How do regulatory proteins interact with dna?
The lacI area of the DNA constitutively expresses a repressor molecule that negatively controls lacZ and lacY expression by blocking RNA primase from synthesizing RNA in order to break down galactose.
When there is high levels of extracellular glucose, galactoside permease is inhibited from letting the cell membrane from being permeable to lactose.
However, when extracellular glucose levels are low, the cell membrane may be more permeable to lactose, which induces transcription of the lacY and lacZ genes to be expressed into proteins and enzymes that promote lactose metabolism.
Moreover when a cAMP-CAP complex is formed within the cell as a result of intercellular lactose, it promotes transcription of the aforementioned genes.
Here's another figure explaining what I just said: