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Information processing using biochemical circuits is essential for survival and reproduction of natural organisms. Construction of synthetic biochemical circuits from simple components provides a useful approach to establish the minimal determinants required for complex logical functions. As stripped-down analogues of genetic regulatory networks in cells, we engineered artificial transcriptional networks consisting of synthetic DNA switches, regulated by RNA signals acting as transcription activators or repressors, and two enzymes, bacteriophage T7 RNA polymerase and Escherichia coli ribonuclease H. The synthetic switch design is modular with programmable connectivity and allows dynamic control of RNA signals through enzyme-mediated production and degradation. The switches support sharp and adjustable thresholds using a competitive hybridization mechanism, analogous to a biological threshold mechanism, "inhibitor ultrasensitivity," thus allowing arbitrary analog or digital circuits to be created in principle. Theoretical correspondence of our biochemical network to neural networks where synaptic weights and thresholds are encoded by concentrations of DNA strands greatly facilitates network design and analysis. Experimentally, we have constructed and analyzed several simple networks: positive and negative autoregulatory circuits, a mutual inhibitory circuit, and oscillators with positive and negative feedback. Reasonable agreement between experimental data and a simple mathematical model was obtained for switch input/output functions, phaseplane trajectories, the bifurcation diagram, and oscillation periods. A systematic quantitative characterization lead to identification of important network properties such as the saturation of degradation machinery and challenges to understand such as the interference by incomplete RNA signals. Construction of larger synthetic circuits provides a unique opportunity for evaluating model inference, prediction, and design of complex biochemical systems and could be used to ontrol nanoscale devices and artificial cells.