Biological systems routinely regulate biomolecular transport with remarkable specificity, low energy input, and simple mechanisms. Here, the biophysical mechanisms of nuclear transport inspire the development of gels for recognition and selective permeation (GRASP). GRASP presents a new paradigm for specific transport and selective permeability, in which binding interactions between a biomolecule and a hydrogel lead to faster penetration of the gel. A molecular transport theory identifies key principles for selective transport: entropic repulsion of noninteracting molecules and affinity-mediated diffusion of multireceptor biomolecules through a walking mechanism. The ability of interacting molecules to walk through hydrogels enables enhanced permeability in polymer networks. To realize this theoretical prediction in a novel material, GRASP is engineered from a poly(ethylene glycol) network (entropic barrier) containing antibody-binding oligopeptides (affinity domains). GRASP is synthesized using simultaneous bioconjugation and polycondensation reactions. The elastic modulus, characteristic pore size, biomolecular diffusivity, and selective permeability are measured in the resulting materials, which are applied to regulate the transport of equally sized molecules by preferentially transporting a monoclonal antibody from a polyclonal mixture. Overall, this work presents rationally designed, nucleopore-inspired hydrogels that are capable of controlling biomolecular transport.