Caveolae as a Thermodynamic Phase Separation of Membrane Proteins

Simple mixtures of two or more material components can reside in a variety of states or "phases". The simplest of these is the mixed state, when all components are evenly mixed throughout the system. However, if molecules of one component have a sufficiently large mutual attraction for one another, or equivalently a sufficiently large repulsion from the remainder, they can "phase separate". There can then be large regions that are rich in this component suspended in a background which contains relatively little of it. It is this effect that causes oil to de-mix from water at room temperature but it is a far more generic effect than is often realized. It is now clear that there are components in the plasma membranes of cells that phase separate, for example, into caveolae. This is a slightly unusual phenomenon in that the phase-separated domains are typically only 100 nm across rather than macroscopic in size, but the principle is the same.

There is now good evidence that caveolin proteins form homo-oligomers [3] containing approximately 15 molecules. Whilst this is not a necessary feature for the generation of bending forces, which require only a molecular asymmetry between the two sides of the membrane, it may act to amplify those forces by increasing the density of interacting cytoplasmic domains (see Fig. 2.4).

It has been suggested [22] that the mechanism by which caveolin homo-oligo-mers form is reminiscent of micellization on a membrane. Usual in spherical surfactant micelles are formed by the aggregation of amphiphilic molecules which experience a mutual attraction between their hydrophobic tails [8]. Such micellar aggregates do not grow indefinitely because of the packing and stretching con-

Fig. 2.4 Sketch indicating the origin of the forces that act to bend the membrane near an asymmetric membrane protein (left), or homo-oligomer thereof (center), with domains extending on one (cytoplasmic) side only. The cytoplasmic domains may be entirely disordered, resembling a random coil (as shown), or may contain some folded structure(s) forced into dense contact within the oligomer. In any case these domains exert forces on the membrane. Even disordered coils have their configurational entropy restricted by the presence of the membrane. This restriction is large for a planar membrane but is reduced if the interface bends away from the coils, leaving more room in which to fluctuate. This means that the proteins exert forces that give rise to "bending moments" (as shown, right).

Fig. 2.4 Sketch indicating the origin of the forces that act to bend the membrane near an asymmetric membrane protein (left), or homo-oligomer thereof (center), with domains extending on one (cytoplasmic) side only. The cytoplasmic domains may be entirely disordered, resembling a random coil (as shown), or may contain some folded structure(s) forced into dense contact within the oligomer. In any case these domains exert forces on the membrane. Even disordered coils have their configurational entropy restricted by the presence of the membrane. This restriction is large for a planar membrane but is reduced if the interface bends away from the coils, leaving more room in which to fluctuate. This means that the proteins exert forces that give rise to "bending moments" (as shown, right).

straints of their tails. The same is true of caveolin. It has been determined that there is an attractive interaction between N-terminal segments of caveolin [3] and, as the aggregate grows, an increasing number of the caveolins enjoy such contacts. Eventually the repulsive forces between the greatly confined cytoplasmic domains is enough just to balance the force of attraction experienced by the next caveolin molecule that seeks to join the aggregate and, at this point, the optimal size has been reached. For caveolin oligomers this size appears to be ~15 molecules. Given the size of the cytoplasmic domains this indicates that the N-terminal attractive domains probably give rise to a substantial attraction, perhaps of the order of 10kBT [22]. This, in turn, results in substantial repulsive forces between the cytoplasmic domains in the oligomers which acts to "amplify" the bending forces indicated in Figure 2.4.

In the same way that there exists a critical micelle concentration (cmc) in surfactant systems there is a similar concentration at which caveolin oligomers will start to form. For simplicity we denote this the cmc. Above this concentration there will be a few single caveolin molecules on the membrane at concentrations equal to that of the cmc, with the remainder forming as many oligomers (micelles) as are required to incorporate all the caveolin. Given that caveolins experience a substantial attraction, their cmc is probably so low that oligomers will always form at physiological concentrations. However, there is another scale of self assembly which also, in its way, involves a concentration that resembles a cmc. This is a critical budding concentration (cbc) which lies above the cmc. Below the cbc all oligomers exist on a roughly flat membrane, whilst above it the flat membrane supports oligomers at, or very close to the cbc, while the remainder of the oligomers form buds that each have an area fraction of oligomers f * > fcbc. As more oligomers are added to the membrane, more N but similar N buds are formed. We are able to establish the cbc by comparing the free energy of a membrane bearing buds to one which does not. In the following section it will be assumed implicitly that the cbc is exceeded and hence buds form.

