The lac Operon: Inducible Regulation
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The lac operon is inducible: when lactose is present, its metabolite binds the lac repressor, releasing it from the operator and permitting transcription.
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Allolactose binds the lac repressor, changing its shape so it can no longer bind the operator, letting polymerase transcribe the lactose genes.
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Repressor release only permits transcription — full activity needs glucose scarce, reverse from the trp operon's shutoff.