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 5, 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 polymer and network elasticity 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 that these filaments can explore. 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 8. For a general review of membrane fluctuations, see Leibler (1989).

Thermal fluctuations in membrane shape

The bending rigidity kb of a phospholipid bilayer lies close to 10−19 J, or 10–20 kBT at ambient temperatures, as summarized in Table 5.3. What does such a small value of kb 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. 6.1(a) as it is deformed into the surface of constant curvature in Fig. 6.1(b).