Michaelis–Menten insight: when substrate concentration [S] equals 0.1 * KM, what is the approximate initial velocity v as a fraction of Vmax for a simple enzyme reaction?

Introduction / Context:
The Michaelis–Menten equation relates initial velocity v to substrate concentration [S], maximum velocity Vmax, and the Michaelis constant KM. Estimating v/Vmax at particular fractions of KM is a practical skill for interpreting kinetic regimes (substrate-limited vs near-saturation).

Given Data / Assumptions:

• Reaction follows v = (Vmax * [S]) / (KM + [S]).
• Given [S] = 0.1 * KM.
• We seek v as a fraction of Vmax.

Concept / Approach:
Substitute [S] = 0.1 KM into the Michaelis–Menten equation and simplify algebraically to get a numeric fraction of Vmax. This illustrates how low-substrate conditions (well below KM) produce velocities much less than Vmax and roughly proportional to [S].

Step-by-Step Solution:

Verification / Alternative check:
At [S] = KM, v = 0.5 Vmax. Since 0.1 KM is ten-fold lower, the velocity should be far below 0.5 Vmax, consistent with ~0.09–0.10 Vmax.

Why Other Options Are Wrong:

• 0.3, 0.5, 0.7, 0.9 Vmax: these correspond to much higher [S] relative to KM; they overestimate v in the low-substrate regime.

Common Pitfalls:
Forgetting to divide by (1 + [S]/KM) after substituting; confusing KM with a saturation threshold rather than the [S] at half Vmax.

Final Answer:
0.1 * Vmax