Membrane Composition Variation and Underdamped Mechanics near Transmembrane Proteins and Coats

We study the effect of transmembrane proteins on the shape, composition, and thermodynamic stability of the surrounding membrane. When the coupling between membrane composition and curvature is strong enough, the nearby membrane composition and shape both undergo a transition from overdamped to underdamped spatial variation, well before the membrane becomes unstable in the bulk. This transition is associated with a change in the sign of the thermodynamic energy and, hence, favors the early stages of coat assembly necessary for vesiculation (budding) and may suppress the activity of mechanosensitive membrane channels and transporters. Our results suggest an approach to obtain physical parameters of the membrane that are otherwise difficult to measure.

Figure 1

Sketch of a membrane inclusion showing the midplane of the bilayer (blue line) at height $u(r,\theta )$. The surface variation of the rigid inclusion in the $\widehat{\mathbf{z}}$ direction is coarse grained out so that the geometry is defined by its radius $r_0$ and two functions describing the height $\mathcal{U}(\theta )$ and contact angle ${\mathcal{U}}^{\prime }(\theta )$ of the hydrophobic belt. These parametrize the protein-membrane interface (red line), where $u(r_0,\theta )=\mathcal{U}(\theta )$ and $\widehat{\mathbf{n}}·\nabla u(r_0,\theta )={\mathcal{U}}^{\prime }(\theta )$, with $\widehat{\mathbf{n}}$ as the inward unit normal vector. We require both the normal force and the torque on the inclusion to vanish. The latter can lead to an equilibrium tilt angle $\psi$ about the axis labeled by $ϵ$.

Figure 2

Membrane profiles induced by an asymmetrical inclusion, where the contact angle is given by ${\mathcal{U}}^{\prime }(\theta )=15°$ if $|\theta |<w/2$ and 0 otherwise, with $\theta$ measured from the $x$ axis (see text). The membrane parameters are here $\alpha r_0=0.1$, $\beta r_0=1.0$, and $\gamma r_0=0.5$, with $r_0$ the radius of the inclusion (not depicted). The compositional asymmetry $\varphi (\mathbf{r})$ is shown as the color map of the surface plots.

Figure 3

(a) Plot of $k_{\pm }^2$ from Eq. (6) against the coupling constant $\gamma$, with $\alpha r_0=0.1$, $\beta r_0=1.0$, and $r_0$ the inclusion radius. This illustrates that both $k_{+}$ (red line) and $k_{-}$ (blue line) are real for $\gamma <|\alpha -\beta |$ and purely imaginary when $\gamma >\alpha +\beta$. The gray shaded area illustrates the region where $k_{\pm }^2$ are complex, while the green line shows the real part only. The domain given by $\gamma >\alpha +\beta$ corresponds to Leibler’s unstable regime [33, 34]. (b) Radial profiles of the bilayer midplane $u(r)$ and the compositional asymmetry $\varphi (r)$ induced by a conical inclusion, with a modest contact angle of 15°, for different values of the coupling constant $\gamma$, where $\alpha$ and $\beta$ have the same values as those in panel (a).

Figure 4

(a) The top sketches, with the bilayer membrane represented by a thick green line, show two idealized schemes for channel gating: the gating-by-tilt model (left) and the dilational gating model (right). The figures below this show the angle for gating by tilt that would account for the absolute value of the entire conformational energy change $\mathcal{F}$ measured for MscL and MscS [69, 70]. The dashed line indicates $\mathcal{F}=0$ separating two domains where the membrane acts to open ($\mathcal{F}>0$) or close ($\mathcal{F}<0$) the gating channel. The uncolored region is not shown, as it corresponds to angles greater than 60°, which are probably unphysical and where our perturbative approach anyway breaks down. (b) The top sketch shows a membrane deformed by the assembly of a protein coat such as clathrin or a viral coat protein. The graphs below this show the radial compositional field $\varphi (r)$ when the coat is of size $r_0=10\mathrm{nm}$ (left) and the change in membrane energy per area of coat monomers $\mathrm{\Delta }{f}_c$ due to coupling to the composition field against the coat radius $r_0$ (right). In both cases, we assume a typical intrinsic coat curvature with ${\mathcal{R}}_c=50\mathrm{nm}$, $\beta =1.0{\mathrm{nm}}^{-1}$, and $\gamma =1.1{\mathrm{nm}}^{-1}$. In the underdamped regime, the energy change $\mathrm{\Delta }{f}_c$ can exhibit an initial decrease, which may drive coat assembly. In both (a) and (b) we use $\kappa =20k_{B}T$.