# Elasticity of cell membranes

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A cell membrane defines a boundary between the living cell and its environment. It consists of lipids, proteins,carbohydrates etc. Lipids and proteins are dominant components of membranes. One of the principal types of lipids in membranes is phospholipid. A phospholipid molecule has a polar hydrophilic head group and two hydrophobic hydrocarbon tails. When a quantity of lipid molecules disperse in water, they will assemble themselves into a bilayer in which the hydrophilic heads shield the hydrophobic tails from the water surroundings because of the hydrophobic forces.

The widely accepted model for cell membranes is the fluid mosaic model proposed by Singer and Nicolson in 1972 [Science 175 (1972) 720]. In this model, the cell membrane is considered as a lipid bilayer where the lipid molecules can move freely in the membrane surface like fluid, while the proteins are embedded in the lipid bilayer. Some proteins are called integral membrane proteins because they traverse entirely in the lipid bilayer and play the role of information and matter communications between the interior of the cell and its outer environment. The others are called peripheral membrane proteins because they are partially embedded in the bilayer and accomplish the other biological functions. Beneath the lipid membrane, the membrane skeleton, a network of proteins, links with the proteins in the lipid membrane.

## Elasticity of lipid vesicles

The first step to study the elasticity of cell membranes is to study lipid bilayers. Usually, the thickness of the lipid bilayer is much smaller than the scale of the whole lipid bilayer. It is reasonable to describe the lipid bilayer by a mathematical surface. In 1973, Helfrich [Z. Naturforsch. C 28 (1973) 693] recognized that the lipid bilayer was just like a nematic liquid crystal at room temperature, and then proposed the curvature energy per unit area of the bilayer

   ${\displaystyle f_{c}={\frac {k_{c}}{2}}(2H+c_{0})^{2}+{\bar {k}}K,}$      (1)


where ${\displaystyle k_{c},{\bar {k}}}$ are bending rigidities. ${\displaystyle c_{0}}$ is called the spontaneous curvature of the membrane. ${\displaystyle H}$ and ${\displaystyle K}$ are the mean and Gaussian curvature of the membrane surface, respectively.

The free energy of a closed bilayer under the osmotic pressure ${\displaystyle \Delta p}$ (the outer pressure minus the inner one) as:

   ${\displaystyle F_{H}=\int (f_{c}+\lambda )dA+\Delta p\int dV,}$      (2)


where ${\displaystyle dA}$ and ${\displaystyle dV}$ are the area element of the membrane and the volume element enclosed by the closed bilayer, respectively. ${\displaystyle \lambda }$ is the surface tension of the bilayer. By taking the first order variation of above free energy, Ou-Yang and Helfrich [Phys. Rev. Lett. 59 (1987) 2486] derived an equation to describe the equilibrium shape of the bilayer as:

   ${\displaystyle \Delta p-2\lambda H+k_{c}(2H+c_{0})(2H^{2}-c_{0}H-2K)+k_{c}\nabla ^{2}(2H)=0.}$     (3)


They also obtained that the threshold pressure for the instability of a spherical bilayer was

   ${\displaystyle \Delta p_{c}\propto k_{c}/R^{3},}$        (4)


where ${\displaystyle R}$ being the radius of the spherical bilayer.

Using the shape equation (3) of closed vesilces, Ou-Yang predicted that there was a lipid torus with the ratio of two generated radii being exactly ${\displaystyle {\sqrt {2}}}$ [Phys. Rev. A 41 (1990) 4517]. His prediction was soon confirmed by the experiment [Phys. Rev. A 43 (1991) 4525]. Additionally, researchers obtained an analytical solution [Phys. Rev. E 48 (1993) 2304] to (3) which explained the classical problem---the biconcave discoidal shape of normal red cells.

