Curvature-driven membrane lipid and protein distribution

Current Opinion in Solid State and Materials Science 17 (2013) 143–150

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Curvature-driven membrane lipid and protein distribution Andrew Callan-Jones a,b, Patricia Bassereau c,d,e,⇑ a

CNRS, UMR 5207, Laboratoire Charles Coulomb, F-34095 Montpellier, France Université Montpellier II, F-34095 Montpellier, France c Institut Curie, Centre de Recherche, F-75248 Paris, France d CNRS, PhysicoChimie Curie, UMR168, F-75248 Paris, France e Université Pierre et Marie Curie, F-75252 Paris, France b

a r t i c l e

i n f o

Article history: Available online 14 September 2013 Keywords: Membrane curvature Lipid sorting Protein sorting In vitro experiments Spontaneous curvature Bending rigidity N-BAR proteins Protein insertion Amphipathic helices Protein scaffold

a b s t r a c t Cellular transport requires that membranes have the ability to recruit specific lipids and proteins to particular positions and at specific times. Here, we review recent work showing that lipids and proteins can be redistributed by spatially varying membrane curvature, without necessarily the need for biochemical targeting signals. We present here an emerging understanding of the various mechanisms by which membrane curvature can sort lipids and proteins, providing the experimental methods in addition to the supporting theoretical concepts. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Exchange of material between and within cells is an essential component of cell life [1]. Trafficking of cellular matter is mediated by membranes, which enclose cells and define the boundaries of intracellular compartments. Membranes are essentially composed of self-assembled bilayers of a huge variety of amphiphilic molecules (lipids) and possess unusual mechanical properties. Owing to the hydrophobic lipid chains and hydrophilic lipid heads, biological membranes are practically incompressible. They, however, cannot resist shear, and are thus a two-dimensional fluid [2], which has important consequences for membrane trafficking. Thus, the mechanical state of a membrane is completely determined by its shape, i.e., curvature, and by its state of tension. The energy needed to shape a membrane is described by its bending elasticity, which is related to the differential strain across the bilayer. The energy scale associated with membrane bending is given by the so-called bending modulus, j, which ranges from 10 to 60 kBT, and depends sensitively on the membrane composition [3]. The relaxed shape of a ⇑ Corresponding author. Address: Laboratoire PhysicoChimie Curie, Centre de Recherche Institut Curie, 26 Rue d’Ulm, 75248 Paris Cedex 05, France. Tel./fax: +33 1 56 24 67 84. E-mail address: [email protected] (P. Bassereau). 1359-0286/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cossms.2013.08.004

membrane, known as the spontaneous curvature, depends on bilayer asymmetry, and, as we will discuss later, can be modulated by the trans-bilayer and lateral lipid and protein distributions. In eukaryotic cells intracellular transport is achieved by membrane-bound structures called transport carriers that shuttle cargo between different cellular compartments, in addition to importing and exporting nutrients and signaling molecules between cells [4]. The generation of these carriers involves deformation and eventually detachment of small vesicles and tubular structures, ranging in diameter from a few tens to a few hundreds of nanometers, from a parent membrane [5]. Membrane deformation leading to vesicle formation results either from the action of molecular motors that pull on membranes when moving along cytoskeletal filaments or from the interaction of membrane–binding proteins with the surrounding bilayer [6]. Upon vesicle formation, specific proteins and lipids are selected as cargos. Lipids are not evenly distributed throughout the organelles of the cells, and cells must therefore possess sorting mechanisms to maintain composition homeostasis [7]. As an example, the percentage of saturated, ceramide-based sphingolipids varies in the cell, from about 0% by mass in the endoplasmic reticulum to about 30% in the plasma membrane [8]. A popular hypothesis for lipid sorting postulated the existence of pre-formed lipid domains, known as lipid rafts, from which lipids could be selectively incorporated into transport vesicles (for a recent review, see [9]).

