Problem 2
Intracellular Concentration of Enzymes To approximate the concentration of enzymes in a bacterial cell, assume that the cell contains equal concentrations of 1,000 different enzymes in solution in the cytosol and that each protein has a molecular weight of 100,000 . Assume also that the bacterial cell is a cylinder (diameter \(1.0 \mu \mathrm{m}\), height \(2.0 \mu \mathrm{m}\) ), that the cytosol (specific gravity \(1.20\) ) is \(20 \%\) soluble protein by weight, and that the soluble protein consists entirely of enzymes. Calculate the average molar concentration of each enzyme in this hypothetical cell.
Problem 3
Rate Enhancement by Urease The enzyme urease enhances the rate of urea hydrolysis at \(\mathrm{pH} 8.0\) and \(20{ }^{\circ} \mathrm{C}\) by a factor of \(10^{14}\). Suppose that a given quantity of urease can completely hydrolyze a given quantity of urea in \(5.0 \mathrm{~min}\) at \(20^{\circ} \mathrm{C}\) and \(\mathrm{pH} 8.0\). How long would it take for this amount of urea to be hydrolyzed under the same conditions in the absence of urease? Assume that both reactions take place in sterile systems so that bacteria cannot attack the urea.
Problem 4
Protection of an Enzyme against Denaturation by Heat When enzyme solutions are heated, there is a progressive loss of catalytic activity over time due to denaturation of the enzyme. A solution of the enzyme hexokinase incubated at \(45{ }^{\circ} \mathrm{C}\) lost \(50 \%\) of its activity in \(12 \mathrm{~min}\), but when incubated at \(45^{\circ} \mathrm{C}\) in the presence of a very large concentration of one of its substrates, it lost only \(3 \%\) of its activity in \(12 \mathrm{~min}\). Suggest why thermal denaturation of hexokinase was retarded in the presence of one of its substrates.
Problem 6
Effect of Enzymes on Reactions Consider this simple reaction: \(\mathrm{S} \underset{\mathrm{k}_{2}}{\stackrel{\mathrm{k}_{1}}{\rightleftharpoons} \mathrm{P}} \quad\) where \(\quad K_{\mathrm{eq}}^{\prime}=\frac{[\mathrm{P}]}{[\mathrm{S}]}\) Which of the listed effects would be brought about by an enzyme catalyzing the simple reaction? a. increased \(k_{1}\) b. increased \(K_{\mathrm{eq}}^{\prime}\) c. decreased \(\Delta G^{\ddagger}\) d. more negative \(\Delta G^{\prime \circ}\) e. increased \(k_{2}\)
Problem 8
Applying the Michaelis-Menten Equation I An enzyme has a \(V_{\max }\) of \(1.2 \mu \mathrm{M} \mathrm{s}^{-1}\). The \(K_{\mathrm{m}}\) for its substrate is \(10 \mu \mathrm{M}\). Calculate the initial velocity of the reaction, \(V_{0}\), when the substrate concentration is a. \(2 \mu \mathrm{M}\) b. \(10 \mu_{M}\) c. \(30 \mu_{\mathrm{M}}\).
Problem 9
Applying the Michaelis-Menten Equation II An enzyme is present at a concentration of \(1 \mathrm{~nm}\) and has a \(V_{\max }\) of \(2 \mu \mathrm{M} \mathrm{s}^{-1}\). The \(K_{\mathrm{m}}\) for its primary substrate is \(4 \mu \mathrm{M}\). a. Calculate \(k_{\text {cat }}\). b. Calculate the apparent (measured) \(V_{\max }\) and apparent (measured) \(K_{\mathrm{m}}\) of this enzyme in the presence of sufficient amounts of an uncompetitive inhibitor to generate an \(\alpha^{\prime}\) of 2 . Assume that the enzyme concentration remains at \(1 \mathrm{~nm}\).
Problem 10
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?
