  We study the contribution of polyelectrolytes in solution to the bending
moduli of charged membranes. Using the Helfrich free energy, and within the
mean-field theory, we calculate the dependence of the bending moduli on the
electrostatics and short-range interactions between the membrane and the
polyelectrolyte chains. The most significant effect is seen for strong
short-range interactions and low amounts of added salt where a substantial
increase in the bending moduli of order $1 k_BT$ is obtained. From short-range
repulsive membranes, the polyelectrolyte contribution to the bending moduli is
small, of order $0.1 k_BT$ up to at most $1 k_BT$. For weak short-range
attraction, the increase in membrane rigidity is smaller and of less
significance. It may even become negative for large enough amounts of added
salt. Our numerical results are obtained by solving the adsorption problem in
spherical and cylindrical geometries. In some cases the bending moduli are
shown to follow simple scaling laws.
