The key to this question is to remember that lacI mutations will be trans-acting as they produce a protein product, and that lacP mutations, like lacO mutations will be cis-acting as they are affecting a binding site for ribonucleic acid (RNA) polymerase. For the purposes of this problem, designate the uninducible lacI mutations as iu mutations, and the uninducible lacP mutations as pu mutations. There are quite a number of satisfactory genotypes, which can serve to distinguish between the iu and pu mutations. Here are a few examples:
| Enzyme activity | ||
| Genotype | β-galactosidase | Permease |
| 1 iu p+ o+ z+ y+ | absent | absent |
| 2 i+ pu o+ z+ y+ | absent | absent |
| 3 iu p+ o+ z+ y- / i+ p+ o+ z-y+ | absent | absent |
| 4 i+ pu o+ z+ y- / i+ p+ o+ z-y+ | absent | inducible |
| 5 iu p+ oc z+ y- / i+ p+ o+ z- y+ | Constitutive | absent |
| 6 i+ pu o c z+ y- / i+ p+ o+ z- y+ | absent | inducible |
Genotypes 1 and 2 are simply symbolic restatements of the phenotypes of the uninducible mutations. Genotypes 3 and 4 are straightforward tests to distinguish the cis-acting pu mutations from the trans-acting iu lesions. The results of genotype 3 reflect the expectation that iu mutations would be trans-acting and dominant to i+. This is expected because the iu-encoded repressor protein molecules would be incapable of being inactivated by binding to inducer; the presence or absence of normal repressor protein is irrelevant. The results of genotype 4, on the other hand, reflect the expectation that pu mutations would only cis-acting.
Hence, any genes in cis to the pu allele would be inactive, while any genes in cis to the normal p+ allele would potentially be transcribed normally that is if all other regulatory functions were normal. In a similar fashion, genotypes 5 and 6 distinguish the cis versus transaction of iu and pu mutations. In genotypes 5, iu remains trans-dominant to i+, but this dominance is overcome by the cis-acting oc mutation, this can be understood by comparing genotypes 3 and 5. In genotype 6, the presence of oc is irrelevant, as it is in a lac operon that contains the pu mutation preventing RNA polymerase binding, this can be understood by comparing genotypes 4 and 6.