Physical Parameters for Biconcave Shape Vesicles

The Helfrich’s shape equation of axisymmetric vesciles is studied. A sufficient condition on the physical parameters and some geometric properties are discovered for the formation of biconcave shape vesicles. The study of the biconcave-discoid shape of a red blood cell has been a continued interest in the last two decades. Theoretically, the shape of the cell is to minimize a certain bending energy. Like minimal surfaces and surfaces with constant mean curvature in geometry, the expected biconcave shape surface is a critical one for the bending energy functional. The early proposal for the bending energy by Canham [1] is purely geometric, which mainly involves the Willmore functional, [15, ch. 7]. It is well known among differential geometers that the unique minimum of the Willmore functional for topologically spherical vesicles is the round sphere, which differs from blood cells observed experimentally. Thus, Willmore functional or Canham’s idea is not a good model for the shape of red blood cells. This is also observed by physicists [2]. It is known that the shape of blood cells and other biological membranes is closely related to the formation of lipid bilayer vesicle in aqueous medium. Based on this and the elasticity of lipid bilayer, Helfrich proposed a modified bending energy, [4]. For a closed surface Σ, this bending energy is a combination of geometric quantities and physical parameters. The geometric quantities are the volume V(Σ) enclosed by Σ, the area A(Σ), the mean curvature H and the Gaussian curvature K of Σ. According to the Gauss-Bonnet Theorem, the integral ofK is a topological constant. Within a certain topological class of Σ, Helfrich’s bending energy can be reduced to, which we call Helfrich functional,