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Alternatively, it has been proposed that the spatial distribution of lipids could be directly coupled to the curvature of the membrane [10,11]. The role of membrane–bound proteins in deforming membrane is crucial. Here, we will review recent developments in understanding membrane shaping by proteins, and the consequences on the spatial organization of membrane lipids and proteins. We will first present the theoretical framework for describing the coupling between membrane curvature and protein organization. Because of the complexity of biological systems in vivo, recent studies using in vitro biomimetic systems have proven invaluable in experimentally probing the relationship between membrane curvature and protein distribution. We will provide a survey of in vitro experimental results. In the second part of the review, we will present theoretical and experimental studies of the role of membrane curvature on lipid sorting. 2. General concepts and methodology 2.1. Membrane energy and interaction between proteins and curved membranes The basic starting point for a description of membrane mechanics is the so-called Helfrich bending energy, that relates the free energy density of the membrane integrated over its area to its total curvature, H, its relaxed curvature C0, and its tension, r:

F Helfrich ¼

Z j j 2

k ð2H  C 0 Þ2 þ r dA

ð1Þ

Interactions between proteins and the membrane affect C0, a key fact that was originally pointed out by Leibler [12]. In the simplest approach, a bare, symmetric lipid bilayer has zero spontaneous curvature, but proteins, whether in the form of inclusions or bound on one of the membrane leaflets, can, due to the asymmetrical membranes stresses they produce across the bilayer, induce a spontaneous curvature in proportion to their areal concentration. Thus, C0 can be supposed to be a linear function of the concentrations, /i, of the N protein species:

C0 ¼

N X C i /

i

ð2Þ

i¼1

This equation reflects spontaneous curvature generated by proteins at the individual protein scale, and neglects protein–protein interactions; we thus expect it to be valid at reasonable low protein area fractions on the membrane. Thus, we see from Eq. (1), that there is a bilinear coupling between membrane curvature and protein concentration of the form /i H. This term, as pointed out by Leibler, implies that the lateral organization of proteins couples directly to the membrane shape [12]. Of course, fluctuations in protein distribution cost entropy, and as result the coupling between membrane shape and protein distribution involves a balance between spontaneous curvature energy and mixing entropy. As illustrated in Fig. 1, the curvature–concentration coupling, expressed in Eqs. (1) and (2), can give rise to membrane shape and protein density instabilities, leading to the formation of tubular structures and membrane buds occasionally seen in cells and also in vitro [13,14]. At equilibrium, the protein-induced spontaneous curvature gives rise to membranes with inhomogeneous curvature and protein distribution, as has been shown theoretically by Bozic et al. [15] and Kralj-Iglic et al. [16]. 2.2. Experimental methods A usual approach for biologists to study protein-induced membrane deformation consists in incubating small liposomes typically

Fig. 1. Membrane instability induced by protein concentration–curvature coupling. A spontaneous membrane fluctuation causes, due to the asymmetric protein shape (here represented as a wedge), an aggregation of protein, which in turn amplifies the initial curvature disturbance. (Based on Ref. [12]).

50–100 nm in diameter containing the ligand (lipid or proteins) with the considered protein, and monitoring by electron microscopy the shape of the liposomes after binding (see for instance [17–20]). Generally, with this method, very high concentrations of protein on the vesicles are needed to detect significant deformations such as membrane tubules. Further understanding of the effect of membrane curvature on protein binding can be obtained using small liposomes and light scattering [21] or quantitative high throughput fluorescence microscopy [22] (Fig. 2A). However, comparison with theoretical models is very limited with such approaches as the mechanical action as a function of protein bulk and surface concentrations cannot be addressed easily. Giant liposomes (also called GUVs, for ‘‘giant unilamellar vesicle’’) are convenient systems to monitor membrane deformations with optical microscopy and control membrane tension at the same time [23]. Additionally, membrane nanotubes with a controlled curvature can be pulled and the mechanical action of proteins on membrane can be deduced from force measurements on the tube and compared to theoretical models [24] (Fig. 2B). Eventually protein distribution as a function of membrane curvature can be tuned by continuously changing tube radius and detected with fluorescence microscopy [24–27] (Fig. 2B). Techniques based on micropipette aspiration and membrane nanotubes pulled from GUVs can also be applied to study curvature-induced lipid sorting [25,28]. Another class of experiment aimed to measure the asymmetry of bilayer lipid composition in small vesicles, ranging in size from 100 to 100 nm [29]. Transverse asymmetry in the curved bilayer was measured in this method by comparing the fluorescence in the two leaflets (Fig. 2C). In yet another approach, Parthasarathy et al. studied lateral segregation of lipid domains in inhomogeneous membranes using supported bilayers [30]. Here, a substrate was etched to have a periodically modulated (‘‘bumpy’’) curvature, onto which a membrane was laid. Upon deposition, the membrane was observed to undergo phase separation, in which softer (more rigid) lipids colocalized with highly (weakly) curved regions (Fig. 2D). 3. How proteins sense/generate curvature The energetic coupling between protein concentration and membrane curvature, as discussed above, has general biological implications regarding the protein distribution depending on membrane shape and on protein-induced membrane deformations. Specifically, if a protein has, due to its shape when embedded  (see Eq. (2)), it will in a lipid environment, a relaxed curvature C

