Question

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)?

Name: Problem 49, Chapter 4: Physical Chemistry for the Life Sciences
Uploaded: 2021-12-24T08:52:50-08:00
Duration: 5 min 24 s
Channel: Mukesh Devi

   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)?

Physical Chemistry for the Life Sciences

Peter Atkins, Julio… 1st Edition

Chapter 4, Problem 49

Instant Answer

Solved by Expert Mukesh Devi

12/24/2021

Step 1

The equilibrium constant $ K_d $ for this reaction is defined as: \[ K_d = \frac{[D]}{[N]} \] Show more…

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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)?

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Key Concepts

Effect of Denaturants on Protein Stability

Denaturants such as guanidinium hydrochloride are chemical agents that destabilize protein structure by interfering with the intermolecular forces that maintain the native conformation. The extent of unfolding in the presence of denaturants is often modeled by a linear relationship between the Gibbs energy of denaturation and the concentration of the denaturant, with the slope (m-value) indicating the sensitivity of the protein to the solvent conditions. This approach links chemical denaturation experiments with the intrinsic thermodynamic stability of the protein.

Melting Temperature (Tm)

The melting temperature is defined as the temperature at which half of the protein population is unfolded. It represents a critical point on the unfolding curve and provides a direct measure of a protein’s thermal stability. The Tm can be determined from the relationship between ?H and ?S, specifically by setting the Gibbs free energy change to zero at Tm (when ?H = Tm?S).

Temperature Dependence and the van’t Hoff Relationship

The temperature dependence of the denaturation equilibrium constant is governed by the enthalpy (?H) and entropy (?S) changes associated with unfolding. This relationship, often derived using the van’t Hoff analysis, illustrates how changes in temperature alter the free energy change (?G = ?H - T?S) of the process, and thus the equilibrium constant, influencing the balance between the folded and unfolded states.

Gibbs Free Energy and its Relationship to Equilibrium

Gibbs free energy (?G) is the key thermodynamic potential that determines the spontaneity of the unfolding process. Its relationship with the equilibrium constant through the equation ?G = -RT ln(K) allows for a quantitative assessment of protein stability under different conditions, and connects macroscopic measurements (such as the fraction unfolded) with molecular-level energetics.

Equilibrium Constant in Protein Denaturation

In the context of protein unfolding, the equilibrium constant quantitatively describes the balance between the native (folded) and denatured (unfolded) forms of a macromolecule. It is derived from the ratio of the concentrations of the unfolded to the folded states and is used to relate the fraction of unfolded protein to the thermodynamic driving forces behind denaturation.

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