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Living systems continuously sense, integrate, and act on chemical information using multiscale biochemical networks whose dynamics are inherently nonlinear, adaptive, and energy-efficient. Yet, most attempts to harness such "wetware" for external computational tasks have centered on neural tissue and electrical interfaces, leaving the information-processing potential of non-neural collectives comparatively underexplored. In this letter, we study a hybrid artificial-living cell network in which programmable artificial cells write time-varying inputs into a biochemical microenvironment, while a living bacterial collective provides the nonlinear spatiotemporal dynamics required for temporal information processing. Specifically, artificial cells transduce an external input sequence into the controlled secretion of attractant and repellent molecules, thereby modulating the "local biochemical context" that bacteria naturally sense and respond to. The resulting collective bacterial dynamics, together with the evolving molecular fields, form a high-dimensional reservoir state that is sampled coarsely (voxel-wise) and mapped to outputs through a trained linear readout within a physical reservoir computing framework. Using an agent-based in silico model, we evaluate the proposed hybrid reservoir on the Mackey-Glass chaotic time-series prediction benchmark. The system achieves normalized root mean square error (NRMSE) values of approximately 0.33-0.40 for prediction horizons H=1 to 5, and exhibits measurable short-term memory as encoded in the distributed spatiotemporal patterns of bacteria and biochemicals. These results motivate the future exploration of non-neural hybrid cell networks for in situ temporal signal processing towards novel biomedical applications.