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* Raman Research Institute, Bangalore, India; and National Centre for Biological Sciences, Tata Institute of Fundamental Research-GKVK Campus, Bangalore, India
Correspondence: Address reprint requests to Madan Rao, E-mail: madan{at}rri.res.in.
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ABSTRACT |
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50–100 nm that are comparable to vesicles involved in endocytosis. Because such chiral and tilted lipid molecules are likely to exist in "rafts", we suggest the involvement of this mechanism in generating membrane buds in the clathrin and dynamin-independent, raft-component mediated endocytosis of glycosylphosphatidylinositol-anchored proteins. We argue that caveolae, permanent cell surface structures with characteristic morphology and enriched in raft constituents, are also likely to be formed by this mechanism. Thus, molecular chirality and tilt, and its expression over large spatial scales may be a common organizing principle in membrane budding of transport carriers. |
INTRODUCTION |
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). A special case of membrane traffic is displayed in endocytosis, the uptake of membrane proteins, lipids, and extracellular ligands from the cell surface. Endocytosis occurs in a wide range of cellular contexts with vastly differing requirements; cells appear to have evolved a diversity of pathways in terms of molecular mechanisms, regulation, cargo specificity, and kinetics (2,3). One such endocytic mechanism is the clathrin mediated (CM) pathway (4) responsible for the internalization of proteins such as transferrin (Tf) and particles such as low-density lipoproteins (LDL) that bind to specific transmembrane receptors on the cell surface. A large number of membrane deforming proteins such as clathrin, epsin, and dynamin, have been reported to be involved in CM endocytosis (3–6). However, even in this well-studied pathway, the physicochemical mechanism of membrane deformation and pinching are poorly understood (5).
Cell surface lipid-anchored proteins such as glycosylphosphatidylinositol (GPI)-anchored proteins (7,8) on the other hand, are endocytosed via an entirely different pathway. This pathway is responsible for the pinocytic (fluid-phase) uptake in many cell types from mammalian to insect cells, and does not involve the membrane deforming proteins of the CM-mediated pathway (7,9). Furthermore unlike the transmembrane cargo of the clathrin-mediated pathway, GPI-anchored proteins do not have any cytoplasmic extension to link with other cytoplasmic proteins involved in the formation of the appropriate carrier. Interestingly, GPI-anchored protein trafficking can be regulated by altering levels of cellular lipids, specifically cholesterol and sphingolipids (10,11).
We have recently shown that lipid-anchored proteins such as GPI-anchored proteins are organized in nanoscale, cholesterol-dependent clusters. This clustering is necessary for GPI-AP endocytosis (12–14). Combined with the experimental evidence that the preexisting lipidic organization of GPI-anchored proteins is actively maintained in the cell, it is likely that these clusters are induced to form larger domains that are endocytosed (14). These active large-scale domains represent specialized lateral heterogeneities in the membranes, similar to the hypothesized membrane rafts, enriched in cholesterol and sphingolipids (15–17).
The absence of any of the conventional membrane deforming proteins (dynamin, clathrin, and caveolin, eps15) (9), raises an important issue regarding the mechanism of endocytosis of GPI-anchored protein containing domains, or rafts. Most importantly, how does initiation of membrane curvature of the desired length scale, a necessary precursor to vesiculation, take place?
In providing a physical mechanism for raft-assisted cellular budding, we need to address the question of the mechanics of membrane deformation at larger than molecular scales, i.e., at mesoscopic scales. Why is membrane deformation a mesoscopic scale phenomena? Consider, for example, a typical domain of diameter 100 nm on a flat membrane, which is subject to mechanical deformation resulting in a bud. Such a patch would consist of 103–104 lipids. At this scale, membrane deformation can be analyzed using continuum elasticity (18). Typical energy scales for membrane deformation, for example, leading to a clathrin-coated bud, are in the order of 10–20 kBT at room temperature. Therefore, to create the required deformation, a collection of force centers is necessary; budding is a result of a collective property of its constituent molecules (which in general include lipids and membrane deforming proteins). One of the aims of our theoretical study is to identify molecular features that are relevant for membrane shape, and consequently to membrane budding.
