Cells use spatial constraints to control and accelerate the flow of information in enzyme cascades and signalling networks. Synthetic silicon-based circuitry similarly relies on spatial constraints to process information. Here, we show that spatial organization can be a similarly powerful design principle for overcoming limitations of speed and modularity in engineered molecular circuits. We create logic gates and signal transmission lines by spatially arranging reactive DNA hairpins on a DNA origami. Signal propagation is demonstrated across transmission lines of different lengths and orientations and logic gates are modularly combined into circuits that establish the universality of our approach. Because reactions preferentially occur between neighbours, identical DNA hairpins can be reused across circuits. Co-localization of circuit elements decreases computation time from hours to minutes compared to circuits with diffusible components. Detailed computational models enable predictive circuit design. We anticipate our approach will motivate using spatial constraints for future molecular control circuit designs.