Question

Cooperative binding to an enzyme. An enzyme has three binding sites: \(A, B\), and \(C\). A ligand can bind to the enzyme at all three sites; however, site \(C\) will not take up ligand until both sites \(A\) and \(B\) have been bound. And, sites \(A\) and \(B\) are independent of each other. This system is a single domain on the protein. (a) Express the binding polynomial for this system. (b) Write an expression for the average number of bound ligands. (c) Express a different binding polynomial, assuming that the enzyme is uncompetitively inhibited by a different inhibitor at site \(C\) and that the overall system consists of three independent domains. (d) What is the average number of bound ligands in this latter case?

Short answer

1. The binding polynomial is: \[ P(y) = 1 + K_A y + K_B y + K_A K_B K_C y^3 \] 2. The average number of bound ligands is: \bar{n} = \frac{K_A y + K_B y + 3 K_A K_B K_C y^3}{1 + K_A y + K_B y + K_A K_B K_C y^3}3. The modified binding polynomial is: \[ P_{mod}(y) = 1 + K_A y + K_B y + \frac{K_A K_B K_C y^3}{1+K_I I} \] 4. The modified average number of bound ligands is: \bar{n}_{mod} = \frac{K_A y + K_B y + \frac{3 K_A K_B K_C y^3}{1 + K_I I}}{1 + K_A y + K_B y + \frac{K_A K_B K_C y^3}{1 + K_I I}}

Step-by-step solution

Define the Binding Polynomials for Part (a)

Given the enzyme has three binding sites labeled as A, B, and C, and site C will not bind ligand until both A and B are bound, while A and B are independent of each other, the binding polynomial for this system can be written considering each site's binding independently and including the cooperative nature of the C site. Let \theta_A, \theta_B, and \theta_C denote the fractional occupancies of sites A, B, and C respectively and K_A, K_B, and K_C are their respective dissociation constants. The binding polynomial is: \[ P(y) = 1 + K_A y + K_B y + (K_A K_B K_C y^3) \]

Write the Expression for Average Number of Bound Ligands for Part (b)

Using the binding polynomial obtained in Step 1, the average number of bound ligands \bar{n} can be calculated using the formula: \[ \bar{n} = y \frac{d \text{ln}(P(y))}{dy} \] Applying this to the binding polynomial: \[ \bar{n} = y \frac{d }{dy} \text{ln}(1 + K_A y + K_B y + K_A K_B K_C y^3) \] \[ \bar{n} = \frac{K_A y + K_B y + 3 K_A K_B K_C y^3}{1 + K_A y + K_B y + K_A K_B K_C y^3} \]

Define the New Binding Polynomial for Part (c)

Assuming an uncompetitive inhibitor binds at site C, and considering the enzyme as three independent domains, the binding polynomial can be modified to reflect the presence of the inhibitor. Let I be the concentration of the inhibitor and K_I be its dissociation constant. The modified binding polynomial is: \[ P_{mod}(y) = 1 + K_A y + K_B y + \frac{K_A K_B K_C y^3}{1 + K_I I} \]

Calculate the Average Number of Bound Ligands for Part (d)

Using the modified binding polynomial from Step 3, the average number of bound ligands \bar{n}_{mod} can be calculated similarly: \[ \bar{n}_{mod} = y \frac{d \text{ln}(P_{mod}(y))}{dy} \] Which simplifies to: \[ \bar{n}_{mod} = \frac{K_A y + K_B y + \frac{3 K_A K_B K_C y^3}{1 + K_I I}}{1 + K_A y + K_B y + \frac{K_A K_B K_C y^3}{1 + K_I I}} \]

Key concepts

binding polynomial

In the context of enzyme binding, a binding polynomial, also known as a partition function, is a mathematical expression that helps us understand how ligands (molecules that can bind to enzymes) interact with binding sites on an enzyme. It accounts for different binding scenarios these ligands can adopt. For the enzyme with three binding sites A, B, and C, the binding polynomial considers the independence of A and B and the cooperative nature of C, which only binds after both A and B are occupied. This specific scenario can be expressed as:
\[ P(y) = 1 + K_A y + K_B y + (K_A K_B K_C y^3) \]
Here, \(K_A, K_B, \text{and} K_C\) represent the dissociation constants for sites A, B, and C respectively, and \(y\) is the ligand concentration. The first term accounts for the unbound state, while subsequent terms account for various binding states.

dissociation constant

The dissociation constant, denoted as \(K_D\), measures the affinity between a ligand and its binding site on the enzyme. It is an equilibrium constant that indicates how easily a complex (enzyme-ligand) dissociates into its components (enzyme and ligand).
A lower value of \(K_D\) means higher affinity, meaning the ligand binds strongly to the enzyme. For our enzyme with three sites, the dissociation constants are \(K_A\), \(K_B\), and \(K_C\).
The equation for the dissociation constant is: \[ K_D = \frac{[E][L]}{[EL]} \] Where:

• \([E]\) is the concentration of the free enzyme
• \([L]\) is the concentration of the free ligand
• \([EL]\) is the concentration of the enzyme-ligand complex

fractional occupancy

Fractional occupancy refers to the fraction of the total enzyme binding sites that are occupied by the ligand at any given concentration. It provides insight into how effectively the ligand binds to the enzyme.
We can determine the fractional occupancy (\(\theta_A, \theta_B, \theta_C\)) for each site using their dissociation constants and ligand concentration (\(y\)). The formula for fractional occupancy of a site (A) is: \[ \theta_A = \frac{[A]y}{K_A + y} \]
Here,

• [A] is the concentration of the enzyme with an occupied site A.
• y is the ligand concentration
• \(K_A\) is the dissociation constant for site A

By considering the cooperative binding and independence of sites A and B, the overall fractional occupancy can be more complex and is calculated using the given binding polynomial.

uncompetitive inhibition

Uncompetitive inhibition is a scenario where an inhibitor binds only to the enzyme-ligand complex, preventing it from catalyzing further reactions. In our case, the inhibitor impacts the binding at site C. This modifies the binding polynomial to account for the inhibitor's presence. The modified binding polynomial is:
\[ P_{mod}(y) = 1 + K_A y + K_B y + \frac{K_A K_B K_C y^3}{1 + K_I I} \]
Where \(K_I\) is the dissociation constant of the inhibitor for site C, and I is the inhibitor concentration.
Uncompetitive inhibition typically results in a decrease in both the maximum reaction rate \(V_{max}\) and the Michaelis constant \(K_m\), ensuring that the inhibition cannot be overcome by simply increasing the ligand concentration.