Three partitions, obtained by FACS, of the low mutation probability trans-element library (corresponding to no expression, intermediate, and high expression phenotypes) were combined with the three equivalent partitions of the high mutation probability cis-element library in the absence of CI. The cis-element library in the absence of CI shows only the effect of mutations on RNAP binding. The DME of the original trans-element library, corresponding to Figure 2E, is shown on top. The DME of the original cis-element library, corresponding to Figure 2—figure supplement 1D, is shown on the right. The arrows illustrate from which category of the original DME (either trans or cis) were the sorted mutants used to make the particular combined library. DMEs of all nine partition-combination libraries were estimated using flow cytometry, with each distribution obtained by pooling two independent measurements of 500,000 cells. Also shown is the sorting accuracy, obtained as the repeated DME measurement of each trans partition following the original FACS sorting (panels J,K,L). The distributions of fluorescence are shown in the presence of CI. The dashed lines separate three categories of phenotypes – ‘no expression’, ‘intermediate’, and ‘high expression’ phenotypes. Numbers are percentage of counts in each category. At least in part, intermolecular epistasis can be qualitatively explained by considering the genetic regulatory structure of the system, as follows. Panels A), (B), and C): no expression cis-element mutants do not bind RNAP sufficiently to lead to expression, so that system mutants containing them remain in the ‘no expression’ bin irrespective of the effect of mutations in trans. (D) No expression trans-element mutants fully repress on wildtype cis. When cis-element mutations of intermediate expression are introduced, some still bind the functional CI mutants leading to repression, while others carry mutations that prevent CI binding, resulting in intermediate expression system mutants. (E) Intermediate expression CI mutants only partially repress on wildtype cis, but fully repress some cis-element mutants that have lowered RNAP binding. Other intermediate expression CI mutants do not bind a mutated cis-element background. (F) High expression CI mutants cannot bind wildtype cis. Similarly, they do not bind intermediate expression cis-element mutants, resulting in intermediate expression in the system. (G), (H): Some high expression cis-element mutants can fully bind mutated but functional CI mutants, others can only partially bind them, while some maintain full RNAP binding while losing all CI binding. (I) non-functional CI mutants do not repress on wildtype cis, hence also not on mutated cis-element backgrounds.