lac operon control: Which condition(s) cause the lac repressor to leave the operator, allowing transcription of the lac genes?

Introduction / Context:
The E. coli lac operon is a classic model of gene regulation. The lac repressor (LacI) blocks transcription by binding the operator. Inducers bind LacI, reduce its DNA affinity, and permit RNA polymerase to transcribe lacZYA.

Given Data / Assumptions:

• Lactose is converted to allolactose, the natural inducer.
• IPTG is a non-metabolizable synthetic inducer.
• Glucose influences cAMP–CAP activation but does not directly displace LacI.

Concept / Approach:
Inducer binding to LacI triggers an allosteric change that releases it from the operator. Lactose (via allolactose) and IPTG both act as inducers; glucose modulates catabolite repression, not LacI binding.

Step-by-Step Solution:
Identify events that physically release LacI from operator → inducer presence. Lactose → allolactose inducer; IPTG → direct inducer. Select the combined option for lactose and IPTG.

Verification / Alternative check:
In lab, IPTG is routinely used to induce expression from lac promoters independent of lactose metabolism.

Why Other Options Are Wrong:
Glucose presence lowers cAMP and CAP binding but does not remove LacI; absence of sugars does not induce and keeps LacI bound.

Common Pitfalls:
Conflating catabolite repression (CAP–cAMP) with repressor–operator control; forgetting that IPTG is non-metabolizable and potent.

Final Answer:
Both (b) and (c).