## Elasticity of open lipid membranes

The opening-up process of lipid bilayers by talin was observed by Saitoh et al. [Proc. Natl. Acad. Sci. 95 (1998) 1026] arose the interest of studying the equilibrium shape equation and boundary conditions of lipid bilayers with free exposed edges. Capovilla et al. [Phys. Rev. E 66 (2002) 021607], Tu and Ou-Yang [Phys. Rev. E 68 (2003) 061915] carefully studied this problem. The free energy of a lipid membrane with an edge ${\displaystyle C}$ is written as

   ${\displaystyle F_{o}=\int (f_{c}+\lambda )dA+\gamma \oint _{C}ds,}$      (5)


where ${\displaystyle ds}$ and ${\displaystyle \gamma }$ represent the arclength element and the line tension of the edge, respectively. The first order variation gives the shape equation and boundary conditions of the lipid membrane:

   ${\displaystyle k_{c}(2H+c_{0})(2H^{2}-c_{0}H-2K)-2\lambda H+k_{c}\nabla ^{2}(2H)=0,}$      (6)

${\displaystyle \left.\left[k_{c}(2H+c_{0})+{\bar {k}}k_{n}\right]\right\vert _{C}=0,}$       (7)

${\displaystyle \left.\left[-2k_{c}{\frac {\partial H}{\partial \mathbf {e} _{2}}}+\gamma k_{n}+{\bar {k}}{\frac {d\tau _{g}}{ds}}\right]\right\vert _{C}=0,}$      (8)

${\displaystyle \left.\left[{\frac {k_{c}}{2}}(2H+c_{0})^{2}+{\bar {k}}K+\lambda +\gamma k_{g}\right]\right\vert _{C}=0,}$       (9)



where ${\displaystyle k_{n}}$, ${\displaystyle k_{g}}$, and ${\displaystyle \tau _{g}}$ are normal curvature, geodesic curvature, and geodesic torsion of the boundary curve, respectively. ${\displaystyle \mathbf {e} _{2}}$ is the unit vector perpendicular to the tangent vector of the curve and the normal vector of the membrane.

## Elasticity of cell membranes

A cell membrane is simplified as lipid bilayer plus membrane skeleton. The skeleton is a cross-linking protein network and joints to the bilayer at some points. Assume that each proteins in the membrane skeleton have similar length which is much smaller than the whole size of the cell membrane, and that the membrane is locally 2-dimensional uniform and homogenous. Thus the free energy density can be expressed as the invariant form of ${\displaystyle 2H}$, ${\displaystyle K}$, ${\displaystyle tr\varepsilon }$ and ${\displaystyle det\varepsilon }$:

   ${\displaystyle f_{cm}=f(2H,K,tr\varepsilon ,det\varepsilon ),}$       (10)


where ${\displaystyle \varepsilon }$ is the in-plane strain of the membrane skeleton. Under the assumption of small deformations, and invariant between ${\displaystyle tr\varepsilon }$ and ${\displaystyle -tr\varepsilon }$, (10) can be expanded up to second order terms as:

   ${\displaystyle f_{cm}={\frac {k_{c}}{2}}(2H+c_{0})^{2}+{\bar {k}}K+\lambda +{\frac {k_{d}}{2}}(tr\varepsilon )^{2}-2\mu (det\varepsilon ),}$       (11)


where ${\displaystyle k_{d}}$ and ${\displaystyle \mu }$ are two elastic constants. In fact, the first two terms in (11) are the bending energy of the cell membrane which contributes mainly from the lipid bilayer. The last two terms come from the entropic elasticity of the membrane skeleton.

## Literature

### Reviews on configurations of lipid vesicles

[1] R. Lipowsky, The Conformation of Membranes, Nature 349 (1991) 475-481.

[2] U. Seifert, Configurations of Fluid Membranes and Vesicles, Adv. Phys. 46 (1997) 13-137.

[3] Z. C. Ou-Yang, J. X. Liu and Y. Z. Xie, Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases (World Scientific, Singapore, 1999).

### Research papers on closed vesicles

[1] W. Helfrich, Elastic Properties of Lipid Bilayers--Theory and Possible Experiments, Z. Naturforsch. C 28 (1973) 693-703.

