61. In a cell, a repressor binds to its cognate operator with a KD = 10-9M. The cellular concentration of the
repressor is 10-8M. The extent of repressor bound to the operator is:
1. 100%
2. 50%
3. 37%
4. 0%

Calculating Repressor Binding to Operator: A KD-Based Approach

In gene regulation, repressors play a crucial role by binding to operator sequences to prevent transcription. The strength of this binding is characterized by the dissociation constant (KD) — a critical value in understanding protein-DNA interactions.

What Is KD?

KD (dissociation constant) is the concentration of the repressor at which 50% of the operator sites are occupied. It reflects the affinity between a repressor and its binding site:

• A lower KD means stronger binding affinity.

• A higher KD implies weaker binding.

The Binding Equation

The fraction of operator bound by the repressor at a given concentration $[L]$ is calculated as:

Fraction bound=[L][L]+KD\text{Fraction bound} = \frac{[L]}{[L] + K_D}Fraction bound=[L]+KD​[L]​

Where:

• $[L]$ = concentration of the repressor

• KDK_DKD​ = dissociation constant

Given in the Question:

• KD=10−9 MK_D = 10^{-9} \, \text{M}KD​=10−9M

• [L]=10−8 M[L] = 10^{-8} \, \text{M}[L]=10−8M

Substitute into the equation:

Fraction bound=10−810−8+10−9=10−81.1×10−8≈0.91\text{Fraction bound} = \frac{10^{-8}}{10^{-8} + 10^{-9}} = \frac{10^{-8}}{1.1 \times 10^{-8}} \approx 0.91Fraction bound=10−8+10−910−8​=1.1×10−810−8​≈0.91

This means approximately 91% of the operator sites are occupied by the repressor.

Closest Matching Option:

None of the options says 91%, but 100% is the closest approximation, indicating nearly complete binding due to high repressor concentration compared to KD.

Final Answer:

1. 100%

This is a reasonable approximation based on the math, where the repressor concentration is much higher than KD, resulting in nearly complete occupancy of operator sites.