Lac Operon Analysis

This solution determines the state of the lac operon based on the intracellular concentration of lac-repressor and its binding affinity to the operator region.

Repressor Concentration Calculation

1. The volume of the E. coli cell is given as $V = 10^{-12} \text{ cm}^3$. Convert this volume to Liters (L):

   $V = 10^{-12} \text{ cm}^3 \times \frac{1 \text{ mL}}{1 \text{ cm}^3} \times \frac{1 \text{ L}}{1000 \text{ mL}} = 10^{-15} \text{ L}$

2. The cell contains 60 molecules of the lac-repressor protein. Calculate the number of moles using Avogadro's number ($N_A \approx 6.022 \times 10^{23} \text{ molecules/mol}$):

   Moles of repressor $= \frac{60 \text{ molecules}}{6.022 \times 10^{23} \text{ molecules/mol}} \approx 9.96 \times 10^{-23} \text{ mol}$

3. Calculate the molar concentration ([Repressor]) of the repressor protein inside the cell:

   $[\text{Repressor}] = \frac{\text{Moles of repressor}}{V (\text{L})} = \frac{9.96 \times 10^{-23} \text{ mol}}{10^{-15} \text{ L}} \approx 1 \times 10^{-7} \text{ M}$

Repressor Binding Affinity Analysis

The question provides binding affinity ($K_a$) values in Molarity (M), which typically represent dissociation constants ($K_d$).

• $K_d$ without lactose = $10^{-9}$ M (Strong binding)
• $K_d$ with lactose = $10^{-8}$ M (Weaker binding)

We compare the intracellular repressor concentration ([Repressor] $\approx 1 \times 10^{-7}$ M) to these $K_d$ values.

Lac Operon State Determination

1. Without Lactose:

   • The repressor concentration ($1 \times 10^{-7}$ M) is significantly higher than the dissociation constant ($K_d = 10^{-9}$ M).
   • Fraction of bound repressor: $\frac{[R]}{[R] + K_d} = \frac{1 \times 10^{-7}}{1 \times 10^{-7} + 10^{-9}} \approx 0.99$.
   • Result: The repressor binds tightly to the operator, keeping the lac operon repressed.

2. With Lactose:

   • Lactose binds to the repressor, increasing the $K_d$ to $10^{-8}$ M.
   • The repressor concentration ($1 \times 10^{-7}$ M) is still higher than this new $K_d$, although the ratio is smaller ($[\text{Repressor}] / K_d = 10$).
   • Fraction of bound repressor: $\frac{[R]}{[R] + K_d} = \frac{1 \times 10^{-7}}{1 \times 10^{-7} + 10^{-8}} = \frac{1 \times 10^{-7}}{1.1 \times 10^{-7}} \approx 0.91$.
   • Result: A high fraction (~91%) of repressor remains bound to the operator, meaning the operon is still effectively repressed and cannot be fully induced under these conditions.

3. Conclusion: Based on the calculated repressor concentration and its binding affinities, the lac operon is repressed and cannot be effectively induced by lactose.