## Physical Modeling of Lipid Membranes

Mathematical models of deformable, fluid membranes have been available for many years [7,8], and have been successfully compared with experimental results, both on artificial [9] and biological [10] membranes. At the most fundamental level these theories rely on the single basic principle underlying statistical mechanics: that the probability of observing a given membrane deformation depends on the energy change involved in making this deformation [11]. The higher the energy, the less likely the deformation. Statistical mechanics tells us that the probability pi of an event i is related to its energy Fi according to:

Fi kBTm

This probability compares the deformation energy Fi to some energy source in the system. This energy is written F to remind us that it is a free energy and therefore includes changes in entropy, as well as internal and chemical energies [12]. Reactions that reduce the entropy of the system are disfavored in the same way as are those that involve a spontaneous increase in the energy by, for example, disrupting chemical bonds. Strictly speaking, Eq. (1) only holds for (sub) systems that are at equilibrium but this can often be a reasonable approximation, for example for small patches of membrane that can move and relax quickly, even though it may be inappropriate for the cell as a whole. In passive systems, the only energy source comes from the thermal fluctuations of energy kBT, where kB is the Boltzmann constant and T is the temperature (in Kelvin). Biological systems are called "active", because chemical energy, coming from, for example, ATP hydrolysis, can be harnessed by specific enzymes (molecular motors) to perform mechanical work. The cell membrane is generally the site of many active processes, including cytos-keleton polymerization and ion pumping. One may adopt the approach that these active processes provide an effective "membrane" temperature Tm > T [13], and it is this that appears in Eq. (1).

In practice, much information can be obtained by the study of membrane deformations that minimize the membrane energy (those having the higher probability to occur). One contribution to membrane energy can arise from any change in the area of the cell, which must act against tension in the membrane. This is reminiscent of the work required to deform a child's balloon, for example by pinching a small patch of its surface between the fingers. As with any interface, a lipid membrane bears a "surface tension" (denoted g throughout), which is the energy cost per unit area associated with decreasing the membrane area. However, while the surface tension observed at a water air interface is of the order 10-1 J m-2, and typically dominates any other type of deformation energy, the surface tension of lipid bilayers can be extremely low (10-8 J m-2 in very floppy artificial systems, and ~10-5 J m-2 for the plasma membrane). As the surface tension is low, other modes of deformation can also play an important role. One such mode is the energy associated with bending the membrane. A symmetrical bilayer membrane prefers to be flat, so that both monolayers have the same structure. Bending the membrane one way or the other breaks this symmetry, and costs an energy which varies quadratically with the membrane curvature (deformation) C. This is fundamentally analogous to the fact that the energy of an ideal spring varies with the square of its extension (known as Hooke's law), and is ultimately the reason why any flexible material that is bent will spring back into its original shape. If the membrane is asymmetrical, and cell membranes are indeed rather asymmetrical, it may prefer a non-zero curvature. This means that the membrane energy might be minimized by, and the membrane therefore most happy with, a non-zero curvature. This curvature is called the "spontaneous curvature" C0. In this case, the deformation energy is again quadratic, but now is the difference between the membrane's (local) curvature and its spontaneous curvature. This can also be identified with a version of Hooke's law. While the membrane tension tells us how much the energy increases when the membrane area is increased, the energy increase caused by a deviation from the preferred membrane curvature is controlled by the "bending rigidity", conventionally denoted k. Adding the curvature energy to the energy of membrane tension, the total energy of a patch of membrane of area S, with a curvature C is

A Typical value for the bending rigidity of biomembranes [10] is k = 20 kBT It is convenient to measure energies in units of the thermal energy scale which, at room temperature is, kBT = 4 x 10-21 J. Thus, 1 kJ mol-1 = 0.4 kBT

In this chapter, we will be mostly concerned by the flask-shaped membrane deformations mimicking the caveolae (Fig. 2.1). For simplicity, we will assimilate the invagination to a spherical cap of constant curvature. In practice, there exists a membrane neck connecting the concave central cap to the flat surrounding membrane. Specialized proteins are likely to be present near the caveolae neck [14], and this is not included in the present models. From Eq. (2), the energy of a spherical membrane (with no spontaneous curvature), is Fsphere = (gS + 8pk). The energy of large patches is dominated by membrane tension, and the energy of small patches by membrane rigidity. Clearly, this has strong consequences for the stability of membrane invaginations in general, and of caveolae in particular. Indeed, small invaginations all have the same energy (~8pk), which is dominated by the bending energy of the membrane. Large invaginations on the other hand, have an energy which increases with their size (~ gS), and are much less likely to be observed. The cross-over size between small and large invaginations in the physical sense corresponds to an area S ~ 8pk/g. Choosing a bending rigidity k ~ 20kBT and a surface tension g ~ 10-5 J m-2, the cross-over size corresponds to a sphere of radius R ~ 120 nm. The fact that this scale is close to the typical size scale of the invaginations is very encouraging for our physical approach. It indicates that even such simple physical arguments can reveal a competition between different physical energies (and hence forces) that could give rise to invaginations with roughly the observed size.

Of course biological membranes have a complexity that is not reflected in the seminal elastic model of Eq. (2). In particular, the complex lipid composition (up to 25 different lipid species), the inclusion of a host of membrane proteins (~30% of the whole genome), and the support of the membrane cytoskeleton. The question of how to incorporate the two former features will form the subject of most of the following text. The cytoskeleton provides a visco elastic scaffold to the cell, and is able to exert direct forces to the membrane. Cytoskeletal anchoring of the plasma membrane is crucial to the membrane's mechanical behavior. The breaking of

Fig. 2.1 Sketch of a spherical membrane cap. The membrane curvature C is roughly constant over the deformed area S which has radius of curvature (the radius of the circle passing through the cap) 1/C, as shown. Inset: the bilayer structure of a lipid membrane.

Fig. 2.1 Sketch of a spherical membrane cap. The membrane curvature C is roughly constant over the deformed area S which has radius of curvature (the radius of the circle passing through the cap) 1/C, as shown. Inset: the bilayer structure of a lipid membrane.

some anchoring sites upon cell deformation and membrane extension is a major component of the energy cost of such deformations. In particular, cytoskeleton anchoring can account for up to 75% of the measured tension of cell membranes [15]. Here, we adopt the philosophy that the cytoskeleton acts to maintain the plasma membrane under tension, but does not exert direct forces to pull flask-shaped invaginations from the membrane. In fact, we will find that a consistent physical explanation of invagination can be constructed without the cytoskeleton playing any direct role in the formation of caveolae.