Final answer:
In the lac operon, the trans-regulatory element is the Lac repressor, which regulates gene expression by blocking the transcription of structural genes in the absence of lactose.
Step-by-step explanation:
In the lac operon, a trans-regulatory element refers to a factor that can regulate gene expression from a different location than the gene it is controlling.
One such trans-regulatory element is the Lac repressor, which is coded by the lacI gene located elsewhere on the Escherichia coli chromosome.
The role of the Lac repressor in gene regulation is to bind to the operator region of the lac operon, physically blocking RNA polymerase from transcribing the structural genes of the operon—specifically the lacZ, lacY, and lacA genes—when lactose is absent. This blockage prevents the unnecessary production of enzymes involved in lactose metabolism.