Here we continue our exploration of the thermodynamics of unfolding of biological macromolecules. Our focus is the thermal and chemical denaturation of chymotrypsin, one of many enzymes that catalyze the cleavage of polypeptides (see Case study 8.1).
(a) The denaturation of a biological macromolecule can be described by the equilibrium
macromolecule in native form $\rightleftharpoons$ macromolecule in denatured form
Show that the fraction $\theta$ of denatured macromolecules is related to the equilibrium constant $K_d$ for the denaturation process by
$$
\theta=\frac{1}{1+K_{\mathrm{d}}}
$$
(b) Now explore the thermal denaturation of a biological macromolecule. (i) Write an expression for the temperature dependence of $K_{\mathrm{d}}$ in terms of the standard enthalpy and standard entropy of denaturation. (ii) $\mathrm{At} \mathrm{pH}=2$, the standard enthalpy and entropy of denaturation of chymotrypsin are $+418 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and +1.32 $kJ$ $\mathrm{K}^{-1} \mathrm{~mol}^{-1}$, respectively. Using these data and your results from parts (a) and (b.i), plot $\theta$ against T. Compare the shape of your plot with that of the plot shown in Fig. 3.16. (iii) The "melting temperature" of a biological macromolecule is the temperature at which $\theta=1 / 2$. Use your results from part (ii) to calculate the melting temperature of chymotrypsin at $\mathrm{pH}=2$. (iv) Calculate the standard Gibbs energy and the equilibrium constant for the denaturation of chymotrypsin at $\mathrm{pH}=2.0$ and $T=310 \mathrm{~K}$ (body temperature). Is the protein stable under these conditions?
(c) We saw in Exercise 3.35 that the unfolding of a protein may also be brought about by treatment with denaturants, substances such as guanidinium hydrochloride ( GuHCl ; the guanidinium ion is shown in 14) that disrupt the intermolecular interactions responsible for the native threedimensional conformation of a biological macromolecule. Data for a number of proteins denatured by urea or guanidinium hydrochloride suggest a linear relationship between the Gibbs energy of denaturation of a protein, $\Delta \mathrm{G}_{\mathrm{d}}$, and the molar concentration of a denaturant [D]:
$$
\Delta \mathrm{G}_{\mathrm{d}}{ }^e=\Delta \mathrm{G}_{\mathrm{d}, \text { water }}-m[\mathrm{D}]
$$
where $m$ is an empirical parameter that measures the sensitivity of unfolding to denaturant concentration and $\Delta \mathrm{G}^{\ominus}{ }_{\mathrm{d} \text {,water }}$ is the Gibbs energy of denaturation of the protein in the absence of denaturant and is a measure of the thermal stability of the macromolecule. (i) At $27^{\circ} \mathrm{C}$ and ${ }^{\mathrm{pH}} 6.5$, the fraction $\theta$ of denatured chymotrypsin molecules varies with the concentration of GuHCl as follows:
FIGURE CAN'T COPY.
$$
\begin{array}{lllllllll}
\theta & 1.00 & 0.99 & 0.78 & 0.44 & 0.23 & 0.08 & 0.06 & 0.01 \\
\text { [GuHCl]/ } & 0.00 & 0.75 & 1.35 & 1.70 & 2.00 & 2.35 & 2.70 & 3.00 \\
(\mathrm{~mol} \mathrm{~L}
\end{array}
$$
Calculate $m$ and $\Delta G^{\ominus}{ }_{d \text { water }}$ for chymotrypsin under these experimental conditions. (ii) Using the same data, plot $\theta$ against [GnHCl]. Comment on the shape of the curve. (iii) To gain insight into your results from part (c.ii), you will now derive an equation that relates $\theta$ to [D]. Begin by showing that $\Delta G_{d, w a t e r}=m[D]_{1 / 2}$, where $[\mathrm{D}]_{1 / 2}$ is the concentration of denaturant corresponding to $\theta=1 / 2$. Then write an expression for $\theta$ as a function of [D], [D] $]_{1 / 2}, m$, and T. Finally, plot the expression using the values of $[D]_{1 / 2}, m$, and $T$ from part (c.i). Is the shape of your plot consistent with your results from part (c.ii)?