Problem 17 - Analytical Chemistry

Question

When an apple is sliced, it turns brown on exposure to air due to catalysis of the oxidation of phenols in the apple by \(o\) -diphenyl oxidase enzyme. An experiment is performed to determine the Michaelis constant of \(o\) -diphenyl oxidase in which fresh pieces of apple are ground up and then centrifuged to produce a supernatant that will serve as the enzyme source (see www.ultranet. com/ -jkimball/BiologyPages/E/EnzymeKinetics.html). Catechol is used as the substrate. A fixed amount of the enzyme preparation is added to a tube containing \(0.300 \mathrm{mM}\) catechol, and the change in absorbance is measured at \(540 \mathrm{nm}\) at 1 -min intervals for several minutes. The experiment is repeated with three other tubes containing \(0.600,1.20,\) and \(4.80 \mathrm{mM}\) catechol. The following results are obtained \((\mathrm{m} M\) catechol/ \(\Delta A / \mathrm{min}): 0.30 / 0.020 ; 0.60 / 0.035\) \(1.20 / 0.048 ; 4.80 / 0.081 .\) Prepare a spreadsheet to determine \(K_{m}\)

Step-by-Step Solution

Answer

Calculate \( K_m \) using the x-intercept from a Lineweaver-Burk plot.

Step 1: Understand the Michaelis-Menten Equation

The Michaelis-Menten equation is a fundamental model used in biochemistry to describe the rate of enzymatic reactions. The equation is given by \( v = \frac{V_{max} \cdot [S]}{K_m + [S]} \), where \( v \) is the rate of reaction, \( V_{max} \) is the maximum rate, \( [S] \) is the substrate concentration, and \( K_m \) is the Michaelis constant. \( K_m \) is the substrate concentration at which the reaction velocity is half of \( V_{max} \).

Step 2: Collect and Organize the Data

From the problem, we have the following substrate concentrations \([S]\) in mM and corresponding reaction rates \( \Delta A / \text{min} \): \(0.300 / 0.020\); \(0.600 / 0.035\); \(1.20 / 0.048\); \(4.80 / 0.081\). These concentrations and rates will be used to determine \( K_{m} \).

Step 3: Transform the Data Using the Lineweaver-Burk Plot

A Lineweaver-Burk plot is a double-reciprocal plot of \( \frac{1}{v} \) against \( \frac{1}{[S]} \). It linearizes the Michaelis-Menten equation: \( \frac{1}{v} = \frac{K_m}{V_{max} \cdot [S]} + \frac{1}{V_{max}} \). Calculate \( \frac{1}{[S]} \) and \( \frac{1}{v} \) using the data: \(\frac{1}{0.020}\), \(\frac{1}{0.035}\), etc. for reaction rate and \(\frac{1}{0.300}\), \(\frac{1}{0.600}\), etc. for substrate concentration.

Step 4: Create a Spreadsheet

Input the reciprocal values of substrate concentration \((\frac{1}{[S]})\) and reaction rates \((\frac{1}{v})\) into a spreadsheet. Plot these values to create the Lineweaver-Burk plot. The x-axis will represent \( \frac{1}{[S]} \) and the y-axis \( \frac{1}{v} \).

Step 5: Determine the Michaelis Constant \(K_m\)

From the Lineweaver-Burk plot, the y-intercept is \( \frac{1}{V_{max}} \) and the x-intercept is \( -\frac{1}{K_m} \). Use these intercepts to determine \( K_m \): \( K_m = -\text{x-intercept} \). Adjust the spreadsheet's best-fit line and identify the intercepts accurately.

Key Concepts

Enzymatic Reaction

Enzymatic reactions involve the transformation of substrates into products with the help of enzymes, which act as biological catalysts. In the context of the problem, the enzyme, o-diphenyl oxidase, plays a crucial role in catalyzing the oxidation of phenols, leading to the browning of sliced apples. The rate of these reactions is dependent on various factors including the concentration of the enzyme and the substrate involved in the process.

Enzymes function by reducing the activation energy needed for the reaction. This allows the reaction to proceed faster at a lower energy cost. Each enzyme is specific to a particular substrate, which fits into the enzyme's active site—a region where the substrate binds and undergoes a chemical change.

• In this particular study, catechol serves as the substrate, whose concentration influences the reaction rate.
• The amount of enzyme present is constant, so variations in reaction rates are primarily due to changes in the concentration of catechol.
• Understanding these variables provides insight into the enzyme's efficiency and behavior under different conditions.

Lineweaver-Burk Plot

The Lineweaver-Burk plot, also known as the double-reciprocal plot, is a graphical representation used to interpret Michaelis-Menten kinetics. It is derived by taking the reciprocal of both sides of the Michaelis-Menten equation. This transformation linearizes the equation, which helps in determining the kinetic parameters: the Michaelis constant ( K_m ) and maximum reaction velocity ( V_{max} ).

The plot is constructed by plotting (1/v) , the reciprocal of reaction velocity, against (1/[S]) , the reciprocal of substrate concentration. This results in a straight line where:

• The y-intercept represents (1/V_{max}) .
• The x-intercept corresponds to (-1/K_m) .

By analyzing the intercepts on the plot, you can easily extract and calculate both V_{max} and K_m . It's an invaluable tool for visualizing kinetic data, especially when working with enzyme-catalyzed reactions, as it simplifies the estimation of these constants.

Substrate Concentration

Substrate concentration is a critical variable in enzymatic reactions. It refers to the amount of substrate available for the enzyme to act upon, which affects the rate at which products are formed. In a typical enzyme-catalyzed reaction, an increase in substrate concentration leads to an increase in reaction rate. However, this continues only up to a certain point.

As substrate concentration increases, there comes a point where all the active sites of the enzyme molecules are occupied, and the reaction reaches a maximum rate ( V_{max} ). Beyond this saturation point, further increases in substrate concentration will not increase the rate of reaction, as the enzyme is working at its full capacity.

• In the described experiment, various concentrations of catechol (e.g., 0.300 mM, 0.600 mM) were used to observe how they affect the enzymatic activity of o-diphenyl oxidase.
• By comparing these conditions, researchers can estimate the saturating concentrations and the efficiency of an enzyme.
• Understanding the behavior of reactions at different substrate concentrations is vital for determining the Michaelis constant ( K_m ).

Michaelis Constant

The Michaelis constant, denoted as K_m , is a fundamental concept in enzyme kinetics, serving as a measure of substrate concentration required to reach half of the maximal reaction velocity ( V_{max}/2 ). K_m is a crucial parameter that provides insight into the affinity of an enzyme for its substrate. A lower K_m value indicates a high affinity, implying that the enzyme can achieve its half-maximal activity at low substrate concentrations.

Understanding K_m allows scientists and researchers to:

• Determine the efficiency and performance of an enzyme under various conditions.
• Compare different enzymes acting on the same or different substrates.
• Apply this knowledge to optimize conditions in biochemical processes, such as in industrial enzyme applications.

The estimation of K_m from experimental data involves plotting kinetic data using methods like the Lineweaver-Burk plot, where the x-intercept provides the inverse of K_m . This helps refine understanding and control of biological reactions catalyzed by enzymes.