The specific lipid content of the raft (15–17,19) is sphingolipids (Sph), glycosphingolipids (GlySph), and cholesterol (Ch); these lipids are associated with the constitutive trafficking of GPI-anchored proteins (10,11). In this context, we can catalog those lipid aspects that may be relevant at the scales of the budding membrane patch: i), the stiffness of the long, saturated acyl chains leading to a high packing fraction below the main transition, Tm, the so called lo (liquid ordered) phase; ii), the presence of dipole moments on the headgroup; iii), the relative area of the head to the tail; iii), the presence of hydrogen bonding centers; and iv), lateral and transbilayer lipid heterogeneity. Here we argue that these molecular properties on their own, when coupled with membrane deformation produce bud sizes much larger than the typical endocytic buds.
Another molecular feature of the lipid constituents in the raft is their chirality. Chirality is the absence of mirror symmetry—a chiral molecule is one whose mirror image is a different molecule albeit of the same chemical composition (20). Indeed most molecules in the plasma membrane are chiral. However, molecular chirality needs to be expressed at larger scales to affect membrane shape.
Here we report that the special constitution and physical characteristics of rafts could promote the presence of a collective "orientation or tilt field" that is responsible for the expression of molecular chirality over the scale of the raft. This in turn leads to membrane shape deformations such as budding and tubulation. Using reasonable parameter estimates, it is possible to obtain bud sizes in the order of 50 nm (21). We suggest that the origin of the orientational field in rafts may be either a single molecular property such as molecular tilt (chain tilt or headgroup orientation) (22) of specific raft lipids, or a collective property such as the formation of chemical aggregates of the raft-associated cholesterol and sphingolipids, or nanoscale clusters of GPI-anchored proteins (13).
In addition to the simple spherical or tubular buds discussed above, we find that the taxonomy of membrane shapes arising from this interplay between orientation, chirality, and membrane elasticity includes novel "flask-like" and "grape-like" structures. These shapes show a remarkable similarity to caveolae that are permanent cellular invaginations at the surface of most eukaryotic cells (23). Caveolae are rich in raft lipids such as cholesterol and sphingolipids. We show that the conditions that promote such morphologies are consistent with the phenomenology of caveolae.
In summary, chirality, a common feature of membrane components, in conjunction with a tilt field can be expressed over large enough scales to induce membrane budding. Such a mechanism can result in bud sizes comparable with typical endocytic buds. We suggest that chirality-induced budding may be a common theme for membrane budding in different cellular contexts.
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INADEQUACY OF CONVENTIONAL MECHANISMS OF BUDDING OF RAFT COMPONENTS ON THE CELL SURFACE |
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–26), is based on the observation that in artificial membranes (freely suspended mono- and bilayers and giant unilamellar vesicles) containing a mixture of raft components Sph/Ch/PC, sphingolipids and cholesterol phase segregate from the rest over a wide range of temperatures (40°C) and composition (1:1:1), leading to macroscopic domains enriched in either Sph/Ch or the unsaturated PC, separated by sharp interfaces (27,28). The domains enriched in Sph/Ch were found to be in the liquid-ordered (lo) phase, characterized by higher packing fraction and stiffening of the hydrocarbon tails.
The tendency of the membrane to reduce the interfacial energy can lead to bud formation (24–26,29). In addition, budding can be facilitated and directed by the presence of a spontaneous curvature, an asymmetry between the two leaves of the bilayer, arising, for instance, because sphingolipids reside only in the outer leaflet of the plasma membrane. Any lateral segregation of these lipids on the outer leaflet, will automatically lead to a transverse lipid heterogeneity resulting in a local spontaneous curvature of the membrane. Local spontaneous curvature effects may also be augmented by the presence of cytosolic membrane bound proteins (e.g., caveolin (30)) and the cytoskeletal cortex.
