Question

Consider the binding of a protein, P, to a DNA segment (gene regulation). Assume that only one binding site for P exists on the DNA and that the concentration of DNA binding sites is much less than the concentration of \(P\). The reaction mechanism for binding can be represented as $$ \mathrm{P}+\mathrm{DNA} \rightleftharpoons \mathrm{P}-\mathrm{DNA} \rightarrow \mathrm{P}-\mathrm{DNA}^{\prime} $$ where the second step represents a conformational change in the protein. Calculate the rate law for the appearance of P-DNA' under the following conditions. A. The first step in the mechanism equilibrates rapidly relative to the rate of the overall reaction and [P-DNA] \ll<[DNA]. B. The intermediate, P-DNA, is in a steady state. C. The first step in the mechanism equilibrates rapidly relative to the rate of the overall reaction and the concentrations of DNA and P-DNA are comparable. Express the rate law in terms of the total concentration of DNA and P-DNA, that is, [DNA] + [P-DNA]. D. The following initial rates were measured with an initial DNA concentration of \(1 \mu \mathrm{M}\). \begin{tabular}{rc} {\([\mathrm{P}](\mu \mathrm{M})\)} & \(10^{4} \operatorname{Rate}(\mathrm{M} / \mathrm{s})\) \\ \hline 100 & \(8.33\) \\ 50 & \(7.14\) \\ 20 & \(5.00\) \\ 10 & \(3.33\) \\ \hline \end{tabular} Which of the rate laws is consistent with the data?

Short answer

Part C is consistent with the data, showing saturation kinetics.

Step-by-step solution

Understanding the Reaction Mechanism

The mechanism involves a protein (P) binding to a DNA segment through a two-step process. The first part is reversible as P binds with DNA to form P-DNA. Subsequently, a conformational change in P occurs, forming P-DNA' irreversibly: \[ \mathrm{P} + \mathrm{DNA} \rightleftharpoons \mathrm{P-DNA} \rightarrow \mathrm{P-DNA}' \] The goal is to derive rate laws under different conditions for the appearance of P-DNA'.

Part A: Rapid Equilibrium and Low [P-DNA]

Since the first step equilibrates rapidly, assume the formation and dissociation of P-DNA is at equilibrium: \[ K_1 = \frac{[\mathrm{P-DNA}]}{[\mathrm{P}][\mathrm{DNA}]} \] The rate of formation of P-DNA' depends on [P-DNA]: \[ \text{Rate} = k_2 [\mathrm{P-DNA}] \] Assuming [P-DNA] << [DNA], the equilibrium assumption simplies to: \[ [\mathrm{P-DNA}] = K_1 [\mathrm{P}][\mathrm{DNA}] \] So, \[ \text{Rate} = k_2 K_1 [\mathrm{P}][\mathrm{DNA}] \] which is a second-order reaction in terms of P and DNA.

Part B: Steady State Assumption for P-DNA

Assume [P-DNA] is in steady state: \[ \frac{d[\mathrm{P-DNA}]}{dt} = k_1[\mathrm{P}][\mathrm{DNA}] - k_{-1}[\mathrm{P-DNA}] - k_2[\mathrm{P-DNA}] = 0 \] Solving for [P-DNA], we get: \[ [\mathrm{P-DNA}] = \frac{k_1[\mathrm{P}][\mathrm{DNA}]}{k_{-1} + k_2} \] Thus, \[ \text{Rate} = k_2 [\mathrm{P-DNA}] = \frac{k_2 k_1 [\mathrm{P}][\mathrm{DNA}]}{k_{-1} + k_2} \] This gives a similar form to Part A.

Part C: Rapid Equilibrium with Comparable [DNA] and [P-DNA]

In this scenario, use the total concentration \([\mathrm{DNA}]_0 = [\mathrm{DNA}] + [\mathrm{P-DNA}]\): \[ K_1 = \frac{[\mathrm{P-DNA}]}{[\mathrm{P}]([\mathrm{DNA}]_0 - [\mathrm{P-DNA}])} \] Solve for [P-DNA] based on this adjustment. Substitute into the rate equation: Rate = \( k_2 [\mathrm{P-DNA}] \).\[ \text{Rate} = \frac{k_2 K_1 [\mathrm{P}][\mathrm{DNA}]_0}{1 + K_1 [\mathrm{P}]} \] This suggests a saturation-like effect when [P] is large.

