Topic: Enzyme Kinetics
The Michaelis-Menten model describes how the initial rate of an enzymatic reaction changes with the concentration of the substrate.
Steps:
Understanding the Question: What is the velocity ($V$) of a reaction when the amount of substrate ($S$) exactly matches the Michaelis constant ($K_m$)?
Key Formulas and Approach: Use the Michaelis-Menten Equation: $V = \frac{V_{max}[S]}{K_m + [S]}$.
Detailed Solution:
Substitute $[S] = K_m$ into the equation:
$V = \frac{V_{max} \cdot K_m}{K_m + K_m}$
$V = \frac{V_{max} \cdot K_m}{2 \cdot K_m}$
The $K_m$ terms cancel out, leaving $V = \frac{V_{max}}{2}$.
This shows that $K_m$ is defined as the substrate concentration at which the reaction rate is exactly half of its maximum.