Question

A dimeric enzyme, glucokinase, has a binding site for glucose in each subunit. The \(K_{\mathrm{D}}\) for the first binding event is \(1 \mathrm{mM}\) and the \(K_{\mathrm{D}}\) for the second event is \(10 \mu \mathrm{M}\). a. What is the Hill coefficient? b. Is this protein positively or negatively cooperative with respect to glucose binding?

Short answer

The Hill coefficient is greater than 1, indicating positive cooperativity.

Step-by-step solution

Understanding the Problem

We have a dimeric enzyme, glucokinase, which binds glucose at two distinct sites. We are given the dissociation constants (\(K_D\)) for each binding site: \(1 \, \text{mM}\) for the first site and \(10 \, \mu\text{M}\) for the second site. We need to determine the Hill coefficient and assess the cooperativity.

Hill Coefficient Formula

The Hill coefficient \( n_H \) is calculated based on the cooperativity of ligand binding. If the \(K_D\) for the second binding event is significantly lower than the first, it indicates positive cooperativity where binding at one site increases the affinity at the other. In this situation, the Hill coefficient can be estimated as greater than 1.

Calculate the Relative Affinity Change

Compare the two \(K_D\) values to determine the cooperativity. The first \(K_D\) is \(1 \, \text{mM} = 1000 \, \mu\text{M}\) and the second \(K_D\) is \(10 \, \mu\text{M}\). The second \(K_D\) is much lower (100 times lower) than the first, suggesting that once the first site is occupied, the second site binds glucose more strongly.

Determine Hill Coefficient

Due to the significant reduction in \(K_D\), the Hill coefficient \(n_H > 1\) indicating positive cooperativity. In practice for dimeric enzymes, this value typically ranges between 1 and 2 when there is positive cooperativity.

Assess Cooperativity

Based on the calculated relative affinity and Hill coefficient, glucokinase exhibits positive cooperativity, as binding at one site reduces the \(K_D\) at the second site, enhancing binding affinity.

Key concepts

Glucokinase Function

Glucokinase is an enzyme that plays a crucial role in glucose metabolism, particularly in liver cells and pancreatic beta cells. It facilitates the conversion of glucose to glucose-6-phosphate, which is the first step in glycolysis and other metabolic pathways. Unlike other hexokinases, glucokinase is not inhibited by its product and has a much higher Km for glucose. This means it only becomes active at higher glucose concentrations, serving as a glucose sensor.

Key functions of glucokinase include:

• Converting glucose to glucose-6-phosphate, crucial for energy production and glycogen synthesis.
• Acting as a sensor to regulate insulin release in response to blood glucose levels.
• Enabling glucose homeostasis in the liver by storing or releasing glucose as needed.

Understanding glucokinase's function helps appreciate its role in maintaining blood sugar levels and its potential implication in diabetes management.

Hill Coefficient

The Hill coefficient is a key concept when evaluating enzyme cooperativity, particularly in systems with multiple binding sites, like glucokinase. It provides insight into whether an enzyme exhibits cooperative binding behavior and to what extent.

• A Hill coefficient of 1 suggests independent binding, with no cooperativity.
• A Hill coefficient greater than 1 indicates positive cooperativity, where binding of one ligand increases the affinity of others.
• A Hill coefficient less than 1 signals negative cooperativity, where binding reduces the affinity.

In the example of glucokinase, a Hill coefficient greater than 1 implies that binding of glucose to one site makes the other site more likely to bind glucose. This enhances the enzyme's responsiveness to changing glucose concentrations.

Dimeric Enzymes

Dimeric enzymes, such as glucokinase, consist of two subunits. Each subunit can independently bind a substrate or ligand, but their structural arrangement can influence overall enzyme activity.

Some characteristics of dimeric enzymes include:

• Possession of multiple active sites that can interact with each other.
• Potential for cooperative behavior due to subunit interactions, which can modify the enzyme's binding affinity.
• Structural flexibility that can lead to allosteric regulation, allowing the enzyme to respond to various metabolic signals.

For glucokinase, being dimeric means that its binding of glucose is not independent; once one site is occupied, it affects the other site, enhancing its affinity for glucose.

Ligand Binding Affinity

Ligand binding affinity refers to the strength of the interaction between a ligand, like glucose, and a protein such as an enzyme. This affinity is often measured using the dissociation constant, or \(K_D\). The lower the \(K_D\), the higher the affinity, meaning the ligand binds more tightly to the protein.

Critical factors influencing ligand binding affinity include:

• The chemical properties and structural fit between ligand and enzyme.
• The presence of other binding sites and their occupancy, as seen in cooperative enzymes.
• Environmental conditions such as pH and temperature, which can alter enzyme structure and binding capabilities.

In glucokinase's case, an initial \(K_D\) of 1 mM decreases to 10 \(\mu\text{M}\) for the second glucose molecule, indicating a stronger binding after the initial site is occupied. This shows how cooperative binding can amplify enzyme response and efficiency.