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'''Membrane curvature''' is the geometrical measure or characterization of the [[curvature]] of [[Membrane (selective barrier)|membrane]]s.
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 membrane]]s.<ref name="The Lipid Chronicles">{{cite web|title=Curvy Biology|url=http://www.samuelfurse.com/2011/12/curvy-biology/|work=The Lipid Chronicles|accessdate=08/01/2012}}</ref> 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 [[carbon]]s in length), such as detergents, will form [[micelles]] if the CMC (critical micelle concentration) is reached.

==Basic Geometry of Curvature==
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.<ref>Spivak, M. '''''A Comprehensive Introduction to Differential Geometry''''' (Brandeis University, Waltham,1970)</ref>
[[File:Curvature radii.JPG|Curvature radii|right|thumb|300px]]

<math>c1 = 1/R1</math>

<math>c2 = 1/R2</math>

The principal curvatures C1 and C2 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 (geometry)|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.
{| class="wikitable" border="1"
|-
! '''Shape'''
! C1
! C2
|-
| '''Plane'''
| 0
| 0
|-
| '''Cylinder'''
| +
| 0
|-
| '''Sphere'''
| +
| +
|-
| '''Saddle'''
| +
| -
|}

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 [[protein]]s embedded and/or bound to membranes.

== Driving forces for membrane Curvature ==
===Lipid Spontaneous Curvature===

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]], [[dioleyl phosphatidylethanolamine]] (DOPE) and [[cholesterol]] exhibit a negative spontaneous curvature.<ref>Martens, S., McMahon, H. T. '''''Nature Reviews'''''. 9, 543-556 (2008)</ref> 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.<ref>Kamal, M et al. Measurement of the membrane curvature preference of phospholipids reveals only weak coupling between lipid shape and leaflet curvature. '''PNAS''' (2009) vol. 106 (52) pp. 22245-50</ref> The table below lists experimentally determined spontaneous curvatures for different lipids in DOPE (dioleyl phosphatidyl ethanolamine).

{| class="wikitable" border="1"
|-
! '''''Lipid'''''
! '''''Js''''' (nm−1)
|-
| '''Lysophospholipids'''
|
|-
| L-lyso PC
| 1/5.8
|-
| O-lyso PC
| 1/3.8
|-
| P-lyso PC
| 1/6.8
|-
| L-lyso PE
| <1/40
|-
| O-lyso PE
| <1/40
|-
| S-lyso PE
| <1/40
|-
| '''Other Lipids'''
|-
| DOPS
| 1/14.4
|-
| DOPC
| -1/20
|-
| PA
| -1/4.6
|-
| DOPE
| -1/3
|-
| Cholesterol
| -1/2.9
|-
| DCG
| -1/1.3
|}

The energy requirements to generate a cylinder shaped cell from an originally flat membrane can be expressed as

'''FCyl =π'''x'''L'''x'''Kb(1/R - 2JB)'''
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.<ref>Polishchuk, R. S. et al. ''Correlative light-electron microscopy reveals the tubular-saccular ultrastructure
of carriers operating between Golgi apparatus and plasma membrane''. '''''Journal of Cell Biology''''' 148, 45–58
(2000).</ref> So, the spontaneous curvature necessary to generate such cylinders equals ~(1/50) nm–1. As JB results from a difference in the spontaneous curvatures of the monolayers, an unusual membrane lipid composition would
be required to produce such curvature. The lipids cholesterol, [[dioleoylphosphatidylethanolamine]](DOPE)
and diacylglycerol are characterized by strongly negative spontaneous curvatures (figure 1) and therefore have the potential to generate a large membrane curvature. However, even for these lipids, the required JB can be reached only if they are extensively concentrated in the internal monolayer.

===Proteins can Induce Curvature===

Some biologically occurring lipids do exhibit spontaneous curvature which could explain the shapes of biological membranes. Nevertheless, calculations show that spontaneous lipid curvature alone is either insufficient or would require conditions that are unrealistic to drive the degree of curvature observed in most [[Cell (biology)|cell]]s. It is now known that lipid curvature is "aided" by protein structures in order to generate complete cellular curvature. A classical example of such interactions is the activity of the protein [[clathrin]]. Clathrin is involved in cellular endocytosis and is sequestrated by specific signaling molecules. Clathrin can attach to adaptor protein complexes on the cellular membrane, and it polymerizes to drive greater curvature, resulting in endocytosis of a vesicular unit.

