Normally, the repressor searches for the operator by rapidly binding and dissociating from non-operator sequences. Even for sequences that mimic the true operator, the dissociation time is only a few seconds or less. Therefore, it is easy for the repressor to find new operators as new strands of deoxyribonucleic acid (DNA) are synthesized.
However, when the affinity of the repressor for the DNA and operator is increased, it takes too long for the repressor to dissociate from sequences on the chromosome that mimic the true operator, and as the cell divides and new operators are synthesized, the repressor never quite finds all of them in time, leading to a partial synthesis of β-galactosidase synthesis.
This explains why, in the absence of inducer isopropyl-β-D-thiogalactoside (IPTG), there is some elevated β-galactosidase synthesis. When IPTG binds to the repressors with increased affinity, it lowers the affinity back to that of the normal repressor without IPTG bound. Then, the repressor can rapidly dissociate from sequences in the chromosome that mimic the operator and find the true operator. Thus, β-galactosidase is repressed in the presence of IPTG in strains with repressors that have greatly increased affinity for operator. In summary, because of a kinetic phenomenon, we see a reverse induction curve.