The pipette aspiration experiment described in the chapter can be used to
measure the bending modulus \(K_{b}\) as well as the area stretch modulus. Lipid
bilayer membranes are constantly jostled about by thermal fluctuations. Even
though a flat membrane is the lowest energy state, fluctuations will cause the
membrane to spontaneously bend The goal of this problem is to use equilibrium
statistic mechanics to predict the nature of bending fluctuations and to use
this understanding as the basis of experimental measurement of the bending
modulus. (NOTE: This problem is challenging and the reader is asked to con
sult the hints on the book website to learn more of our Fourier transform
conventions, how to handle the relevant delta functions, the subtleties
associated with the limits of integration, etc.).
(a) Write the total free energy of the membrane as an integral over the area
of the membrane. Your result should have a contribution from membrane bending
and a contribution from membrane tension. Write your result using the function
\(h(\mathbf{r})\) to characterize the height of the membrane at position
\(\mathbf{r}\).
(b) The free energy can be rewritten using a decomposition of the membrane
profile into Fourier modes. Our Fourier transform convention is
$$h(\mathbf{r})=\frac{A}{(2 \pi)^{2}} \int h(\mathbf{q})
\mathrm{e}^{-\mathrm{i} \mathbf{q} \mathbf{r}} \mathrm{d}^{2} \mathbf{q.}$$
where \(A=L^{2}\) is the area of the patch of membrane of inter est. Plug this
version of \(h(\mathbf{r})\) into the total energy you derived above (that is,
bending energy and the energy related to tension) to convert this energy in
real space to an energy in \(q\) -space. Note that the height field in \(q\)
-space looks like a sum of harmonic oscillators.
(c) Use the equipartition theorem in the form \(\langle E(\mathbf{q})\rangle=\)
\(k_{B} T / 2,\) where \(E(\mathbf{q})\) is the energy of the \(q\) th mode and the
free energy can now be written as
$$F[\hbar(\mathbf{q})]=\frac{A}{(2 \pi)^{2}} \int E(\mathbf{q}) \mathrm{d}^{2}
\mathbf{q}.$$
Use this result to solve for
\(\left(\left.\mathrm{h}(\mathbf{q})\right|^{2}\right\rangle\) which will be
used in the remainder of the problem.
(d) We now have all the pieces in place to compute the relation between
tension and area and thereby the bending modulus. The difference between the
actual area and the projected area is
$$A_{a c t}-A=\frac{1}{2} \int(\nabla h(\mathbf{r}))^{2} \mathrm{d}^{2}
\mathbf{r}.$$
This result can be rewritten in Fourier space as
$$\left(A_{a c t}-A\right)=\frac{1}{2} \frac{A^{2}}{(2 \pi)^{2}} \int_{\pi /
\sqrt{A}}^{\pi / \sqrt{\alpha_{0}}} q^{2}\left(\left.\pi
\tilde{h}(\mathbf{q})\right|^{2}\right) 2 \pi q d q.$$
Work out the resulting integral which relates the areal strain to the bending
modulus, membrane slze, temperature, and tension. The limits of integration
are set by the overall size of the membrane (characterized by the area \(A\) )
and the spacing between lipids \(\left(a_{0}\right.\) is the area per lipid),
respectively. This result can be directly applied to micropipette experiments
to measure the bending modulus. In particular, using characteristic values for
the parameters appearing in the problem suggested by Boal (2002) such as
\(r=10^{-4} \mathrm{J} / \mathrm{m}^{2}, K_{b} \approx 10^{-1} \mathrm{J}\) and
\(a_{0} \approx 10^{-20} \mathrm{m}^{2}\) show that
$$\frac{\Delta A}{A} \approx \frac{k_{B} T}{8 \pi K_{B}} \ln \left(\frac{A
r}{K_{D} \pi^{2}}\right).$$
and describe how this result can be used to measure the bending modulus. (For
an explicit comparison with data, see Rawicz et al., 2000 . For further
details on the analysis, see Helfrich and Servuss, \(1984 .\) An excellent
account of the entire story covered by this problem can be found in Chapter 6
of 30 al, 2002 .)
