Introduction

Increasing attention has been devoted to study the behaviour of assemblies of lipid molecules with biological relevance, such as vesicles and tubules (for an update review, see [1]). Each lipid molecule consists of a polar head and a hydrocarbon tail: the former is hydrophilic, whereas the latter is hydrophobic. Thus, in an aqueous environment these molecules tend to be organized in bilayers, each consisting of two molecular layers, with the heads outside and the tails inside. (see Figure 1.1).

Figure 1. Sketch of a bilayer. The hydrophobic tails hide themselves from the surrounding medium.

Vesicles are usually composed of a single bilayer and can take various equilibrium shapes, all described by closed surfaces: a complete catalogue of vesicles shapes can be found either in [2] or in [3]. Tubules are formed by many bilayers and are open structures, approximately cylindrical. Both vesicles and tubules have a prescribed area, as molecules are not likely to be added to or removed from them. Moreover, in the case of vesicles, the volume enclosed is usually prescribed.

A variational model for the equilibrium of lipid membrane was independently proposed by Canham in [4] and Helfrich in [5]. They described the elasticity of a lipid membrane by a quadratic, but not necessarily homogeneous functional of the curvatures of the membrane. In slightly more general terms, we can use the following elastic energy for a vesicle:

$${\cal F}_{e}[{\cal S}]:=\int _{{\cal S}}\psi (\sigma _{1},\sigma _{2}){\mathrm d}a ,$$ (1)

where a is the area measure. The elastic free-energy density $\psi$ is a non-negative smooth function of the principal curvatures $\sigma _{1}$ and $\sigma _{2}$ of the surface ${\mathcal S}$ that models the vesicle. Adhesion of vesicles to either a rigid wall or to another vesicle was studied by Seifert and Lipowsky [6]-[7] by employing a simple model to describe the intermolecular forces responsible for the contact. They proposed an adhesion energy ${\cal F}_{a}$ proportional to the area of contact:

$${\cal F}_{a}=-w\enskip \mathrm{area}({\cal S}_{*}),$$ (2)

where w>0 is the adhesion potential and ${\cal S}_{*}$ is the portion of the membrane that adheres to the wall.