Modularly structured signaling networks coordinate the fate and function of complex biological systems. Each component in the network performs a discrete computational operation, but when connected to each other intricate functionality emerges. Here we study such an architecture by connecting auxin signaling modules and inducible protein biotinylation systems with transcriptional control systems to construct synthetic mammalian high-detect, low-detect and band-detect networks that translate overlapping gradients of inducer molecules into distinct gene expression patterns. Guided by a mathematical model we apply fundamental computational operations like conjunction or addition to rewire individual building blocks to qualitatively and quantitatively program the way the overall network interprets graded input signals. The design principles described in this study might serve as a conceptual blueprint for the development of next-generation mammalian synthetic gene networks in fundamental and translational research.