Question

A mutation of core residues in the E. coli protein Rop speeds up the rate of protein folding from \(0.013 \mathrm{sec}^{-1}\) to \(1.3 \mathrm{sec}^{-1}\). a. What is the effect of the mutation on the free energy of the transition state relative to the unfolded state? b. Given that a simple mutation can increase the folding rate, what does this result suggest about the relationship between the protein folding rate and natural selection? (Data based on M. Munson, K.S. Anderson, and L. Regan, Structure: Folding and Design, 2: 77-87, 1997.)

Short answer

The mutation decreases the free energy of the transition state by about \(-11.4 \text{ kJ/mol}\). This suggests that protein folding efficiencies can evolve through natural selection.

Step-by-step solution

Understand the problem

We are given that a mutation changes the rate of protein folding from \(0.013 \text{ sec}^{-1}\) to \(1.3 \text{ sec}^{-1}\). We need to calculate the change in the free energy of the transition state relative to the unfolded state as a result of this mutation.

Use the Arrhenius equation concept

The change in the free energy of activation \(\Delta\Delta G^\ddagger\) can be calculated from the rate constants using the Arrhenius equation. The relationship is given by: \[ \Delta\Delta G^\ddagger = -RT \ln\left(\frac{k_{\text{mutant}}}{k_{\text{wild}}} \right) \]where \(R\) is the universal gas constant and \(T\) is the temperature in Kelvin. Assuming standard conditions at room temperature \(298 \text{ K}\) and \(R = 8.314 \text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}\).

Calculate the rate ratio

Calculate the ratio of the mutant rate constant \(1.3 \text{ sec}^{-1}\) to the wild-type rate constant \(0.013 \text{ sec}^{-1}\): \[ \frac{k_{\text{mutant}}}{k_{\text{wild}}} = \frac{1.3}{0.013} = 100 \]

Calculate the change in free energy

Now, substitute the values into the equation to find \(\Delta\Delta G^\ddagger\):\[ \Delta\Delta G^\ddagger = -8.314 \times 298 \times \ln(100) \]\[ \Delta\Delta G^\ddagger = -8.314 \times 298 \times 4.605 \approx -11.4 \text{ kJ/mol} \]

Analyze the relationship with natural selection

The increase in folding rate due to the mutation suggests that protein folding efficiency can be enhanced through mutations and may be subject to natural selection, optimizing proteins for faster and more efficient folding. Efficient folding can be advantageous for organisms as it ensures quick and proper protein function.

Key concepts

Free Energy of Activation

The free energy of activation is a concept in chemistry and biology that refers to the energy barrier that must be overcome for a reaction to proceed. In the context of protein folding, this relates to the energy required to convert an unfolded protein into its correct, functional folded form. The energy barrier determines the rate of protein folding, with higher energy barriers resulting in slower folding rates.

Free energy of activation is symbolized as \(\Delta G^\ddagger\)\. The effect of a mutation on protein folding can be quantified by changes in this energy barrier. When a mutation decreases the free energy of activation, it speeds up the reaction, meaning the protein folds more quickly. On the other hand, when this energy increases, the folding slows down. By calculating these changes, scientists can better understand and predict the impacts of genetic mutations on protein function.

Mutant Rate Constant

A rate constant indicates how quickly a reaction proceeds. For protein folding, the rate constant describes how fast the protein transitions from its unfolded to folded state. When a mutation occurs, it may change the protein structure, affecting its rate constant.

In our given exercise, the mutant rate constant increased from 0.013 sec⁻¹ to 1.3 sec⁻¹. This indicates a much faster protein folding rate in the mutant compared to the wild-type protein. Such dramatic changes in rate constants can reveal the role of specific residues—parts of the protein sequence—in facilitating or hindering folding. Understanding these roles can be crucial for developing therapies for diseases caused by protein misfolding.

Arrhenius Equation

The Arrhenius equation is a critical formula in chemical kinetics used to describe how temperature and activation energy affect the rate of a reaction. It allows us to calculate the energy barrier, or free energy of activation, from observed rate constants. The equation is:\[ k = A e^{-\Delta G^\ddagger / RT} \]where:

• \( k \): rate constant of the reaction
• \( A \): pre-exponential factor, related to the frequency of collisions
• \( \Delta G^\ddagger \): free energy of activation
• \( R \): universal gas constant
• \( T \): temperature in Kelvin

In the context of protein folding, it allows scientists to determine the effect of mutations on the energy barrier of folding under specific conditions. When the mutant protein folds more quickly, a lower \(\Delta G^\ddagger\) is calculated, implying a lower energy barrier and hence more efficient folding.

Natural Selection

Natural selection is a fundamental concept in biology, describing how species evolve and adapt over time due to environmental pressures. In the context of protein folding, natural selection can favor mutations that improve protein folding efficiency because proteins need to fold correctly and quickly to function properly.

If a mutation results in a protein that folds faster, as seen in the E. coli protein Rop example, this could confer an advantage to the organism. Efficient protein folding might enhance cellular functions, support better stress responses, or streamline metabolic processes, thus being positively selected over generations. Through this lens, protein folding rates and mechanisms are not just biochemical curiosities but integral aspects of evolutionary biology, influencing how organisms adapt to their environments.