Progress in synthetic biology tools has transformed the way we engineer living cells. Applications of circuit design have reached a new level, offering solutions for metabolic engineering challenges that include developing screening approaches for libraries of pathway variants. The use of transcription-factor-based biosensors for screening has shown promising results, but the quantitative relationship between the sensors and the sensed molecules still needs more rational understanding. Herein, we have successfully developed a novel biosensor to detect pinocembrin based on a transcriptional regulator. The FdeR transcription factor (TF), known to respond to naringenin, was combined with a fluorescent reporter protein. By varying the copy number of its plasmid and the concentration of the biosensor TF through a combinatorial library, different responses have been recorded and modeled. The fitted model provides a tool to understand the impact of these parameters on the biosensor behavior in terms of dose-response and time curves and offers guidelines to build constructs oriented to increased sensitivity and or ability of linear detection at higher titers. Our model, the first to explicitly take into account the impact of plasmid copy number on biosensor sensitivity using Hill-based formalism, is able to explain uncharacterized systems without extensive knowledge of the properties of the TF. Moreover, it can be used to model the response of the biosensor to different compounds (here naringenin and pinocembrin) with minimal parameter refitting.