Question

When a native protein in solution is heated to a high enough temperature, its polypeptide chain will unfold to become the denatured protein. The temperature at which a large portion of the protein unfolds is called the melting temperature. The melting temperature of a certain protein is found to be \(46^{\circ} \mathrm{C},\) and the enthalpy of denaturation is \(382 \mathrm{~kJ} / \mathrm{mol}\). Estimate the entropy of denaturation, assuming that the denaturation is a twostate process; that is, native protein \(\longrightarrow\) denatured protein. The single polypeptide protein chain has 122 amino acids. Calculate the entropy of denaturation per amino acid. Comment on your result.

Short answer

Entropy of denaturation per amino acid is approximately 9.8 J/mol·K.

Step-by-step solution

Understanding the Problem Setup

We are given a native protein with a melting temperature \(T_m = 46^{\circ} \mathrm{C}\), which needs to be converted to Kelvin for calculations. The enthalpy of denaturation is \(\Delta H = 382 \mathrm{~kJ/mol}\). We need to estimate the entropy of denaturation \(\Delta S\) assuming a two-state process.

Convert Celsius to Kelvin

The melting temperature needs to be in Kelvin: \(T_m = 46 + 273.15 = 319.15\, \text{K}\).

Calculate the Entropy of Denaturation

For a two-state denaturation process, use the formula \(\Delta G = \Delta H - T_m \Delta S = 0\) at the melting temperature. Therefore, \(\Delta S = \frac{\Delta H}{T_m}\).\[\Delta S = \frac{382 \times 10^3\, \text{J/mol}}{319.15\, \text{K}} \approx 1196.25\, \text{J/mol}\cdot\text{K}.\]

Calculate Entropy per Amino Acid

Divide the total entropy of denaturation by the number of amino acids (122): \[\Delta S_{\text{per amino acid}} = \frac{1196.25}{122} \approx 9.8\, \text{J/mol}\cdot\text{K} \text{ per residue}.\]

Comment on the Result

The entropy per amino acid residue calculated is about \(9.8\, \text{J/mol}\cdot\text{K}\). This indicates that there is a significant increase in disorder per amino acid upon denaturation, corroborating the idea that protein denaturation significantly increases molecular entropy in the system.

Key concepts

Melting Temperature

When we talk about melting temperature in the context of proteins, we're referring to the specific temperature at which half of the protein molecules are in their folded, native state, and the other half are unfolded, or denatured.
This is a critical point in understanding protein stability.
The melting temperature is often denoted as \( T_m \), and it provides insights into how a protein will behave under different thermal conditions.

• At temperatures below \( T_m \), proteins generally remain stable and retain their functional shape.
• At \( T_m \) and higher, proteins begin to lose their structure, leading to denaturation.

In the exercise, the melting temperature of the protein is given as \( 46^{\circ} \mathrm{C} \), which is equivalent to \( 319.15 \mathrm{K} \) in Kelvin.
Understanding this temperature helps us predict when a protein will become denatured under heat.
Proteins with higher \( T_m \) values are typically more stable at elevated temperatures.

Entropy of Denaturation

Entropy is a measure of disorder within a system. When a protein denatures, it goes from a structured, ordered state to a more disordered state.
The entropy of denaturation, denoted as \( \Delta S \), quantifies this increase in disorder.
For the process of protein denaturation, entropy increases because the unfolded protein has more ways to arrange itself compared to the folded protein.
Thus, the system becomes more chaotic.

• The formal equation used to estimate entropy of denaturation is \( \Delta S = \frac{\Delta H}{T_m} \).
• This equation arises from the fact that at \( T_m \), the Gibbs free energy change (\( \Delta G \)) equals zero, so \( \Delta G = \Delta H - T_m \Delta S \).

The calculation showed an entropy of denaturation as approximately \( 1196.25 \; \text{J/mol}\cdot\text{K} \). This larger value indicates significant disorder increases upon protein unfolding.

Enthalpy of Denaturation

Enthalpy changes during protein denaturation represent the heat absorbed or required to unfold the protein.
The enthalpy of denaturation, represented as \( \Delta H \), reflects the energy needed to break the non-covalent interactions like hydrogen bonds and van der Waals forces that stabilize the protein structure.
In this problem, the enthalpy is given as \( 382 \mathrm{kJ/mol} \).

• This value is derived from the input energy needed to disrupt the structured state of a protein and shift it to a conformation where its polypeptide chain is relaxed and interaction-free.
• Higher \( \Delta H \) values imply that more energy is required to unfold the protein.

Understanding \( \Delta H \) offers deep insight into protein stability – proteins with larger enthalpy changes are typically more thermodynamically stable in their native forms.

Two-State Process

When considering protein denaturation as a two-state process, we imply that the protein exists in only two forms: the native state and the denatured state, with no intermediate states.
This simplification assumes an all-or-nothing transition.

• The native state refers to the folded, functional form of the protein.
• The denatured state is the unfolded, non-functional form.

This assumption helps simplify calculations of thermodynamic parameters like entropy and enthalpy, as the transition between these two states can be described using the same formulas typically applied to simpler physical systems.
The two-state model is applicable to our problem and is often used because it simplifies the analysis without significantly losing accuracy in many situations.

Amino Acids Calculation

Amino acids are the basic building blocks of proteins. Each protein is composed of a unique sequence of amino acids, which determines its overall structure and function.
In the problem at hand, we performed a calculation to find the entropy change per amino acid during denaturation.
This was done by dividing the total entropy change (\( \Delta S = 1196.25 \; \text{J/mol}\cdot\text{K} \)) by the number of amino acids in the protein chain, which is 122.

• The entropy per amino acid was found to be approximately \( 9.8 \; \text{J/mol}\cdot\text{K} \).
• This value suggests that even individual amino acids contribute to the overall increase in disorder during protein unfolding.

Such calculations highlight the contribution of each amino acid to the thermodynamic properties of a protein, providing more detailed insight into protein stability and folding mechanisms.