Another example of protein interactions that directly affect membrane curvature is that of the [[BAR domain|BAR]] (Bin, amphiphysin, Rvs’) domain. The BAR domain is present in a large family of proteins. This domain is rigid, relative to the cellular lipid bilayer, and it exhibits a "banana" shape. Upon binding, the membrane's curvature is increased by the rigid domain.<ref>Martens, S., McMahon, H. T. '''''Nature Reviews'''''. 9, 543-556 (2008)</ref>

One more case of protein interaction that induces and/or aids curvature is the class of proteins such as [[epsin]]. Epsin has several alpha helices that possess amphipathic properties, which allows it to partition between the hydrophobic core of the membrane and the hydrophilic aqueous environment. Another interesting characteristic of epsin and other proteins that bind to membranes is the fact that it shows high binding affinity for a fairly common membrane lipid, [[phosphatidylinositol 4,5-bisphosphate]] (PI-4,5-P2). Unlike other proteins that simply bend the membrane through sheer rigidity, epsin is a globular soluble protein and thus not rigid. The insertion of its helices into the membrane force the neighboring lipids of the leaflet that has been bound to expand laterally. This displacement of lipids on only one of the leaflets increases the bilayer's curvature.

[[File:mechanisms.jpg|Mechanisms of curvature induction by proteins|right|thumb|200px]]

The figure illustrates the different mechanisms through which proteins can aid and/or induce membrane curvature. In '''A''', an illustration of a BAR domain present in a number of proteins. The curvature is induced by the very shape of this proteic region. This domain attaches to the lipid bilayer through strong coulombic interactions. This idea is supported by the existence of positively charged [[amino acid]] residues in the concave region of the BAR domain.<ref>Zimmerberg, J. & McLaughlin, S. ''Membrane curvature: how BAR domains bend bilayers''. '''''Current Biology''''' 14, R250–R252 (2004).</ref> These amino acids would come into contact with the negatively charged polar head groups of lipids in the bilayer. This form phenomenon is also referred to as the "scaffold mechanism".

'''B''' shows a protein coating that induces curvature. As mentioned above, proteins such as [[clathrin]] are recruited to the membrane through signaling molecules and assemble into larger polymeric structures that form a rigid structure which serves as a frame for the membrane. Clathrin binds to its receptors that are present in the membrane.

'''C''' illustrates a slightly different mechanism. In this case, the membrane-bending protein does not exhibit intrinsic rigidity. Instead they are often [[globular]] and soluble. The protein epsin is an example. Epsin has an ENTH (epsin N-terminal homology) domain which inserts its amphipathic [[alpha helix]] into the membrane. Epsin has high binding affinity for the membrane if PI-4,5-P2 is present.<ref>Stahelin, R. V. et al. ''Contrasting membrane interaction mechanisms of AP180 N-terminal homology (ANTH) and epsin N-terminal homology (ENTH) domains''. '''''Journal Biological Chemistry''''' 278, 28993–28999 (2003).</ref>

==References==
{{Reflist}}
*Martens, S., McMahon, H. T. '''''Nature Reviews'''''. 9, 543-556 (2008)
*Spivak, M. ''A Comprehensive Introduction to Differential Geometry'' (Brandeis University, Waltham,1970)
*Polishchuk, R. S. et al. ''Correlative light-electron microscopy reveals the tubular-saccular ultrastructure of carriers operating between Golgi apparatus and plasma membrane''. '''''Journal of Cell Biology''''' 148, 45–58
(2000).
*Zimmerberg, J. & McLaughlin, S. ''Membrane curvature: how BAR domains bend bilayers''. '''''Current Biology''''' 14, R250–R252 (2004).
*Stahelin, R. V. et al. ''Contrasting membrane interaction mechanisms of AP180 N-terminal homology (ANTH) and epsin N-terminal homology (ENTH) domains''. '''''Journal Biological Chemistry''''' 278, 28993–28999 (2003).

[[Category:Membrane biology]]

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en>Sun Creator: /* View 1: total states = states * elapsed times */Typo fixing and checking, typos fixed: , → , using AWB (8097)