Membrane Curvature

All cells are bounded by a plasma membrane consisting of a bilayer of dual-chain lipid molecules within which are embedded proteins and other molecules such as cholesterol. Bilayers are serf-assembled structures whose equilibrium configuration is spatially flat if the molecular composition is the same within both layers. Such symmetric bilayers resist bending with an energy cost per unit area e whose simplest parameterization is

where the constants κ (bending rigidity) and κ_G (Gaussian curvature modulus) have units of energy [for a review of more complete descriptions of bilayer bending, building on the original approach of Helfrich (1973), see Lipowsky (1991)]. The quantities R₁ and R₂ are the two principal radii of curvature displayed in Figure 2. As an illustration, a sphere of radius R has R₁ = R₂ = R, while a cylinder has an infinite radius of curvature along the axis of cylindrical symmetry. To find the bending energy of a particular surface, one simply integrates ε over the entire surface: for example, a spherical shell has a bending energy of 8πκ + 4πκ_G, independent of the shell's radius.

Figure 2.

Principal radii of curvature for a saddle-like surface.

What is the magnitude of the bending energy for typical cells? Lipid bilayers in terrestrial cells are found to have κ= 10-25 k_BT, where k_B is Boltzmann's constant and T is the temperature [see Evans and Rawicz (1990) and references therein]. The value of κ_G is much less well known, but is expected to have a similar magnitude as κ. With κ = κ_G, the energy of a spherical shell is 12πκ. Considering only the contribution from k, the bending energy of a spherical cell would be 250-600 k_BT. Although this is not really a large amount of energy (recall that k_BT is roughly the kinetic energy of an atom in a gas), why would nature expend this energy to form a closed surface from an open bilayer sheet? To answer this question, we examine how a bilayer might rupture.

Membrane Rupture

The fluid membrane not only resists bending, but also resists in-plane stretching. Under tensile stress, the membrane first stretches then ruptures once the area has expanded a few percent beyond its unstressed value. The creation of a hole in a membrane likely involves reconfiguring the lipid molecules around the boundary of the hole in order to reduce contact between the aqueous medium surrounding the bilayer and the water-avoiding hydrocarbon chains of the lipid molecules, which are normally buried within the bilayer. In general, the orientation of the lipids at the hole boundary is energetically unfavorable compared to that of an intact bilayer, so that there is an energy penalty if the membrane has a hole or a free edge.

The boundary of the hole can be characterized by an edge tension λ, (energy per unit length along the boundary), which has been measured to be in the 10^-11 J/m range (for example, Fromherz, 1983); the measured values are larger than the minimum edge tension for membrane stability estimated from computer simulations of membrane rupture (Boal and Rao, 1992). For example, the edge energy of a fiat disk of radius R_disk and perimeter 2πR_disk is E_disk = 2πR_diskλ. A membrane having this shape will be energetically favored over the closed sphere considered above (E_sphere = 12πκ for κ = κ_G) if R_disk < 6κ/λ. If the disk and the sphere have the same surface area then R_sphere = R_disk/2 (see Figure 3). Thus we expect R_sphere > 3κ/λ (after Fromherz, 1983). Using typical values of κ ˜ 15 k_BT and λ = 10^-11 J/m leads to R_sphere > 20 nm, a bound whose exact value depends upon the membrane composition. Experimentally, one finds that pure bilayer vesicles (simple artificial cells in some sense) can be produced in the lab with radii as small as 30 nm (Fromherz, 1983; Frisken, 1998, private communication). Once the membrane has adopted a closed shape, the configuration could be further stabilized by the addition of lipids to the outer layer, thus reducing the strain in the bilayer.

Figure 3.

Energetics of disks and spheres. The two shapes have the same areas if R_sphere = R_disk/2.

Experiments on membrane failure find that typical bilayers rupture at tensile stresses of 1 × 10^-2 J/m² on laboratory time scales (Needham and Hochmuth, 1989). In cells, a (two-dimensional) surface stress P can result from the osmotic pressure difference P between the cell's interior and its external environment. For a spherical shell of radius R, the stress and pressure are related by (Fung, 1994)

Thus, a spherical shell of radius 1-micron can support a pressure difference of up to 2 × 10⁴ J/m³, if the two-dimensional bursting stress is 1 × 10^-2 J/m² on laboratory timescales. However, many bacteria operate at much higher internal pressures, ranging up to many atmospheres, where 1 atmosphere = 10⁵ J/m³. Most varieties of bacteria accommodate this pressure by the use of a cell wall.