A free inclusion, as shown in Figure 2.4 (left and center panels), is one which is not anchored to any external structure such as the cytoskeleton. In this case it is further possible to prove that there can be no net overall force acting to move the membrane up or down, nor torques which tilt it left or right. This is a direct consequence of Newton's third law: The membrane cannot experience a force (or torque) without another body or structure experiencing one that is equal and opposite. If there is no such structure there can be no such forces. This leaves the bending moment shown in Figure 2.4 (right panel) as the dominant mechanism for local membrane deformation [23]. The inclusion pushes down on the membrane with its "arms" and pulls up with its "body", but it is not connected to any other structure.

We propose to investigate how an asymmetric inclusion such as the protein caveolin (or an oligomer thereof) can generate a local curvature in the membrane, and how this curved membrane can then provide an environment preferred by other identical curvature-sensitive caveolin molecules. Whilst this approach has the advantage of providing a formal method for calculating the size of a caveolae bud directly from physical arguments, it suffers from the limitation that several parameters are known only to within an order of magnitude. Its utility should therefore be understood in the following terms:

• It represents a check on whether invaginations of 100 nm diameter might possibly be driven by the physical process described. In particular, since we find that this is indeed plausible, it provides a mechanism for the formation of caveolae that explicitly does not involve cytoskeletal forces playing any significant role.

• Such a model is then able to predict how the stability and size of the invaginations will vary with the control parameters, for example, surface tension, membrane rigidity and the tendency of any inclusions to curve the membrane. We are then able to compare these predictions with experiments involving several mutant caveolins.

Whilst an exact calculation of these bending forces is difficult there is one calculation that can at least give us an indication of the magnitude of such forces. This, involves treating the cytoplasmic tails as random coils. In this case there are well-established rules from the theory of polymers [24,25] that allow these bending forces to be calculated exactly [20,23,26]. The basic principle is that the chains gain more configurational entropy if they locate next to a convex surface rather than a flat one. This is formally equivalent to saying that each caveolin oligomer imprints a local spontaneous curvature C0 on the membrane, which is of order C0 ~ f0/K, where f is a characteristic force exerted on the membrane by the cytoplasmic tails ("arms") of the caveolin oligomer, of the order 10pN [22].

One final physical "ingredient" is required in order to complete our model for the formation of caveolae. It is necessary to include the effect of mixing n caveolin oligomers, each of areas s, and hence area fraction f = ns/S, on the surface of a caveolae. Simplistically, we view the surface of the caveolae as being made up of two components, caveolin oligomers (with fraction f) surrounded by the rest of the caveolar membrane (with fraction 1-f). The energy of the membrane deformation is given by Eq. (2), where the spontaneous curvature of the bud increases with the density of caveolin, and is equal to G)f. Assuming that these components have no interactions of longer range than a, then this is a classical two-component ideal mixture. The free energy of mixing of this fluid is well known [12,22], contains similar contributions from the oligomer and non-oligo-mer membrane patches, and has the natural feature that it is very costly to remove all of either component. Indeed, the energy required to do this actually increases without bound as f or 1 - fw 0. For inclusions that interact with one another the interaction energy includes a contribution that scales like the density of oligomer -oligomer interactions (~ f2).

The formation of buds is controlled by a variety of physical processes that have been introduced above. These can be combined into a single equation for the free energy per caveolin oligomer on a caveolae of radius R containing oligomers with area fraction f. This merely encodes mathematically all of the physical contributions to the energy discussed earlier in this section. These are:

• The existence of a spontaneous curvature, indicating that energy is gained when

Lipid Membranes Origins

Fig. 2.5 The origin of the contributions to total free energy. (a) Energy is gained by bending the membrane in response to the bending moments exerted by the caveolin oligomers. (b) There is an energy cost when the membrane is bent away from its preferred shape: flat in the absence of caveolin and at precisely the preferred (spontaneous)

Fig. 2.5 The origin of the contributions to total free energy. (a) Energy is gained by bending the membrane in response to the bending moments exerted by the caveolin oligomers. (b) There is an energy cost when the membrane is bent away from its preferred shape: flat in the absence of caveolin and at precisely the preferred (spontaneous)

curvature when they are present. (c) There is an energy cost when the area of the sphere is removed from the cell membrane against surface tension. (d) There is an energy cost associated with both mixing of and interactions between inclusions on the sphere surface. The latter increases with the area fraction of buds.

the bud, with curvature 1/R bends in response to the bending moments of the oligomers; this is the only effect which drives bending of the membrane.

• The energy cost of bending a membrane away from its preferred shape. If there are no oligomers, the membrane would like to remain flat. When it forms a bud it need not always have a curvature exactly equal to its spontaneous curvature and this, similarly, costs bending energy.

