Question

The probability of promoter occupancy can be computed using both statistical mechanics and thermodynamics (that is, using equilibrium constants). These two perspectives were already exploited for simple ligand-receptor binding in Sections 6.1 .1 and 6.4 .1 (a) Write an expression for the probability of finding RNA polymerase bound to the promoter as a function of the equilibrium constants for specific and nonspecific binding. (b) In vitro, the dissociation constant of RNA polymerase binding to nonspecific DNA is \(10 \mu \mathrm{M}\) approximately and the dissociation constants of RNA polymerase to the lac \(P I\) and \(T 7 \mathrm{A} 1\) promoters are \(550 \mathrm{nM}\) and \(3 \mathrm{nM}\) respectively. Use these constants and the results from part (a) to estimate the in vivo binding energies of RNA polymerase to lac \(P\) I and T 7 Al promoters.

Short answer

The probability of finding RNA polymerase bound to the promoter can be represented by calculating dissociation constants in a specific or non-specific binding situation. By taking the negative logarithmic ratio of the dissociation constants and an energy constant (thermal energy constant), the binding energy can be estimated.

Step-by-step solution

Derive an Expression for the Probability of RNA Polymerase Binding

To write an expression for the probability of finding RNA polymerase bound to the promoter, one must consider both specific and nonspecific binding. For this, the system is assumed to be in equilibrium. The total number of polymerase molecules is the sum of those bound specifically, those bound non-specifically, and those free in solution. This can be represented by:\(P = P_s + P_{ns} + P_0\)where \(P\) is total polymerase, \(P_s\) is specifically bound polymerase, \(P_{ns}\) is non-specifically bound polymerase, and \(P_0\) is free polymerase.The probability of finding the polymerase bound specifically to the promoter (Pr) can be expressed as a function of the equilibrium constants for specific and nonspecific binding (Ks and Kn respectively) as:\(Pr = P_s / P = K_s \cdot [P_0] \cdot [S] / P\)where \([S]\) is the concentration of specific binding sites.

Use the Dissociation Constants to derive binding constants

Dissociation constants are inversely related to binding constants. Therefore, one can write the binding constants as follow:\(K_s = 1 / Kd_s\) and \(K_{ns} = 1 / Kd_{ns}\)where \(Kd_s\) and \(Kd_{ns}\) are the dissociation constants for specific and non-specific binding respectively.

Estimate the in vivo Binding Energies

Given the dissociation constants, the in vivo binding energies can be estimated by using the equation:\(\Delta G = -RT \cdot \ln(K_s)\)where \(R\) is the gas constant, \(T\) is the temperature and \(K_s\) is the apparent binding constant for specific binding. This equation can be used to estimate the binding energies for RNA polymerase to the lac PI and T7A1 promoters respectively.

Key concepts

Understanding RNA Polymerase Binding

RNA polymerase is a critical enzyme in the process of transcription, where genetic information from DNA is transcribed into messenger RNA. This process begins when RNA polymerase binds to a specific region of DNA known as the promoter. The efficiency of transcription relies on the probability of RNA polymerase binding to these promoters.

The probability that RNA polymerase will be found bound to a promoter depends on the equilibrium between bound and free RNA polymerase molecules in the cell. Generally, there are two types of binding - specific and nonspecific. Specific binding occurs when RNA polymerase binds to promoters with high affinity, which are sequences that are optimally positioned to initiate transcription. Nonspecific binding, on the other hand, involves RNA polymerase attaching to DNA at sites other than promoters, which usually happens with lower affinity.

To understand this concept fully, imagine a crowded party where RNA polymerase is a guest looking for a particular friend—the promoter. When it finds its friend, that's specific binding, leading to transcription. Other random conversations it may engage in represent nonspecific binding. Now, the probability of RNA polymerase finding the promoter in this busy room can be likened to its binding probability within a cell.

Deciphering Equilibrium Constants

Equilibrium constants play a pivotal role in various biochemical interactions, including RNA polymerase binding. To put it simply, an equilibrium constant is a number that describes the ratio of the concentration of products to reactants when a reaction has reached equilibrium. In the context of RNA polymerase binding, we mainly consider two types of equilibrium: one for specific binding and one for nonspecific binding.

Returning to our party analogy: the equilibrium constant is like the likelihood of RNA polymerase sticking to the conversation with its friend, the promoter, vs. mingling with others (nonspecifically bound guests). High equilibrium constants for specific binding mean that RNA polymerase prefers to stay with the promoter, usually leading to a high probability of transcription initiation.

To compute these constants accurately, it's crucial to understand that they are not static values. They can differ depending on several factors such as the environmental conditions within the cell (like pH and temperature), the presence of other molecules that can influence the binding, and the actual concentration of the RNA polymerase and DNA.

Explaining the Dissociation Constant

The dissociation constant, often denoted by Kd, is inversely related to the affinity between a ligand (like RNA polymerase) and a binding site (such as the DNA promoter). Think of it as a measure of a couple's (RNA polymerase and promoter) desire to stay together or split up. A low Kd means that the binding affinity is high and they prefer staying together, whereas a high Kd implies that binding is weak and they are likely to go their separate ways.

In terms of transcription, a small Kd for specific binding indicates strong interactions between RNA polymerase and the promoter, hence a higher promoter occupancy probability. On the contrary, a large Kd for nonspecific binding signifies a lower affinity, reducing the likelihood that RNA polymerase will remain bound to these sites.

It's important for students to grasp that while Kd provides invaluable information about the binding strength, it doesn't tell the whole story. The true physiological scenario involves a dynamic balance influenced by multiple factors, including the concentration of the molecules involved, the presence of competing molecules, and the actual conditions in the cell where the binding takes place.