Elasticity of cell membranes

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.    (1)    (2)    (3)    (4)    (5)    (6)    (7)    (8)    (9)    (10)    (11) 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. 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. 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 k c , k ¯ {displaystyle k_{c},{ar {k}}} are bending rigidities, c 0 {displaystyle c_{0}} is the spontaneous curvature of the membrane, and H {displaystyle H} and K {displaystyle K} are the mean and Gaussian curvature of the membrane surface, respectively. The free energy of a closed bilayer under the osmotic pressure Δ p {displaystyle Delta p} (the outer pressure minus the inner one) as: 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 derived an equation to describe the equilibrium shape of the bilayer as: