Question

The \(E\). coli nickel-binding protein binds to its ligand, \(\mathrm{Ni}^{2+}\), with a \(K_{\mathrm{d}}\) of \(100 \mathrm{~nm}\). Calculate the \(\mathrm{Ni}^{2+}\) concentration when the fraction of binding sites occupied by the ligand \((Y)\) is (a) \(0.25\), (b) \(0.6\), (c) \(0.95 .\)

Short answer

To find \\([ ump{Ni}^{2+}]\\), use \\([ ump{Ni}^{2+}] = \frac{YK_d}{1-Y} \\).

Step-by-step solution

Understanding the Binding Equation

To tackle this problem, we need to use the equation for fractional occupancy \( Y \) in terms of the dissociation constant \( K_d \) and the ligand concentration \( [ump{Ni}^{2+}] \): \[ Y = \frac{[ump{Ni}^{2+}]}{K_d + [ump{Ni}^{2+}]} \].This equation describes the fraction of binding sites occupied \( Y \) in terms of the concentration of the ligand and the dissociation constant \( K_d \). We will solve for the concentration of \( [ump{Ni}^{2+}] \) at given occupancy levels.

Rearrange the Equation for [ ump{Ni}^{2+}]

We need to rearrange the formula \( Y = \frac{[ump{Ni}^{2+}]}{K_d + [ump{Ni}^{2+}]} \) to solve for \([ump{Ni}^{2+}]\). Multiplying both sides by \(K_d + [ump{Ni}^{2+}]\) and simplifying gives us: \[ Y(K_d + [ump{Ni}^{2+}]) = [ump{Ni}^{2+}], \],which can be further rearranged to:\[ YK_d + Y[ump{Ni}^{2+}] = [ump{Ni}^{2+}]. \]Finally, isolating \([ump{Ni}^{2+}]\) gives:\[ (1-Y)[ump{Ni}^{2+}] = YK_d \]\[ [ump{Ni}^{2+}] = \frac{YK_d}{1-Y} \].

Key concepts

Ligand Binding

Ligand binding is a fundamental concept in biochemistry. It describes the process where molecules, called ligands, interact with specific proteins or receptors. This interaction often occurs at a binding site characterized by its shape, charge, and hydrophobic or hydrophilic properties that complement the ligand. Ligands can be various types of molecules, such as ions, small molecules, or even other proteins. When a ligand binds to a protein, it can trigger a biological response, which is crucial for many cellular processes. For example, hormones binding to their receptors to initiate signaling pathways.

In the context of our exercise, the ligand in question is the nickel ion \(\mathrm{Ni}^{2+}\). The specific protein in \(E. coli\) binds this ion with a certain affinity, measured by how tightly or weakly it holds onto the \(\mathrm{Ni}^{2+}\). Understanding this binding helps us solve how much of the ligand is required to reach certain levels of protein activation, also known as fractional occupancy.

Dissociation Constant

The dissociation constant, \(K_d\), is a critical parameter in the study of ligand-protein interactions. It provides a quantitative measure of the affinity between a ligand and its binding site on a protein. The \(K_d\) value is defined as the concentration of the ligand at which half of the available binding sites are occupied.

A low \(K_d\) indicates strong binding affinity, meaning the ligand binds tightly to the protein. Conversely, a high \(K_d\) suggests weaker binding affinity. For the \(E. coli\) nickel-binding protein, the \(K_d\) of 100 nm tells us that at this concentration of \(\mathrm{Ni}^{2+}\), half of the protein's binding sites are occupied. This knowledge aids in calculating \(\mathrm{Ni}^{2+}\) concentrations for different levels of site occupancy.

Fractional Occupancy

Fractional occupancy, or \(Y\), refers to the proportion of binding sites on a protein that are occupied by ligand molecules. It is a dimensionless number ranging from 0 to 1, where 0 indicates no occupied sites and 1 indicates full occupation. This concept is central to understanding how effectively a ligand can engage a protein at given concentrations.

The equation \( Y = \frac{[\mathrm{Ni}^{2+}]}{K_d + [\mathrm{Ni}^{2+}]}\) allows us to compute \(Y\) based on the known \(K_d\) and the concentration of the ligand. By rearranging this equation, as shown in the solution, we find the necessary ligand concentration to achieve a specific fractional occupancy. For example, for \(Y = 0.95\), the ligand concentration reflects the scenario where most of the binding sites are occupied, indicating strong binding conditions. This calculation is critical for applications such as drug design, where target saturation is often required for therapeutic efficacy.