8.6. Self-assembly in cells[[Student version, January 17, 2003]] 287
The cost to bend the membrane into asphericalpatch of radiusRis four times as great as in
Idea 8.36, because each head group gets stretched in two directions, and so ∆ais twice as great.
Thus bending the layer into a spherical shape with radius of curvatureRcosts free energy per unit
area 2κ/R^2. The total bending energy to wrap a membrane into a spherical vesicle is then 8πκ.
This is already an important result: The total bending energy of a sphere is independent of the
sphere’s radius.
Tounderstand the significance of the free energy cost of bending a bilayer (Idea 8.36), we need
an estimate of the numerical value ofκ. Consider first a single layer at an oil-water interface.
Bending the layer into a spherical bulge, with radius of curvatureRcomparable to the length (^) tail
of the hydrocarbon tails, will spread the heads apart, exposing the tails to water. Such a large
distortion will incur a hydrophobic free energy cost per unit area, Σ, comparable to that at an
oil-water interface. The corresponding cost for a bilayer in water will be roughly twice this.
Wethus have two different expressions for the bending energy of a spherical patch of bilayer,
namely 2κ/( (^) tail)^2 and 2Σ. Equating these expressions lets us estimateκ.Taking typical values
Σ≈ 0. 05 J/m^2 and (^) tail≈ 1. 3 nmgives our estimate:κ≈ 0. 8 · 10 −^19 J.Our estimate is crude, but it’s
not too far from the measured valueκ=0. 6 · 10 −^19 J=15kBTrfor dimyristoyl phosphatidylcholine
(DMPCDMPC (dimyristoyl phosphatidylcholine)). The total bending energy 8πκof a spherical
vesicle of DMPC is then around 400kBTr.
Wecan extract a simple lesson from the measured value ofκ.Suppose we take a flat membrane
of areaAand impose on it a corrugated (washboard) shape, alternating cylindrical segments of
radiusR. The free energy cost of this configuration is^12 κA/R^2 .TakingAto be 1000μm^2 ,a
value corresponding to a typical 10μmcell, we find that the bending energy cost greatly exceeds
kBTrfor any value ofRunder 10μm.Thusthe stiffness of phospholipid bilayer membranes has
been engineered to prevent spontaneous corrugation by thermal fluctuations. At the same time, the
bending energy needed for gross, overall shape changes (for example, those needed for cell crawling)
is only a few hundred timeskBTr,and so such changes require the expenditure of only a few dozen
ATPmolecules (see Appendix B). Phospholipid bilayer membrane stiffness is thus in just the right
range to be biologically useful.
Not only are cells themselves surrounded by a bilayer plasma membrane. Many of the organelles
inside cells are separate compartments, partitioned from the rest by a bilayer. Products synthesized
on one part of the cell (the “factory”) are also shipped to their destinations in special-purpose trans-
port containers, themselves bilayer vesicles. Incoming complex food molecules awaiting digestion
to simpler forms are held in still other vesicles. And as we will see in Chapter 12, the activation of
one neuron by another across a synapse involves the release of neurotransmitters, which are stored
in bilayer vesicles until needed. Self-assembled bilayers are ubiquitous in cells.
T 2 Section 8.6.1′on page 295 mentions some elaborations to these ideas.
8.6.2 Vista: Macromolecular folding and aggregation
Protein folding Section 2.2.3 on page 45 sketched a simple-sounding answer to the question
of how cells translate the static, one-dimensional data stream in their genome into functioning,
three-dimensional proteins. The idea is that the sequence of amino acid residues determined by
the genome, together with the pattern of mutual interactions between the residues, determines a
unique, properly folded state, called thenative conformation.Evolution has selected sequences
that give rise to useful, functioning native conformations. We can get a glimpse of some of the