Statistical Mechanics of Membranes

The past decade has seen a surge of interest in the statistical mechanics of two-dimensional surfaces, coming from directions as disparate as fundamental particle physics, biophysics, and chemical engineering. Models for subatomic particles in which the elementary constituents of matter are string-like (and hence sweep out two-dimensional trajectories in space and time) hold out the tantalizing prospect of finally reconciling gravitation with quantum mechanics, a problem as old as quantum mechanics itself. On the other hand, physical membranes composed of fatty chain molecules are a basic architectural element of biological systems. The study of structural phase transformations of such membranes, driven by thermal fluctuations, may shed light on simple biological processes. Finally, quite apart from these fundamental applications, the study of membranes is of practical importance for understanding microemulsions and other ``complex fluids''.

To picture the physical membranes described above, imagine mixing oil and water to get salad dressing. As soon as we stop stirring the mixture it rapidly begins to separate; it consists of two immiscible fluids with a large surface tension for the interface between them. The tension means that the system is happiest with the smallest possible surface area, so two bubbles of oil in the water will tend to fuse, making a single bubble with less area than the two parent bubbles.

To find something more interesting, suppose that we add some egg to the mixture. Now we obtain mayonnaise, a long-lived suspension of tiny droplets of one fluid in the other. What has changed? The egg is playing the role of a ``surfactant''. More generally, detergents let us mix oil and water because they are ``amphiphilic'' molecules, long chains with one end liking oil and the other end liking water (i.e. the ends are nonpolar and polar, respectively). In a mixture they migrate to the interface, where each end can be happy (minimize its free energy) by touching the fluid it prefers.

With this picture in mind, one can ask, what happens when we omit the oil, instead just mixing surfactant molecules with water? One might think that there would be no way now for the molecules to be happy, but Nature is more clever than this. Instead the molecules form a double layer, with their water-loving ends pointing outwards to the water and their oil-loving ends pointing inwards at each other. The long, narrow molecules align perpendicular to the surface, as in a field of wheat. Such a double layer, or ``fluid bilayer membrane'', can form a thin (20-50 Å thick) two-dimensional surface of almost unlimited extent. The word ``fluid'' emphasizes that the individual molecules may slosh around each other freely, provided they always form a seamless sheet. The membrane maintains its integrity not because the molecules are chemically bonded to each other, but simply by the free energy cost of tearing it and letting the water touch the interior. Thus amphiphilic molecules will spontaneously self-assemble into huge, thin, flexible surfaces nearly impermeable to water, freely suspended in the surrounding water.

In fact the membranes of biological cells seem to be constructed in basically this way. They are mainly composed of amphiphiles called ``lipids'' (for example, lecithin), and the basic structure is a bilayer, although many additional elements such as embedded proteins complicate the picture. Thus it is of considerable interest to see if basic cell membrane behavior, including the shapes of normal and abnormal cells, the budding of sacs off a main cell, and so on, can be understood from a simple generic model of such a membrane. This is not such a tall order. In fact, chemists and physicists have long been familiar with the properties of polymers, enormous molecules in the form of long chains of atoms. Many important mechanical properties of polymers are independent of the detailed chemistry of the atoms. For example, simple scaling relations describe how the diffusion of a polymer depends on the length of the chain. These scaling relations can be derived from simple models of the polymers as long strings with a certain resistance to bending, some thermal excitation, and similar generic physical assumptions. Can we understand two-dimensional membranes in an equally simple way?

In the simplest model we describe a membrane as a mathematical surface in space, neglecting the fact that it is really composed of constituent molecules, which may themselves have interesting dynamics. Since the van der Waals (and other) forces between molecules are of short range, we suppose that these complicated forces can all be summarized by assigning to each surface an effective free energy cost computed as a local functional of the shape. Remarkably there is very little freedom in choosing such a functional; thus even if we cannot compute the effective free energy from first principles, only a handful of phenomenological constants determine it completely. [Similarly, in the ``string'' models of elementary particles mentioned in the introduction, the highly constrained form of the free energy functional lies at the heart of these models' success at dealing with the divergences which plague other quantum-mechanical models of gravitation.]

