A largely unsolved problem in soft materials is how surface reconstruction competes with the rate of adsorption. Here, supported phospholipid - bilayers of DMPC (1,2-dimyristoyi-sn-glycero-3-phosphocholine) were employed as substrates for the adsorption of a weak polyelectrolyte, polymethacrylic acid, whose time dependent ratio of charged to uncharged functional groups served to probe the local dielectric environment. Chains that encountered sparsely covered surfaces spread to maximize the number of segment-surface contacts at rates independent of the molar mass (which was varied by a factor of 30), but dependent on the phase of the lipid bilayer, gel or liquid crystal. Surface reconstruction rather than molar mass of the adsorbing molecules seemed to determine the rate of spreading. The significance of these findings is the stark contrast with well-known views of polymer adsorption onto surfaces having structures that are 'frozen' and unresponsive, and is relevant not just from biological and biophysical standpoints, but also in the formulation of many cosmetics and pharmaceutical products.