A mutant strain of E. coli is grown in a culture medium containing...

The $lac$ operon consists of three structural genes: $lacZ$, $lacY$, and $lacA$.

The $lacZ$ gene codes for $\beta$-galactosidase, which is responsible for the hydrolysis of the disaccharide lactose into its monomeric units, galactose and glucose.

The $lacY$ gene codes for permease, which increases the permeability of the cell to $\beta$-galactosides (lactose), allowing it to enter the cell.

Since lactose efficiently enters the bacterial cells, the $lacY$ gene is functional. However, the inability of the cells to hydrolyse lactose indicates that $\beta$-galactosidase is non-functional or absent. This points to a missense mutation primarily in the $lacZ$ gene.

Answer: $lacZ$ gene