Final answer:
An increase in lacI repressor protein makes inducement of the lac operon more difficult because more of the repressor is available to bind to the operator region, preventing transcription. The repression can occur despite the presence of lactose and the positive influence of the CAP-cAMP complex, as the lacI binding is dominant.
Step-by-step explanation:
An increase in lacI protein, which is the repressor protein of the lac operon, makes it more difficult to induce the lac operon in the presence of lactose because the lacI protein binds to the operator region. When the repressor is bound, it prevents RNA polymerase from transcribing the structural genes necessary for lactose metabolism. Therefore, a higher concentration of the lacI protein translates to more repressor molecules that can bind to the operator, making it less likely that lactose will induce the operon by removing the repressor.
The expression of the lac operon is also influenced by the presence of glucose. When glucose levels are low, cyclic AMP (cAMP) accumulates and binds to the catabolite activator protein (CAP). This complex then enhances the transcription of the lac operon. But even with CAP acting as an activator, if the levels of repressor protein (lacI) are high, this can still hinder the induction of the operon because repressor binding overrides the positive effect of the CAP-cAMP complex.
Overall, the presence of lactose results in the formation of allolactose, which binds to lacI, preventing it from binding to the operator region and thus allowing transcription. However, with an increased amount of lacI protein, more allolactose is required to inhibit all the repressor molecules, leading to difficulty in inducing the operon.