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Genes are regulated by transcription factors that bind to regions of genomic DNA called enhancers. Considerable effort is focused on identifying transcription factor binding sites, with the goal of predicting gene expression from DNA sequence. Despite this effort, general, predictive models of enhancer function are currently lacking. Here we combine quantitative models of enhancer function with manipulations using engineered transcription factors to examine the extent to which enhancer function can be controlled in a quantitatively predictable manner. Our models, which incorporate few free parameters, can accurately predict the contributions of ectopic transcription factor inputs. These models allow the predictable 'tuning' of enhancers, providing a framework for the quantitative control of enhancers with engineered transcription factors.