Adhesion by Curvature

We now turn attention to thin tubules, whose cross-section is simply modelled as a curve c, whose length 2L is fixed. We aim at obtaining the equilibrium condition that holds at the points like p^* in Figure 7, where c detaches itself from a rigid wall, making no assumptions on the symmetry of both c and the wall.

Figure 7. The contour of a lipid tubule adhering to a curved wall.

The elastic energy is modelled according to (1), with $\sigma _{2}\equiv 0$and $\sigma _{1}$ set equal to the curvature $\sigma$ of c. The adhesion energy is modelled by

$${\cal F}_{a}=-w\enskip \mathrm{length}(c_{*}),$$ (11)

where c_* is the portion of c that adheres to the wall. It is shown in [12] that the detachment equilibrium condition is the following:

$$[\hspace{-0.05cm}[\frac{{\mathrm d}\psi}{{\mathrm d}\sigma}]\... ...rac{{\mathrm d}\psi}{{\mathrm d}\sigma}]\hspace{-0.05cm}]-w=0,$$ (12)

where $\sigma_{*}$ is the curvature of the wall at a detachment point p^*. The jump $[\hspace{-0.05cm}[g ]\hspace{-0.05cm}]$ of a function g is defined as

$$[\hspace{-0.05cm}[g ]\hspace{-0.05cm}]:=\lim_{p \stackrel{p\i... ...ow}p^{*}}g- \lim_{p \stackrel{p\in c^{*}}{\rightarrow}p^{*}}g,$$ (13)

where c^* is the free part of c. When $\psi$ is taken to be a quadratic function of $\sigma$, that is,

$$\psi (\sigma )=\frac{K}{2}\sigma ^{2},$$ (14)

equation (12) gives the following adhesion condition

$$\sigma ^{*}-\sigma _{*}=\sqrt{\frac{2w}{K}},$$ (15)

where $\sigma ^{*}$ is the curvature of c^* at p^*. Equation (15) generalizes the adhesion condition found by Seifert & Lipowsky in [6] for flat walls. As an application, we have studied the adhesion of tubules to a groove modelled as a hollow half-cylinder with radius R (see Figure 8).

Figure 8. The contour of a tubule attached to a groove.

The existence of equilibrium solutions depends on the values of the dimensionless parameters $\sigma ^{*}R$ and $\sigma ^{*}L$. It turns out (see [12]) that there exists precisely one equilibrium solution whenever $\sigma ^{*}L$ is such that

$$\pi \le \sigma ^{*}L\le \rho _{M},$$ (16)

where $\rho _{M}$ is a function of $\sigma ^{*}R$. In [11], we studied by a similar method the adhesion of a lipid tubule to a flat wall, and we only found a lower bound L₀ on L, above which there is precisely one equilibrium contour for an adhering tubule:

$$L\ge L_{0}:=\pi \sqrt{\frac{K}{2w}}.$$

Since $\sigma ^{*}=\frac{1}{R}+\sqrt{\frac{2w}{K}}$, it turns out that for a tubule adhering to a groove the lower bound on L is smaller than L₀. This is one way the curvature of the wall acts: it promotes the adhesion of narrower tubules. Another way is again suggested by (16): the upper bound on L prevents large tubules from adhering to a groove. This is, however a less surprising result, as it essentially reflects a geometric obstruction. Figures 9a and 9b show, respectively, some equilibrium contours of tubules adhering either to a flat wall or to a groove. These contours were obtained by means of the geometric construction illustrated in the previous section, which relies upon the knowledge of the curvature of the tubule's contour as a function of the angle $\vartheta$.

a)

b)

Figure 9. (a) Equilibrium contours of the same tubule for different values of L₀: from left to right, $\frac{L}{L_{0}}=\frac{4}{\pi}$, $\frac{L}{L_{0}}=\frac{8}{\pi}$, and $\frac{L}{L_{0}}=\frac{15.4}{\pi}$. In this sequence the adhesion potential w increases, and so does the length of the adhesion segment. (b) Equilibrium contours of a tubule adhering to a groove, for different values of $\sigma ^{*}R$: from left to right, $\sigma^{*}R=2$, $\sigma^{*}R=4$, and $\sigma^{*}R=8$. All these figures are drawn for $\frac{L}{R}=2$.