Consider the simplest case of a raft domain P of area A = 
R2 and perimeter L on the outer leaflet of a tensionless membrane (Fig. 1); this domain contains specific lipids that are distinct from the rest of the membrane P'. The energy of such a membrane can be written as,
 | (1) |
0 is the line tension separating regions P and P', 
and ' are the corresponding bend elastic moduli and H, the local mean curvature (definitions in Appendix A). For simplicity, we have ignored a possible Gaussian curvature contribution. The spontaneous curvature H0 is a measure of the asymmetry in the lipid composition of the inner and outer leaflets in the region of the patch. We allow the conformations of the membrane to vary from a flat membrane with a circular domain of perimeter L = 2R to a spherical bud attached to the rest of the flat membrane via an infinitesimal neck, keeping the area A fixed.
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= '. Ignoring the negligibly small curvature energy contribution coming from the neck, we find, as first shown in Lipowsky (24), that if R is greater than a threshold, the membrane forms a spherical bud; this provides a minimum bud size rbud = /0. To estimate its magnitude, we need to determine the values of the elastic parameters in the raft region of the plasma membrane. A more practical approach is to take the values measured in artificial systems that best resemble the lipid composition of rafts on the cell surface.
The section "Estimation of parameters" contains a discussion of estimates of the parameters involved in membrane deformation, from which we take the following values: = 4 x 10–19 J for a mixture of dimyristoylphosphatidylcholine (DMPC) with 50 mol % cholesterol (resembling the local concentration of cholesterol in the putative rafts) at 40°C, and 0 
10–13 N in giant unilamellar vesicles (GUVs) containing unimolar mixtures of Sph/Ch/PC.
These estimates give a minimum bud size rbud 4 µm, at least two orders larger than in vivo bud sizes! In fact, it could be argued that this is an underestimate, because: a), we expect "compatible" nonraft lipids to organize proximal to the raft boundary, thus reducing 0; b), the coupling of the plasma membrane to the cortical proteins such as actin or other coat proteins would stiffen the membrane further; and c), the special lo nature of the segregated raft lipids would be accompanied by an increase in membrane thickness (31) and an enhanced splay stiffness; both these effects would lead to an increase in .
The estimate of the bud size could be reduced to some extent by transbilayer membrane asymmetries leading to a spontaneous curvature H0 or alternatively to a relative extension of the inner membrane leaflet with respect to the outer. Spontaneous curvature can arise from the transverse asymmetry of raft lipids, coupling to a variety of raft proteins and receptors (e.g., the GPI-anchored proteins), or a strong coupling to cytoplasmic proteins. However, given that there are several integral and peripheral proteins that bind onto either side of the membrane raft (32), it is difficult to ascribe a unique nonzero magnitude and sign to the spontaneous curvature. Relative areal extension of the two leaves of the bilayer can arise from incorporation of excess lipids onto one leaf, e.g., (33), as a result the membrane can form a high curvature bud to accommodate this increase in relative tension.
In the context of curvature generation in caveolae that share the same raft composition, it has been argued (30) that the binding of the cytosolic membrane protein caveolin to the inner leaflet membrane via cholesterol, produces bending moments on the membrane leading to a spontaneous curvature. Starting with a tension-bearing membrane, these authors explicitly compute the deformation of a membrane arising from a model of force distribution generated by the binding of the caveolin oligomer to the membrane (30). With their numerical estimates, they find that rbud 60 nm, comparable to the radius of caveolae. However, they do not differentiate between spherical buds and flask-shaped invaginations, nor do they account for the grape-like or tubular morphologies that are unique features of caveolae (34).
One problem with these mechanisms and estimates is that they largely ignore the special molecular features of the raft constituents, namely its lo organization. Both an increase in the local bilayer thickness (31), and an increase in the splay energy arising from the lo nature of raft lipids should go against the tendency to bud, since both effects lead to an enhancement of the effective .
Undeniably, contributions from these mechanisms are present in any budding context that involves lateral and transverse lipid heterogeneity. However, the numbers that emerge suggest that these mechanisms on their own cannot produce buds of the required dimension (50 nm) and morphology (e.g., grapes and tubules). This suggests that we need to look for additional bulk contributions to membrane deformation energy that are specific to the lipid composition of rafts. Moreover this mechanism should produce different morphologies observed in the context of specific raft lipid containing caveolae. In the following sections, we provide an explanation of why the interplay between an orientational field and chirality, characteristic features of raft components, may produce membrane deformation leading to a bud (21). We also present a detailed study of the morphology of membrane shapes that are generated by these interactions.
