Coupling between Inclusions and Membranes at the Nanoscale

The activity of cell membrane inclusions (such as ion channels) is influenced by the host lipid membrane, to which they are elastically coupled. This coupling concerns the hydrophobic thickness of the bilayer (imposed by the length of the channel, as per the hydrophobic matching principle) but also its slope at the boundary of the inclusion. However, this parameter has never been measured so far. We combine small-angle x-ray scattering data and a complete elastic model to measure the slope for the model gramicidin channel and show that it is surprisingly steep in two membrane systems with very different elastic properties. This conclusion is confirmed and generalized by the comparison with recent results in the simulation literature and with conductivity measurements.

Figure 1

Interaction energy $V$ between two gramicidin channels in a DLPC bilayer as a function of their center-to-center distance $d$. The red shading visualizes the contact between the inclusions. The parameters of the model correspond to our best fit of the experimental data. Upper inset: corresponding shape of the membrane for $d=2d_0=43\AA$. The channels are represented as cylinders. The relative deformation $u$ is magnified by a factor of 3. Lower inset: coordinates for the multipolar expansion.

Figure 2

Structure factors $S$ as a function of the scattering vector $q$ for gramicidin channels in DLPC bilayers at different surface fractions $\eta$ of inclusions. Red dots: experimental data from Ref. [10]. Black curves: Monte Carlo simulations for the parameters of the model corresponding to our best fit of the experimental data (see text). Blue dashed lines: HNC approximations. Curves are shifted downwards by unit increments.

Figure 3

Experimental structure factors and fits for gramicidin channels in ${\mathrm{C}}_{12}{\mathrm{E}}_5$ bilayers. Same notation as in Fig. 2.

Figure 4

Interaction energy $V$ between two gramicidin channels in ${\mathrm{C}}_{12}{\mathrm{E}}_5$ bilayers for the model parameters corresponding to our best fit of the experimental data. Shaded area: contact between the inclusions. Inset: zoom for $d>20\AA$.

Figure 5

PMF between two gramicidin channels, simulated by Yoo and Cui (symbols; redrawn from Fig. 2A of Ref. [33]) and calculated using the present model (lines) in DMPC (black solid dots and solid line), DPPC (red squares and dashed line), and DSPC (blue triangles and dotted line) bilayers, with no adjustable parameters.

Figure 6

Logarithm of the formation rate $f$ of gramicidin channels in DOPC bilayers as a function of the membrane tension $\sigma$. Solid dots: experimental data retrieved from Fig. 6(b) of Ref. [4] (after subtraction of a constant baseline, see Ref. [13] for the detailed analysis). Solid red line and red shaded area: our prediction using the elastic model and the parameter values discussed in the text (with no adjustable parameters). Black dashed line: best linear fit.