Problem 11
Applying the Michaelis-Menten Equation IV Researchers discover an enzyme that catalyzes the reaction \(\mathrm{X} \rightleftharpoons \mathrm{Y}\). They find that the \(K_{\mathrm{m}}\) for the substrate \(\mathrm{X}\) is \(4 \mu \mathrm{M}\), and the \(k_{\text {cat }}\) is \(20 \mathrm{~min}^{-1}\). a. In an experiment, \([\mathrm{X}]=6 \mathrm{mM}\), and \(V_{0}=480 \mathrm{nM} \mathrm{min}^{-1}\). What was the \(\left[\mathrm{E}_{\mathrm{t}}\right]\) used in the experiment? b. In another experiment, \(\left[\mathrm{E}_{\mathrm{t}}\right]=0.5 \mu \mathrm{M}\), and the measured \(V_{0}=5 \mu \mathrm{M} \mathrm{min}^{-1}\). What was the \([\mathrm{X}]\) used in the experiment? c. The researchers discover that compound \(Z\) is a very strong competitive inhibitor of the enzyme. In an experiment with the same \(\left[E_{t}\right]\) as in (a), but a different \([\mathrm{X}]\), they add an amount of \(\mathrm{Z}\) that produces an \(a\) of 10 and reduces \(V_{0}\) to \(240 \mathrm{nM} \mathrm{min}^{-1}\). What is the \([\mathrm{X}]\) in this experiment? d. Based on the kinetic parameters given, has this enzyme evolved to achieve catalytic perfection? Explain your answer briefly, using the kinetic parameter(s) that define catalytic perfection.
Problem 12
Estimation of \(V_{\max }\) and \(\boldsymbol{K}_{\mathrm{m}}\) by Inspection Graphical methods are available for accurate determination of the \(V_{\max }\) and \(K_{\mathrm{m}}\) of an enzyme-catalyzed reaction. However, these quantities can sometimes be estimated by inspecting values of \(V_{0}\) at increasing [S]. Estimate the \(V_{\max }\) and \(K_{m}\) of the enzyme-catalyzed reaction for which the data in the table were obtained. \begin{tabular}{cc} {\([\mathbf{S}](\mathrm{M})\)} & \(V_{0}(\mu \mathrm{M} / \mathrm{min})\) \\ \hline \(2.5 \times 10^{-6}\) & 28 \\ \(4.0 \times 10^{-6}\) & 40 \end{tabular} \begin{tabular}{ll} \(1 \times 10^{-5}\) & 70 \\ \(2 \times 10^{-5}\) & 95 \\ \(4 \times 10^{-5}\) & 112 \\ \(1 \times 10^{-4}\) & 128 \\ \(2 \times 10^{-3}\) & 140 \\ \(1 \times 10^{-2}\) & 139 \\ \hline \end{tabular}
Problem 13
Properties of an Enzyme of Prostaglandin Synthesis Prostaglandins are one class of the fatty acid derivatives called eicosanoids. Prostaglandins produce fever and inflammation, as well as the pain associated with inflammation. The enzyme prostaglandin endoperoxide synthase, a cyclooxygenase, uses oxygen to convert arachidonic acid to \(\mathrm{PGG}_{2}\), the immediate precursor of many different prostaglandins (prostaglandin synthesis is described in Chapter 21 . Ibuprofen inhibits prostaglandin endoperoxide synthase, thereby reducing inflammation and pain. The kinetic data given in the table are for the reaction catalyzed by prostaglandin endoperoxide synthase in the absence and presence of ibuprofen. a. Based on the data, determine the \(V_{\max }\) and \(K_{\mathrm{m}}\) of the enzyme. \(\begin{array}{ccc}\begin{array}{c}\text { [Arachidonic } \\ \text { acid] }(\mathrm{mM})\end{array} & \begin{array}{c}\text { Rate of formation of } \\\ \mathrm{PGG}_{2}\left(\mathrm{mM} \mathrm{min}^{-1}\right)\end{array} & \begin{array}{c}\text { Rate of formation of } \\ \mathrm{PGG}_{2} \text { with } 10 \mathrm{mg} / \mathrm{mL}\end{array}\end{array}\) \begin{tabular}{ccc} ibuprofen & \(\left(\mathrm{mM}^{-1} \mathrm{~min}^{-1}\right)\) \\ \hline \(0.5\) & \(23.5\) & \(16.67\) \\ \(1.0\) & \(32.2\) & \(30.49\) \\ \(1.5\) & \(36.9\) & \(37.04\) \\ \(2.5\) & \(41.8\) & \(38.91\) \\ \(3.5\) & \(44.0\) & 25 \\ \hline \end{tabular} b. Based on the data, determine the type of inhibition that ibuprofen exerts on prostaglandin endoperoxide synthase.