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"RAFTS": A MEMBRANE PATCH INVOLVING ORIENTATION AND CHIRALITY |
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,14). In this article we do not discuss the mechanism by which a "raft" membrane domain arises; instead we wish to understand the properties of raft lipids that could induce membrane curvature. For this we need to understand in greater detail, the molecular specificity and the nature of interactions between the raft components. Sphingolipids have long saturated acyl chains (as does the GPI anchor) attached to a small sphingosine head that has an amide group and a (zwitter-)dipole moment. Cholesterol is a short stiff amphiphile with a hydroxyl group at the head. Glycosphingolipids, another raft component, is a type of sphingolipid attached to a large sugar group oriented along the plane of the membrane. All these molecules are strongly chiral.
Although the organization of raft components in live cells has not yet been elucidated, several experiments on artificial membrane systems containing ternary mixtures of Sph/Ch/PC, over a range of temperatures, pressures, and composition (27,28,35), suggest that membrane regions enriched in sphingolipids and cholesterol may be identified with a liquid-ordered (lo) phase with high packing density (27,28). This is supported by x-ray diffraction (36) and NMR studies (37), which suggest that the sphingolipid acyl chains in the lo phase are stretched out, thus reducing chain-entropy and increasing the local packing density. Atomic force microscopy of suspended mono-/bilayers (31), has revealed that membrane regions identified with the lo phase have larger membrane thickness by 0.8 nm.
We suggest that active processes on the cell surface (14), primarily arising from cortical actin and other coat proteins, can give rise to a collective orientational field within the raft domain at the cell surface. For instance, cortical actin or coat proteins associated with raft regions can produce lateral stresses on the membrane bilayer, and thus modulate (decrease) the local bilayer thickness, inducing a tilt of the stiff acyl chains of the lo-raft lipids (S. Mayor and M. Rao, unpublished data). As is customary practice in liquid crystal physics, we denote the tilt version of the liquid-ordered phase by lo'. Alternatively, one may assign a tilt or bond orientation field with the cortical actin or coat proteins associated with rafts.
Whatever the origin of tilt or orientation, its presence on the raft domain immediately implies that local shape of the membrane should be governed by the coupling between tilt and curvature. This is borne out from numerous theoretical and experimental studies on artificial membranes (we provide relevant references as we go along). In addition, since the raft constituents are chiral, the existence of a well-defined orientational field allows this chirality to be expressed over the scale of the raft domain. This implies that local shape of the membrane should be governed by an interplay between chirality, tilt, and curvature. We will show that this is indeed the case; the interplay between chirality and orientation-curvature coupling (21) gives rise to a variety of membrane shapes such as buds, tubules, flasks, and grapes.
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DESCRIPTION OF A MEMBRANE CONTAINING ORIENTATION AND CHIRALITY |
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). However, (free)-energy considerations, a combination of hydrophobicity, van der Waals, and "hydrophobic shielding", constrain the center of mass of the molecules to lie on the two-dimensional (2D) membranal surface. Further the projection of the long axis of the molecule onto the 2D plane will have a fixed magnitude, since deviations of the projection from this fixed value cost a similar energy. Thus owing to strong uniaxial anisotropy, the orientational field at every point on the raft-patch may be described by a 2D polar vector m with unit magnitude (XY spin) (38). We will assume that within the raft-patch, the center of mass density 
(x, y) is uniform.
The raft-components interact with each other, and with the molecules outside the patch, both sterically (purely repulsive) and via short-range (e.g., van der Waals) attractive interactions. Both these effects contribute to chiral interactions; the former via the Straley picture of interlocking screws (39), the latter via a generalization of the Van der Waals dispersion to chiral molecules (40). In the continuum limit, these short-range interactions can be written as the usual Frank energy (41), modified to include the effects of chirality.
