So far, we have two possible mechanisms by which invaginated membrane domains may form at the plasma membrane. The raft model of Section 2.3 tells us that a phase separation in the membrane promotes membrane curvature because of chemical immiscibility, the driving force is the domain line tension. The curvature instability model of Section 2.5, takes the point of view that membrane curvature promotes the phase separation because of the aggregation of proteins into patches of preferred curvature, the driving force being the protein spontaneous curvature. In both cases, the tendency to membrane invagination must overcome the membrane tension, which favors a smooth, flat membrane, and in both cases, increasing the membrane tension may lead to the flattening of the invagination. Figure 2.7 shows the effect of membrane tension on the energy of flat and invaginated domains for the raft model. At low tension, chemical incompatibility and the asymmetry of the membrane conspire to promote domain invagination, the budded state being the most stable state. At large tension, budding the domain costs too much energy, and invaginated domains flatten. Flat caveolae have indeed been reported in the literature , although they appear much less common than their invaginated counterparts. Figure 2.7 shows that an additional complexity arises from the fact that there is an energy barrier between the flat and budded states. This means that intermediate states (such as the hemispherical state - b = 0.5 of Fig. 2.7) are very unfavorable. This fact has two very important physical consequences. On the one hand, this means that the transition from invaginated
Fig. 2.7 Variation of the energy of a domain as a function of the domain shape for different membrane tension. The shape is characterized by a single parameter b, which vanishes for a flat domain and is equal to unity for a fully budded domain (corresponding to a closed sphere, a state that can be attained only if the domain endocytoses). The energy plot shows the domain energy (in kBT unit) for a domain of size R = 100 nm, of line energy a = kBT nm, and of bending rigidity k = 20 kBT The energy is shown for three different membrane tensions. Under low tension (red), the domain tends to bud to minimize contact with the surrounding phase, and the
Fig. 2.7 Variation of the energy of a domain as a function of the domain shape for different membrane tension. The shape is characterized by a single parameter b, which vanishes for a flat domain and is equal to unity for a fully budded domain (corresponding to a closed sphere, a state that can be attained only if the domain endocytoses). The energy plot shows the domain energy (in kBT unit) for a domain of size R = 100 nm, of line energy a = kBT nm, and of bending rigidity k = 20 kBT The energy is shown for three different membrane tensions. Under low tension (red), the domain tends to bud to minimize contact with the surrounding phase, and the minimum of energy is for b = 1. As the membrane tension increases (green), the energy of the budded domain increases with respect to the energy of the flat domain. At even higher tension, the flat domain becomes the minimum of energy. Flattening the invagination requires an energy barrier (AF) to be exceeded. The transition over an energy barrier (inset) is activated by thermal fluctuations, and requires a time that increases exponentially with the height of the barrier (see text). The higher the membrane tension, the lesser the time needed to operate the flattening of the invagination.
to flat domains does not occur continuously. Instead, the domains will abruptly snap open when the membrane tension is raised to a sufficient value. We argue below that interesting biological functions for caveolae could stem from this physical fact. Furthermore, we have seen in Eq. (1) that states of high energy are exponentially unlikely, which means that the passage of a barrier AF requires an appropriate thermal fluctuation. This is a rare event which occurs only after a time proportional to eAF/kBTm. This means that the response of caveolae to a mechanical perturbation is sensitive to the time scale over which this perturbation occurs.
A new putative function of caveolae at the plasma membrane emerges from these physical considerations, in addition to their supposed role in cell signaling and cholesterol transport (to cite only a few). Specifically, this is a possible role in cell mechano-sensitivity and mechano-regulation. Although direct experimental evidence of such role is lacking at this time, Figure 2.7 provides a clear picture of the effect that membrane tension should have on caveolae. The one remaining question at this stage is the level of membrane tension required to affect caveolae morphology. We are able to relate this tension to relevant physical parameters of the caveolar membrane, such as the line tension, bending rigidity, and spontaneous curvature [22,30]. The expected values for these parameters leads to the identification of a characteristic membrane tension that can be observed in cells. At present, however, these parameters are not known with sufficient precision for the theory to produce quantitatively precise predictions.