Referring to the above description of the physical membrane we can easily see that curved surfaces will pay an energy cost: as we bend a surface some molecule heads get squeezed together while those on the other side of the surface get pulled apart from their preferred spacing. Mathematically, at each point imagine the tangent plane to the surface at that point. As we move away from the point the surface in general bends away from the tangent plane. The distance, a function of how far we have moved in the surface, determines a second-rank ``curvature tensor'', somewhat analogous to the strain tensor of elasticity theory. The free energy cost must be some invariant function of this tensor, integrated over the whole surface. In fact one can show that there is precisely one such function, up to irrelevant terms, with just one unknown coefficient. (Technically this term is the square of the ``mean curvature''.) Thus one phenomenological parameter, the ``bending stiffness'' parameter tex2html_wrap_inline27, fully determines this simplified ``curvature model''.

Of course a membrane in the form of a bag (or ``vesicle'') enclosing fluid cannot have everywhere zero curvature; to do that it would have to rupture, which would expose the nonpolar tails of the amphiphiles to water. Instead it seeks a compromise, minimizing its total mean curvature at fixed enclosed volume. The best compromise shape is a theoretical prediction for the shape of membranes with given stiffness, total area, and enclosed volume. Remarkably, this extremely simple model (and generalizations almost as simple, in which the inner and outer layers have different total area) manages to predict nontrivial shapes actually seen in nature, such as the famous double-concave red blood cell shape, and even shape transformations induced by changes in temperature, such as budding of smaller vesicles from a larger parent.

But there is more to the story than equilibrium shape. Red blood cells in particular have been known since the nineteenth century to exhibit gigantic fluctuations in shape, visible even in the light microscope. This ``flicker'' phenomenon was originally attributed to some biological process, but eventually understood as an analog of Brownian motion for flexible two-dimensional surfaces in a fluid at room temperature. To understand it we need to progress from simply minimizing the free energy of a surface to computing statistical sums over all possible surfaces, weighted according to the rules of statistical mechanics by the free energy. Once again the almost trivially simple curvature model gives a good account of these shape fluctuations.

Even more dramatic manifestations of thermal fluctuations are possible. For large enough membranes, or great enough temperatures, thermal fluctuations can disrupt altogether the simple picture of a more or less flat surface, leading instead to a crumpled mass of lipid. In fact a rich array of possible phase transitions between different structures emerges once we begin to make the curvature model more realistic. In general we should not ignore the fact that a real surface is made of molecules, which can pack in various inequivalent ways. For instance, often the long narrow molecules do not lie perpendicular to the plane, but rather snuggle closer to each other by tilting off the normal. The direction of tilt can change from place to place; in the above field-of-wheat analogy imagine a wind blowing, but not everywhere in the same direction. We now have a statistical-mechanical system with variable membrane shapes and variable order within the plane itself. (Another form of in-plane order, ``hexatic'' order, is harder to visualize but has similar effects.) Another well-studied possibility is that the molecules actually do link to their neighbors, forming a surface more akin to a rubbery sheet of chain mail than a field of wheat. The fixed relationships between neighbors again gives such ``tethered'' membranes greater in-plane order than the fluid membranes above.

The effect of such in-plane order can be dramatic, imparting an additional stiffness to surfaces and, in the tethered case, preventing the above crumpling phenomenon altogether at low enough temperatures. At some critical temperature fluctuations still win, leading to a ``crumpling transition'' now apparently seen in experiments (though many complications still leave this unclear) and numerical simulations. The case of tilt or hexatic order falls somewhere in between; theory predicts a novel new ``crinkled'' phase less compact than the crumpled one and with continuously variable fractal dimension.

Are any of these exotic phases really relevant to biology? Maybe not, but our experience in the field of random surfaces makes it abundantly clear that new ideas in one domain have a way of proving crucial to some other distant domain. For example the crinkled phase just mentioned may show the way to resolving an old problem in string theory, that of finding realistic models of elementary particles in three spatial dimensions instead of the nine seemingly required by earlier versions of the theory. The richness of the field goes far beyond what one could have imagined starting from a naive analogy to polymers.