Problem 2
Studies of the inhibition of enzymes by various compounds often provide information about the nature of the binding site and the mechanism. Competitive inhibition is when the inhibitor, I, competes with the substrate for the catalytic site. This mechanism can be written as $$ \begin{gathered} \mathrm{E}+\mathrm{S} \rightleftharpoons \mathrm{ES} \rightarrow \mathrm{E}+\mathrm{P} \\ \mathrm{E}+\mathrm{I} \rightleftharpoons \mathrm{EI} \end{gathered} $$ Derive the steady-state rate law for this mechanism and show that it follows Michaelis-Menten kinetics when the inhibitor concentration is constant. Assume the inhibitor concentration is much greater than the enzyme concentration.
Problem 3
In the text, the steady-state rate law was derived with the assumption that the reaction is irreversible and/or only the initial velocity was determined. Derive the steady-state rate law for the reversible enzyme reaction: $$ \mathrm{E}+\mathrm{S} \rightleftharpoons \mathrm{X} \rightleftharpoons \mathrm{E}+\mathrm{P} $$ Show that the rate law can be put into the form $$ v=\frac{\left.V_{\mathrm{S}} / K_{\mathrm{S}} \mid \mathrm{S}\right]-V_{\mathrm{p}} / K_{\mathrm{p}}[\mathrm{P}]}{1+[\mathrm{S}] / K_{\mathrm{S}}+[\mathrm{P}] / K_{\mathrm{p}}} $$ where \(V_{\mathrm{S}}\) and \(V_{\mathrm{P}}\) are the maximum velocities for the forward and reverse reactions, and \(K_{\mathrm{s}}\) and \(K_{\mathrm{p}}\) are the Michaelis constants for the forward and reverse reactions. When equilibrium is reached, \(v=0\). Calculate the ratio of the equilibrium concentrations of \(S\) and \(P,[P] /[S]\), in terms of the four steady-state parameters. This relationship is called the Haldane relationship and is a method for determining the equilibrium constant of the overall reaction.
Problem 5
Consider the binding of a protein, P, to a DNA segment (gene regulation). Assume that only one binding site for P exists on the DNA and that the concentration of DNA binding sites is much less than the concentration of \(P\). The reaction mechanism for binding can be represented as $$ \mathrm{P}+\mathrm{DNA} \rightleftharpoons \mathrm{P}-\mathrm{DNA} \rightarrow \mathrm{P}-\mathrm{DNA}^{\prime} $$ where the second step represents a conformational change in the protein. Calculate the rate law for the appearance of P-DNA' under the following conditions. A. The first step in the mechanism equilibrates rapidly relative to the rate of the overall reaction and [P-DNA] \ll<[DNA]. B. The intermediate, P-DNA, is in a steady state. C. The first step in the mechanism equilibrates rapidly relative to the rate of the overall reaction and the concentrations of DNA and P-DNA are comparable. Express the rate law in terms of the total concentration of DNA and P-DNA, that is, [DNA] + [P-DNA]. D. The following initial rates were measured with an initial DNA concentration of \(1 \mu \mathrm{M}\). \begin{tabular}{rc} {\([\mathrm{P}](\mu \mathrm{M})\)} & \(10^{4} \operatorname{Rate}(\mathrm{M} / \mathrm{s})\) \\ \hline 100 & \(8.33\) \\ 50 & \(7.14\) \\ 20 & \(5.00\) \\ 10 & \(3.33\) \\ \hline \end{tabular} Which of the rate laws is consistent with the data?
Problem 6
Many biological reactions are very sensitive to \(\mathrm{pH}\). This can readily be incorporated into the rate laws because protolytic reactions can be assumed to be much faster than other rates in most cases. For example, in enzyme mechanisms the ionization states of a few key protein side chains are often critical. Suppose that two ionizable groups on the enzyme are critical for catalytic activity and that one of them needs to be protonated and the other deprotonated. The protolytic reactions can be written as $$ \mathrm{EH}_{2} \rightleftharpoons \mathrm{EH}+\mathrm{H}^{\prime} \rightleftharpoons \mathrm{E}+2 \mathrm{H}^{\prime} $$ If only the species \(\mathrm{EH}\) is catalytically active and the protolytic reactions are much more rapid than the other steps in the reaction, all of the rate constants that multiply the free enzyme concentration in the rate law have to be multiplied by the fraction of enzyme present as EH. A. Calculate the fraction of free enzyme present as EH at a given pH. Your answer should contain the concentration of \(\mathrm{H}^{+}\)and the ionization constants of the two side chains, \(\left.K_{\mathrm{E} 1}=[\mathrm{E}]\left[\mathrm{H}^{+}\right] / \mathrm{EH}\right]\) and \(\left.K_{\mathrm{E} 2}=[\mathrm{EH}]\left[\mathrm{H}^{+}\right] / \mathrm{EH}_{2}\right]\). B. Assume that ES in the Michaelis-Menten mechanism (Eq. 5-2) also exists in three protonation states, \(\mathrm{ESH}_{2}\), ESH, and ES, with only ESH being catalytically active. Calculate the fraction of the enzyme-substrate complex present as EHS. Designate the ionization constants as \(K_{\mathrm{ESI}}\) and \(K_{\mathrm{ES} 2}\). C. Use the results of parts \(\mathrm{A}\) and \(\mathrm{B}\) to derive equations for the \(\mathrm{pH}\) dependence of \(V_{\mathrm{m}^{\prime}} K_{\mathrm{M}}\), and \(V_{\mathrm{m} /} K_{\mathrm{M}}\). Measurement of the \(\mathrm{pH}\) dependence of the steady. state parameters permits determination of the ionization constants, and sometimes identification of the amino acid side chains.