We may however investigate some biological consequences of the disruption of caveolae at high membrane tension. In that respect, the scenarios presented in Sections 2.5 and 2.3 are somewhat different. If the very formation of the caveolin raft is coupled with membrane curvature (which is what the thermodynamic model of Section 2.5 predicts), increasing the membrane tension will lead not only to flattening of the invagination but also to dispersion of the caveolin aggregate. One can relate this phenomenon to a putative function of caveolae in cell signaling, which is to hold signaling components inactive until they are released and activated by an appropriate stimulus . The thermodynamical model of Section 2.5 suggests that an increase in the mechanical tension of the cell membrane could provide such a stimulus. If, on the other hand, the phase separation leading to caveolin aggregates is independent of the shape of the membrane, the caveolin rafts will remain regardless of the membrane mechanical tension. Their morphology will however change upon tension increase, as is described Figure 2.7.
The morphological changes of caveolae with the tension of the plasma membrane provides the basis for mechanical regulation at the cell membrane. It is known that the tension of a cell increases if it is mechanically perturbed , and that cells have developed regulatory mechanisms to accommodate mechanical perturbations [32,33]. Caveolae might play a role in this mechanism, and many pathologies associated with caveolin seem to involve the mechanical behavior of the cell. One can cite their involvement in various muscular diseases , and defects in vascular relaxation and contractility in mice deficient of caveolin-1 . There is an increase of the number of caveolae in Duchenne muscular dystrophy , and such an increase has also been observed in cells subjected to long-lasting shear stress . Furthermore, there exists evidence that caveolin can contribute significantly to cell-cycle regulation , and cell entry into mitosis can be inhibited by artificially maintaining a high level of caveolin prior to mitosis. One can argue on the basis of physical arguments that the coexistence of flat and invaginated membrane domains regulates cell membrane tension . Figure 2.8 shows that the mechanism of a perturbation of the cell membrane area can be buffered by the flattening and invagination of domains. By this mechanism, caveolin expression could regulate and buffer the membrane tension of cells by controlling the number of caveolae at the cell membrane. Along with many other factors, this could be one reason why caveolin is down-regulated prior to mitosis , as a means low-
Fig. 2.8 The possible involvement of caveolae in mechanical regulation at the cell membrane. (Left) A mechanical perturbation is said to be positive if membrane area is taken out, and negative if it is put in. In the absence of invaginations, such perturbation would influence the membrane tension (visualized as a spring of various length), increasing it for positive perturbation, and decreasing it for negative perturbation. The presence of membrane invaginations, for which caveolae are a good candidate, allow
Fig. 2.8 The possible involvement of caveolae in mechanical regulation at the cell membrane. (Left) A mechanical perturbation is said to be positive if membrane area is taken out, and negative if it is put in. In the absence of invaginations, such perturbation would influence the membrane tension (visualized as a spring of various length), increasing it for positive perturbation, and decreasing it for negative perturbation. The presence of membrane invaginations, for which caveolae are a good candidate, allow buffering of the changes in membrane tension, as positive perturbations flatten the invagination (P1) prior to tension increase (P2), and negative perturbation leads to more invagination (N1), before it starts affecting the tension (N2). (Right) Variation of membrane tension with perturbation in the absence of invagination (dashed line), and with invagination (solid line). The membrane reservoir in the invagination helps keep the tension unaltered.
ering membrane tension by releasing the membrane area stored in caveolae, thereby assisting cell division. This is consistent with the fact that caveolin-1 knockout mice show an increased rate of cellular proliferation .
Compensatory pathways exist for cells lacking caveolin that do not exhibit cav-eolae at their plasma membranes . In order to investigate further the possible role of caveolae in tension regulation, one potentially promising line of research might be to perturb (e. g. mechanically) cells that do have caveolae, and to observe the effect of that perturbation on caveolae before any alternative regulatory mechanisms can have a significant effect.
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