Undulations of a straight polymer chain embedded in a membrane

We consider a sinusoidal undulation of a straight, linear polymer embedded in a flat membrane exposed to a lateral tension. We determine the shape of the membrane and compute the elastic energy associated with the undulation of the polymer. Analysing this energy at different wavelengths of the undulation, we find the criteria of stability of the straight shape of the polymer, and show them to be controlled by the difference of moduli of Gaussian curvature, Δκ̄, of the polymerized and unpolymerized membrane. We also discuss formal analogies and differences between the behaviour of a polymer embedded into a membrane and a charged bilayer membrane immersed in a solution of electrolyte.