Membranes of the cell are characterized by several elastic parameters, such as the area compression modulus, that reflect the membrane’s quasi-two- dimensional structure. As described in Chapter 7, these parameters have small values for a lipid bilayer just 4–5 nm thick, yet they properly describe the energetics of membrane deformation at zero temperature where thermal fluctuations in shape are unimportant. But what happens at finite temperature? In the discussion of polymers and networks in Part I, we saw that the entropic contribution to the elasticity of very flexible filaments is significant at ambient temperatures, owing to the large configuration space available to these filaments. Do we expect similar behavior for flexible sheets? In this chapter, we develop a mathematical description of surfaces, and explore the characteristics of membrane undulations. Membranes are treated in isolation here, and in interaction with other surfaces in Chapter 9. A more extensive treatment of membrane defects and fluctuations can be found in Leibler (1989) and Chaikin and Lubensky (1995).

Thermal fluctuations in membrane shape

The bending rigidity κb of a phospholipid bilayer lies close to 10−19 J, or 10−20 kBT at ambient temperatures, as summarized in Table 7.3. What does such a small value of κb imply about the undulations of a membrane with the dimensions of a cell? For illustration, we calculate the change in energy of the flat, disk-shaped membrane in Fig. 8.1(a) as it is deformed into the surface of constant curvature in Fig. 8.1(b). Configuration (b) has radius of curvature R and energy density 2κb/R2, where the Gaussian rigidity is neglected (see Sections 7.4 and 8.2). Taking the disk diameter as 2 μm and κb ~ 10 kBT, the deformation energy is 20πkBT/R2, where R must be quoted in microns. What is the typical value of R when the disk flexes at non-zero temperature? The thermal energy scale is set by kBT and the disk energy rises to this value (from zero for a flat disk with R = ∞) when R declines to ~8 μm. This radius of curvature corresponds to θ ~ 15o in Fig. 8.1(b), meaning that undulations may have measurable effects in common cells.