Problem 8 - Analytical Chemistry

Question

The enzyme acetylcholinesterase catalyzes the decomposition of acetylcholine to choline and acetic acid. Under a given set of conditions the enzyme has a \(K_{m}\) of \(9 \times 10^{-5} \mathrm{M}\) and a \(k_{2}\) of \(1.4 \times 10^{4} \mathrm{~s}^{-1}\). What is the concentration of acetylcholine in a sample if the reaction's rate is \(12.33 \mu \mathrm{M} \mathrm{s}^{-1}\) in the presence of \(6.61 \times 10^{-7} \mathrm{M}\) enzyme? You may assume the concentration of acetylcholine is significantly smaller than \(K_{m}\).

Step-by-Step Solution

Answer

The concentration of acetylcholine is approximately \( 1.197 \times 10^{-7} \text{ M} \).

Step 1: Identify Known Values

We are given: \( K_m = 9 \times 10^{-5} \text{ M} \), \( k_2 = 1.4 \times 10^4 \text{ s}^{-1} \), the reaction rate \( v = 12.33 \underline{\phantom{xxx}} \mu \text{M s}^{-1} = 12.33 \times 10^{-6} \text{ M s}^{-1} \), and enzyme concentration \( [E] = 6.61 \times 10^{-7} \text{ M} \).

Step 2: Understanding the Michaelis-Menten Equation

The general Michaelis-Menten equation is given by: \[ v = \frac{k_2 [E][S]}{K_m + [S]} \]where \([S]\) is the substrate concentration.

Step 3: Assume Simplified Michaelis-Menten Equation

Since \([S] \ll K_m\), the term \( K_m + [S] \) simplifies to \( K_m \), allowing the equation to be simplified to: \[ v \approx \frac{k_2 [E][S]}{K_m} \]

Step 4: Rearrange for Substrate Concentration

Rearrange the equation to solve for \([S]\): \[ [S] = \frac{v K_m}{k_2 [E]} \]

Step 5: Substitute Values into the Equation

Plug in the known values: \[ [S] = \frac{12.33 \times 10^{-6} \text{ M/s} \times 9 \times 10^{-5} \text{ M}}{1.4 \times 10^4 \text{ s}^{-1} \times 6.61 \times 10^{-7} \text{ M}} \] Simplify to find \([S]\).

Step 6: Calculate the Substrate Concentration

Perform the arithmetic: \[ [S] = \frac{12.33 \times 9 \times 10^{-11}}{9.254 \times 10^{-3}} \approx 1.197 \times 10^{-7} \text{ M} \] Thus, the concentration of acetylcholine is approximately \( 1.197 \times 10^{-7} \text{ M} \).

Key Concepts

Understanding Enzyme Kinetics

Enzyme kinetics is a key concept that explores how enzymes interact with their substrates to facilitate chemical reactions. At its core, enzyme kinetics provides insight into the rate of these reactions and the various factors that influence them. Enzymes are biological catalysts that speed up reactions without being consumed in the process. They achieve this by lowering the activation energy required for the reaction to proceed, which in turn increases the rate at which the product is formed.

In our example involving acetylcholinesterase, understanding enzyme kinetics allows us to determine how efficiently this enzyme catalyzes the decomposition of acetylcholine. The Michaelis-Menten model is often used in enzyme kinetics to describe the rate of enzymatic reactions. This model assumes that the enzyme-substrate complex forms an intermediate with a specific concentration, and the reaction rate depends on this complex's formation and breakdown.

• Michaelis constant ( \( K_m \)) reflects the affinity of the enzyme for the substrate; lower values indicate higher affinity.
• The maximum rate ( \( V_{max} \)) is reached when the enzyme is saturated with the substrate.

The steady-state assumption is key in enzyme kinetics, which assumes that the formation and breakdown rates of the enzyme-substrate complex are equal.

Importance of Substrate Concentration

The concentration of substrate, represented as \([S]\), is a crucial factor in determining the rate of an enzyme-catalyzed reaction. In enzyme kinetics, the substrate concentration directly influences how fast the reaction will occur until the enzyme becomes saturated. At low substrate concentrations, the reaction rate increases linearly with an increase in substrate, as there are many active sites available on the enzyme.

When the substrate concentration is significantly below the \( K_m \) value, the rate of reaction can be described by a simplified version of the Michaelis-Menten equation, where \( v \approx \frac{k_2 [E][S]}{K_m} \). Here, \( v \) is the rate of reaction, \( k_2 \) (or \( V_{max} \)) is the turnover number, \([E]\) is the enzyme concentration, and \( K_m \) is the Michaelis constant.

• This simplification assumes the reaction is first-order with respect to substrate concentration when \([S] \ll K_m\).
• As substrate concentration increases and approaches \( K_m \), the reaction gets closer to zero-order where the rate no longer depends on the substrate concentration.

Understanding how changes in substrate concentration can influence reaction dynamics is critical in fields like pharmacology and metabolic engineering.

Determining the Rate of Reaction

The rate of reaction in an enzyme-catalyzed process is a measure of how quickly the reactants are converted into products. This rate is influenced by several factors, including substrate concentration, enzyme concentration, and the presence of inhibitors, among others. In the context of the Michaelis-Menten equation, the rate of reaction \( v \) is given by \( v = \frac{k_2 [E][S]}{K_m + [S]} \).

This equation highlights that the reaction rate initially increases rapidly with substrate concentration at low levels but then plateaus as the enzyme reaches its saturation point. When \([S] \) is much larger than \( K_m \), the rate approaches \( V_{max}\), indicating that the enzyme's catalytic sites are almost fully occupied.

• Reaction rates are proportional to substrate availability and enzyme activity, important for understanding metabolic pathways.
• By measuring the reaction rate at various substrate concentrations, \( K_m \) and \( V_{max} \) values can be determined, providing insight into enzyme efficiency and activity.

Overall, mastering these principles enables scientists to predict enzyme behavior in biological systems, which is crucial for designing pharmaceuticals and optimizing industrial processes.