Of course, in addition to these short-range interactions there could be long-range dipole-dipole (or higher multipole) interactions between the tilt molecules carrying a permanent dipole moment. The long-ranged quadrupolar (or higher multipolar) interactions may also have independent chiral contributions. However, in this article, we will largely ignore the contribution of dipolar interactions, which we justify in "Estimation of parameters" by demonstrating that they are smaller than the Frank energy contributions.
Though the system of rafts embedded in the cell membrane may not be in thermodynamic equilibrium, we will assume that a single raft, taken to be a stable circular region of area A on the membrane, attains a conformation minimizing the free energy of that single raft (Fig. 1). This assumption tacitly entails another: variations in the size of the raft due to molecules leaving and entering the raft, either via diffusion or exo/endocytosis, are small compared to A. Furthermore, all macroscopic quantities associated with the raft, such as its energy, its texture, or its shape, are evaluated not at a single instant of time but are averaged over a timescale long compared to the timescale of variations in A, but shorter than endocytic or domain coalescence timescales of seconds to tens of seconds.
Energy functional describing the raft
Recalling that the raft components are on the outer leaflet of the cell membrane, our description of the bilayer membrane thus starts with a membrane patch of area A on the outer leaflet decorated by an orientation field m, the inner lipid leaflet being structureless. We then project these variables onto the neutral surface of the membrane (42), represented as a mathematical surface 
Each leaflet has its own elastic stiffness; combining the sheets, the elastic stiffnesses simply add (for this asymmetric bilayer). The raft will thus be a (simply or multiply connected) domain with perimeter L (which is allowed to vary) on this neutral surface. The conformation of the domain is described by the local texture m, the local membrane shape 
and the boundary 
(We will consistently denote 2-vectors with boldface and 3-vectors with an over-arrow.)
The effective energy-functional written in terms of m and the local membrane curvature Kij (see Appendix A for mathematical definitions) may be divided into contributions from within the patch (P), the boundary (
), and outside the patch (P'),
 | (2) |
The energy functional within the patch has contributions from distortions of the orientation m (written as a generalized Frank energy), deformations of the shape of the membrane (written as a Helfrich energy), and a coupling between the curvature and the orientation.
 | (3) |
The form of the energy follows from general symmetry arguments (21,43–47); here we retain terms up to quadratic order in fields and to lowest order in spatial derivatives. The former restriction assumes that the field values are small, the latter says that we are interested in mesoscopic scale physics, at the scale of the bud. To ensure that we have accounted for all contributions to this order, we write the energy in a covariant form (21,45). The generalized Frank energy can be written as,
 | (4) |
The generalized splay and bend terms are defined via the covariant divergence (Div) and curl (Curl) of a vector field m on a curved surface (Appendix A). For simplicity, we will assume the equal-constants approximation where k1 = k2 = k. Note that for a 2D vector field m, Curl m is a pseudoscalar: the kc and 2 terms are chiral and so are dependent on the density of the chiral molecular component.
The membrane deformation energy is written in the usual Helfrich form (48),
 | (5) |
(Appendix A). For convenience, we have assumed that the membrane has zero bare surface tension. The coupling between the texture and curvature is given by
 | (6) |
ij (Appendix A), and is referred to as the Helfrich-Prost interaction (49). In addition, there are anisotropic bending terms, such as (m·K·K·m) and (m·K·m)2 (50,51), which can lead to the formation of spherical buds and tubules on their own, i.e., without the help of chirality. We have however ignored such contributions since they are higher order in wavenumber and fields.
The contribution from outside the patch P' is given by
 | (7) |
In general, the elastic moduli 
are different in regions P and P'. In our variational calculation we will for the most part assume that membrane in P' is flat (or asymptotically flat) and that all shape variations are restricted to the region P. We will also ignore the contribution of the Gaussian curvature term.
The boundary energy is proportional to the perimeter of the boundary 
with a line tension 0,
 | (8) |
Note that the total derivative terms Div and Curl in Slepnev and de Camilli (4) can be integrated to the boundary via a generalized Gauss and Stokes law (52); this will give rise to an anisotropic line tension. For simplicity we will fix the boundary to be a circle on the flat membrane surface P', take only the isotropic tension, and, ignore a potential geodesic curvature contribution to the boundary energy.
