Experimental Support for Tilt-Dependent Theory of Biomembrane Mechanics

Recent simulations have indicated that the traditional model for topographical fluctuations in biomembranes should be enriched to include molecular tilt. Here we report the first experimental data supporting this enrichment. Utilizing a previously posited tilt-dependent model, a height-height correlation function was derived. The x-ray scattering from a liquid crystalline stack of oriented fluid phase lipid bilayers was calculated and compared with experiment. By fitting the measured scattering intensity, both the bending modulus $K_c=8.3\pm 0.6\times {10}^{-20}\mathrm{J}$ and the tilt modulus $K_{\theta }=95\pm 7\mathrm{mN}/\mathrm{m}$ were determined for DOPC lipid bilayers at $30°\mathrm{C}$.

Figure 1

A diagram of the bilayer stack is labeled to illustrate the two fluctuation fields and various definitions given in the text. The lighter colored region in the $j$th bilayer is shown expanded in the right-hand side.

Figure 2

A diagram of the experimental geometry and a CCD image of the scattered intensity from a stack of DOPC bilayers at $30°\mathrm{C}$ with repeat spacing $D=63.0\AA$ are shown. On the left, the broader gray slab is the Si substrate with the DOPC bilayer stack depicted as the narrower lighter gray parallelogram. On the right, the gray scale depicts increasing intensity from black to white. The red boxes indicate regions of data fitted by the theory. The highly attenuated direct beam is visible at $k_x=k_z=0{\AA }^{-1}$, and the black region along the meridian for $k_z\lesssim 0.2{\AA }^{-1}$ is the shadow of a molybdenum attenuator through which the first two lamellar orders are visible. The narrow streak at $k_z\approx 0.26{\AA }^{-1}$ and $k_x\approx 0{\AA }^{-1}$ is specular reflectivity from rotating the substrate angle $\alpha$.

Figure 3

Scattered intensity as a function of $k_x$ is shown for several typical values of $k_z$, and at the bottom is shown an average over all $\sim 200$ fitted $k_z$ values, ${\overline{k}}_z$ (see Fig. 2). Solid and dashed lines are best fits to the data for tilt-dependent and tilt-independent theories, respectively. The data (open symbols) and fits have been vertically offset to improve visibility. Representative error bars for $k_z=0.35{\AA }^{-1}$ correspond to $\pm 1$ standard deviation.

Figure 4

Theoretical height fluctuation spectra for a single membrane are plotted for $K_{\theta }=95\mathrm{mN}/\mathrm{m}$ and $K_{\theta }=\infty$, solid and dashed lines, respectively, for $K_c=8.3\times {10}^{-20}\mathrm{J}$. For $Q_z=2\pi /D$, Eq. (2) reduces to the single membrane spectrum.

Figure 5

For $K_c=8.3\times {10}^{-20}\mathrm{J}$, height-height pair correlation functions $C_j(r)$ Eq. (3) (linear-log, left) and pair scattering correlation functions $G_j(r)$ (log-log, right) are plotted for $K_{\theta }=95\mathrm{mN}/\mathrm{m}$ and $K_{\theta }=\infty$, solid and dashed lines, respectively.