The enzyme $\alpha$-amylase used in starch hydrolysis has an affinity constant ($K_m$) value of $0.005$ M. To achieve one-fourth of the maximum rate of hydrolysis, the required starch concentration in mM (rounded off to two decimal places) is____.
Enzyme Kinetics Calculation for $\alpha$-Amylase
This problem involves enzyme kinetics, specifically using the Michaelis-Menten model to determine the substrate concentration ($[S]$) required to achieve a specific fraction of the maximum reaction rate ($V_{max}$). The enzyme is $\alpha$-amylase, and its affinity constant ($K_m$) is given.
Applying Michaelis-Menten Kinetics
The Michaelis-Menten equation relates the reaction rate ($V$) to the maximum rate ($V_{max}$), substrate concentration ($[S]$), and the Michaelis constant ($K_m$):
$V = \frac{V_{max} \cdot [S]}{K_m + [S]}$
Determining Required Starch Concentration
We need to find the starch concentration ($[S]$) when the rate ($V$) is one-fourth of the maximum rate ($V_{max}/4$). Setting up the equation:
$\frac{V_{max}}{4} = \frac{V_{max} \cdot [S]}{K_m + [S]}$
To solve for $[S]$, we can simplify the equation:
- Divide both sides by $V_{max}$:
$\frac{1}{4} = \frac{[S]}{K_m + [S]}$
- Cross-multiply:
$K_m + [S] = 4 \cdot [S]$
- Rearrange the terms to isolate $[S]$:
$K_m = 4 \cdot [S] - [S]$
$K_m = 3 \cdot [S]$
- Solve for $[S]$:
$[S] = \frac{K_m}{3}$
Calculation and Conversion
Now, substitute the given $K_m$ value ($0.005$ M) into the derived formula:
$[S] = \frac{0.005 \text{ M}}{3}$
$[S] \approx 0.001666... \text{ M}$
The question asks for the concentration in millimolar (mM). Since $1$ M $= 1000$ mM:
$[S] \approx 0.001666... \times 1000 \text{ mM}$
$[S] \approx 1.666... \text{ mM}$
Final Answer Formatting
Rounding the result to two decimal places:
$[S] \approx 1.67 \text{ mM}$
The required starch concentration is approximately 1.67 mM.
Important Questions from Enzyme Kinetics and Michaelis Menten Equation
- An enzyme (E) catalyzes the biochemical reaction $A \rightarrow B$ with $k_{cat}$ equal to $500 s^{-1}$. If the initial reaction velocity ($V_0$) is $10 \mu M.s^{-1}$ at the total enzyme concentration $[E_t]$ of 30 nM and substrate concentration $[A]$ of $40 \mu M$, the value of $K_m$ (in $\mu M$) is ________
Within the Michaelis-Menten framework, the ratio of $v_0/V_{max}$
when $[S] = 20 \times K_m$ is _________.
(Round off to two decimal places)
An enzymatic reaction exhibits Michaelis-Menten kinetics. For this reaction, on doubling the concentration of enzyme while maintaining [S] >> [$E_o$],
- In an assay of the type II dehydroquinase of molecular mass 18 kDa, it is found that the $V_{max}$ of the enzyme is $0.0134 \ \mu mol.min^{-1}$ when $1.8 \ \mu g$ enzyme is added to the assay mixture. If the $K_m$ for the substrate is $25 \ \mu M$, the $k_{cat}/K_m$ ratio will be ____________________ $\times 10^4 \ M^{-1}.s^{-1}$.
- A single subunit enzyme converts 420 µmoles of substrate to product in one minute. The activity of the enzyme is __________ $ \times 10^{-6} $ Katal.
