**5. Adhesion of Lipid Vesicles to Rigid Surface**

In this section, we describe the interplay of membrane elasticity, geometrical constraints, and adhesive forces between the lipid bilayer and charged solid surface in the adsorption of lipid vesicles to the solid surface. The numerically calculated shapes of adhered lipid vesicles based on the system free-energy minimization are presented.

 The shape of a vesicle upon adsorption to a surface is determined by the interplay of adhesion, bending, and geometrical constraints. This interplay is theoretically studied starting from a simple model in which the membrane experiences a contact potential arising from the attractive surface. Let us recall the free energy *F* expression of an adsorbed vesicle in terms of a simple model that takes into account the local bending energy terms, the adhesion energy and two geometrical constraints [130,131]:

$$F = \frac{1}{2}\kappa \oint \left(\mathbf{C}\_1 + \mathbf{C}\_2 - \mathbf{C}\_0\right)^2 \mathbf{d}A + \kappa\_\mathbf{G} \oint \mathbf{C}\_1 \mathbf{C}\_2 \mathbf{d}A - W\mathbf{A}\_\varepsilon + PV + \Sigma A\_\prime \tag{17}$$

where *κ* is the local bending modulus; *κ<sup>G</sup>* is the Gaussian curvature modulus; *C*1, *C*2, and *C*<sup>0</sup> denote the two principal curvatures and the (effective) spontaneous curvature, respectively; and d*A* is an infinitesimal membrane area element. In the third term, *W* is the strength of adhesion and *A<sup>c</sup>* is the contact area of the membrane and the surface. The last two terms represent the volume (*V*) and area (*A*) constraints with corresponding Lagrange multipliers *P* and Σ. The normalization of the membrane free energy (Equation (17)) by the bending energy of a sphere for zero spontaneous curvature 8*πκ* leads to the expression for the reduced free energy *f* = *Fb*/8*πκ*: = <sup>ଵ</sup> ଶ ∮(<sup>ଵ</sup> + <sup>ଶ</sup> − ) <sup>ଶ</sup>d + ீ ∮ <sup>ଵ</sup> ଶd − + + Σ ீ <sup>ଵ</sup> <sup>ଶ</sup> d 8

$$f \underset{f}{=} \frac{1}{4} \oint (c\_1 + c\_2 - c\_0)^2 \mathrm{d}a - \frac{w}{2} \left(\frac{A\_c}{A}\right) + p \oint \mathrm{d}v + \sigma \oint \mathrm{d}a,\tag{18}$$

where *v* = *V*/ 4*πR* 3 *s*/3 is the reduced volume (see, for example, [131,132]); *a* = *A*/4*πR* 2 *<sup>s</sup>* = 1 is the reduced area; *c*<sup>0</sup> = *C*0*R<sup>s</sup>* , *c*<sup>1</sup> = *C*1*R<sup>s</sup>* , and *c*<sup>2</sup> = *C*2*R<sup>s</sup>* are the reduced curvatures; *p* and *σ* are the reduced Lagrange multipliers; and = /(4<sup>௦</sup> <sup>ଷ</sup>/3) = /4<sup>௦</sup> <sup>ଶ</sup> = 1 = <sup>௦</sup> <sup>ଵ</sup> = ଵ<sup>௦</sup> <sup>ଶ</sup> = ଶ<sup>௦</sup> 

$$w = \mathsf{WR}\_{\mathsf{s}}^2 / \mathsf{x}\_{\mathsf{s}} \tag{19}$$

ଶ

is a dimensionless parameter, where *R<sup>s</sup>* = √ *A*/4*π*. The ratio *Ac*/*A* = *Ac*/4*πR* 2 *s* is the reduced contact area and varies between zero for a spherical vesicle with reduced volume *v* = 1.0 and 0.5 for pancake-shaped vesicles for a very small reduced (zero) volume *v*. The above energy expression (Equation (18)) is minimized numerically, as described in [133]. Note that, in Figure 12 (for *c*<sup>0</sup> = 0), the calculated nonadhered vesicle shapes corresponding to minimal bending energy and reduced volumes *v* ≤ 0.591 are stomatocytic, while the shapes for 0.592 ≤ *v* ≤ 0.651 are oblate and, for *v* ≥ 0.652, prolate (see also [134,135]). It can be seen in Figures 12 and 13 that for high reduced adhesion strength *w*, the calculated shapes of adhered vesicles approach the limiting shapes composed of the sections of spheres corresponding to the maximal reduced contact area at a given reduced volume *v*. <sup>௦</sup> = ඥ/4 / = /4<sup>௦</sup> = 1.0 0.5 = 0 ≤ 0.591 0.592 ≤ ≤ 0.651 ≥ 0.652 

**Figure 12.** The calculated shapes of free (nonadsorbed) and adsorbed vesicles obtained by the minimization of the free energy given by Equation (18), determined for *c*<sup>0</sup> = 0 and different values of reduced volume: *v* = 0.5 (**A**), 0.6 (**B**), 0.8 (**C**), and 0.95 (**D**), and different values of reduced adhesion strength *w*, defined by Equation (18).

= 0.5 0.6 0.8 0.95

 = = 0.8 **Figure 13.** The calculated pear vesicle shapes of free (nonadsorbed) and adsorbed vesicles obtained by the minimization of the free energy given by Equation (18), determined for *c*<sup>0</sup> =2.4, *v* = 0.8, and different values of reduced adhesion strength *w*, defined by Equation (19).

= 0