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Fig. 2. Techniques used to study protein and lipid curvature-based sorting. (A) Single liposome curvature assay. Small liposomes of varying radii are tethered to a substrate. Protein sorting as a function of radius is measured from the ratio of protein to lipid fluorescence intensities. (B) Membrane nanotubes pulled from GUVs. Micropipette-optical tweezer set-up is used to extract membrane tethers of controlled radius. Protein and lipid sorting are measured using confocal microscopy; the action of sorting on membrane mechanics is inferred from changes in tube radius and pulling force [24,25]. (C) Small liposomes containing fluorescently-labeled lipids of differing shape. The fluorescence of the outer leaflet of the liposome is quenched (gray); the ratio of intensities before and after quenching is measured as a function of liposome curvature. In the absence of sorting, geometry dictates that I2/I1  1 + 2h/R, where h is the bilayer thickness and R is the inner liposome radius. Inverted cone-shaped lipids (red) are found to be weakly enriched in the outer leaflet; cone-shaped lipids (blue) are found to be weakly enriched in the inner leaflet [29]. (D) Supported bilayer on corrugated substrate induces membrane phase separation. Membrane curvature varies between zero, on hilltops, and 4 lm1 along grooves. More rigid, liquid-ordered phase (black) colocalize on hilltops, while softer, liquid-disordered phase (red) aggregates elsewhere, including grooves [30].

preferentially accumulate in curved membrane areas with the same sign of curvature. The energetic arguments presented above are purely generic, and all of the molecular–biological details are contained in the  in Eq. (2). There are three classes of proteins that parameters C we will consider here, and each has a distinct mechanism leading to protein relaxed curvature. The emphasis here is on proteins coupling to curved membranes via spontaneous curvature; we do not consider here membrane deformations generated by molecular

motors pulling on membrane or by cytoskeletal filaments, such as actin bundles producing filopodia. First of all, proteins that insert amphipathic or hydrophobic structures from one side of the membrane create stresses across a flat membrane bilayer, thus producing a non-zero, or spontaneous, curvature in the relaxed membrane [31]. For example, proteins such as Epsin, which is involved in endocytosis, contain unfolded N-terminal groups when in solution. Upon binding to the lipid membrane, however, they undergo a conformational

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change in which these groups fold into helices, such that the hydrophobic face is inserted into the bilayer [32]. The spontaneous curvature created by the insertion mechanism is related to three parameters: the typical size of the insertion; the depth of insertion into the bilayer; and the number of proteins per unit area of membrane (the density); see illustration in Fig. 3A. Secondly, proteins that possess a curved, rigid backbone can create spontaneous curvature if this backbone can bind strongly to the lipid membrane. In this case, spontaneous curvature is believed to arise due to a scaffolding action of the protein on the membrane; see Fig. 3B-left. As an example, it has been shown in vitro and in vivo that proteins containing the banana-shaped BAR (Bin, Amphiphysin, Rvs) domain can generate membrane curvature [33–35]. Specifically, the proteins centaurin and the sorting nexins, involved in vesicle trafficking, produce membrane tubules upon membrane binding [20,36]. Next, it has now been established that proteins, such as Amphiphysin and Endophilin, which contain a BAR domain and amphipathic helices, referred to as an N-BAR domain, can generate spontaneous curvature through the dual action of helix insertion into the lipid bilayer and the scaffold mechanism, introduced above (Fig. 3B-right). We note that proteins in the BAR superfamily can generate membrane curvature of both signs [34], through the

formation of a rigid coat that can impose its curvature on the membrane. The way proteins in the BAR family interact with curved membranes is a very rich problem, owing to the anisotropic, curved shape of the protein backbone and due to the capacity for its amphipathic helices to insert in the membrane. Theoretical [20,27] and experimental [24,37] work is currently being undertaken to shed light on this problem. Atomistic computer simulations have revealed protein-induced membrane deformation at the scale of a single protein [38,39]. Work done by the Voth group showed the importance of the electrostatic interactions between the protein backbone and the membrane in order for the membrane deformation by protein scaffolding to be effective [40]. To study the relation between protein organization and membrane shape at the level of several (i.e., 10–100 or more) proteins, a multi-scale approach has been used based on coarse-grained simulations that use data from the atomistic scale as input parameters [41]. Coarse-grained simulations of Endophilin on membranes revealed the importance of lateral interactions between helices of neighboring proteins for the stability of a protein scaffold structure and its effectiveness in curving the membrane [42,43]. In addition to helix insertions and scaffolding, membrane– bound proteins can produce spontaneous curvature through steric interactions between proteins and between protein and membrane

