Question

An enzyme is discovered that catalyzes the chemical reaction CALM \(\rightleftharpoons\) ANGER. A team of motivated researchers sets out to study the enzyme, which they called Angerase. They find that the \( \text{K}_{cat} \) (constant) for Angerase is \( 600 \, \text{s}^{-1} \). Some experiments were done. In one such experiment when [\( \text{E}_t \)] (total enzyme concentration) \( = 20 \, \text{nM} \) and [CALM] \( = 40 \, \mu\text{M} \), the reaction velocity, \( \text{V}_0 \) is \( 9.6 \, \mu\text{M s}^{-1} \), calculate Km (Michaelis constant) for the substrate 'CALM'.

• A. 90 \(\mu\)M
• B. 200 \(\mu\)M
• C. 10 \(\mu\)M
• D. 0.8 \(\mu\)M

Hint:

Always ensure concentration units match before plugging them into the Michaelis-Menten equation. Converting nM (Total Enzyme) to \(\mu\)M prevents massive calculation errors.
\( 20 \, \text{nM} = 0.02 \, \mu\text{M} \). Then \( V_{max} = 600 \times 0.02 = 12 \, \mu\text{M s}^{-1} \). Much faster!

Answer 1

The Correct Option is C

Step 1: Concept:
We are given enzyme kinetics parameters (\(K_{cat}\), Total Enzyme \([E_t]\), Substrate concentration \([S]\), and initial velocity \(V_0\)) and asked to calculate the Michaelis-Menten constant (\(K_m\)).

Step 2: Key Formula or Approach:
This requires two foundational equations of Michaelis-Menten kinetics:
1) Maximum velocity: \( V_{max} = K_{cat} \times [E_t] \)
2) Michaelis-Menten equation: \( V_0 = \frac{V_{max} \times [S]}{K_m + [S]} \)

Step 3: Step-by-step Explanation:

• First, identify the given variables and standardize units:
\( K_{cat} = 600 \, \text{s}^{-1} \)
\( [E_t] = 20 \, \text{nM} = 20 \times 10^{-9} \, \text{M} \)
\( [S] = [\text{CALM}] = 40 \, \mu\text{M} \)
\( V_0 = 9.6 \, \mu\text{M s}^{-1} \)

• Calculate \( V_{max} \):
\[ V_{max} = K_{cat} \times [E_t] \]
\[ V_{max} = 600 \, \text{s}^{-1} \times (20 \times 10^{-9} \, \text{M}) \]
\[ V_{max} = 12000 \times 10^{-9} \, \text{M s}^{-1} = 12 \times 10^{-6} \, \text{M s}^{-1} \]
Convert to \(\mu\text{M}\) to match the other units:
\[ V_{max} = 12 \, \mu\text{M s}^{-1} \]

• Substitute into the Michaelis-Menten equation:
\[ V_0 = \frac{V_{max} \times [S]}{K_m + [S]} \]
\[ 9.6 = \frac{12 \times 40}{K_m + 40} \]

• Solve for \( K_m \):
\[ 9.6 \times (K_m + 40) = 480 \]
\[ 9.6 \, K_m + 384 = 480 \]
\[ 9.6 \, K_m = 480 - 384 \]
\[ 9.6 \, K_m = 96 \]
\[ K_m = \frac{96}{9.6} = 10 \, \mu\text{M} \]

Step 4: Final Answer:
The calculated \( K_m \) for the substrate CALM is \( 10 \, \mu\text{M} \), corresponding to Option (C).