Given the total energy functional, we obtain the optimal conformation of the membrane shape and texture that minimizes this energy, subject to two constraints. One is that the orientation m is a unit vector—this may either be ensured by a "hard-spin" version of the model (where we explicitly set |m| = 1, by suitable parameterization) or a soft-spin potential of the form V(m) = –
(m·m) + ß(m·m)2, which makes deviations of |m| from unity hard to obtain.
A note of caution—our restriction to terms with lowest order in spatial derivatives is valid only when the length scale over which the deformation occurs is large. To check whether this restriction is valid over scales corresponding to the bud size, we have explicitly considered the contribution of symmetry allowed terms containing higher order spatial derivatives such as, (m·K·m)(Div m), 
mkKkimj(Djmi), and a chiral contribution (
)(Curl m). We find them to be smaller than the terms retained; indeed the effect of these terms (except the chiral term) is to renormalize the spontaneous curvature c0 and ß, favoring the formation of a bud.
Before ending this section, we restate that the parameters in front of the chiral terms in the energy functional, principally kc and 
are nonzero only when the constituent molecules are chiral. They are phenomenological parameters that may vary with temperature, concentration, and surface pressure, and may even change sign (40).
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PHASE DIAGRAM: TEXTURE AND SHAPE |
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) to obtain the optimal shape and texture—this involves: i), guessing the right conformation; ii), expressing the conformation by a few parameters; and iii), obtaining the optimal values of the parameters. Most often our guesses are based on symmetry considerations and a general understanding of chiral structures; in some cases, however, they are guided by Monte Carlo simulations. Because our aim is to understand the nature of budding induced by chirality, we will simplify our energy functional and focus particularly on the effects of chirality.
Without loss of generality, we may set our units of length and energy such that k = 1 and 0 = 1.
Before we examine the effects of chirality on a deformable membrane, it is instructive to study finite chiral textures in the rigid ( 
) limit where the membrane is a flat 2D plane.
Texture on a flat membrane
In the case of a flat membrane, the form of the energy functional is considerably simplified (21,53–55). Keeping only the isotropic tension, we can rewrite the Frank energy functional (4) as
 | (9) |
Increasing the chiral strength, kc > 1 (in units of Frank constants), the raft would assume a texture with a high curl and a divergence equal and opposite to the curl. Such a condition is satisfied by the Archimedes spiral texture (Fig. 2 a), where the lines of m diverge from the center C. In polar coordinates (r, 
) with the origin being at the center of the raft, the spiral described by m 
(mr, m
),
 | (10) |
 | (11) |
 | (12) |
where the lines of m diverging from the center C, subtend an angle /4 with respect to the local radial direction. The energy of this optimal texture is
 | (13) |
is the radius of the raft, rc and 
c are the core radius and core energy of this spiral defect. The chiral energy density is large (and negative) in the vicinity of the core, and falls off as r–2.
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,55), we showed that a chiral tweed texture, with the above characteristic (Fig. 3), wins over the spiral defect phase. We were then able to parameterize this texture and calculate its energy analytically. This gives the phase diagram Fig. 4.
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) when the membrane is deformable; the chiral interactions are now represented by two terms kc and 
Before exhibiting a detailed phase diagram (21), we provide a qualitative understanding of the effects of these chiral terms on the shape and texture of the membrane.
Start with = 0: we have just shown that the optimal texture of a circular domain of radius R on a flat membrane when kc > 1, is an Archimedes spiral diverging from the center of the domain (Fig. 2 a). If the membrane is made flexible, then the spiral can close itself on the opposite pole of a sphere, producing two centers of chirality, C1 and C2, instead of one (Fig. 2 b)—this conformation gains in both bulk chiral energy (Appendix B) and line tension energy. A spherical bud would be produced if the kc contribution is sufficiently strong to overcome the rigidity of the membrane.
Now start with kc = 0: as shown in (49) and explicitly demonstrated in Appendix B, a sufficiently large value of would prefer to wrap the texture in a helix around a narrow cylinder, the pitch of the helix being proportional to the radius of the cylinder.
