Membrane curvature

[2] A biological membrane is commonly described as a two-dimensional surface, which spans a three-dimensional space.

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.

The table below lists common geometric shapes and a qualitative analysis of their two principal curvatures.

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.

This is because, depending on their chemical structures, lipids tend to curve with a slight spontaneously negative or positive curvature.

[5] 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

where L is the length of the cylinder, JB is the difference between the spontaneous curvature, Js, for the lipids in the inner and outer leaflet divided by two, and Kb is the bending modulus of the bilayer.

The radii of membrane cylinders that form in intracellular membrane-transport pathways are typically ~25–30 nm.

However, even for these lipids, the required JB can be reached only if they are extensively concentrated in the internal monolayer.

Multiple factors influence whether a lipid will exhibit positive or negative curvature.

[8] In the figure, the different shape of lipids with a double bond - also known as unsaturated - can be visualized.

However, a single conically shaped lipid will not induce curvature across an entire region of the membrane.

Instead, clustering of similarly shaped lipids in one leaflet compared to the other is required to induce curvature.

[10] When the membrane does curve, a higher number of lipids will be required to be present on the positive curvature side of the membrane to cover the increased surface area that is present compared to the negatively curved side.

Transmembrane proteins with an inherently conical shape will be more stable in, and induce curvature in membranes.

Epsin has several amphipathic alpha helices that allows it to partition between the hydrophobic core of the membrane and surrounding aqueous, hydrophilic environment.

The insertion of its helices into the membrane force the neighboring lipids of the leaflet that has been bound to expand laterally.

This figure shows membrane bending by insertion of a hydrophobic protein motif into a lipid bilayer.

[12] Another example of protein interactions that directly affect membrane curvature is that of the BAR (Bin, amphiphysin, Rvs’) domain.

Relative to the cellular lipid bilayer, this domain is rigid and exhibits a "banana" shape.

It has been postulated that the positively charged amino acid residues in the concave region of the BAR domain would come into contact with the negatively charged polar head groups of lipids in the bilayer, thus allows the binding process.

This idea is supported by the existence of positively charged amino acid residues in the concave region of the BAR domain.

[13] These amino acids would come into contact with the negatively charged polar head groups of lipids in the bilayer.

Clathrin can attach to adaptor protein complexes on the cellular membrane, and it polymerizes into lattices to drive greater curvature, resulting in endocytosis of a vesicular unit.

[2] This requires the membrane to be fluid enough to do so in a stable manner, and is often stabilized by the other mechanisms listed in this article, in particular lipid composition.

[17] Although contribution of this mechanism remains unclear, multiple experimental and computation evidences have shown its potential in bending membrane.

Curvature radii
Different changes to lipid structure, such as tail saturation, affect the overall shape of the lipid. A change in shape such as the one shown, when disproportionately in higher concentration on one side of the membrane, allows the membrane to curve.
Transmembrane proteins with inherent curvature inducing curvature in a membrane.
Insertion of a piece of a protein into one leaflet of the membrane induces curvature.
A BAR domain of a protein inducing and stabilizing the curvature of a membrane.
Cage-like structure of clathrin . When this structure forms around a membrane, it pulls the membrane into a tight curvature until eventual vesicle budding.
Cytoskeletal structure inducing membrane curvature. The inherent shape of a cell - as controlled by its cytoskeleton - requires that the bilayer membrane curve around it.
This figure illustrates membrane bending caused by protein crowding. When a high local concentration of proteins (shown in green) are present on the membrane surface (shown in black), membrane curvature can be induced. This hypothesis reasoned that the high protein concentration increases the likelihood of repulsions between proteins, therefore generates steric pressure between proteins. To relieve such pressure, lipid membrane has to bend in order to decrease protein repulsions.