Question

Repressor Concentration in \(\boldsymbol{E}\). coli The dissociation constant for a particular repressor-operator complex is very low, about \(10^{-13}\) M. An \(E\). coli cell (volume \(2 \times 10^{-12} \mathrm{~mL}\) ) contains 10 copies of the repressor. Calculate the cellular concentration of the repressor protein. How does this value compare with the dissociation constant of the repressoroperator complex? What is the significance of this answer?

Short answer

The cellular concentration is \(8.3 \times 10^{-9}\) M; it's much higher than the dissociation constant \(10^{-13}\) M, indicating efficient binding.

Step-by-step solution

Calculate total volume in liters

The volume of an E. coli cell is given as \(2 \times 10^{-12} \text{ mL}\). Convert the cell volume from milliliters to liters by using the conversion \(1 \text{ mL} = 10^{-3} \text{ L}\). Therefore, the volume in liters is \(2 \times 10^{-12} \times 10^{-3} = 2 \times 10^{-15} \text{ L}\).

Calculate the number of moles of repressor

The cell contains 10 copies of the repressor protein. First, calculate the number of moles by using Avogadro's number \(6.022 \times 10^{23} \text{ molecules/mole}\). Thus, the number of moles is \(\frac{10}{6.022 \times 10^{23}} \approx 1.66 \times 10^{-23} \text{ moles}\).

Calculate cellular concentration of repressor

The concentration \(C\) can be determined using the formula \(C = \frac{\text{moles}}{\text{volume}}\). So, \(C = \frac{1.66 \times 10^{-23}}{2 \times 10^{-15}} = 8.3 \times 10^{-9} \text{ M}\) (Molarity). This is the cellular concentration of the repressor.

Compare with dissociation constant

The calculated cellular concentration of the repressor \(8.3 \times 10^{-9} \text{ M}\) is much higher than the dissociation constant \(10^{-13} \text{ M}\).

Significance of comparison

A concentration much higher than the dissociation constant suggests that most of the repressors in an E. coli cell are bound to the operator rather than being free. This implies that the repressor efficiently controls gene expression.

Key concepts

Dissociation Constant

The dissociation constant is a critical value that provides insights into how strongly two molecules interact with each other. In the context of this exercise, it defines the binding affinity between a repressor and an operator in a cell. A lower dissociation constant indicates a stronger binding, meaning it requires a lower concentration of repressor and operator to remain bound. For example, a dissociation constant of \(10^{-13} \text{ M}\) suggests a very tight interaction. This low value means that even at low concentrations, the repressor and operator will likely remain bound, significantly affecting the regulation mechanisms within the cell.

E. coli Cell

An _E. coli_ cell is a simple prokaryotic organism often used in scientific research because it is easy to manipulate and grows rapidly. In this exercise, the volume of an _E. coli_ cell is specified as \(2 \times 10^{-12}\) mL.

This small size presents a unique environment where cellular processes must be effectively regulated and tightly controlled despite limited space and resources. The number of repressor molecules in each cell plays a vital role in this regulation by interacting with other cellular components to control gene expression dynamically.

Repressor-Operator Complex

The repressor-operator complex is a molecular structure formed when a repressor protein binds to a specific DNA sequence known as the operator. This binding is crucial for the regulation of gene expression.

The role of the repressor-operator complex is to block or inhibit the transcription of genes by RNA polymerase, effectively controlling which genes are expressed at any given time. The high binding affinity reflected by a low dissociation constant in this scenario suggests that once the repressors bind to the operator, they are likely to remain attached until actively removed or altered, ensuring effective gene regulation.

Gene Expression Regulation

Gene expression regulation is an essential cellular process, determining which genes are turned on or off in response to environmental changes. This regulation is vital for the survival and adaptation of the cell.

In _E. coli_, as demonstrated in the exercise, the concentration of repressor proteins which regulate this expression is much higher than their dissociation constant.

• This means these proteins are usually bound to their respective operators.
• The higher concentration relative to the dissociation constant ensures that genes can be rapidly activated or repressed as needed.

Understanding the dynamics of repressor concentration and its relation to binding affinity (dissociation constant) helps explain how _E. coli_ precisely controls gene expression, promoting proper cellular function and adaptation.