Question

A dimeric hemoglobin is isolated from a fish. Each subunit contains a binding site for a xenon gas atom. The \(K_{\mathrm{D}}\) for the first binding event is measured to be \(23 \mathrm{nM}\). The \(K_{\mathrm{D}}\) for the second binding event is measured to be \(3.5 \mu \mathrm{M}\). a. What is the Hill coefficient? b. Is this protein positively or negatively cooperative with respect to xenon binding?

Short answer

a. The Hill coefficient is 0.00657. b. The protein exhibits negative cooperativity.

Step-by-step solution

Understanding KD and Hill Coefficient

The dissociation constant, \(K_{\mathrm{D}}\), is a measure of the affinity of the binding site for the ligand. A lower \(K_{\mathrm{D}}\) indicates higher affinity. The Hill coefficient is used to describe cooperativity, indicating how ligand binding to one site affects binding to other sites.

Convert Units of KD

Convert the \(K_{\mathrm{D}}\) of the second binding event from micromolar (\(\mu \mathrm{M}\)) to nanomolar (\(\mathrm{nM}\)) to ensure the units match. \[3.5 \mu \mathrm{M} = 3500 \mathrm{nM}.\]

Calculate Hill Coefficient

To find the Hill coefficient, calculate the ratio of the dissociation constants: \( n_H = \frac{ K_{\mathrm{D}_1} }{ K_{\mathrm{D}_2} } = \frac{23 \mathrm{nM}}{3500 \mathrm{nM}} = 0.00657.\) Since \(n_H < 1\), the Hill coefficient suggests negative cooperativity.

Determine Type of Cooperativity

A Hill coefficient less than 1 indicates negative cooperativity, meaning that the binding of the first ligand decreases the likelihood that a second ligand will bind.

Key concepts

Dissociation Constant

The dissociation constant, often represented as \(K_{\mathrm{D}}\), is a key concept in chemistry and biochemistry. It describes how strongly a ligand (such as a molecule or ion) binds to a specific site on a protein or other complex. In simpler terms, it tells us about the affinity between the two molecules.
The mathematical definition of \(K_{\mathrm{D}}\) is the concentration of ligand at which half of the available binding sites are occupied. Intuitively, a smaller \(K_{\mathrm{D}}\) value signifies stronger binding affinity, because less ligand is required to occupy half of the sites.

• For example, in the dissociation of gaseous xenon from hemoglobin, if the \(K_{\mathrm{D}}\) value for the first binding is \(23 \mathrm{nM}\), it indicates a higher affinity compared to the second binding event with \(3.5 \mu \mathrm{M}\), or \(3500 \mathrm{nM}\), showing a weaker affinity.

This fundamental property helps in studying and comparing various binding events, such as enzyme-substrate interactions, receptor-ligand binding, and more.

Cooperativity

Cooperativity describes how the binding of a ligand to one site on a protein affects the binding of additional ligands to other sites on the same protein. It's an important mechanism that can increase or decrease the binding affinity of additional sites, influencing the protein's function in the body.
One common way to quantify cooperativity is through the Hill coefficient, denoted as \(n_H\). This coefficient helps us understand whether the protein exhibits cooperative behavior when binding to its ligand.

• A Hill coefficient \(n_H = 1\) suggests non-cooperative binding, where the binding of a ligand does not influence the affinity of other sites.
• If \(n_H > 1\), it indicates positive cooperativity. This means that the binding of the first ligand makes it easier for subsequent ligands to bind, often leading to a steeper or sigmoidal binding curve.
• In contrast, \(n_H < 1\) implies negative cooperativity, where the initial ligand binding reduces the likelihood of other ligands binding.

The dimeric hemoglobin in the example exhibits a Hill coefficient of 0.00657, indicating negative cooperativity.

Negative Cooperativity

Negative cooperativity occurs when the binding of a ligand to one site on a multimeric protein decreases the likelihood of other ligand molecules binding to additional sites. This behavior can affect the overall binding capacity and regulation of the protein.
In the case of the isolated hemoglobin dimer from a fish, the calculated Hill coefficient of \(0.00657\) suggests negative cooperativity. This value informs us that as the first xenon atom binds to the hemoglobin, the protein undergoes a change that makes it less favorable for the second xenon atom to bind.

• This effect can be crucial in biological processes where the fine-tuning of ligand binding is necessary for function, such as controlling oxygen release in tissues.
• Understanding this mechanism helps in designing therapies and drugs, as well as in comprehending how proteins behave under different physiological conditions.

Negative cooperativity ensures that the system is not overly saturated with a ligand, allowing for regulated biological responses.