Question

A protein molecule \(\mathrm{P}\) of molar mass \(\mathscr{M}\) dimerizes when it is allowed to stand in solution at room temperature. A plausible mechanism is that the protein molecule is first denatured (i.e., loses its activity due to a change in overall structure) before it dimerizes: \(\mathrm{P} \stackrel{k}{\longrightarrow} \mathrm{P}^{*}(\) denatured \() \quad\) (slow) $$ 2 \mathrm{P}^{*} \longrightarrow \mathrm{P}_{2} $$ (fast) where the asterisk denotes a denatured protein molecule. Derive an expression for the average molar mass (of \(\mathrm{P}\) and \(\left.\mathrm{P}_{2}\right), \bar{U},\) in terms of the initial protein concentration \([\mathrm{P}]_{0}\) and the concentration at time \(t,[\mathrm{P}]_{t},\) and \(\mathscr{M} .\) Describe how you would determine \(k\) from molar mass measurements.

Short answer

\( \bar{U} = \mathscr{M} + \frac{\mathscr{M}[\mathrm{P}_2]}{[\mathrm{P}]_0} \). To find \( k \), fit molar mass data to the decay model for \( [\mathrm{P}]_t \).

Step-by-step solution

Understand the Process

The protein molecule \( \mathrm{P} \) dimerizes via an initial slow denaturation stage followed by a fast dimerization stage. The given mechanism shows that first, \( \mathrm{P} \) becomes \( \mathrm{P}^* \) (denatured), and then two \( \mathrm{P}^* \) molecules quickly form \( \mathrm{P}_2 \).

Express Mass Conservation

The total concentration of all forms of the protein remains constant. Initially, we have only \( \mathrm{P} \) with concentration \([\mathrm{P}]_0\). At time \( t \), we need to consider concentrations of \( \mathrm{P} \), \( \mathrm{P}^* \), and \( \mathrm{P}_2 \). This can be expressed as \([\mathrm{P}]_0 = [\mathrm{P}]_t + [\mathrm{P}^*] + 2[\mathrm{P}_2]\).

Define the Average Molar Mass

The average molar mass \( \bar{U} \) accounts for the molar masses of \( \mathrm{P} \), \( \mathrm{P}^* \), and \( \mathrm{P}_2 \). Since \( \mathrm{P} \) and \( \mathrm{P}^* \) have the same molar mass \( \mathscr{M} \), and \( \mathrm{P}_2 \) has mass \( 2\mathscr{M} \), the weighted average is \[ \bar{U} = \frac{[\mathrm{P}]_t \mathscr{M} + [\mathrm{P}^*] \mathscr{M} + [\mathrm{P}_2] (2\mathscr{M})}{[\mathrm{P}]_0}. \]

Substitute Expressions for Intermediate Concentrations

Since \( 2[\mathrm{P}_2] = [\mathrm{P}]_0 - [\mathrm{P}]_t - [\mathrm{P}^*] \) from mass conservation, it follows that \([\mathrm{P}^*] = \frac{1}{2}([\mathrm{P}]_0 - [\mathrm{P}]_t - 2[\mathrm{P}_2])\). This allows us to simplify the expression for \( \bar{U} \) as \[ \bar{U} = \mathscr{M} + \frac{1}{2}\frac{2\mathscr{M}[\mathrm{P}_2]}{[\mathrm{P}]_0}. \]

Determine Rate Constant \( k \)

To find \( k \), note that \( \mathrm{P} \to \mathrm{P}^* \) is slow and limits the formation of \( \mathrm{P}_2 \). Measure \( \bar{U} \) over time. The change in \( \bar{U} \) reflects the denaturation and dimerization progress, thereby giving information about \( k \). Use kinetic data to fit this into the exponential decay model for \( [\mathrm{P}]_t = [\mathrm{P}]_0 e^{-kt} \) assuming first-order kinetics for the denaturation.

Key concepts

Molar Mass

Molar mass is a key concept when studying protein dimerization. It represents the weight of one mole of a substance, expressed in grams per mole (g/mol). In the context of proteins, understanding molar mass helps in analyzing the behavior of proteins under different conditions. For the protein in question, both the native and denatured forms, denoted as \( \mathrm{P} \) and \( \mathrm{P}^* \) respectively, have the same molar mass \( \mathscr{M} \). Meanwhile, the dimerized form \( \mathrm{P}_2 \) possesses a molar mass of \( 2\mathscr{M} \). This is because it is composed of two protein molecules. The average molar mass \( \bar{U} \) during the dimerization process provides insights into the proportion of monomers \( \mathrm{P} \), denatured monomers \( \mathrm{P}^* \), and dimers \( \mathrm{P}_2 \) present in the solution.

Denaturation

Denaturation is a process where a protein loses its functional conformation, affecting its activity. This change is often due to external factors like temperature or pH, disrupting the protein's structure. In our dimerization context, denaturation is a critical first step. The protein \( \mathrm{P} \) is converted to its denatured form \( \mathrm{P}^* \), which lacks the functional properties of the original structure. Interestingly, this denaturation step is described as slow, indicating it is the rate-limiting step in the process of forming \( \mathrm{P}_2 \). Understanding denaturation is important because it initiates the path leading to protein interactions and eventual dimerization.

Kinetics

Kinetics deals with the rate of reactions and is crucial in understanding protein dimerization. In this scenario, the transformation of the protein from its initial state \( \mathrm{P} \) to the denatured form \( \mathrm{P}^* \) is the slow step. As the denaturation progress controls the speed of the entire reaction, it determines how quickly \( \mathrm{P}_2 \) forms. By measuring the average molar mass \( \bar{U} \) over time, we can gain valuable data to determine the rate constant \( k \). The rate of change in \( \bar{U} \) reflects the underlying kinetics of the reaction. Specifically, the relationship \( [\mathrm{P}]_t = [\mathrm{P}]_0 e^{-kt} \) follows first-order kinetics for denaturation, useful in calculating the rate constant from experimental data.

Concentration

Concentration is a term that defines the amount of a protein or a molecule in a given volume of solution. It's a core aspect when exploring chemical reactions and is especially vital for characterizing protein dimerization. Initially, we start with a known concentration of \( \mathrm{P} \), denoted as \([\mathrm{P}]_0\). As the reaction progresses, the concentration of \( \mathrm{P} \) decreases, while \( \mathrm{P}^* \) and \( \mathrm{P}_2 \) emerge. By tracking these changes in concentration, researchers can assess how the protein transitions from monomers to dimers. The crucial relationship, \([\mathrm{P}]_0 = [\mathrm{P}]_t + [\mathrm{P}^*] + 2[\mathrm{P}_2]\), ensures mass conservation and provides a foundation for further analysis.

Protein Structure

Protein structure refers to the three-dimensional arrangement of amino acids that dictates a protein's function. Proteins can undergo structural changes, leading to denaturation. Initially, the protein \( \mathrm{P} \) possesses a specific structure essential for its activity. When it denatures into \( \mathrm{P}^* \), it loses this structure, subsequently affecting its biological function. This structural change is necessary for the protein to dimerize into \( \mathrm{P}_2 \). Understanding a protein's structural changes provides insights into its interaction dynamics and potential to form complexes such as dimers. Recognizing the link between structure and function allows researchers to predict and control protein behavior during dimerization.