A cell membrane defines a boundary between the living cell and its environment. The primary constituent of a membrane is a phospholipid bilayer that forms in a water-based environment due to the hydrophilic nature of the lipid head and the hydrophobic nature of the two tails. In addition there may be other lipids and proteins in the membrane, the latter typically in the form of isolated rafts.
Contents
- Elasticity of closed lipid vesicles
- Elasticity of open lipid membranes
- Elasticity of cell membranes
- References
Of the numerous models that have been developed to describe the deformation of cell membranes, a widely accepted model is the fluid mosaic model proposed by Singer and Nicolson in 1972. In this model, the cell membrane surface is modeled as a two-dimensional fluid-like lipid bilayer where the lipid molecules can move freely. The proteins are partially or fully embedded in the lipid bilayer. Fully embedded proteins are called integral membrane proteins because they traverse the entire thickness of the lipid bilayer. These communicate information and matter between the interior and the exterior of the cell. Proteins that are only partially embedded in the bilayer are called peripheral membrane proteins. The membrane skeleton is a network of proteins below the bilayer that links with the proteins in the lipid membrane.
Elasticity of closed lipid vesicles
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. Therefore the lipid bilayer can be represented by a two-dimensional mathematical surface. In 1973, based on similarities between lipid bilayers and nematic liquid crystals, Helfrich proposed the following expression for the curvature energy per unit area of the closed lipid bilayer
where
The free energy of a closed bilayer under the osmotic pressure
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 surface tension of the bilayer. By taking the first order variation of above free energy, Ou-Yang and Helfrich derived an equation to describe the equilibrium shape of the bilayer as:
They also obtained that the threshold pressure for the instability of a spherical bilayer was
where
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
Elasticity of open lipid membranes
The opening-up process of lipid bilayers by talin was observed by Saitoh et al. arose the interest of studying the equilibrium shape equation and boundary conditions of lipid bilayers with free exposed edges. Capovilla et al., Tu and Ou-Yang carefully studied this problem. The free energy of a lipid membrane with an edge
where
where
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
where
where