Fig. 3. Mechanisms for spontaneous curvature generation by proteins. (A) Curvature generation by protein insertion, for instance, by amphipathic helices (represented in red). (B) Membrane sculpting by protein scaffold and hybrid action of scaffold and helix insertion. (C) Steric repulsion between bound proteins causes crowding and membrane bending (left). Entropic pressure of grafted flexible chain deforms membrane away from graft site (right). (D) Interactions between proteins at high densities results in oligomerization and coat formation around membrane. Here, proteins are represented as red crescents, which form a cylindrical coat.

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(Fig. 3C). Recent work by Stachowiak et al. demonstrated that Epsin, even in the absence of amphipathic helices, when binding specifically to PIP2 lipids in PIP2 rich domains, generates membrane tubes [44]. This result is evidence of a crowding mechanism, in which steric interactions between proteins create a lateral pressure and a spontaneous curvature (Fig. 3C left). Based on the theory of polymer brushes grafted to interfaces [45,46], the brush contributes to the spontaneous curvature and to the bending rigidity, thus favoring deformation of an initially flat interface, with an energy release that increases with the polymer coverage. This is consistent with the crowding hypothesis of Stachowiak et al., whose work showed that the extent of membrane deformations increases with protein coverage [44]. Steric interactions between bound proteins and the membrane can also result in spontaneous curvature. The membrane deformation due to a single grafted polymer has been studied theoretically [47,48] and arises due to the entropic based pressure of the polymer on the membrane (Fig. 3C right). Experimentally, it has been shown that polymers grafted on membranes at very low density – in the mushroom regime – can generate spontaneous curvature, without lateral repulsive interactions between polymers and without insertions into the bilayer [49]. So far, we have discussed the ability of single proteins to generate membrane curvature, either through helix insertion or by scaffolding. However, another class of proteins – coat proteins – are known to deform membranes through protein self-assembly into a rigid, curved, shell-like structure which, bound to the membrane, imposes its curvature on the bilayer. For instance, dynamin or FBAR, when adsorbed on flaccid membranes, polymerize in a helical fashion and can generate highly curved membrane tubes [18,50], whose radius depends only on the coat geometry, and not on the mechanical parameters of the membrane, namely its tension and bending rigidity (Fig. 3D). We note that N-BAR domain-containing proteins, such as Endophilin and Amphiphysin, can deform membrane at the single protein and multi-protein level, either through the insertion/scaffold mechanism at low protein density on the membrane or by forming a loosely assembled coat over the membrane at higher densities [24,35]. In contrast, proteins such as COPI, COP-II, and clathrin, implicated in membrane trafficking, bind to membranes, oligomerize, and form spherical membrane buds ranging in size between 50 and 100 nm, with the protein coat forming a cage around the bud [51,52]. Owing to the greater degrees of freedom of proteins forming a two-dimensional cage than, say, dynamin forming a tubular coat, different cage radii for a given protein type can arise, and with different protein organization in the coat in order to accommodate various cargo sizes. It remains an open question to what extent the cage mechanics depends on the underlying lipid membrane.

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4.1. Mechanisms of curvature-induced lipid sorting The connection between the membrane bending energy and the lipid composition has its origin at the microscopic scale. First, the spontaneous curvature of a membrane is related to the lipid shape, packing, and the difference in lipid composition across the two leaflets; see Fig. 4A for an illustration. Second, the bending rigidity is a material property of the membrane and is related to lipid properties such as the bilayer thickness and the lipid compression modulus [56], and thus depends on the lipid composition (Fig. 4B) [3,56,57]. For instance, ternary mixtures containing cholesterol, saturated and unsaturated lipids are known to display different phases, notably a more rigid, saturated lipid-rich liquid ordered (Lo) phase and a softer, unsaturated lipid-rich liquid-disordered (Ld) phase [58–60]. While the simple picture of curvature-induced sorting of lipids based on molecular shape and flexibility is visually quite appealing, experiments and theory have shown that this is not an effective sorting mechanism against the homogenizing effect of entropy [25,55,61]. A simple order of magnitude calculation highlights the importance of entropy. The gain in bending energy in transferring a sphingolipid from a tube of radius R = 20 nm to a flat reservoir, given a liberal estimate of the difference in bending stiffness between the tube and reservoir of 40 kBT, is 1 DE ¼ 12 DRj2 a ¼ 40 kB T, for an area per lipid equal to a = 0.5 nm2. We see that this energy is much smaller than kBT, the scale of thermal energy, meaning that entropy will win out over any gain made in making the transfer. The small value of DE is due to the smallness of the lipid dimension compared with the radius of the tube. The above calculation is valid, however, only if we consider the mem-

