Analytical solution of lipid membrane morphology subjected to boundary forces on the edges of rectangular membranes

We develop a complete analytical solution predicting the deformation of rectangular lipid membranes resulting from boundary forces acting on the perimeter of the membrane. The shape equation describing the equilibrium state of a lipid membrane is taken from the classical Helfrich model. A linearized version of the shape equation describing membrane morphology (within the Monge representation) is obtained via a limit of superposed incremental deformations. We obtain a complete analytical solution by reducing the corresponding problem to a single partial differential equation and by using Fourier series representations for various types of boundary forces. The solution obtained predicts smooth morphological transition over the domain of interest. Finally, we note that the methods used in our analysis are not restricted to the particular type of boundary conditions considered here and can accommodate a wide class of practical and important edge conditions.