 Therefore, the lipid bilayer can be represented by a two-dimensional mathematical surface.In 1973, based on similarities between lipid bilayers and nematic liquid crystals, Helfrich  proposed the following expression for the curvature energy per unit area of the closed lipid bilayer 

where formula_1 are bending rigidities, formula_2 is the spontaneous curvature of the membrane, and formula_3 and formula_4 are the mean and Gaussian curvature of the membrane surface, respectively.The free energy of a closed bilayer under the osmotic pressure formula_5 (the outer pressure minus the inner one) as