Q.54 An enzyme (E) catalyzes the biochemical reaction A → B with k_cat equal to 500 s⁻¹. If the initial reaction velocity (V₀) is 10 μM.s⁻¹ at the total enzyme concentration [E_t] of 30 nM and substrate concentration [A] of 40 μM, the value of K_m (in μM) is ____________.

Km Calculation from Michaelis–Menten Kinetics

The Michaelis-Menten equation relates initial velocity V₀ to substrate concentration, enzyme amount, and rate constants:

V₀ = Vmax[A] / (Km + [A])

Given values:

• kcat = 500 s⁻¹
• V₀ = 10 μM s⁻¹
• [Et] = 30 nM = 0.03 μM
• [A] = 40 μM

Step 1: Compute Vmax

Vmax = kcat × [Et] = 500 × 0.03 = 15 μM s⁻¹

Step 2: Substitute into Michaelis-Menten

10 = 15 × 40 / (Km + 40)

Cross multiply:

10(Km + 40) = 600

Divide both sides by 10:

Km + 40 = 60

Solve:

Km = 20 μM

Final Answer

Km = 20 μM

Common Pitfalls

• Failing to convert nM → μM before calculating Vmax
• Plugging Vmax incorrectly into the equation
• Assuming [A] ≫ Km without checking

Introduction

Enzyme kinetics problems often require calculating Km from experimental parameters.
This GATE Biotechnology–style question demonstrates how to use the Michaelis-Menten equation to determine Km accurately.

Using the Michaelis-Menten Equation

The general form:

V₀ = Vmax [A] / (Km + [A])

First compute Vmax from turnover number and enzyme concentration:

Vmax = kcat × [Et]

Km Computation

Insert values:

Vmax = 15 μM/s
V0 = 10 μM/s
10 = 15×40 / (Km + 40)
Km = 20 μM

Why This Matters

Km indicates how efficiently enzymes bind substrates.
Low Km means tight binding; high Km indicates weak affinity.
Here, Km = 20 μM means substrate concentration must reach ~20 μM to reach half Vmax.

Exam Insights

• Always convert units before multiplying (nM → μM)
• Check whether assumptions like [A] ≫ Km apply
• Understand that Vmax = kcat × [Et] only when enzyme is saturated

Final Takeaway: Km = 20 μM