DNA supercoiling acts as a basal transcriptional regulator, which contributes to the quick and global transcriptional response of bacteria to many environmental changes. In spite of this importance, mechanistic models explaining the differential response of promoters to global topological variations of the chromosome remain essentially lacking. Here, we present the first quantitative transcriptional regulatory model by DNA supercoiling, focusing on the specific step of promoter opening during transcription initiation. Based on the known physico-chemical properties of DNA denaturation, it involves only one global adjustable parameter and is yet able to predict the global supercoiling response of promoters in a wide range of bacteria, based on the sequence content of their "discriminator" element. We first show that it quantitatively predicts both \emph{in vitro} and \emph{in vivo} data from transcription assays focusing on individual model promoters. We then assess the universality of the mechanism by analyzing transcriptomes of phylogenetically distant bacteria under conditions of supercoiling variation: (1) by gyrase-inhibiting antibiotics, (2) by biologically relevant environmental stresses, (3) naturally acquired and inherited in the longest-running evolution experiment. The model robustly predicts a significant contribution of the entire transcriptomic response to transient or inherited supercoiling variations in various species. This study strongly suggests that the proposed physical mechanism is used as an ubiquitous regulatory mechanism in the whole prokaryotic kingdom, based on the fundamental mechanical properties of the double-helix. Importance: DNA supercoiling acts as a global yet underestimated transcriptional regulator in bacteria. We propose the first quantitative model of this regulation mode, based on the specific step of promoter opening during transcription initiation, explaining the differential response of promoters to global topological variations of the chromosome. In contrast to classical mechanisms requiring dedicated regulatory molecules to bind target promoters, we show that global deformations of the DNA template itself underpin a selective response of each particular promoter, according to its "discriminator" sequence, by modulating the ability of RNA Polymerase to initiate transcription. This study defines the first systematic rule underpinning the ubiquitous regulatory action of DNA supercoiling on the core transcriptional machinery, in particular in response to quick environmental changes.