Curvature instability of membranes near rigid inclusions

In multicomponent membranes, internal scalar fields may couple to membrane curvature, thus renormalizing the membrane elastic constants and destabilizing the flat membranes. Here, a general elasticity theory of membranes is considered that employs a quartic curvature expansion. The shape of the membrane and its deformation energy near a long rod-like inclusion are studied analytically. In the limit where one can neglect the end effects, the nonlinear response of the membrane to such inclusions is found in exact form. Notably, exact shape solutions are found when the membrane is curvature unstable, manifested by a negative rigidity. Near the instability point (i.e., at vanishing rigidity), the membrane is stabilized by the quartic term, giving rise to a different length scale and scale exponents for the shape and its energy profile than those found for stable membranes. The contact angle induced by an applied force at the inclusion provides a method to experimentally determine the quartic curvature modulus.

Figure 1

Diagram of a biomembrane near a rigid inclusion (e.g., actin bundles or colloidal nanocylinders absorbed on the membrane surface) that is infinitely long in one direction, denoted by the $y$ axis. This induces a shape deformation along the $x$ axis, and its geometry is described within the Monge gauge, where its surface is given by a height function $u(x,y)$ above a flat reference plane.

Figure 2

Membrane profiles near an inclusion for a few values of the contact angles at vanishing spontaneous curvature ($C_0=0$). Here, we take $x_0=1\mathrm{nm}$, $\kappa =20k_{B}T$, and $\sigma =0.5\mathrm{mN}/\mathrm{m}$. The solid curves are the exact solutions from Eq. (7), while the dashed curves are their corresponding linearized versions. The bottom curve (red) corresponds to a contact angle of ${15}^{\circ }$, whereas the second (green), the third (orange), and the one on the top (blue) are the curves given by ${30}^{\circ }$, ${45}^{\circ }$, and ${60}^{\circ }$, respectively. The inset plot shows the deformation free-energy per unit length $\mathrm{\Delta }\mathcal{F}$ against the angles ${tan}^{-1}\left(v_0\right)$ for a few values of $C_0$, where the solid line is the result in Eq. (9), while the dashed curve is the associated linearized energy. The first solid curve from the top (purple) is the energy profile when $C_0=0.50n{m}^{-1}$, while the second (orange), third (blue), fourth (red), and bottom (green) curves correspond to spontaneous curvatures of 0.25, 0.5, 0, $-0.25$, and $-0.5n{m}^{-1}$, respectively.

Figure 3

Semi-log plot of the height deformation profiles of a quartic membrane for a few values of $\varepsilon$ [see Eq. (13)]. The solid and dotted lines correspond to the regimes of $\tilde{\kappa }>0$ and $\tilde{\kappa }<0$, respectively. The dashed curves are the asymptotic solutions as $\varepsilon \to \infty$, as given by Eq. (14). The bottom curve (red) corresponds to $\varepsilon \to 0$, whereas the second (blue), third (orange), and top (green) curves are given by $\varepsilon =20$, 35, and 50, respectively. Here, the contact angle is ${80}^{\circ }$, while $x_0=1$ nm, $\sigma =0.5\mathrm{mN}/\mathrm{m}$, and $\left|\tilde{\kappa }\right|=20k_{B}T$. The inset plot shows the relative error $\delta \mathcal{E}=\left(u\left[\varepsilon \right]-u[\varepsilon =0]\right)/u\left[\varepsilon \right]$ at $x=x_0$, when the renormalized rigidity $\tilde{\kappa }>0$, as a function of $ln\left(\varepsilon \right)$ and the contact angle ${tan}^{-1}\left(v_0\right)$.

Figure 4

(a) The deformation free energy per unit length, $\mathrm{\Delta }{\mathcal{F}}_4$, in the quartic curvature theory of membranes. By setting $\sigma =0.5\mathrm{mN}/\mathrm{m}$, $\mathrm{\Delta }{\mathcal{F}}_4$ versus the contact angle is shown for a few values of $\varepsilon$ when $\tilde{\kappa }=20k_{B}T$ (solid lines) and $\tilde{\kappa }=-25k_{B}T$ (dashed lines). In terms of their values at ${90}^{\circ }$, the top dashed curve (green) is given by $\varepsilon =2$, while the second (blue) and last dashed curve (red) from the top correspond to $\varepsilon =6$ and 12, respectively. On the other hand, the solid curves from the bottom to top are given by $\varepsilon =0$ (black), 2 (green), 6 (red), and 12 (blue), respectively. (b) The induced contact angle at the inclusion-membrane interface as a function of the pulling force per unit length ${\mathrm{\Pi }}_0$ on the rigid inclusion, which is normalized here by the surface tension $\sigma$. Again, in terms of their values at ${90}^{\circ }$, from the bottom curve to the one on the top, the respective curves correspond to $\varepsilon =0$ (black), 2 (green), 6 (red), and 12 (blue).