Question

How the dielectric constant affects electrostatic binding. An anesthesiologist has isolated two proteins \(A\) and \(B\) that bind to each other by electrostatic attraction. \(A\) has net positive charge and \(B\) has net negative charge. In water, which has a dielectric constant of \(D=80\) at room temperature, their binding constant is \(K_{1}=1000\). What is their binding constant if the medium is trifluoroethanol with a dielectric constant of \(D=27 ?\) (Assume that the charges on each protein and the distance between them are the same in water and trifluoroethanol.)

Short answer

The binding constant in trifluoroethanol is 2963.

Step-by-step solution

- Understand the relationship between binding constant and dielectric constant

The binding constant, which is a measure of the affinity between two molecules, depends on the electrostatic interaction between them. This interaction is influenced by the dielectric constant of the medium. As the dielectric constant decreases, the electrostatic interaction between the charged molecules increases, which typically increases the binding constant.

- Use the proportional relationship

For binding constants in different media, the relationship between the binding constants and the dielectric constants can be given by ewline ewline \[ K_{2} = K_{1} \left( \frac{D_1}{D_2} \right) \] ewline ewline where ewline ewline \( K_{1} = 1000 \) is the binding constant in water, ewline ewline \( D_1 = 80 \) is the dielectric constant of water, ewline ewline \( D_2 = 27 \) is the dielectric constant of trifluoroethanol.

- Substitute values into the formula

Using the values provided: ewline ewline \[ K_{2} = 1000 \left( \frac{80}{27} \right) \] ewline ewline Calculate the value: ewline ewline \[ K_{2} = 1000 \cdot 2.96 = 2963 \]

Key concepts

binding constant

In chemistry and biochemistry, the binding constant is an essential concept when studying interactions between molecules. The binding constant, often symbolized as \( K \), quantifies the strength of the interaction between two molecules, such as two proteins or an enzyme and its substrate. A higher binding constant indicates a stronger affinity between the molecules. For example, if proteins \(A\) and \(B\) have a binding constant of 1000 in water, it means they bind quite strongly in this medium.
Binding constants are crucial for understanding biological processes and drug efficacy. If a drug binds strongly to its target, it is more effective. Hence, calculating and understanding binding constants helps researchers design better drugs and therapies.

electrostatic interaction

Electrostatic interactions are forces between charged particles. In the case of the proteins \(A\) and \(B\), protein \(A\) is positively charged, and protein \(B\) is negatively charged. These opposite charges attract, causing the proteins to bind together.
Electrostatic interactions are critical in many biological functions, including protein-protein interactions, enzyme-substrate binding, and DNA-protein binding.
Factors affecting electrostatic interactions include the magnitude of the charges, the distance between them, and the medium in which the charges are located. The medium's properties can significantly influence the strength of these interactions, leading us to consider the dielectric constant.

dielectric constant

The dielectric constant, also known as the relative permittivity, is a measure of a substance's ability to insulate charges from each other. A higher dielectric constant means a greater ability to diminish the electrostatic force between the charges.
For the proteins \(A\) and \(B\), water’s dielectric constant is 80, meaning it can effectively reduce the electrostatic attraction between them. On the other hand, trifluoroethanol has a dielectric constant of 27, indicating a lower ability to insulate these charges, hence increasing the electrostatic interaction.
This relationship can be represented in the formula:
\[ K_{2} = K_{1} \frac{D_1}{D_2} \]
Given \(K_1 = 1000\), \(D_1 = 80\), and \(D_2 = 27\), substituting these values gives:
\[ K_{2} = 1000 \frac{80}{27} = 2963 \]
Thus, the binding constant in trifluoroethanol is 2963, showing higher binding affinity due to lower dielectric constant.