Individual cells direct non-equilibrium processes through coordinated signal transduction and gene expression, allowing for dynamic control over multicellular, system-wide behavior. This behavior extends to remodeling the extracellular polymer matrix that encases biofilms and tissues, where constituent cells dictate spatiotemporal network properties including stiffness, pattern formation, and transport properties. The majority of synthetic polymer networks cannot recreate these phenomena due to their lack of autonomous centralized actuators (i.e., cells). In addition, non-living polymer networks that perform computation are generally restricted to a few inputs (e.g., light, pH, enzymes), limiting the logical complexity available to a single network chemistry. Toward synergizing the advantages of living and synthetic systems, engineered living materials leverage genetic and metabolic programming to establish control over material-wide properties. Here we demonstrate that a bacterial metal respiration mechanism, extracellular electron transfer (EET), can control metal-catalyzed radical cross-linking of polymer networks. Linking metabolic electron flux to a synthetic redox catalyst allows dynamic, tunable, and predictable control over material formation and bulk polymer network mechanics using genetic circuits. By programming key EET genes with transcriptional Boolean logic, we rationally design computational networks that sense-and-respond to multiple inputs in biological contexts. Finally, we capitalize on the wide reactivity of EET and redox catalyses to predictably control another class of living synthetic materials using copper(I) alkyne-azide cycloaddition click chemistry. Our results demonstrate the utility of EET as a bridge for controlling abiotic materials and how the design rules of synthetic biology can be applied to emulate physiological behavior in polymer networks.