In enzyme kinetics, according to the Michaelis–Menten framework, what is the correct rate expression for competitive inhibition (define symbols clearly and choose the appropriate formula)?

Introduction / Context:
Competitive inhibition is a classic regulatory mode in biochemical kinetics where an inhibitor competes with the substrate for the same active site. This question checks whether you can recall and apply the modified Michaelis–Menten equation that accounts for a competitive inhibitor at concentration I with inhibition constant Ki.

Given Data / Assumptions:

• Single-substrate, steady-state Michaelis–Menten kinetics.
• Inhibitor competes reversibly for the active site (no catalysis by inhibitor-bound enzyme).
• Symbols: Vmax is the maximum rate, Km is the Michaelis constant without inhibitor, S is substrate concentration, I is inhibitor concentration, Ki is the dissociation constant for E·I.

Concept / Approach:
In competitive inhibition, the apparent affinity for substrate decreases because inhibitor occupancy reduces the fraction of free enzyme available for S. The measurable effect is an increased apparent Km (Km,app = Km * (1 + I / Ki)) while Vmax remains unchanged at saturating S. Therefore, only the denominator of the Michaelis–Menten expression is modified by the multiplicative factor (1 + I / Ki) applied to Km.

Step-by-Step Solution:
1) Start with v = (Vmax * S) / (Km + S) for the uninhibited reaction.2) For competitive inhibition, replace Km with Km,app = Km * (1 + I / Ki).3) Hence v = (Vmax * S) / (Km * (1 + I / Ki) + S).4) Check boundary conditions: as S → ∞, v → Vmax (unchanged), and as I increases, low-S slope decreases, consistent with increased Km,app.

Verification / Alternative check:
Lineweaver–Burk: 1/v = (Km/Vmax)(1/S) * (1 + I / Ki) + 1/Vmax. Slopes increase with I while y-intercept 1/Vmax remains the same, confirming unchanged Vmax and increased apparent Km.

Why Other Options Are Wrong:
Vmax * E in the numerator (option B) is dimensionally inconsistent for the rate equation form and not a standard expression.Vmax * I in the numerator (option C) falsely suggests rate increases with inhibitor concentration.Omitting + S in the denominator (option D) removes the saturable term and gives incorrect limiting behavior.

Common Pitfalls:
Confusing competitive with noncompetitive or uncompetitive changes; mixing up which parameter (Km vs Vmax) is affected; dropping the S term in the denominator; or misplacing the (1 + I / Ki) factor on S rather than on Km.

Final Answer:
v = (Vmax * S) / (Km * (1 + I / Ki) + S)