In the lac operon the operator overlaps the promoter and the 5 end of the first structural gene

The answer is that repression never completely shuts down transcription of the lac operon. Even with active repressor bound to the operator, there is a low level of transcription and a few molecules of p-galactosidase, permease, and transacetylase are synthesized. When lactose appears in the medium, the permease that is present transports a small amount of lactose into the cell. There, the few molecules of p-galactosidase that are present convert some of the lactose into allolactose. The allolactose then attaches to the repres-sor and alters its shape so that the repressor no longer binds to the operator. When the operator site is clear, RNA poly-merase can bind and transcribe the structural genes of the lac operon.

Several compounds related to allolactose also can bind to the lac repressor and induce transcription of the lac operon. One such inducer is isopropylthiogalactoside (IPTG). Although IPTG inactivates the repressor and allows the transcription of lacZ, lacY, and lacA, IPTG is not metabolized by p-galactosidase; for this reason it is often used in research to examine the effects of induction, independent of metabolism.

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