4. Lipid sorting by membrane curvature Lipid homeostasis – controlled lateral lipid inhomogeneities across different membrane compartments – is a critical aspect of proper cell functioning [8]. Thus, as with proteins, cells require mechanisms to couple lipid organization to membrane shape. Much recent attention has focused on the role that membrane curvature plays in sorting lipids. In general, the two parameters describing the membrane bending energy, the bending rigidity j and the spontaneous curvature C0 depend on the local lipid composition of a curved membrane patch. If this patch is more curved than its surroundings, the bending energy can be lowered by adjusting the composition to reduce j and/or to have C0 match C – this results in lipid sorting. This idea has been explored theoretically, with special emphasis on the interplay between lipid inhomogeneity and the mechanical properties of vesicles [25,53–55].

Fig. 4. Mechanisms for lipid sorting. (A) Depending on the relative sizes of the lipid head group to the lateral extent of the hydrophobic chains, lipids can be represented as inverse cones (red), cylinders (green), or cones (blue). In general, the membrane spontaneous curvature depends on the relative proportions of inverse-cone, cylindrical, and cone-shaped lipids. (B) Curvature-sorting by spontaneous curvature. Membranes with zero curvature have, on average, a symmetrical distribution of lipids across the bilayer (left). Curved membranes, as shown on the right, have, on average, a relative enrichment of inverse cone-shaped lipids in the upper leaflet, and a relative enrichment of cone-shaped lipids in the lower leaflet. (C) Curvature-sorting by bending rigidity. Tubes, or other highly curved structures, formed from membranes consisting of multiple lipid types, will be enriched in lipids forming a softer bilayer (red), whereas lipids forming a more rigid bilayer will be sorted to the flatter membrane regions (green).

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brane to be an ideal solution of lipids. Membranes consisting of a mixture of lipids are characterized not only by elastic bending but also by the short-range interactions among the different lipid species, which can offset the entropic resistance to sorting. Below the miscibility temperature, these interactions give rise to largescale domains, as seen, for example, by the coexistence of Lo and Ld phases [58–60]. Sorre et al. measured the importance of lipid–lipid interactions on sorting using an integrated tube pulling through optical tweezers and confocal fluorescence imaging setup [25]. The GUVs used contained the saturated lipid sphingomyelin (SM), cholesterol, and the unsaturated lipid DOPC, as representative of the plasma membrane. They observed a relative enrichment of unsaturated over saturated lipids in the tube, consistent with earlier observations that stiffer saturated lipid-rich Lo-phase domains are expelled from tubes in phase separated vesicles [62]. This, however, was not detected for arbitrary mixtures of the three lipid types: proximity to phase separation, and hence lipid–lipid interactions, were critical for observing sorting. Tian et al., using a related technique, also observed lipid sorting for membrane mixtures close to phase separation [28]. In the case of highly curved spherical vesicles, the lipid compositions of the two leaflets can adjust to accommodate their respective curvatures. In the study by Kamal et al. [29], some asymmetry in the transverse distribution of the dilute amount of fluorescent lipid markers was detected, from which they were able to measure the spontaneous curvature of the marker, but found that the asymmetry in the distribution on the two leaflets only reached a maximum of about 20% for the smallest vesicles, of 20 nm in radius. The key conclusion from these works was that lipid shape alone is not enough to drive measurable lipid sorting: lipid–lipid interactions or lipid–protein interactions are necessary to amplify the curvature–based sorting. 4.2. Protein binding enhances lipid sorting Membrane–bound proteins interact differently with the various lipid species in the membrane and thus can strongly influence lipid distributions. First, when a protein binds to multiple lipid receptors, it clusters these lipids and has the capacity to sort more easily than an individual lipid due to the reduced entropy per lipid. For example, when cholera toxin binds to GM1 lipids on a vesicle, the resulting cluster has a natural curvature opposite to that of the vesicle; once a tube is pulled, the protein–lipid clusters will be excluded from the tube. Importantly, the curvature-based sorting of remaining lipids that interact non-specifically with those in the cluster is enhanced by cluster partitioning [25]. If, however, the cluster curvature matches that of the highly curved membrane, such as a tube, the cluster will be enriched, with the potential to colocalize non-cluster lipids that, in the absence of the protein, would be excluded [63]. 4.3. Lipid and protein sorting in bacteria So far we have discussed spatial organization of lipids and proteins in cell membranes in eukaryotic cells, though analogous questions are relevant to bacteria as well. Localization of proteins to the cell poles or mid-region in bacteria is critical, for example, for symmetric cell division (see Refs. [3,4] of [64]). Recent experimental work by Renner et al. [65] has confirmed theoretical work [64,66] predicting that, as in eukaryotic cells, bacterial lipids and proteins can be spatially sorted by membrane curvature. Renner et al. used a novel setup consisting of polymerbased microchambers of defined shapes; upon confining Escherichia coli spheroplasts (cell wall removed) in the microchambers, they observed sorting of the lipid cardiolipin and, in addition, that

