Mathematical model of a cell membrane

Bimolecular cell membranes play a crucial role in many biological processes and possess a unique set of physical properties. Bimolecular membranes and monomolecular films can be considered as a "two-dimensional fluid" because the diffusion of molecules along the membrane or film is a hydrodynamic process. On the other hand, the bending of the cell membrane is controlled by its stiffness and elastic tension. The aim of this work is to adapt the Navier-Stokes hydrodynamic equations, obtained using the classical Chapman-Enskog method, to the case of two-dimensional membranes. The hydrodynamic equation system is complemented by an elasticity equation for the bending oscillations of the membrane. The obtained system of equations for the dynamics of the cell membrane is linearized for the case of disturbances with small amplitude. Dispersion equations for stable and unstable linear oscillations of cell membranes are investigated, and conditions for the onset of instabilities are derived.