Are The Curves in all the Right Places?

Membranes bend as they move. Endoplasmic reticulum tubes and cisternae, Golgi sacs, transport tubules and vesicles, endosomes, and plasma membrane, all have their own curvatures. When membranes undergo fusion or fission, their curvatures become extreme in the fusion pores or the fission necks [1]. Non-bilayer intermediates have even greater curvatures [2]. The tight curves result in each element of a membrane changing shape. But membranes are not made up of static building blocks, because lipid tails are fluid. However, all lipids are not the same. Different lipids affect fusion in different ways [3], and fission machines contain enzymes that alter lipid chemical structure [4,5]. Thus, an understanding of membrane fusion and fission requires an understanding of membrane elasticity and structure and the stresses that result from lipid–lipid interactions and depend on lipid composition. The first investigations to invoke lipid shape were a combination of electron microscopy and X-ray diffraction on a series of lipid/water mixtures [6]. They showed two main structures: lamellar phases in dilute systems and hexagonal phases, predominating as water, is withdrawn from polar head groups ( Fig. 1A). They introduced the term ‘polymorphic’ for lipid structure, because the same lipid occupied two different shapes as the packing constraints for a hexagon are different to those for a plane. The same X-ray data clearly indicated liquid, disordered hydrocarbon chains. These are fluid phases in which lipid molecules have rotations of many bonds and, on average, occupy a volume that is different for different phases. A) Lipid molecules can have different effective shapes. For example, some phosphatidylethanolamine molecules can form either a lamellar phase in which each molecule occupies, time averaged, an effective shape that is cylindrical, or, when water is removed, an inverted haxagonal phase in which it occupies an effective shape that is conical. B) A phospholipid vesicle has overall positive bilayer curvature, but the outer and inner monolayers have opposite curvatures, one positive and the other negative. C) If it doesn't break, a tree branch can be bent, resulting in an actual curvature like a bent spring, only to return to its unstressed state (intrinsic curvature) when the stress is removed. D) Different lipids spontaneously form monolayers of different intrinsic curvature, within which each molecule, averaged over time, occupies a different effective shape. Illustration by Katherine Sutliff. A language for the polymorphism of lipids is that of monolayer curvature [7]. Monolayer curvature is defined from the perspective of an observer in the hydrophobic interior looking out at the headgroups: positive when concave, negative when convex. Monolayer curvature is not bilayer curvature, defined for the whole membrane from the perspective of an observer in the aqueous interior of a vesicle (also positive when concave). So while the vesicle has positive bilayer curvature, and its outer monolayer has positive monolayer curvature, its inner monolayer has a negative curvature because its headgroups are convex from the viewpoint of the hydrocarbon ( Fig. 1B). A planar membrane has zero curvature and each of its monolayers has zero curvature. So when a planar membrane becomes a vesicle, the bilayer and outer monolayers are bent positively, and the inner monolayer is bent negatively. The amount of work a protein carries out to bend a membrane or monolayer depends on its unstressed curvature. You can pull on a branch of a tree and bend it, but when you release it, the branch returns to its original state – its original curvature. That original curvature may be stressed or not, depending on the other factors bending the branch. Only when a curved branch falls to the ground and no longer experiences its own weight and that of its fruit, can it relax to its unstressed state – its intrinsic curvature ( Fig. 1C). For any actual curvature, the magnitude of curvature stress on a branch, membrane, or monolayer goes as the difference between the actual curvature and the intrinsic curvature. If an element of planar bilayer membrane has the same number and type of components in each monolayer, it will have zero intrinsic bilayer curvature, by symmetry. So it will take no work to keep such a membrane flat, and it will take work to bend it. If this membrane were bent into a sphere, it would be under considerable stress, because its intrinsic curvature is still zero. The pinching-off of such a vesicle would trap that stress, because boundary conditions (hydrophobic continuity) would constrain it to a sphere. A spherical membrane whose intrinsic bilayer curvature was positive would resist flattening, even if unconstrained. The