Question

Applying the Michaelis-Menten Equation III A research group discovers a new version of happyase, which they call happyase *, that catalyzes the chemical reaction HAPPY \(\rightleftharpoons\) SAD. The researchers begin to characterize the enzyme. a. In the first experiment, with \(\left[E_{t}\right]\) at \(4 \mathrm{~nm}\), they find that the \(V_{\max }\) is \(1.6 \mu \mathrm{M} \mathrm{s}^{-1}\). Based on this experiment, what is the \(k_{\text {cat }}\) for happyase*? (Include appropriate units.) b. In the second experiment, with \(\left[E_{t}\right]\) at \(1 \mathrm{~nm}\) and [HAPPY] at \(30 \mu \mathrm{M}\), the researchers find that \(V_{0}=300 \mathrm{nM} \mathrm{s}^{-1}\). What is the measured \(K_{\mathrm{m}}\) of happyase* for its substrate HAPPY? (Include appropriate units.) c. Further research shows that the purified happyase * used in the first two experiments was actually contaminated with a reversible inhibitor called ANGER. When ANGER is carefully removed from the happyase * preparation and the two experiments are repeated, the measured \(V_{\max }\) in (a) is increased to \(4.8 \mu \mathrm{M} \mathrm{s}^{-1}\), and the measured \(K_{\mathrm{m}}\) in (b) is now \(15 \mu_{\mathrm{M}}\). Calculate the values of \(a\) and \(\alpha^{\prime}\) for ANGER. d. Based on the information given, what type of inhibitor is ANGER?

Short answer

a) 400 s^{-1}, b) 130 μM, c) a = 3, α' ≈ 8.67, d) mixed inhibitor.

Step-by-step solution

Calculate k_cat for Experiment (a)

To find the catalytic constant (turnover number) \( k_{\text{cat}} \), use the formula:\[ k_{\text{cat}} = \frac{V_{\max}}{[E_t]}\]Substitute the given values: \( V_{\max} = 1.6 \mu M \cdot s^{-1} \) and \( [E_t] = 4 \text{ nm} \) (convert to \( \mu M \): \( 4 \text{ nm} = 0.004 \mu M \)).\[ k_{\text{cat}} = \frac{1.6 \mu M \cdot s^{-1}}{0.004 \mu M} = 400 s^{-1}\]Thus, \( k_{\text{cat}} \) is 400 \( s^{-1} \).

Calculate K_m for Experiment (b)

The Michaelis-Menten equation is given by:\[ V_0 = \frac{V_{\max} \cdot [S]}{K_m + [S]}\]Reorganize to solve for \( K_m \):\[ K_m = \frac{V_{\max} \cdot [S]}{V_0} - [S]\]Substitute \( V_0 = 300 \text{ nM} \cdot s^{-1} = 0.3 \mu M \cdot s^{-1} \), \( V_{\max} = 1.6 \mu M \cdot s^{-1} \), and \( [S] = 30 \mu M \).\[ K_m = \frac{1.6 \mu M \cdot s^{-1} \cdot 30 \mu M}{0.3 \mu M \cdot s^{-1}} - 30 \mu M = 160 \mu M - 30 \mu M = 130 \mu M\]The measured \( K_m \) is 130 \( \mu M \).

Calculate Inhibition Constant a for ANGER

From Step 1, \( V_{\max} \) increased from \( 1.6 \mu M \cdot s^{-1} \) to \( 4.8 \mu M \cdot s^{-1} \) after removing the inhibitor, suggesting that ANGER affects \( V_{\max} \). The factor \( a \) is calculated as:\[ a = \frac{4.8}{1.6} = 3\]

Calculate Inhibition Constant \( \alpha' \) for ANGER

From Step 2, \( K_m \) changed from \( 130 \mu M \) to \( 15 \mu M \) when ANGER was removed. The factor \( \alpha' \) is:\[ \alpha' = \frac{130}{15} \approx 8.67\]

Determine the Type of Inhibitor

Since \( V_{\max} \) changed, ANGER is not a non-competitive inhibitor, and since both \( V_{\max} \) and \( K_m \) were altered, it suggests mixed inhibition, as competitive inhibition only changes \( K_m \). Therefore, ANGER is a mixed inhibitor.

