Membrane curvature
Membrane curvature is the geometrical measure or characterization of the curvature of membranes. The membranes can be naturally occurring or man-made (synthetic). An example of naturally occurring membrane is the lipid bilayer of cells, also known as cellular membranes. Synthetic membranes can be obtained by preparing aqueous solutions of certain lipids. The lipids will then "aggregate" and form various phases and structures. According to the conditions (concentration, temperature, ionic strength of solution, etc.) and the chemical structures of the lipid, different phases will be observed. For instance, the lipid POPC (palmitoyl oleyl phosphatidyl choline) tends to form lamellar vesicles in solution, whereas smaller lipids (lipids with shorter acyl chains, up to 8 carbons in length), such as detergents, will form micelles if the CMC (critical micelle concentration) is reached.
A biological membrane is commonly described as a two-dimensional surface, which spans a three-dimensional space. So, to describe membrane shape, it is not sufficient to determine the membrane curling that is seen in a single cross-section of the object, because in general there are two curvatures that characterize the shape each point in space. Mathematically, these two curvatures are called the principal curvatures, c1 and c2, and their meaning can be understood by the following thought experiment. If you cross-section the membrane surface at a point under consideration using two planes that are perpendicular to the surface and oriented in two special directions called the principal directions, the principal curvatures are the curvatures of the two lines of intercepts between the planes and the surface which have almost circular shapes in close proximity to the point under consideration. The radii of these two circular fragments, R1 and R2, are called the principal radii of curvature, and their inverse values are referred to as the two principal curvatures.
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