Introduction to Computational Statistical Mechanics of Tethered Membranes

Interface physics is of considerable interest to the condensed-matter theory community, an interface being defined as a boundary between two phases and controlled by a surface tension. Because of surface tension, the interface is essentially flat. Most recently, there has been a growing interest in "membranes". While surface-like, membranes do not necessarily separate two distinct phases and are not composed of the molecules of the imbedding medium. It is typical that the membrane surface tension is vanishingly small, hence they execute "wild" fluctuations. This, in turn, makes for different physics, requiring new ideas and theoretical tools [1].

Membranes may exist in various phases, and we will distinguish between solid and liquid phases. Most biological membranes are found in the fluid phase where the molecules comprising the bilayer can diffuse freely and their hydrocarbon chains are disordered. Figure 1a is a caricature of a red blood cell [1]. The cell wall (drawn) is a membrane, composed of a bilayer of amphiphillic molecules each with one or more hydrophobic hydrocarbon tails and a polar head group. The membrane has a spherical topology and exhibits negligible (but finite) shear modulus at biological temperatures. Another example of a fluid membrane is the oil/water microemulsion system made possible because of the addition of significant amounts of an amphiphile-like SDS (sodium dodecyl sulfate), which exists at the interface between oil and water and reduces the surface tension to almost zero (Fig. 1b) [1].

Figure 1 Figure 1 (13 kB). Examples of liquid-like membranes: (a) red blood cell and (b) microemulsion (taken from Ref. 1, Chapter 1).

Our interest is in the study of solid membranes, such as flexible sheet polymers, or "tethered surfaces". Tethered surfaces can be synthesized by polymerizing Langmuir-Blodgett films or amphiphillic bilayers [2]. Figure 2 shows a schematic picture of the plasma membrane of the human red blood cell [3]. We can see that in addition to lipids, proteins, etc., this membrane also consists of a "spectrin network", which is attached to the inner layer through proteins. This protein network is an example of a solid membrane and is important in determining the elastic and mechanical properties of the red blood cell membrane; e.g., the shear modulus of is largely due to spectrin.

Figure 2 Figure 2 (28 kB). A schematic picture of the plasma membrane of human red blood cells [3]. Note the network of proteins attached to the bilayer. This is an example of a cytoskeleton. For human RBCs the cytoskeleton is two-dimensional and does not extend inside the cells. A red blood cell is a simple model system to study the elastic properties of biomembranes.

Solid membranes are being regarded as two-dimensional generalizations of linear polymer chains. However, flexible membranes should exhibit a greater complexity in physical phenomena and have properties which are strikingly different from conventional linear polymer chains.

Polymer chains in a good solvent crumple into a fractal object [4] whose characteristic size or "radius of gyration" varies as a nontrivial power of the linear dimension , . Linear polymers fold up on scales larger than a "persistence length" which is typically only a few monomer units in size. The statistical mechanics of two-dimensional polymer networks [5] has attracted intense theoretical interest, in part because, unlike linear polymers, they are expected to exhibit a low-temperature flat phase with an infinite persistence length. The flat phase arises because the resistance to inplane shear deformation leads to an anomalous stiffening of the surface in the presence of thermal fluctuations [6]. This flattening results from a delicate interplay between geometry and statistical mechanics which has no analog in conventional polymer solutions.

The simplest model of polymerized membranes is a "tethered surface", composed of hard spheres each tied to six near neighbors to form a planar triangulated network; the network is then equilibrated at a finite temperature by allowing it to bend and possibly crumple in three dimensions. Although the first simulations of such tethered surfaces were interpreted in terms of a high-temperature crumpled phase [5], extensive simulations of much larger surfaces with a very similar potential revealed that these objects were in fact flat [8], with very large fluctuations in the direction parallel to the average surface normal.

Ref. [7] presents this finding. For more information check out its abstract and the visualization of the flat phase.

Ref. [9] presents extensive computer simulation results of the flat phase and discusses the origin of the "entropic" bending rigidity introduced by distant neighbor self-avoidance interactions. By introducing an attractive distant interaction, one can produce a compact or collapsed self-avoiding tethered membrane. It also studies the internal structure of membranes in the flat phase. For more details look at the abstract and the visualizations of the flat, the collapsed and the recovery of the flat phases.

Ref. [10] shows that an attraction between monomers of the tethered surface leads also to a well defined sequence of folding transitions on lowering temperature. Using insights gained from a Landau theory and simulations of bimembranes, it concludes that the folding transitions are intimately linked to the unbinding of membranes. For more information look at the abstract and the visualization of the folded phase.


1) D. R. Nelson, T. Piran and S. Weinberg, editors, Statistical Mechanics of Membranes and Interfaces. (World Scientific, Singapore, 1989).

2) A. Blumstein, R. Blumstein, T. H. Vanderspurt, J. Colloid Interface Sci.. 31, 236 (1969); S. L. Regen, J.-S. Shin, J. F. Hainfield, J. S. Wall, J. Am. Chem. Soc.. 106, 5756 (1984); N. Beredjick and W. J. Burlant, J. Polymer Sci.. A8, 2807 (1970); J. H. Fendler and P. Tundo, Acc. Chem. Res.. 17, 3 (1984).

3) B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and J. D. Watson, The Molecular Biology of the Cell. (Garland, New York, 1983). The best example of a biological tethered surface is probably the spectrin protein skeleton of eurythrocytes, separated from its natural lipid environment; see A. Elgsaeter, B. Stokke, A. Mikkelsen and D. Branton, Science. 234, 1217 (1986).

4) P.-G. de Gennes, Scaling Concepts in Polymer Physics. (Cornell Univ. Press, Ithaca, 1979).

5) Y. Kantor, M. Kardar and D. R. Nelson, Phys. Rev. Lett.. 57, 791 (1986); Phys. Rev.. A35, 3056 (1987).

6) D. R. Nelson and L. Peliti, J. Phys. (Paris). 48, 1085 (1987).

7) F. F. Abraham, W. E. Rudge, M. Plischke, Phys. Rev. Lett.. 62, 1757 (1989).

8) For earlier work along these lines, see M. Plischke and D. Boal, Phys. Rev.. A38, 4943 (1988). In our view, it is difficult to distinguish between the isotropic crumpling hypothesis of Kantor et al. [5] and the hypothesis of a flat, but very rough phase in these simulations.

9) F. F. Abraham and D. R. Nelson, J. Phys. France. 51, 2653 (1990).

10) F. F. Abraham and M. Kardar, Science. 252, 419 (1991).

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