This question requires calculating the Michaelis constant ($K_m$) using the Michaelis-Menten equation parameters provided.

Solution Steps

Step 1: Calculate Maximum Velocity ($V_{max}$)

The maximum velocity ($V_{max}$) is related to the turnover number ($k_{cat}$) and the total enzyme concentration ($[E_t]$) by the formula:

$V_{max} = k_{cat}[E_t]$

Given:

• $k_{cat} = 500 s^{-1}$
• $[E_t] = 30 \text{ nM} = 30 \times 10^{-3} \mu M$ (converting nM to $\mu M$ for unit consistency)

Calculation:

$V_{max} = (500 s^{-1}) \times (30 \times 10^{-3} \mu M)$

$V_{max} = 15 \mu M.s^{-1}$

Step 2: Calculate Michaelis Constant ($K_m$)

The Michaelis-Menten equation relates initial velocity ($V_0$), maximum velocity ($V_{max}$), substrate concentration ($[S]$), and $K_m$:

$V_0 = \frac{V_{max}[S]}{K_m + [S]}$

We need to rearrange this equation to solve for $K_m$. First, let's isolate the term containing $K_m$:

$K_m + [S] = \frac{V_{max}[S]}{V_0}$

Now, solve for $K_m$:

$K_m = \frac{V_{max}[S]}{V_0} - [S]$

Given values for this step:

• $V_{max} = 15 \mu M.s^{-1}$ (from Step 1)
• $[S] = [A] = 40 \mu M$
• $V_0 = 10 \mu M.s^{-1}$

Calculation:

$K_m = \frac{(15 \mu M.s^{-1}) \times (40 \mu M)}{10 \mu M.s^{-1}} - 40 \mu M$

$K_m = \frac{600}{10} \mu M - 40 \mu M$

$K_m = 60 \mu M - 40 \mu M$

$K_m = 20 \mu M$

Conclusion

The calculated value of $K_m$ is $20 \mu M$. This value falls within the specified range.