286 Chapter 8. Chemical forces and self-assembly[[Student version, January 17, 2003]]
d dahead
ahead
aheadR +∆aab
Figure 8.9:(Schematic.) Bilayer membrane bending.dis one half the total membrane thickness. The cylindrical
objects represent individual phospholipid molecules (a second layer on the other side of the dashed line is not shown).
(a)The relaxed, flat, membrane conformation gives each head group its equilibrium areaahead.(b)Wrapping the
membrane about a cylinder of radiusRseparates the headgroups on the outer layer to occupy areaahead+∆a,
where ∆a=aheadd/R.
Abilayer membrane’s state of lowest free energy is that of aflat(planar) surface. Because both
layers are mirror images of each other (see Figure 2.24 on page 53), there is no tendency to bend
to one side or the other. Because each layer is fluid, there is no memory of any previous, bent,
configuration (in contrast to a small patch snipped from a rubber balloon, which remains curved).
In short, while it’s not impossible to deform a bilayer to a bent shape (indeed it must so deform in
order to close onto itself and form a bag), still bending will entail some free-energy cost. We would
like to estimate this cost.
Figure 2.24 suggests that the problem with bending is that on one side of the membranes the
polar heads get stretched apart, eventually admitting water into the nonpolar core. In other words,
each polar headgroup normally occupies a particular geometric areaahead;adeviation ∆afrom
this preferred value will incur some free-energy cost. To get the mathematical form of this cost, we
assume that it has a series expansion: ∆F=C 0 +C 1 ∆a+C 2 (∆a)^2 +···.The coefficientC 0 is a
constant and may be dropped. Since the cost is supposed to be minimum when ∆a=0,weshould
takeC 1 to equal zero; for small bends, the higher terms will be negligibly small. Renaming the one
remaining coefficient as^12 k,then, we expect that the free energy cost will be given by
elastic energy per phospholipid molecule =^12 k(∆a)^2. (8.35)The value of the spring constantkis an intrinsic property of the membrane. This relation should
hold as long as ∆ais much smaller thanaheaditself. To apply it, suppose we wrap a small patch of
membrane around a cylinder of radiusRmuchbigger than the membrane’s thicknessd(Figure 8.9).
Examining the figure shows that bending the membrane requires that we stretch the outer layer by
∆a=aheadd/R.Thusweexpect a bending energy cost per headgroup of the form^12 k(aheadd/R)^2.
Because the layer is double, the number of such headgroups per area is 2/ahead.Introducing the
new symbolκ≡kd^2 ahead(thebending stiffness), we can compactly summarize the discussion by
saying:
The free energy cost per unit area to bend a bilayer membrane into a cylinder
of radiusRis of the form^12 κ/R^2 ,whereκis an intrinsic parameter describing
the membrane. Thus the dimensions ofκare those of energy.