Key concepts

Catalytic Constant (k_cat)

The catalytic constant, commonly denoted as \( k_{\text{cat}} \), is a crucial parameter in enzyme kinetics. It represents the turnover number, or the number of substrate molecules converted into product by an enzyme molecule in a unit of time, when the enzyme is fully saturated with the substrate. This metric gives us an insight into the enzyme's efficiency.

To calculate \( k_{\text{cat}} \), you can use the formula: \[ k_{\text{cat}} = \frac{V_{\text{max}}}{[E_t]} \] where \( V_{\text{max}} \) is the maximum rate of the reaction and \( [E_t] \) is the total enzyme concentration.

• For instance, if \( V_{\text{max}} = 1.6 \, \mu M \cdot s^{-1} \) and \( [E_t] \) your enzyme concentration is given in \( \text{nm} \) like 4 nm, remember to convert it to \( \mu M \) which is 0.004 \( \mu M \).
• Applying these values, the calculation becomes \( k_{\text{cat}} = \frac{1.6 \, \mu M \cdot s^{-1}}{0.004 \, \mu M} = 400 \, s^{-1} \).

A higher \( k_{\text{cat}} \) indicates a more efficient enzyme, meaning it rapidly transforms substrates to products, given optimal conditions.

Michaelis Constant (K_m)

The Michaelis constant, \( K_m \), is another fundamental concept in enzyme kinetics. It is defined as the substrate concentration at which the reaction velocity is half of \( V_{\text{max}} \). \( K_m \) provides insights into the enzyme's affinity for the substrate; a low \( K_m \) means high affinity, and vice versa.

This constant can be found using the rearranged Michaelis-Menten equation:\[ K_m = \frac{V_{\text{max}} \cdot [S]}{V_0} - [S] \]Here, \( V_0 \) is the initial reaction velocity, \( V_{\text{max}} \) is the maximum rate, and \( [S] \) is the substrate concentration.

• For example, with \( V_0 = 300 \, \text{nM} \, s^{-1} \) (converted to \( 0.3 \, \mu M \, s^{-1} \)), \( V_{\text{max}} = 1.6 \, \mu M \, s^{-1} \), and \( [S] = 30 \, \mu M \), we apply the equation: \( K_m = \frac{1.6 \, \mu M \, \cdot \ 30 \, \mu M}{0.3 \, \mu M \, s^{-1}} - 30 \, \mu M = 160 \, \mu M - 30 \, \mu M = 130 \, \mu M \).

Understanding \( K_m \) helps in elucidating how effective an enzyme is under various substrate concentrations.

Enzyme Inhibition Types

Enzyme inhibition is another important concept and can be classified into several types: competitive, non-competitive, uncompetitive, and mixed inhibition. These inhibitions describe how an inhibitor interacts with an enzyme and affects its action.

In the context of the original exercise, the reversible inhibitor ANGER affects both \( V_{\text{max}} \) and \( K_m \). This signifies a mixed type of inhibition. Mixed inhibitors bind to the enzyme at a site other than the active site, and they can bind to both the enzyme-substrate complex and the free enzyme.

This dual interaction results in changes to both the enzyme's substrate affinity and maximal catalytic activity, hence altering both \( K_m \) and \( V_{\text{max}} \):

• For instance, after removing ANGER, \( V_{\text{max}} \) increased from \( 1.6 \, \mu M \, \cdot s^{-1} \) to \( 4.8 \, \mu M \, \cdot s^{-1} \), suggesting that ANGER was indeed affecting the maximal rate.
• Similarly, \( K_m \) changed from \( 130 \, \mu M \) to \( 15 \, \mu M \), indicating its role in altering substrate affinity.

Through understanding how these different types of inhibitors work, we can better understand and possibly control enzymatic reactions, which is crucial in drug development and treatment of enzyme-related diseases.