Question

The Turnover Number of Carbonic Anhydrase Carbonic anhydrase of erythrocytes \(\left(M_{\mathrm{r}} 30,000\right)\) has one of the highest turnover numbers known. It catalyzes the reversible hydration of \(\mathrm{CO}_{2}\) : $$ \mathrm{H}_{2} \mathrm{O}+\mathrm{CO}_{2} \rightleftharpoons \mathrm{H}_{2} \mathrm{CO}_{3} $$ This is an important process in the transport of \(\mathrm{CO}_{2}\) from the tissues to the lungs. If \(10.0 \mu \mathrm{g}\) of pure carbonic anhydrase catalyzes the hydration of \(0.30 \mathrm{~g}\) of \(\mathrm{CO}_{2}\) in \(1 \mathrm{~min}\) at \(37^{\circ} \mathrm{C}\) at \(V_{\max }\), what is the turnover number \(\left(k_{\text {cat }}\right)\) of carbonic anhydrase (in units of \(\min ^{-1}\) )?

Short answer

The turnover number (\(k_{\text{cat}}\)) is approximately \(2.05 \times 10^7\, \text{min}^{-1}\).

Step-by-step solution

Calculate Moles of Carbonic Anhydrase

First, determine the number of moles of carbonic anhydrase (\( ext{C.A.}\)) present. Use the given mass of carbonic anhydrase and its molecular weight:\[\text{moles of C.A.} = \frac{10.0 \, \mu\mathrm{g}}{30,000 \, \mathrm{g/mol}} \]Convert \(10.0 \, \mu\mathrm{g}\) to grams:\[10.0 \, \mu\mathrm{g} = 10.0 \times 10^{-6} \, \mathrm{g} \]Thus:\[\text{moles of C.A.} = \frac{10.0 \times 10^{-6}}{30,000} \approx 3.33 \times 10^{-10} \, \mathrm{mol} \]

Calculate Moles of CO2 Hydrated

Convert the mass of \(\text{CO}_2\) to moles. Given:\[0.30 \, \mathrm{g} \,\text{CO}_2 \text{ and molecular weight of \(\text{CO}_2 = 44.01 \, \mathrm{g/mol} \)}\]Calculate:\[\text{moles of \(\text{CO}_2 \)} = \frac{0.30}{44.01} \approx 6.82 \times 10^{-3} \, \mathrm{mol} \]

Calculate Turnover Number (kcat)

The turnover number \(k_{\text{cat}}\) is calculated as the moles of substrate converted per minute per mole of enzyme alive at \(V_{\max}\):\[k_{\text{cat}} = \frac{\text{moles of substrate converted}}{\text{moles of enzyme} \times \text{time in minutes}} \]Substitute the values:\[k_{\text{cat}} = \frac{6.82 \times 10^{-3}}{3.33 \times 10^{-10} \times 1} \approx 2.05 \times 10^7 \, \text{min}^{-1} \]

Conclusion: Interpretation of kcat Value

The calculated turnover number \(k_{\text{cat}}\) \(= 2.05 \times 10^7\, \text{min}^{-1}\), indicates that each molecule of carbonic anhydrase can convert approximately \(2.05 \times 10^7\) molecules of CO2 into bicarbonate per minute at its maximum rate (\(V_{\max}\)).

Key concepts

Turnover Number

The turnover number, often symbolized as \(k_{\text{cat}}\), is a key parameter in enzyme kinetics. It expresses the number of substrate molecules an enzyme can convert into product per unit of time. Specifically, it's useful for understanding how efficiently an enzyme performs at its highest possible activity. In simpler terms, \(k_{\text{cat}}\) tells us how fast an enzyme works.

In the context of the exercise, you learned that carbonic anhydrase has a turnover number of approximately \(2.05 \times 10^7 \, \text{min}^{-1}\). This means each carbonic anhydrase molecule can convert that many carbon dioxide molecules into bicarbonate every minute when \(V_{\max}\) is reached. Knowing this number helps explain why certain enzymes like carbonic anhydrase are incredibly effective in facilitating biochemical reactions.

Carbonic Anhydrase

Carbonic anhydrase is a fascinating enzyme, especially due to its extraordinarily high turnover number. Its primary function in the body is to catalyze the reversible reaction that converts carbon dioxide and water into carbonic acid. This reaction is essential for maintaining acid-base balance and helps in transporting carbon dioxide from tissues to the lungs, where it can be expelled from the body.

Erythrocytes, or red blood cells, contain a significant amount of carbonic anhydrase to quickly facilitate these reactions, ensuring efficient gas exchange and maintaining proper pH levels in the blood. The speed at which it operates is crucial. Without carbonic anhydrase's catalytic efficiency, the rate of these vital reactions would be too slow to sustain normal physiological function.

Catalysis

Catalysis involves accelerating a chemical reaction using a substance known as a catalyst—in biological systems, enzymes serve as these catalysts. Enzymes like carbonic anhydrase do not get consumed in the reaction; instead, they temporarily bind to substrates to lower the activation energy, thus speeding up the process.

This increased rate of reaction is essential for sustaining life, as many biochemical reactions wouldn't proceed at a significant rate without catalysts. Carbonic anhydrase showcases how effective enzyme catalysis can be; it facilitates the rapid conversion of carbon dioxide, making it one of the fastest enzymes known.

Vmax

\(V_{\max}\) represents the maximum rate of an enzyme-catalyzed reaction when the enzyme is saturated with substrate. It reflects the utmost velocity an enzyme can achieve under specific conditions. Understanding \(V_{\max}\) is crucial for studying enzyme effectiveness and the influence of various factors, like temperature and pH, on enzyme activity.

For carbonic anhydrase, reaching \(V_{\max}\) means it is converting carbon dioxide to bicarbonate at its fastest possible rate, with the highest turnover number. This concept is vital in understanding the capacity and efficiency of enzymes, aiding in the design of drugs or modification of enzyme activity for therapeutic purposes.