of the cell.
In 1973, based on similarities between lipid bilayers and nematic liquid crystals, Helfrich [2] proposed the following expression for the curvature energy per unit area of the closed lipid bilayer
where dA and dV are the area element of the membrane and the volume element enclosed by the closed bilayer, respectively, and λ is the Lagrange multiplier for area inextensibility of the membrane, which has the same dimension as surface tension.
By taking the first order variation of above free energy, Ou-Yang and Helfrich [3] derived an equation to describe the equilibrium shape of the bilayer as:
Using the shape equation (3) of closed vesicles, Ou-Yang predicted that there was a lipid torus with the ratio of two generated radii being exactly
[4] His prediction was soon confirmed by the experiment [5] Additionally, researchers obtained an analytical solution [6] to (3) which explained the classical problem, the biconcave discoidal shape of normal red blood cells.
In the last decades, the Helfrich model has been extensively used in computer simulations of vesicles, red blood cells and related systems.
[7] The opening-up process of lipid bilayers by talin was observed by Saitoh et al.[8] arose the interest of studying the equilibrium shape equation and boundary conditions of lipid bilayers with free exposed edges.
This line tension is a function of dimension and distribution of molecules comprising the edge, and their interaction strength and range.
[11] The first order variation gives the shape equation and boundary conditions of the lipid membrane:[12]
The skeleton is a cross-linking protein network and joints to the bilayer at some points.
In fact, the first two terms in (11) are the bending energy of the cell membrane which contributes mainly from the lipid bilayer.
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[6] U. Seifert, K. Berndl, and R. Lipowsky, Shape transformations of vesicles: Phase diagram for spontaneous- curvature and bilayer-coupling models, Phys.
[7] L. Miao, et al., Budding transitions of fluid-bilayer vesicles: The effect of area-difference elasticity, Phys.
[1] A. Saitoh, K. Takiguchi, Y. Tanaka, and H. Hotani, Opening-up of liposomal membranes by talin, Proc.
[5] T. Umeda, Y. Suezaki, K. Takiguchi, and H. Hotani, Theoretical analysis of opening-up vesicles with single and two holes, Phys.
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[1] Y. C. Fung and P. Tong, Theory of the Sphering of Red Blood Cells, Biophys.
[2] S. K. Boey, D. H. Boal, and D. E. Discher, Simulations of the Erythrocyte Cytoskeleton at Large Deformation.
[3] D. E. Discher, D. H. Boal, and S. K. Boey, Simulations of the Erythrocyte Cytoskeleton at Large Deformation.
Bausch and L. Vonna, Physics of Composite Cell Membrane and Actin Based Cytoskeleton, in Physics of bio-molecules and cells, Edited by H. Flyvbjerg, F. Julicher, P. Ormos And F. David (Springer, Berlin, 2002).
[5] G. Lim, M. Wortis, and R. Mukhopadhyay, Stomatocyte–discocyte–echinocyte sequence of the human red blood cell: Evidence for the bilayer–couple hypothesis from membrane mechanics, Proc.
[6] Z. C. Tu and Z. C. Ou-Yang, A Geometric Theory on the Elasticity of Bio-membranes, J. Phys.
[7] Z. C. Tu and Z. C. Ou-Yang, Elastic theory of low-dimensional continua and its applications in bio- and nano-structures,arxiv:0706.0001.