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) was reached. There are five commonly proposed mechanisms by which membrane curvature is created, maintained, or controlled: lipid composition, shaped transmembrane proteins, protein motif insertion/BAR domains, protein scaffolding, and cytoskeleton scaffolding.
Geometry 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, c 1 {\displaystyle c_{1}} and c 2 {\displaystyle c_{2}} , 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, R 1 {\displaystyle R_{1}} and R 2 {\displaystyle R_{2}} , are called the principal radii of curvature, and their inverse values are referred to as the two principal curvatures.
c 1 = 1 / R 1 {\displaystyle c_{1}=1/R_{1}}
c 2 = 1 / R 2 {\displaystyle c_{2}=1/R_{2}}
The principal curvatures c 1 {\displaystyle c_{1}} and c 2 {\displaystyle c_{2}} can vary arbitrarily and thereby give origin to different geometrical shapes, such as cylinder, plane, sphere and saddle. Analysis of the principal curvature is important, since a number of biological membranes possess shapes that are analogous to these common geometry staples. For instance, prokaryotic cells such as cocci, rods, and spirochette display the shape of a sphere, and the latter two the shape of a cylinder. Erythrocytes, commonly referred to as red blood cells, have the shape of a saddle, although these cells are capable of some shape deformation. The table below lists common geometric shapes and a qualitative analysis of their two principal curvatures.
Even though often membrane curvature is thought to be a completely spontaneous process, thermodynamically speaking there must be factors actuating as the driving force for curvature to exist. Currently, there are some postulated mechanisms for accepted theories on curvature; nonetheless, undoubtedly two of the major driving forces are lipid composition and proteins embedded and/or bound to membranes.
Induced by lipids
Dynamics Perhaps the most simple and intuitive driving force in membrane curvature is the natural spontaneous curvature exhibited by some lipids. This is because, depending on their chemical structures, lipids tend to curve with a slight spontaneously negative or positive curvature. Lipids such as DOPC (dioleoyl phosphatidyl choline), diacyl glycerol, dioleoyl phosphatidyl ethanolamine (DOPE) and cholesterol exhibit a negative spontaneous curvature. On the other hand, lipids with smaller acyl chain area to polar head group area ratio tend to curve positively; in other words they exhibit positive spontaneous curvature. The table below lists experimentally determined spontaneous curvatures for different lipids in DOPE.
The energy requirements to generate a cylinder shaped cell from an originally flat membrane can be expressed as
F C y l = π L K b ( 1 R − 2 J B ) {\displaystyle {F_{Cyl}}=\pi LK_{b}({\frac {1}{R}}-2J_{B})}
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