Simple enzyme reaction rate law: for a Michaelis–Menten system, which expression correctly gives the rate of product formation rp in terms of maximum rate rmax and substrate concentration Cs?

Introduction / Context:
The Michaelis–Menten equation is a cornerstone of enzyme kinetics, relating reaction velocity to substrate concentration under quasi-steady-state conditions. Recognizing the correct algebraic form prevents common mistakes when fitting kinetic data or interpreting saturation behavior.

Given Data / Assumptions:

• Single-substrate irreversible scheme in initial-rate conditions.
• Parameters: rmax (Vmax), Km (Michaelis constant), Cs (substrate concentration).
• Product inhibition and reverse reactions are negligible at initial times.

Concept / Approach:
From QSSA and enzyme conservation, the initial rate of product formation is rp = (rmax * Cs) / (Km + Cs). This captures first-order dependence at low Cs (Cs « Km) and zero-order behavior at high Cs (Cs » Km), aligning with saturation kinetics observed experimentally.

Step-by-Step Solution:

Verification / Alternative check:
Plotting rp vs Cs yields a rectangular hyperbola approaching rmax; linear transformations (e.g., Eadie–Hofstee) confirm the parameter relationships.

Why Other Options Are Wrong:

• Expressions using CES or Cp in the denominator are not standard forms for initial-rate Michaelis–Menten.
• rmax * (Km + Cs) / Cs inverts the dependence and gives incorrect limits.

Common Pitfalls:
Confusing Vmax with kcat or substituting product concentration into the denominator; forgetting units consistency (rate vs concentration/time).

Final Answer:
rp = rmax * Cs / (Km + Cs)