Analyzing Initial Rate Data

Given the rate data, analyze how rate depends on [P]:\[ \begin{align*} [P] = 100, & \quad \text{Rate} = 8.33 \ [P] = 50, & \quad \text{Rate} = 7.14 \ [P] = 20, & \quad \text{Rate} = 5.00 \ [P] = 10, & \quad \text{Rate} = 3.33 \end{align*} \] Observing that rate increases with [P] and approaches a plateau, this is consistent with the saturation kinetics described in Part C. In Part C, for large [P], \( 1 + K_1 [\mathrm{P}]\) dominates, leading to a near-constant rate.

Conclusion: Consistent Rate Law

Part C is consistent with observed data since the rate increases with [P] but shows saturation behavior at high [P] values, aligning with the rate plateau. This matches the observed initial rates suggestive of saturation kinetics.

Key concepts

Protein-DNA binding

Protein-DNA binding is a crucial interaction in biological systems, particularly for processes like gene regulation. In the given reaction, a protein denoted as "P" binds to a DNA segment. This process occurs in a specific site on the DNA, meaning there is only one binding site available for the protein. Understanding this interaction involves recognizing that DNA contains sequences that proteins can specifically bind to, often influencing the expression of genes.
In a biological context, protein-DNA binding is essential for controlling which genes are turned on or off. This binding can either promote or inhibit the transcription of genes, depending on the function of the protein. Given the low concentration of DNA binding sites compared to the concentration of protein, this interaction is often examined through a lens of saturation, where the maximal binding capacity of DNA can become a limiting factor.

Reaction mechanism

A reaction mechanism describes the step-by-step sequence of elementary reactions by which overall chemical change occurs. In the exercise, the binding of protein P to DNA is a two-step process:

• First, the protein (P) reversibly binds to DNA to form a complex (P-DNA).
• Second, the protein-DNA complex undergoes a conformational change to produce P-DNA', an irreversible step.

The initial reversible step indicates that the binding of the protein to DNA can reach equilibrium rapidly. This implies that the rates of formation and dissociation of the P-DNA complex are balanced. The subsequent irreversible step, where a conformational change occurs, marks the completion of the binding process as the protein locks into a specific form when attached to the DNA.

Steady-state approximation

The steady-state approximation is a common assumption used in enzyme kinetics, where the concentration of intermediates remains constant over the time course of the reaction. This is the scenario in Part B of the exercise, where it is assumed that the intermediate P-DNA complex does not change significantly over time.
Mathematically, this means that the rate of formation of P-DNA from P and DNA is equal to the rate at which P-DNA is consumed in forming the product P-DNA'.
Using this approximation simplifies the derivation of rate laws, providing a clear picture of how the system proceeds to equilibrium. It is especially useful when dealing with reactions that have unstable intermediates or when direct measurement is complex.

Saturation kinetics

Saturation kinetics refers to a characteristic behavior seen in enzyme-catalyzed reactions and other biological processes where increasing the substrate concentration results in a maximum reaction rate that cannot be exceeded. This is observed when the available sites (like the DNA binding site) become fully occupied, and further increases in substrate (protein, in this case) concentration have no effect on the reaction rate.
In Part C of the exercise, this concept is applied where the concentration of DNA and P-DNA are comparable. As the concentration of protein increases, the rate of P-DNA' formation rises but reaches a plateau. This plateau indicates that all DNA binding sites are occupied and the reaction rate has reached its maximum possible value, a hallmark of saturation kinetics.
Such behavior is critical in biological systems as it ensures that reactions can proceed quickly when needed without endlessly consuming resources, allowing the organism to maintain efficient control over metabolic processes.