intrinsic, or spontaneous, monolayer curvature refers to the spontaneous curvature of a piece of monolayer without any edge conditions. Unlike intrinsic bilayer curvature, which depends on component number, intrinsic monolayer curvature depends on the composition of the monolayer and head group–solution interactions. For lipids, it can be taken from the curvature of a monolayer in whatever phase develops spontaneously at equilibrium for any solution and lipid mixture ( Fig. 1D). Some lipids, like lysolipids, form spherical micelles or cylindrical micelles in solution. These monolayers have positive spontaneous curvature. Other lipids, such as phosphatidylethanolamine, form monolayers of negative intrinsic curvature (in inverted hexagonal phases). Constraining a lipid molecule into a membrane with monolayer curvature different from its intrinsic curvature produces curvature stress, as discussed above. Another language to express the frustration of lipids trapped in stressful geometries is that of ‘effective’ molecular shape, defined as its average shape in the monolayer of its spontaneous curvature [8]. Lysolecithin in a micelle, averaged over time, occupies a conical volume, while lysolecithin in a planar bilayer on average occupies a cylindrical volume. However, lysolecithin ‘wishes’ to occupy a conical volume, so its ‘effective’ shape is conical in both environments. The effective shape of a lipid molecule is a simplification that can be a great aid to the conceptualization of monolayer curvature, but we cannot determine the actual shapes of individual lipid molecules in fluid phases, which are very dynamic [9], only the average envelope that fits the time-averaged volumes they occupy. Also, since many structures have negative curvature in one dimension and positive curvature in another dimension, a single effective shape cannot suffice. Ongoing work is refining our intuition on predicting lipid architectures [10], but for now only a retrospective assignment of effective shapes and intrinsic monolayer curvatures from structural data on spontaneous phases is reliable. For spherical vesicles, if the outer monolayer had positive spontaneous curvature and the inner monolayer had negative spontaneous curvature, then that would stabilize the positive curvature of the vesicle. Note that a planar membrane may be composed of either two monolayers having negative (or positive) intrinsic monolayer curvature, or two monolayers having zero intrinsic monolayer curvature. Because of symmetry, the net bilayer intrinsic curvature will be zero for both membranes, but the membranes composed of lipids with non-cylindrical effective shapes will be stressed. That stress can be relieved through a pathway for formation of a bent intermediate more to that lipids liking [2]. Lipid composition clearly can control membrane fusion, shown by the experimental tool of adding or removing exogenous lysolipids, or the activity of adding phospholipases to create lysolipid or diacylglyceride products [3,11]. Does biology use the same tools? Amazingly, two different fission machines have crucial components, endophilin and brefeldin A ribosylated substrate (BARS), which are lysophosphatidic acid acyl transferases that change the lipid composition of membranes much as experimentalists have, transmuting lysophosphatidic acid into phosphatidic acid [4,5]. By keeping the head group constant and doubling the volume of the tails, such lysoacyltransferases should make the intrinsic curvature of any ‘monolayer’ containing substrate more negative. But it is not clear why that leads to fission. It implies an intermediate of fission with more negative monolayer curvature. During constriction, the neck may develop actual bilayer curvatures that are overall negative, but that would depend on the exact geometry of the neck (obtained by integrating the curvatures in two different radial dimensions). Of course, linking components already in the bilayer will probably not significantly change any curvature. But incorporation and retention of material previously partitioning between membrane and solution, such as lysolipids and acyl CoA, should increase the number of molecules in the monolayer facing the enzyme, increasing positive bilayer curvature of that entire membrane. This was controlled for in [4], but might explain [5] and the facilitation by endophilin of the budding of vesicles facing cytoplasm [12]. The fact that fission machines evolved to alter effective lipid shape means that a bent monolayer intermediate may play a crucial role in the pathway of vesicle formation, as they are thought to do twice during membrane fusion, once for hemifusion and once again for fusion pore formation [3]. Many thanks to V. Adrian Parsegian and Leonid V. Chernomordik for discussions.