Thus the interplay between kc and will produce a combination of spherical caps and cylinders. Appendix B contains detailed calculations of the combined effects of kc and for textures on prescribed surfaces such as the sphere, cylinder, and saddle. These calculations help us in constructing general variational shapes (obtained by patching these surfaces) and textures (smoothly connecting the lines of m), which we optimize to obtain a detailed phase diagram. To highlight the effects of chirality we have ignored the spontaneous curvature c0 of the raft. Including the effects of c0 and ß (Eq. 6) would enhance the tendency to form buds even further.
We parameterize the spherical bud by a spherical cap of radius rbud attached to the rest of the membrane by an infinitesimal neck of radius r0. Using the parameterization of the texture m as given in Appendix B, we have calculated the optimum energy (texture+shape) for = 0 variationally,
 | (14) |
c = rc/rbud is the angle subtended by the defect core of radius rc at the center of the bud, and 0 = r0/rbud is the angle subtended by the neck at the center of the bud. The contributions k and c represent the energies of the neck and the defect core, respectively. Because the area of the domain is the same, before and after, the formation of the bud, we have,
 | (15) |
The chiral bulk energy kc prefers to have zero neck radius, as seen from the variational calculation. This is because an infinitesimal neck allows the spherical bud to have two defects, resulting in a gain in chiral energy. Moreover, as 
the neck energy 
(26,56). As we will see later, the Helfrich-Prost contribution, 
reduces the energy cost of the neck even further.
As we increase the value of the bud is stretched into a prolate shape, with the defect drawn away from the neck. We represent this prolate bud by a cylinder of length l capped by two hemispheres of radius rbud on either side, one of which joins the rest of the (flat) membrane via an infinitesimal neck. The m texture on the cylinder is the helix described in Appendix B, whereas the m texture on the sphere is the spiral described in Appendix B (and above). Note that the helical lines of m on the cylinder smoothly join the spiral lines on each hemisphere. The energy of this prolate bud is
 | (16) |
We have numerically obtained the optimum shape and texture of the bud, with the constraint that the area of the bud
 | (17) |
becomes of the order /R (Fig. 5 b).
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to be positive everywhere; had they been negative we would simply reflect the optimal texture shown in Fig. 5 a on a mirror passing through the axis of the bud.
The variational calculation just outlined produces the phase diagram (Fig. 6), showing how a domain of size R on a flat membrane can give rise to a spherical/prolate bud or tubule by turning on the strength of chirality; the transitions are discontinuous. For instance, a domain of size R = 0.01 (corresponding to 10 nm) on a flat membrane can be induced to form a spherical bud as soon as 
for = 10 and kc = 2 (this, as we will see in the section "Caveolae: a consequence of tilt and chirality?" are perfectly reasonable estimates). Recall the lower bound rbud = 4 µm in the section "Inadequacy of conventional mechanisms of budding of raft components on the cell surface"; the tendency to bud via bulk chirality preempts budding induced by line tension alone.
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That anything unusual should happen for larger domains may be gauged by the following argument (55). Consider a chiral tilt domain of radius R on a flat membrane with . Because increasing the strength of kc beyond unity produces a spiral defect at the center of the domain, we expect that when kc > 1, the texture would prefer to maximize the number of spiral defect points. One way to achieve this is for the domain to split into multiple domains. To study the conditions under which such breakup is favorable, we calculate the energy 
of n circular domains of equal area, each bearing the same spiral texture and compare it to the energy 
of a single circular domain with the same total area and texture. The total energy of this configuration is
 | (18) |
For small values of R, a single domain has the least energy. As R increases, 
becomes smaller than : chirality in the bulk wins over interfacial energy. As R increases further, multidomain splitting is favored. This tendency to split holds when kc is large enough; for a fixed value of 0rc, there is a critical kc beyond which chirality-induced splitting would manifest. The relevance of this analysis to the observed domain repulsion in lipid domains on tense GUVs consisting of two lipid components has been discussed in Sarasij and Rao (55).