of MinD protein which colocalizes with cardiolipin. Using epifluorescence microscopy, they found that cardiolipin microdomains, which have a non-zero intrinsic curvature, are enriched in the inner membrane leaflet at the curved bacterial poles [65]. Importantly, short-ranged attractive interactions between lipids were found to be essential to form these microdomains, which were sorted by curvature at the micrometer scale; however, individual lipids, due to their small size, could not be sorted by such small membrane curvature. This work supports the curvature sorting hypothesis that lipids and proteins can be spatially segregated for membrane energetic reasons. 4.4. Summary Experiments and theory show that curvature-based lipid sorting is generally ineffective against the de-mixing resistance of entropy. In cell membranes, it is very unlikely that individual lipids, unassisted by interactions with themselves or with proteins, can be enriched in curved regions simply based on their shape alone. Rather, larger-scale effects, resulting from lipid–lipid interactions or from lipid–protein interactions, appear to be essential in making the membrane susceptible to curvature-driven sorting. 5. Concluding remarks In the last decade, there has been growing recognition of the importance of cell membrane curvature in modulating its lipid and protein compositions. Quantitative, in vitro studies have generally focused on equilibrium properties; lipid and protein concentrations and membrane shape are determined by free energy minimization. However, membrane trafficking, occurring, for instance, during lipid sorting in the Golgi apparatus or protein-mediated endocytosis, are dynamic processes, and it remains unclear whether transient membrane shape and concentration inhomogeneities always equilibrate on physiological timescales of interest. In future developments, it will be essential to account for the dynamics of protein–induced membrane deformations and curvature-based lateral redistribution of the membrane constituents to understand the timing of sequential protein–membrane binding events that underlie intra-cellular transport. References Paper of particular interest have been highlighted as: * of special interest, ** of outstanding interest. [1] Conner SD, Schmid SL. Regulated portals of entry into the cell. Nature 2003;422:37–44. **[2] Helfrich W. Elastic properties of lipid bilayers: theory and possible experiments. Zur Naturforsch 1973;28c:693–703 [This seminal work laid the theoretical foundation for the study of fluid membrane mechanics. In particular, a free energy functional in terms of the membrane curvature was derived, containing two material parameters, the bending rigidity and the spontaneous curvature.]. [3] Marsh D. Elastic curvature constants of lipid monolayers and bilayers. Chem Phys Lipids 2006;144:146–59. [4] Bonifacino JS, Glick BS. The mechanisms of vesicle budding and fusion. Cell 2004;116:153–66. [5] McMahon HT, Gallop JL. Membrane curvature and mechanisms of dynamic cell membrane remodelling. Nature 2005;438:590–6. [6] Zimmerberg J, Kozlov MM. How proteins produce cellular membrane curvature. Nat Rev Mol Cell Biol 2006;7:9–19. **[7] Simons K, Van Meer G. Lipid sorting in epithelial-cells. Biochem-US 1988;27:6197–202 [This paper was the first to recognize the importance of lipid and protein sorting in cell membranes in maintaining composition gradients among different compartments of the cell. The existence of lipid microdomains – lipid rafts – were first hypothesized here as a necessary requirement for lipid sorting.]. [8] van Meer G, Voelker DR, Feigenson GW. Membrane lipids: where they are and how they behave. Nat Rev Mol Cell Biol 2008;9:112–24. [9] Lingwood D, Simons K. Lipid rafts as a membrane-organizing principle. Science 2010;327:46–50.

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