Step 1: Understanding the Concept:
Restriction endonucleases are bacterial enzymes that cleave double-stranded DNA at or near specific recognition sequences.
They are classified into Types I, II, III, and IV based on their subunit composition, cofactor requirements, and cleavage sites.
Step 2: Detailed Explanation:
Let us compare the cofactor requirements of the listed enzymes:
1. EcoRI (Type II): Type II restriction endonucleases are simple, homodimeric enzymes.
They require only divalent magnesium ions (\( \text{Mg}^{2+} \)) as a cofactor to catalyze phosphodiester bond hydrolysis at highly specific, palindromic target sites.
They do not require ATP hydrolysis or S-adenosylmethionine (SAM) for their endonuclease activity, making them primary tools in cloning.
2. EcoK and EcoB (Type I): Type I restriction enzymes are complex, multisubunit enzymes.
They require \( \text{Mg}^{2+} \), S-adenosylmethionine (SAM), and ATP.
They cleave DNA at non-specific sites far from their recognition sequence.
3. EcoP1 (Type III): Type III restriction enzymes are heterodimeric proteins that require \( \text{Mg}^{2+} \) and ATP for cleavage.
They do not strictly require SAM for cleavage, though it can stimulate the reaction.
Therefore, EcoRI is the classic representative that relies exclusively on \( \text{Mg}^{2+} \) as its active cofactor for site-specific cleavage.
Step 3: Final Answer:
The enzyme that requires only \( \text{Mg}^{2+} \) is EcoRI, corresponding to option (B).