The above argument can be extended to bud splitting when the membrane is deformable. We find that as long as the chiral parameters kc and are large enough, the bud will prefer to split into two beyond a critical size. Assuming that the neck of the spherical bud attached to the parent membrane is infinitesimally small, we have for the total energy of n equal buds with the same texture,
 | (19) |
This form assumes that the buds do not interact with each other. With a large chiral strength 
we find that for small values of R, a single bud 
has the least energy. As R increases, 
becomes smaller than : bulk chirality prefers the bud to split into two, when R > R* as seen in Fig. 7.
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CAVEOLAE: A CONSEQUENCE OF TILT AND CHIRALITY? |
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The surface of most mammalian cells have stable cellular invaginations known as caveolae (34). Caveolae are rich in cholesterol and sphingolipids (57), and other raft constituents. They are morphologically distinct: large flask-shaped or "grape-like" invaginations on the plasma membrane, with a diameter nearly an order of magnitude larger than the size of the raft-assisted buds discussed earlier.
A defining feature of caveolae is the presence of caveolin, coat proteins that striate the cytoplasmic surface of caveolae. Caveolin binds to cholesterol and glycosphingolipids and is firmly anchored to the membrane by a palmitoyl chain (Fig. 8). Caveolins oligomerize on the membrane forming the characteristic spiral striations observed in freeze-fracture images. It is likely that this ability to bind cholesterol and form oligomers helps sequester "rafts" into larger structures (58), thus stabilizing the caveolar pits (59). In our view, this binding due to caveolin oligomerization is the additional molecular mechanism needed to hold the raft together and make a domain larger. This novel role for caveolin is in addition to other effects that membrane-bound caveolin might have such as generating bending moments to curve the membrane (30).
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(Fig. 10); ii), large raft area R (Fig. 11); and iii), high bending modulus (Fig. 12)—conditions that are characteristically met in caveolae.
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a neck is the favored conformation, i.e., 
We model the neck by patching together a saddle and a cylinder (Fig. 9). We have described the saddle geometry in Appendix A, and have seen that the texture favored by chirality (Appendix B) is the one in which the lines of m at any point bisect the right angle between the transverse and the longitudinal sections of the saddle passing through that point (see Fig. 15). The neck begins at the smallest cross section of the saddle, the circle 
of radius Rß and angle = 0 and fans out to the maximum angle = max, where the radius of the cross section is Rß + R(1 – cos max) (Fig. 15).
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The first neck subtends an angle 0 at the center of the sphere (Fig. 9), thus 
further, as the neck joins up with the cylinder of radius RC, we have Rß(1) = RC. If RS is the radius of the sphere then from geometry,
 | (20) |
The energy of the first neck is
 | (21) |
 | (22) |
The second neck has to join the cylinder smoothly to a flat membrane, thus 
and as before, Rß(2) = RC (Fig. 9). The domain boundary has a length 2(R(2) + RC). The energy of the second neck is
 | (23) |
 | (24) |
The total energy of the flask can now be written as,
 | (25) |
0 is given by Stryer (20). We have numerically obtained the optimum shape of the flask (i.e., the values of {RS, RC, R(1), R(2), LC} that minimize Eflask), subject to the constraint of constant total area,
 | (26) |
The optimal shapes fall into two broad classes (Fig. 10): (A) a spherical bud, with no neck, i.e., LC = 0, and (B) a flask shape, with LC > 0. Every bud has 
and 
while every flask has and 
Therefore the necks are narrow and the shape of the flask is almost entirely determined by the dimensions of the spherical and the cylindrical parts.
We now study how changing R, and affect the shape parameters of the flask. For fixed values of R, kc, and , flask shapes are obtained only when the chiral strength crosses a threshold, any smaller value will produce only a bud (Fig. 10). This threshold increases with increase in kc (Fig. 11). More interestingly, the threshold decreases with an increase in R (Fig. 11), implying that larger (stable) domains favor flask formation. Thus for a given kc and there is a minimum size, Rmin, for a raft to be a flask (Fig. 12), consistent with observations of caveolae supporting cells. The transition from bud to flask is discontinuous—keeping R and kc fixed, the length of the cylindrical part jumps sharply from zero beyond a threshold (Fig. 10).
We comment on
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ABSTRACT
INTRODUCTION
on the outer membrane. The unit vectors n and t represent the normal and tangent to the boundary 
as a function of 
