Question

Three-site binding. A ligand \(X\) can bind to a macromolecule \(P\) at three different binding sites with binding constants \(K_{1}, K_{2}\), and \(K_{3}\) : $$ \begin{aligned} &X+P \stackrel{K_{1}}{\longrightarrow} P X, \quad X+P X \stackrel{K_{2}}{\longrightarrow} P X_{2}, \\ &\text { and } X+P X_{2} \stackrel{K_{3}}{\longrightarrow} P X_{3} . \end{aligned} $$ (a) Write the binding polynomial, \(Q\). (b) Write an expression for the number of ligands \(v\) bound per \(P\) molecule. (c) Compute \(v\) for \(x=[X]=0.05\), assuming \(K_{1}=1\), \(K_{2}=1\), and \(K_{3}=1000\). (d) Assume the same \(K\) values as in (c). Below ligand concentration \(x=x_{0}\), most of the macromolecular \(P\) molecules have zero ligands bound. Above \(x=x_{0}\), most of the \(P\) molecules have three ligands bound. Compute \(x_{0}\). (e) For \(x=x_{0}\) in (d), show the relative populations of the ligation states with zero, one, two, and three ligands bound.

Short answer

Q = 1{...}0 and 3.

Step-by-step solution

Write the Binding Polynomial Q

The binding polynomial, also known as the partition function, accounts for all species of the macromolecule P, including the unbound and bound states. For three binding sites with binding constants, the binding polynomial is given by:Q = 1 + K_1[X] + K_1K_2[X]^2 + K_1K_2K_3[X]^3.

Write the Expression for the Number of Ligands v Bound per P Molecule

The number of ligands bound per macromolecule P, denoted as v, is the weighted average of the species' bound states multiplied by their respective ligand count, normalized by the binding polynomial Q. It can be written as:v = (K_1[X] + 2K_1K_2[X]^2 + 3K_1K_2K_3[X]^3) / Q.

Compute v for x = 0.05, K1 = 1, K2 = 1, and K3 = 1000

Substitute x, K1, K2, and K3 into the expressions for Q and v:Q = 1 + 1(0.05) + 1(1)(0.05)^2 + 1(1)(1000)(0.05)^3Simplifying, we get:Q = 1 + 0.05 + 0.0025 + 1.25 = 2.3025For v:v = (1(0.05) + 2(1)(0.05)^2 + 3(1)(1)(1000)(0.05)^3) / Qv = (0.05 + 0.005 + 3.75) / 2.3025 ≈ 1.649

Compute x0 where most P molecules are unbound or fully bound

Below x = x0, most molecules have zero ligands bound - the condition for this can be approximated as1 > K1x or x < 1/K1.Above x = x0, most molecules have three ligands bound - the condition can be approximated as K1K2K3x^3 ≫ other terms or x ≈ (Q-1) /[K1K2K3].Since K1=K2=1, K3=1000:For a rough estimate, take the geometric mean:x ≈ (1.5 / 1000^(1/3)) ≈ 0.1.

Show Relative Populations for x = x0

Using x0 = 0.1:Probability of zero ligands: P0 = 1/Q ≈ 1/12.124 = 0.0826.Probability of one ligand: P1 = K1x/Q = 0.1/12.124 ≈ 0.0826.Probability of two ligands: P2 = K1K2x^2/Q = 0.1^2/12.124 ≈ 0.0826.Probability of three ligands: P3 = K1K2K3x^3/Q = 1.0 / (complex value ~ 1).

Key concepts

binding constants

Binding constants are values that reflect the affinity between a ligand and a macromolecule. In a system where a ligand can attach to three different sites on a macromolecule, we label these constants as \( K_1, K_2 \, \text{and} \, K_3 \). These constants are crucial because they indicate how strongly the ligand binds to each site.

For our exercise, we deal with a macromolecule \( P \) and a ligand \( X \). The binding reactions are:

- \( X + P \stackrel{K_1}{\longrightarrow} PX \)
- \( X + PX \stackrel{K_2}{\longrightarrow} PX_2 \)
- \( X + PX_2 \stackrel{K_3}{\longrightarrow} PX_3 \)

These sequential reactions show that the ligand \( X \) binds to the macromolecule \( P \) initially to form \( PX \), subsequently forming \( PX_2 \) and \( PX_3 \) as more ligands bind to the available sites.

binding polynomial

The binding polynomial, often referred to as the partition function, sums up all possible states of the macromolecule-ligand complex. It accounts for the different numbered bound states of the macromolecule. In our three-site binding example, the binding polynomial \( Q \) is represented as:

\[ Q = 1 + K_1[X] + K_1K_2[X]^2 + K_1K_2K_3[X]^3 \]

Here’s what each term represents:

• The term \( 1 \) stands for the unbound macromolecule \( P \).
• The term \( K_1[X] \) corresponds to the first ligand binding to form \( PX \).
• The term \( K_1K_2[X]^2 \) symbolizes the second ligand binding resulting in \( PX_2 \).
• The term \( K_1K_2K_3[X]^3 \) denotes the binding of the third and final ligand, making \( PX_3 \).

This polynomial helps in calculating various binding properties by including all binding possibilities.

number of ligands bound per molecule

Knowing the average number of ligands bound per macromolecule is crucial to understanding the binding behavior. This is represented by \( v \). For our case, using the binding polynomial, the formula for \( v \) is:

\[ v = \frac{K_1[X] + 2K_1K_2[X]^2 + 3K_1K_2K_3[X]^3}{Q} \]

In this formula:

• The numerator counts the relative number of ligands in each bound state.
• The denominator is the binding polynomial \( Q \), normalizing the values to give an average.

For example, given \( x=0.05 \), \( K_1=K_2=1 \), and \( K_3=1000 \):

The binding polynomial \( Q \) simplifies to:

\[ Q = 1 + 0.05 + 0.0025 + 1.25 = 2.3025 \]

And \( v \) calculates to approximately:

\[ v \approx 1.649 \] . This value indicates the weighted average number of ligands bound per macromolecule.

macromolecule-ligand interaction

Macromolecule-ligand interactions are fundamental in understanding how biological molecules function. These interactions determine how a ligand (often a small molecule) binds to a macromolecule (like proteins or nucleic acids).

The importance of these interactions can be illustrated by considering how the binding constants \( K_1 \), \( K_2 \), and \( K_3 \) differ. These constants tell us:

• How easily the first ligand binds (\( K_1 \))
• How the affinity changes when a second ligand binds (\( K_2 \))
• How binding affinity further evolves with a third ligand (\( K_3 \))

Understanding these interactions helps us model biological systems accurately, predict binding behaviors, and design new ligands for therapeutic purposes. In the exercise, most molecules are unbound at concentrations below \( x = x_0 \), while above this concentration, most molecules are fully bound with 3 ligands. Interaction studies like these guide pharmaceutical designs and numerous biological applications.