Because the surface stress is proportional to R in Equation (2), a smaller cell would experience a lower stress for a given osmotic pressure P. In fact, a bilayer alone could handle an osmotic pressure of 4 atmospheres for a cell with a radius of just 50 nm, so that very small cells would not need a cell wall to function at moderate osmotic pressures. The absence of a cell wall would reduce the functional tasks of the cell and hence eliminate that part of DNA required to produce the proteins associated with cell wall construction. Alternatively, a small cell could choose to have a cell wall and increase the osmotic pressure at which it operates. Because the osmotic pressure is directly proportional to the concentration of proteins, ions, etc., then small cells could have a higher concentration of chemical reactants. Given that the rate of chemical reactions is proportional to the product of the reactant concentrations, an increase in the concentrations would result in an increase of the chemical reaction rates.

Flexible Filaments

The most evolutionarily advanced cells—eucaryotic cells—contain a filamentous cytoskeleton, which helps maintain the cell's shape, along with its other duties. Components of the cytoskeleton frequently include actin, intermediate filaments, and microtubules, with diameters in the range of 10 to 25 nm. Compared to a typical eucaryotic cell diameter of 10 microns or more, the transverse dimension of a cytoskeletal filament is trivial. Smaller cells such as bacteria, whose evolutionary origin predates eucaryotes, do not contain a cytoskeleton, but may instead possess a strong cell wall surrounding the pressurized bag bounded by a fluid membrane. Even bacteria, with a typical diameter of 1 micron, could accommodate the size of cytoskeletal filaments found in eucaryotes. However, cells with a radius as small as 50 nm would probably not have sufficient interior volume to permit a conventional cytoskeleton.

The absence of a cytoskeleton within a small cell does not imply that there are no filaments present. Cells must have some means of carrying hereditary information; the earliest cells may have used RNA but today's cells use DNA, both of which are linear molecules. Now, the visual appearance of a flexible rope, string, or linear molecule depends on the length scale of observation. For example, a human hair may be curly as seen by the eye on a length scale of centimeters, but a segment of the hair would seem straight if viewed through a microscope on a length scale of less than a millimeter. A quantity called the persistence length can be used to describe the straightness of a linear molecule. Figure 4 illustrates two linear objects; part [a] is convoluted with a short persistence length while [b] is much straighter with a long persistence length. Mathematically, the persistence length is a measure of the length scale over which a curve undergoes a significant change in direction. The arrows in Figure 4b are about a persistence length apart, as measured along the curve.

Figure 4.

Schematic representation of strings with short [a] and long [b] persistence lengths. The arrows in [b] are about a persistence length apart, as measured along the contour of the string.

Now, double-stranded DNA has a persistence length of about 50 run (Bustamante et al., 1994), meaning that a 100 nm filament of DNA might look like the configuration in Figure 4b: it would appear to be neither a straight rod, nor a tangled ball of thread. At 0.34 nm per base pair, a 100 nm filament of DNA traversing the cell once would contain just 300 base pairs, not a lot of genetic information. This means that cells probably would have to be larger than 50 nm in radius to accommodate a moderate amount of DNA if it were present as a random chain. It is more likely that small cells would use RNA or another flexible molecule to carry genetic information, consistent with the idea that RNA predated DNA in evolution. Many biopolymers display a persistence length that varies as the square of the mass per unit length along the polymer, a scaling behavior consistent with the theoretical expectation that the persistence length varies as the fourth power of the radius for uniform cylindrical rods (Doi and Edwards, 1986; Landau and Lifshitz, 1986). Thus, a molecule with the same mass density as double-stranded DNA, but only half the mass per unit length, would have a persistence length of one-quarter that of DNA, just 13 nm. With a persistence length closer to 10 nm, a long molecule could be balled up in a cell of 100-nm diameter. Self-interactions along the molecule's length, as might be expected for RNA, would reduce the size of the genetic ball even further.