In this work we have examined in detail by means of fully atomistic molecular dynamics simulations, physical hydrogels formed by a polymer electrolyte, poly(acrylic acid), and graphene oxide, at two different charging states of the polymer and two different graphene oxide concentrations. It was found that variations of these parameters incurred drastic changes in general morphological characteristics of the composite materials, the degree of physical adsorption of polyelectrolyte chains onto the graphene oxide surface, the polymer dynamic response at local and global length scales, in the charge distributions around the components, and in the mobility of the counterions. All these microscopic features are expected to significantly affect macroscopic physical properties of the hydrogels, such as their mechanical responses and their electrical behaviors.