• The energy cost of drawing the area of the bud away from the remainder of the cell membrane into the bud; this involves doing work against the membrane tension g.

• Finally, the repulsion between caveolin oligomers and the mixing energy must be included. The origin of all of these contributions is sketched in Figure 2.5.

The preferred caveolae state can be obtained by identifying the minimum of this free energy which, in turn, yields predictions for the preferred caveolae bulb radius R* and oligomer density f* (see Fig. 2.6). The invagination radius decreases with increasing caveolin density, so that the curvature of the bud approaches the spontaneous curvature of the caveolar membrane: C* = C0f*.

These results follow from the same model for the bud as a nearly-closed sphere attached to the membrane by a small neck, as was introduced in Section 2.4. We find that a physical description of caveolae can yield predictions for such observable quantities as the caveolae radius R that are in good agreement with observations. In this model we find that the primary reason why caveolae form is because of the coupling between membrane curvature and protein density: proteins accu-

Curvature Caveolae

Fig. 2.6 Variation of the preferred radius of the caveolar bulb R with surface tension of the membrane. The size of the caveolae reduces with increasing tension for small tensions, but at larger tensions reaches a preferred value that is rather insensitive to tension. The two sketches of the calveolar bulb indicate this, with darker shading representing higher density of caveolin. Inset: The variation of the critical budding concentration of inclusions, expressed as the area fraction fcbc, and the area fraction of oligomers on buds, f*. For all but the smallest tensions our simple two-component model predicts that the caveolae should be almost entirely covered with caveolin oligomers, at a concentration f * s- that is always above that of the surrounding membrane between caveolae ffcbc s-.

Fig. 2.6 Variation of the preferred radius of the caveolar bulb R with surface tension of the membrane. The size of the caveolae reduces with increasing tension for small tensions, but at larger tensions reaches a preferred value that is rather insensitive to tension. The two sketches of the calveolar bulb indicate this, with darker shading representing higher density of caveolin. Inset: The variation of the critical budding concentration of inclusions, expressed as the area fraction fcbc, and the area fraction of oligomers on buds, f*. For all but the smallest tensions our simple two-component model predicts that the caveolae should be almost entirely covered with caveolin oligomers, at a concentration f * s- that is always above that of the surrounding membrane between caveolae ffcbc s-.

mulate to a curved membrane (bud), stabilizing its shape and thereby attracting more proteins. If the formation of a curved membrane is too energetically costly because of a high surface tension, the domains are destabilized [22].

Also, we can again return to make contact with caveolae as lipid rafts. The consensus seems to be that these are the only rafts for which the identity and stability is not controversial. They are large, comparatively easy to observe, and very stable. We take this as evidence that the particular membrane curvature of caveolae stabilizes the phase separation into buds, a mechanism which is entirely consistent with the results of our physical analysis of their stability. Further evidence for a coupling between caveolae and membrane curvature can be found in the fact that an increase of the number of caveolae can enhance other types of membrane deformation, namely endothelial capillary tubule formation [6].

Finally, our theory provides a framework with which to seek to understand caveolae formation in mutant caveolin systems [27]. Mutants lacking the mutually attractive domain of the N-terminus are still able to drive membrane invagination, but with a much larger size R ~ 1 mm. This is consistent with the fact that the force exerted by isolated proteins should be an order of magnitude smaller than the force exerted by oligomers which results in a 10-fold increase in bud radius. Other mutants lacking the mutually attractive C-terminus also form larger buds. We would understand this as being due to a weaker oligomer - oligomer attraction, resulting in a lower density of caveolin in caveolae andtherefore larger buds.

The origin of the striated texture observed on the surface of caveolae [1], which superficially resembles tree bark, is still not understood. It may be that such structures are due to the spatial organization of caveolin oligomers on the surface of these "gnarly buds". If this is the case it may not be immediately obvious how circularly symmetric oligomers can organize themselves into asymmetric phases consisting of long, linear structures. However, it is now known that caveolin proteins also interact via the distal regions of their C-termini tails [3]. This attraction, together with a purely physical, membrane-mediated, longer-range repulsion that it is possible o estimate [23] could lead to a phase separation of the caveolin oligomers with caveolin-dense regions (stripes) coexisting with a caveolin-poor surface. Such phenomena have been studied in physically similar systems [28], and a reasonable conclusion would be that these structures may occur naturally as a result of a balance between a short ranged (C-terminal) attraction and a longer-ranged (membrane deformation-mediated) repulsion.

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