Question

Within the Michaelis-Menten framework, the ratio of \( v_0 / V_{{max}} \) when [S] = 20 × \( K_m \) is ________.

Hint:

When the substrate concentration is much higher than the Michaelis constant, the reaction velocity approaches \( V_{{max}} \), and the ratio \( v_0 / V_{{max}} \) approaches 1.

Answer 1

Step 1: Recall the Michaelis-Menten equation. The Michaelis-Menten equation is: \[ v_0 = \frac{V_{{max}} [S]}{K_m + [S]}, \] where:
\( v_0 \) is the initial velocity,
\( V_{{max}} \) is the maximum velocity,
\( [S] \) is the substrate concentration,
\( K_m \) is the Michaelis constant.
The ratio \( v_0 / V_{{max}} \) is given by: \[ \frac{v_0}{V_{{max}}} = \frac{[S]}{K_m + [S]}. \] Step 2: Substitute \( [S] = 20 \times K_m \).
Substitute \( [S] = 20 \times K_m \) into the equation: \[ \frac{v_0}{V_{{max}}} = \frac{20 \times K_m}{K_m + 20 \times K_m} = \frac{20}{1 + 20} = \frac{20}{21}. \] Step 3: Calculate the value.
\[ \frac{v_0}{V_{{max}}} = \frac{20}{21} \approx 0.95. \] Thus, the ratio \( v_0 / V_{{max}} \) is \( \boxed{0.95} \).