Transcriptional regulation in all organisms relies on weak protein-DNA and protein-protein interactions. The quantitative strength of these interactions is critically important for understanding regulatory mechanisms, but measuring such interactions to the necessary precision in living cells has presented major experimental difficulties. I will describe a general experimental/modeling strategy that overcomes these difficulties. This approach involves making two-dimensional expression measurements on synthetic regulatory sequences. If the underlying molecular mechanisms are properly understood, these measurements should fall along a one-dimensional “expression manifold” within this two-dimensional space. Energetic parameters are then learned by quantitatively modeling this manifold. This strategy has important advantages over standard statistical modeling. In particular, it is far more transparent, allowing one to quantify key parameters and to assess potential model misspecification, both by direct visual inspection.