Question

Polymerase binding to the promoter revisited 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 approximately \(10 \mu \mathrm{M}\) and the dissociation constants of RNA polymerase to the lac \(P 1\) and T7A1 promoters are \(550 \mathrm{nM}\) and \(3 \mathrm{nM}\), respectively. Use these constants and the results from (a) to estimate the in vivo binding energies of RNA polymerase to \(\operatorname{lac} P 1\) and \(\mathrm{T} 7 \mathrm{Al}\) promoters.

Short answer

The probability of finding RNA polymerase bound to the promoter is \(P_{\text{bound}} = \frac{K_{\text{s}}[P]}{1 + K_{\text{s}}[P] + K_{\text{ns}}[D]}\). The in vivo binding energies of RNA polymerase to lac P1 and T7A1 promoters can be calculated using the Boltzmann's relation \(E = -kT \cdot \ln(K)\) and the given dissociation constants.

Step-by-step solution

Derive Probability Expression

The probability of finding RNA polymerase bound to the promoter can be calculated using equilibrium constants for specific and nonspecific binding. In general, the probability \(P_{\text{bound}}\) can be written as the ratio of the concentration of the RNA polymerase-promoter complex to the total concentration of the RNA polymerase. Given \(K_{\text{s}}\) and \(K_{\text{ns}}\) as the equilibrium constants for specific and nonspecific binding respectively, the expression would be: \(P_{\text{bound}} = \frac{[P \cdot R]}{[R]} = \frac{K_{\text{s}}[P]}{1 + K_{\text{s}}[P] + K_{\text{ns}}[D]}\)

Compute Binding Energies

In vitro, the dissociation constant of RNA polymerase binding to nonspecific DNA is approximately \(10 µM\), and the dissociation constants of RNA polymerase to the lac P1 and T7A1 promoters are \(550 nM\) and \(3 nM\) respectively. The equilibrium constant is the reciprocal of the dissociation constant and can be used to calculate the energy involved in a process at equilibrium using the Boltzmann's relation: \(E = -kT \cdot \ln(K)\). Applying this equation, the in vivo binding energies of RNA polymerase to lac P1 and T7A1 promoters in Joules can be computed.

Conclusion

Thus, the probability of finding RNA polymerase bound to the promoter is a function of the equilibrium constants for specific and nonspecific binding. And the binding energies can be calculated using Boltzmann's relation

Key concepts

Statistical Mechanics

Statistical mechanics provides a molecular-level understanding of how physical systems come to be in the state that we observe, particularly in equilibrium. This field of physics explains macroscopic phenomena based on the behavior of individual atoms and molecules and their interactions.

In the context of RNA polymerase binding to a promoter, statistical mechanics allows us to predict the probability of a polymerase molecule being bound to a specific site on the DNA by considering the sum over all possible states of the system. Each state has a certain probability of occurring, which is based on its energy level and the temperature of the environment. This is captured by the Boltzmann distribution, which relates the probability of a state to its energy.

Application to RNA Polymerase Binding

In calculating the probability of RNA polymerase binding to a promoter, the system might be in one of two states: bound or unbound. The probability of each state is governed by statistical mechanics principles, where the lower-energy state (specific binding) is more likely than the higher-energy state (nonspecific binding). This is because molecules tend toward lower energy configurations at thermal equilibrium, based on the distribution of energies in the system.

Therefore, the exercise solutions introduce statistical mechanics implicitly through the use of equilibrium constants, which are a macroscopic representation of the molecular behaviors governed by the underlying statistical mechanics.

Thermodynamics

Thermodynamics is the study of heat, energy, and work. It provides a macroscopic perspective on how energy transfer affects matter, without needing to consider the atomic or molecular specifics of the system. It's crucial in understanding the flow and conversion of energy.

In biological systems, including the study of RNA polymerase binding, thermodynamics can predict the direction and extent of chemical reactions. The second law of thermodynamics states that systems tend toward increased entropy, or disorder. However, in living organisms, highly ordered structures and processes are common, and these are maintained through continual energy input.

Relevance to RNA Polymerase Promoter Binding

The binding of RNA polymerase to DNA is an energetically favorable event, under certain conditions, because it leads to a decrease in the system's free energy. The change in Gibbs free energy, a thermodynamic property, indicates the spontaneity of this process. In the exercise, the binding energies of RNA polymerase to promoters are determined using thermodynamics through the calculation of equilibrium constants. The smaller the dissociation constant, the stronger the binding, and the more negative the free energy change, reflecting a spontaneous and favorable binding process.

Equilibrium Constants

Equilibrium constants are fundamental to both the studies of statistical mechanics and thermodynamics. They are quantitative measures of the position of equilibrium in a reversible chemical reaction. In essence, they give a ratio of the concentration of products to reactants once the reaction has reached its equilibrium state.

For the RNA polymerase and promoter binding scenario, the equilibrium constants for specific and nonspecific binding reflect the different strengths of the interactions. Specific binding involves more favorable interactions between the polymerase and a particular promoter sequence, leading to a higher equilibrium constant and thus a lower dissociation constant.

Impact on Binding Probability

For example, if the equilibrium constant for specific binding to a segment of DNA is large, it indicates that the RNA polymerase prefers to be bound than unbound (strong affinity). Conversely, a small equilibrium constant for nonspecific binding indicates a weak affinity, which is less likely to be observed at equilibrium. In the solution to the exercise, the equilibrium constants derived from dissociation constants are crucial for calculating both the probability of binding and the binding energies, relating physical observations with the mathematical framework that describes them.