= Solution
\b[Strong lac expression requires relief of lactose-responsive repression together with a favourable glucose-responsive activation state.] In <Escherichia coli>, the <lac operon> encodes <beta-galactosidase>, <lactose permease> and a transacetylase. <LacI> binds operator <DNA> and represses transcription; binding of the inducer <allolactose> shifts its conformation and reduces operator occupancy. Basal expression provides enough permease and beta-galactosidase to import and process some <lactose>, including formation of <allolactose>. This explains induction without requiring a completely silent system somehow to manufacture its own detector.
<Glucose> availability affects positive regulation. When <glucose> is scarce, the physiological state of the <phosphotransferase system> favours activation of adenylate cyclase and increased <cyclic AMP>. Binding of <cyclic AMP> activates <catabolite activator protein>, also called CRP, which binds near the lac promoter and helps recruit <RNA polymerase>. In glucose-rich conditions this activation is reduced. Operator repression and activator-dependent recruitment act at distinct sites, so relief of repression does not guarantee maximum expression. The promoter behaves approximately as a requirement for <lactose> availability AND <glucose> scarcity, but its quantitative output is graded rather than a perfect digital gate.
<Glucose> also regulates inducer entry. During <glucose> uptake through the <phosphotransferase system>, the glucose-specific <EIIA glucose> <protein> tends to be dephosphorylated. That form can bind and inhibit <lactose permease>, reducing <lactose> entry and hence <allolactose> formation. This <inducer exclusion> reinforces repression while a preferred substrate is being used. Mutations in the central cytoplasmic loop of the permease that abolish regulation and binding provide a way of separating transport inhibition from promoter regulation.<Involvement of the central loop of the lactose permease of Escherichia coli in its allosteric regulation by the glucose-specific enzyme IIA of the phosphoenolpyruvate-dependent phosphotransferase system by C. Hoischen and others>[Hoischen and colleagues, 1996, abstract and permease-loop mutagenesis results]{ref} It is therefore incomplete to attribute every <glucose> effect to <cyclic AMP> alone.
The four limiting conditions make the integration clear. Without <lactose>, <LacI> represses whether <glucose> is high or low; an active CRP cannot simply substitute for inducer. With <lactose> and abundant <glucose>, induction is limited by weak activation and often by inducer exclusion. With <lactose> and little <glucose>, repression is relieved and CRP-dependent recruitment is favoured, giving the highest expression. With both sugars initially available, <glucose> consumption followed by adaptation to <lactose> can produce <diauxic growth>; the timing depends on concentrations, transport and culture conditions.
Genetic and physiological controls distinguish the mechanisms. A loss-of-function <LacI> mutation should remove inducer dependence and can be complemented by a functional repressor supplied in trans. An operator mutation acts primarily in cis, on the linked operon. A superrepressor that cannot respond appropriately to inducer can remain repressive despite <lactose>. Compare promoter reporters and lactose-uptake measurements separately, because an uptake defect and a transcription defect can give similar growth phenotypes. Adding <cyclic AMP> tests one activation arm, but cannot necessarily overcome a permease blocked by inducer exclusion. \b[The cell combines a local substrate signal with a global carbon-use signal, acting through both transcription and transport.]
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