[2] O.-Y. Zhong-Can and W. Helfrich, Instability and Deformation of a Spherical Vesicle by Pressure, Phys. Rev. Lett. 59 (1987) 2486-2488.

[3] O.-Y. Zhong-Can, Anchor Ring-Vesicle Membranes, Phys. Rev. A 41 (1990) 4517-4520.

[4] H. Naito, M. Okuda, and O.-Y. Zhong-Can, Counterexample to Some Shape Equations for Axisymmetric Vesicles, Phys. Rev. E 48 (1993) 2304-2307.

[5] U. Seifert, Vesicles of toroidal topology, Phys. Rev. Lett. 66 (1991) 2404-2407.

[6] U. Seifert, K. Berndl, and R. Lipowsky, Shape transformations of vesicles: Phase diagram for spontaneous- curvature and bilayer-coupling models, Phys. Rev. A 44 (1991) 1182-1202.

[7] L. Miao etal., Budding transitions of fluid-bilayer vesicles: The effect of area-difference elasticity, Phys. Rev. E 49 (1994) 5389-5407.

### Research papers on open membranes

[1] A. Saitoh, K. Takiguchi, Y. Tanaka, and H. Hotani, Opening-up of liposomal membranes by talin, Proc. Natl. Acad. Sci. 95 (1998) 1026-1031.

[2] R. Capovilla, J. Guven, and J.A. Santiago, Lipid membranes with an edge, Phys. Rev. E 66 (2002) 021607.

[3] R. Capovilla and J. Guven, Stresses in lipid membranes, J. Phys. A 35 (2002) 6233-6247.

[4] Z. C. Tu and Z. C. Ou-Yang, Lipid membranes with free edges, Phys. Rev. E 68, (2003) 061915.

[5] T. Umeda, Y. Suezaki, K. Takiguchi, and H. Hotani, Theoretical analysis of opening-up vesicles with single and two holes, Phys. Rev. E 71 (2005) 011913.

### Numerical solutions on lipid membranes

[1] J. Yan, Q. H. Liu, J. X. Liu and Z. C. Ou-Yang, Numerical observation of nonaxisymmetric vesicles in fluid membranes, Phys. Rev. E 58 (1998) 4730-4736.

[2] J. J. Zhou, Y. Zhang, X. Zhou, Z. C. Ou-Yang, Large Deformation of Spherical Vesicle Studied by Perturbation Theory and Surface Evolver, Int J Mod Phys B 15 (2001) 2977-2991.

[3] Y. Zhang, X. Zhou, J. J. Zhou and Z. C. Ou-Yang, Triconcave Solution to the Helfrich Variation Problem for the Shape of Lipid Bilayer Vesicles is Found by Surface Evolver, In. J. Mod. Phys. B 16 (2002) 511-517.

[4] Q. Du, C. Liu and X. Wang, Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions, J. Comput. Phys. 212 (2006) 757.

[5] X. Wang and Q. Du, physics/0605095.

### Selected papers on cell membranes

[1] Y. C. Fung and P. Tong, Theory of the Sphering of Red Blood Cells, Biophys. J. 8 (1968) 175-198.

[2] S. K. Boey, D. H. Boal, and D. E. Discher, Simulations of the Erythrocyte Cytoskeleton at Large Deformation. I. Microscopic Models, Biophys. J. 75 (1998) 1573-1583.

[3] D. E. Discher, D. H. Boal, and S. K. Boey, Simulations of the Erythrocyte Cytoskeleton at Large Deformation. II. Micropipette Aspiration, Biophys. J. 75 (1998) 1584-1597.

[4] E. Sackmann, A.R. 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. Natl. Acad. Sci. 99 (2002) 16766-16769.

[6] Z. C. Tu and Z. C. Ou-Yang, A Geometric Theory on the Elasticity of Bio-membranes, J. Phys. A: Math. Gen. 37 (2004) 11407-11429.

[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.