Question

Drug binding to a protein. A drug \(D\) binds to a protein \(P\) with two different equilibrium binding constants, \(K_{1}\) and \(K_{2}\) : $$ D+P \stackrel{K_{1}}{\longrightarrow} P D_{1}, \quad \text { and } \quad D+P D_{1} \stackrel{K_{2}}{\longrightarrow} P D_{2}, $$ where \(K_{1}=1000 \mathrm{M}^{-1}\), and \(K_{2}=2000 \mathrm{M}^{-1}\) at \(T=300 \mathrm{~K}\). (a) Write an algebraic expression for the fraction of protein molecules that have two drug molecules bound as a function of drug concentration \(x=[D]\). (b) If the drug concentration is \(x=10^{-3} \mathrm{M}\), what is the fraction of protein molecules that have one drug molecule bound? (c) If the drug concentration is \(x=10^{-3} \mathrm{M}\), what is the fraction of protein molecules that have zero drug molecules bound? (d) If the drug concentration is \(x=10^{-3} \mathrm{M}\), what is the fraction of protein molecules that have two drug molecules bound?

Short answer

a) \( \frac{K_{1}K_{2}x^2}{1 + K_{1}x + K_{1}K_{2}x^2} \), b) 0.25, c) 0.25, d) 0.5

Step-by-step solution

- Establish Definitions and Equilibrium Equations

Define the equilibrium constants provided: \(K_{1}=1000 \, \mathrm{M}^{-1}\) and \(K_{2}=2000 \, \mathrm{M}^{-1}\). The equilibrium reactions are: \[D + P \stackrel{K_{1}}{\longrightarrow} PD_{1}\] and \[D + PD_{1} \stackrel{K_{2}}{\longrightarrow} PD_{2}\]

- Expression for Fraction of Protein Molecules with Two Drug Molecules Bound

Express the fraction of protein molecules with two drug molecules bound: Let \(x = [D]\) represent the drug concentration. The fraction of protein bound with two drug molecules, \( f_{PD_{2}} \), is given by: \[ f_{PD_{2}} = \frac{K_{1}K_{2}x^2}{1 + K_{1}x + K_{1}K_{2}x^2} \]

- Calculate the Fraction with One Drug Bound at \(x=10^{-3} \mathrm{M}\)

The fraction of protein molecules with one drug molecule bound, \( f_{PD_{1}} \), can be expressed as: \[ f_{PD_{1}} = \frac{K_{1}x}{1 + K_{1}x + K_{1}K_{2}x^2} \] Plug in \(K_{1}=1000 \mathrm{M}^{-1}\), \(K_{2}=2000 \mathrm{M}^{-1}\), and \(x=10^{-3} \mathrm{M}\): \[ f_{PD_{1}} = \frac{1000 \times 10^{-3}}{1 + 1000 \times 10^{-3} + 1000 \times 2000 \times (10^{-3})^2} = \frac{1}{1 + 1 + 2} = \frac{1}{4} = 0.25 \]

- Calculate the Fraction with Zero Drug Bound at \(x=10^{-3} \mathrm{M}\)

The fraction of protein molecules with zero drug molecules bound, \( f_{P} \), can be expressed as: \[ f_{P} = \frac{1}{1 + K_{1}x + K_{1}K_{2}x^2} \] Plug in \(K_{1}=1000 \mathrm{M}^{-1}\), \(K_{2}=2000 \mathrm{M}^{-1}\), and \(x=10^{-3} \mathrm{M}\): \[ f_{P} = \frac{1}{1 + 1000 \times 10^{-3} + 1000 \times 2000 \times (10^{-3})^2} = \frac{1}{1 + 1 + 2} = \frac{1}{4} = 0.25 \]

- Calculate the Fraction with Two Drugs Bound at \(x=10^{-3} \mathrm{M}\)

Using the expression from Step 2 and plugging in \(x=10^{-3} \mathrm{M}\): \[ f_{PD_{2}} = \frac{1000 \times 2000 \times (10^{-3})^2}{1 + 1000 \times 10^{-3} + 1000 \times 2000 \times (10^{-3})^2} = \frac{2}{1 + 1 + 2} = \frac{2}{4} = 0.5 \]

Key concepts

equilibrium binding constants

Equilibrium binding constants, often denoted as K, are fundamental in understanding how drugs interact with proteins.
They quantify the affinity between two molecules.
In our problem, a drug D binds to a protein P with two different binding constants, \(K_{1}\) and \(K_{2}\):
D + P \(\stackrel{K_{1}}{\longrightarrow}\) PD₁ and D + PD₁ \(\stackrel{K_{2}}{\longrightarrow}\) PD₂.
The higher the value of K, the stronger the binding affinity.
In this case:
\(K_{1} = 1000 \, \mathrm{M}^{-1}\)
\(K_{2} = 2000 \, \mathrm{M}^{-1}\)
These constants tell us that the second binding event has a higher affinity (or is more favorable) than the first binding event.
Understanding these constants enables us to predict the likelihood of the drug binding at different stages of protein interaction.

fractional occupancy

Fractional occupancy describes the proportion of protein molecules occupied by the drug.
It's an important concept in drug-binding studies as it helps in understanding the extent to which a drug can interact with its target.
There are different fractional occupancies for protein with no drug (\(f_P\)), one drug molecule bound (\(f_{PD_{1}}\)), and two drug molecules bound (\(f_{PD_{2}}\)).

Fraction with two drug molecules:
\[f_{PD_{2}} = \frac{K_{1}K_{2}x^2}{1 + K_{1}x + K_{1}K_{2}x^2}\]

Fraction with one drug molecule:
\[f_{PD_{1}} = \frac{K_{1}x}{1 + K_{1}x + K_{1}K_{2}x^2}\]

Fraction with no drug:
\[f_P = \frac{1}{1 + K_{1}x + K_{1}K_{2}x^2}\]

Plugging in values of K and x (drug concentration) allows us to calculate these probabilities.
This process helps us understand how drug concentration impacts the binding and occupancy of protein molecules.

protein-ligand interactions

The interaction between a protein and a ligand, such as a drug, is a key concept in biochemistry.
These interactions often determine the biological activity of drugs.
In our scenario, the drug D binds sequentially to a protein P, forming complexes PD₁ and PD₂.
Each binding event has its own equilibrium constant, reflecting the affinity and specificity of the interaction.
This sequential binding can be plotted and analyzed to deliver insights into how a drug functions.
By examining protein-ligand interactions, researchers can design more effective drugs with desired binding properties.
Complexities like the number of binding sites, binding constants, and drug concentrations need to be carefully considered to fully understand and predict these interactions.
This approach helps in drug development and therapeutic applications.