Biochemistry · Enzyme kinetics
Michaelis-Menten practice problems
The Michaelis-Menten equation, v = Vmax[S] / (Km + [S]), gives the starting rate of an enzyme reaction at a given substrate concentration. Km is the [S] that gives half of Vmax.
What each term means
- Vmax: the rate when the enzyme is saturated with substrate. It depends on how much enzyme is present.
- Km: the substrate concentration at half of Vmax. A lower Km means the enzyme reaches half speed at lower [S]. It is often used as a rough sign of affinity, but that holds only under some conditions.
- kcat = Vmax / [E]total: substrate molecules turned over per enzyme per unit time.
- kcat/Km: catalytic efficiency, used to compare substrates.
Worked example
An enzyme has Vmax = 100 µM/min and Km = 5 µM. What is v at [S] = 20 µM?
v = 100 × 20 / (5 + 20) = 80 µM/min, which is 80% of Vmax.
Useful landmarks: at [S] = Km, v = 50% of Vmax. At [S] = 9 Km, v = 90%.
The Lineweaver-Burk plot
Taking reciprocals gives a straight line: 1/v = (Km/Vmax)(1/[S]) + 1/Vmax.
- y-intercept = 1/Vmax
- x-intercept = −1/Km
- slope = Km/Vmax
The plot is handy for reading inhibition patterns, though it exaggerates errors at low [S].
Practice problems
1. A Lineweaver-Burk plot has a y-intercept of 0.02 min/µM and an x-intercept of −0.25 µM−1. Find Vmax and Km.
Show answer
Vmax = 1/0.02 = 50 µM/min. Km = 1/0.25 = 4 µM.
2. With Vmax = 50 µM/min and a total enzyme concentration of 0.01 µM, what is kcat?
Show answer
kcat = 50 ÷ 0.01 = 5,000 per minute, about 83 per second.
3. At what [S], in terms of Km, does an enzyme run at 75% of Vmax?
Show answer
0.75 = [S]/(Km + [S]) gives [S] = 3 Km.
Frequently asked questions
Does Km change with enzyme concentration?
No. Km is a property of the enzyme and substrate under given conditions. Vmax rises with more enzyme.
Why use initial rates?
Early in the reaction, product is scarce and [S] is close to its starting value, so the reverse reaction and substrate depletion can be ignored.
Reference: OpenStax Biology 2e section 6.5, and standard biochemistry texts (Berg et al., Biochemistry, chapter 8).