Question

The binding of NADH to human liver mitochondrial isozyme was studied [Biochemistry 28 (1989): 5367] and it was determined that only a single binding site is present with \(K=2.0 \times 10^{7} \mathrm{M}^{-1} .\) What concentration of NADH is required to occupy \(10 \%\) of the binding sites?

Short answer

A concentration of approximately \(2.22 \times 10^{-8} \mathrm{M}\) NADH is required to occupy 10% of the binding sites on human liver mitochondrial isozyme.

Step-by-step solution

Write down the formula for fractional occupancy (θ).

The formula for the fractional occupancy (θ) of a binding site by a ligand is given by: \[ θ = \frac{[L]}{[L] + K^{-1}}\] where θ is the fractional occupancy, [L] is the concentration of the ligand (NADH in this case), and K is the binding constant.

Substitute the given values in the formula.

We are given that 10% of the binding sites have to be occupied, which means θ = 0.1. The given binding constant K is \( 2.0 \times 10^{7} \mathrm{M}^{-1} \). Now, let's substitute these values in the formula: \[ 0.1 = \frac{[L]}{[L] + 2.0 \times 10^{-7}}\]

Solve the equation for the concentration of NADH (L).

To solve for [L], first multiply both sides of the equation by the denominator: \[ 0.1([L] + 2.0 \times 10^{-7}) = [L]\] Now, distribute and simplify the equation: \[ 0.1L + 2.0 \times 10^{-8} = L\] Subtract 0.1L from both sides of the equation: \[ 2.0 \times 10^{-8} = 0.9L\] Finally, divide both sides by 0.9 to isolate [L]: \[ [L] = \frac{2.0 \times 10^{-8}}{0.9}\]

Calculate the concentration of NADH (L).

Now, let's use a calculator to find the value of [L]: \[ [L] \approx 2.22 \times 10^{-8} \mathrm{M}\] So, a concentration of approximately \(2.22 \times 10^{-8} \mathrm{M}\) NADH is required to occupy 10% of the binding sites on human liver mitochondrial isozyme.

Key concepts

Fractional Occupancy

Understanding fractional occupancy is crucial when studying the interactions between ligands and binding sites on proteins such as enzymes or receptors. Fractional occupancy (\theta) represents the proportion of total binding sites occupied by a ligand at a given concentration. When all the sites are occupied, \theta equals 1 or 100%, indicating full saturation. Conversely, \theta equals 0 when no sites are occupied. The relationship of fractional occupancy to ligand concentration (\text{[L]}) is elegantly described by the formula:
\[ \theta = \frac{\text{[L]}}{\text{[L]} + K^{-1}} \]
In this equation, \text{[L]} is the ligand concentration and K is the binding constant, which is a measure of the affinity between the ligand and the binding site. A high K value indicates a strong affinity, leading to higher fractional occupancy at lower ligand concentrations. By mastering this concept, students can predict how changes in ligand concentration affect the occupancy of binding sites, which is pivotal in fields like pharmacology and biochemistry.

Binding Constant

The binding constant (K) is a fundamental quantitative measure of the strength of the interaction between a ligand and a binding site. It's expressed in units of concentration inverse (\text{M}^{-1}) and reflects the affinity of the ligand for the binding site: the higher the K, the stronger the interaction.
The binding constant can be derived experimentally and is used in the equation for calculating fractional occupancy. In our exercise, the provided K value is exceptionally high at \(2.0 \times 10^{7} \text{M}^{-1}\), indicating a potent interaction between NADH and the enzyme's binding site. Understanding how to manipulate and interpret this constant is vital for students interested in enzyme kinetics and drug design, as it helps predict how much ligand is required to achieve a desired level of binding site occupancy.

NADH Concentration

The concentration of a ligand, such as NADH (nicotinamide adenine dinucleotide hydride), is a practical aspect of biochemistry with implications in metabolic processes and enzymatic reactions. NADH serves as a coenzyme in redox reactions, transferring electrons in metabolic pathways. The availability of NADH can greatly influence the rate of these reactions.
In our textbook exercise, the task was to determine the NADH concentration required to reach a specific fractional occupancy of binding sites. This teaches students the importance of calculating precise concentrations needed in assays and therapeutic contexts. Practical application of these calculations is essential in experimental biochemistry and pharmacology, where the efficacy and potency of compounds are often determined by their concentrations.

Enzyme-Ligand Interaction

Enzyme-ligand interactions are the basis of enzyme catalysis and regulation. When a ligand, such as a substrate, inhibitor, or activator, binds to an enzyme's active site or regulatory site, it can alter the enzyme's activity and function.
In the context of our exercise, students examine the interaction of NADH with a specific isozyme of human liver mitochondria, a process fundamental to cellular respiration and energy production. By delving into enzyme-ligand interactions, students build a foundation for understanding more complex biochemical pathways and the effects of various compounds on those pathways. That foundational knowledge aids in the development of therapeutics and the elucidation of disease mechanisms at the molecular level.