The spatial coordination of cellular differentiation enables functional organogenesis. How coordination results in specific patterns of differentiation in a robust manner is a fundamental question for all developmental systems in biology. Theoreticians such as Turing and Wolpert have proposed the importance of specific mechanisms that enable certain types of patterns to emerge, but these mechanisms are often difficult to identify in natural systems. Therefore, we have started using synthetic biology to ask whether specific mechanisms of pattern formation can be engineered into a simple cellular background. In this talk, I will show several examples of emergent spatial patterning that results from the insertion of synthetic signalling pathways and transcriptional logic into E. coli. In all cases, we use computational modelling to initially design circuits with a desired outcome, and improve the selection of biological components (DNA sub-sequences) that achieve this outcome according to a quantifiable measure. In the specific case of Turing patterns, we have yet to produce a functional system in vivo, but I will describe new analytical tools that are helping to guide the design of synthetic circuits that can produce a Turing instability.