Final answer:
If a single cortisol-responsive transcriptional activator regulates a set of genes, the genes should have shared DNA sequences in their regulatory regions for the activator to bind, and the transcriptional regulators should be the same. Genes do not need to be in an operon and might respond to other hormones as well.
Step-by-step explanation:
If you're studying mouse genes that increase their expression in response to cortisol, and you suspect that the same cortisol-responsive transcriptional activator regulates all of these genes, then certain key characteristics should be true if your hypothesis is correct.
(a) The cortisol-responsive genes share a DNA sequence in their regulatory regions that binds the cortisol-responsive transcriptional activator - This statement should be true, as transcriptional activators typically bind to specific DNA motifs or response elements in the regulatory regions of genes they control. For cortisol-responsive genes, this means there would be a shared sequence that the cortisol-bound receptor recognizes and binds to, initiating transcription.
(b) The genes must all be in an operon - This statement is not necessarily true in eukaryotes such as mice, as operons are primarily found in prokaryotes. In eukaryotes, genes regulated by the same transcriptional activator do not need to be physically linked as in an operon.
(c) The transcriptional regulators that bind to the regulatory regions of the cortisol-responsive genes must all be the same - This statement should be true if the regulatory mechanism is via the same cortisol-responsive transcriptional activator. The specificity of this interaction is likely due to the presence of cortisol-binding domains on the regulators that recognize the hormone-receptor complex.
(d) The genes must not be transcribed in response to other hormones - This statement is not necessarily true, as genes can be responsive to multiple hormones and have several regulatory elements that allow them to be controlled by different transcriptional activators.
Overall, if the same transcriptional activator regulates a set of cortisol-responsive genes, then these genes will indeed share specific DNA sequences in their regulatory regions that the activator binds to, and the transcriptional regulators involved will be the same or very similar. However, these genes do not need to be part of an operon, and their expression might still be influenced by other hormones due to the complexity of gene regulation mechanisms in eukaryotic cells.