Enzyme Kinetics: Initial Velocity Calculation
This solution details the calculation of the initial reaction velocity ($v_0$) relative to the maximum velocity ($V_{max}$) for an enzyme-catalyzed reaction. The calculation is performed under specific conditions where the substrate concentration ([S]) is four times the Michaelis constant ($K_m$).
Michaelis-Menten Equation Fundamentals
The core principle governing enzyme kinetics under these conditions is the Michaelis-Menten equation:
$ v_0 = \frac{V_{max}[S]}{K_m + [S]} $
This equation relates the initial reaction velocity ($v_0$) to the maximum possible velocity ($V_{max}$), the substrate concentration ([S]), and the enzyme's affinity for the substrate, represented by $K_m$.
Applying the Specific Substrate Concentration
The problem specifies that the substrate concentration is four times the $K_m$. This can be written as:
$ [S] = 4 K_m $
Step-by-Step Velocity Calculation
Substitute the given substrate concentration into the Michaelis-Menten equation:
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Replace $[S]$ with $4 K_m$ in the equation:
$ v_0 = \frac{V_{max}(4 K_m)}{K_m + (4 K_m)} $
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Simplify the denominator:
$ v_0 = \frac{V_{max}(4 K_m)}{5 K_m} $
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Cancel the $K_m$ terms:
$ v_0 = \frac{4}{5} V_{max} $
Determining Velocity as a Percentage of $V_{max}$
To express the initial velocity as a percentage of $V_{max}$, calculate the fraction $\frac{v_0}{V_{max}}$ and multiply by 100%:
$ \frac{v_0}{V_{max}} = \frac{4}{5} $
$ \text{Percentage of } V_{max} = \frac{4}{5} \times 100\% = 0.8 \times 100\% = 80\% $
Final Result
Under the condition where the substrate concentration is four times the $K_m$, the observed initial velocity ($v_0$) is 80% of the maximum velocity ($V_{max}$).
Important Questions from Enzyme Kinetics Michaelis Menten K_m V_{max}
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