Behavior of a polymer chain confined by a membrane

We consider a single polymer chain, grafted on a flat and rigid subtrate, covered by a membrane. The membrane presents some affinity for the surface, caused by non specific adhesive interactions. The challenge is to anticipate and describe the different possible relative configurations of the membrane and the polymer chain, depending on parameters such as the surface tension and curvature of the membrane, or the chain gyration radius. We propose for this system a phase diagram of the different regimes, as well as quantitative predictions for comparison with some recent experiments.