Eventually it was discovered that two additional operators are involved in lac regulation.  One (O 3 ) lies about -90 bp upstream of O 1 in the end of the lacI gene, and the other (O 2 ) is about +410 bp downstream of O 1 in the early part of lacZ . These two sites were not found in the early work because they have redundant functions and individual mutations do not affect repression very much. Single mutations to either O 2 or O 3 have only 2 to 3-fold effects. However, their importance is demonstrated by the fact that a double mutant defective in both O 2 and O 3 is dramatically de-repressed (by about 70-fold).