Wrapping conformations of a polymer on a curved surface

The conformation of a polymer on a curved surface is high on the agenda for polymer science. We assume that the free energy of the system is the sum of bending energy of the polymer and the electrostatic attraction between the polymer and surface. As is also assumed, the polymer is very stiff with an invariant length for each segment so that we can neglect its tensile energy and view its length as a constant. Based on the principle of minimization of free energy, we apply a variation method with a locally undetermined Lagrange multiplier to obtain a set of equations for the polymer conformation in terms of local geometrical quantities. We have obtained some numerical solutions for the conformations of the polymer chain on cylindrical and ellipsoidal surfaces. With some boundary conditions, we find that the free energy profiles of polymer chains behave differently and depend on the geometry of the surface for both cases. In the former case, the free energy of each segment distributes within a narrower range and its value per unit length oscillates almost periodically in the azimuthal angle. However, in the latter case the free energy distributes in a wider range with larger value at both ends and smaller value in the middle of the chain. The structure of a polymer wrapping around an ellipsoidal surface is apt to dewrap a polymer from the endpoints. The dependence of threshold lengths for a polymer on the initially anchored positions is also investigated. With initial conditions, the threshold wrapping length is found to increase with the electrostatic attraction strength for the ellipsoidal surface case. When a polymer wraps around a sphere surface, the threshold length increases monotonically with the radius without the self-intersection configuration for a polymer. We also discuss potential applications of the present theory to DNA/protein complex and further researches on DNA on the curved surface.