Question

The following data were obtained for an enzyme-catalyzed reaction. Determine \(\mathrm{V}_{\max }\) and \(K_{m}\) by inspection. Plot the data using the Eadie- Hofstee method and determine these constants graphically. Explain the discrepancy in your two determinations. The initial rate data for the enzyme- catalyzed reaction are as follows: \begin{tabular}{cc} \hline\({[\mathrm{S}] }\) \(\mathrm{mol} / 1\) & \(v\) \(\mu \mathrm{mol} / \mathrm{min}\) \\ \hline \(5.0 \times 10^{-4}\) & 125 \\ \(2.0 \times 10^{-4}\) & 125 \\ \(6.0 \times 10^{-5}\) & 121 \\ \(4.0 \times 10^{-5}\) & 111 \\ \(3.0 \times 10^{-5}\) & \(96.5\) \\ \(2.0 \times 10^{-5}\) & \(62.5\) \\ \(1.6 \times 10^{-5}\) & \(42.7\) \\ \(1.0 \times 10^{-5}\) & \(13.9\) \\ \(8.0 \times 10^{-6}\) & \(7.50\) \\ \hline \end{tabular} Do these data fit into Michaelis-Menten kinetics? If not, what kind of rate expression would you suggest? Use graphical methods.

Short answer

Graphically determine \( V_{\text{max}} \) and \( K_m \) from Eadie-Hofstee plot. Compare with values by inspection to explain discrepancies. Assess Michaelis-Menten fit.

Step-by-step solution

Arrange Data for Plotting

Organize the provided data in a table format, listing substrate concentration \( [\text{S}] \) and initial reaction rate \( v \).

Calculate v/[S]

For each data point, calculate \[ \frac{v}{[\text{S}]} \].

Plot Eadie-Hofstee Diagram

Plot \( v \) on the y-axis and \( \frac{v}{[\text{S}]} - v \) on the x-axis. This will help in determining \( V_{max} \) and \( K_m \) graphically.

Determine Vmax and Km Graphically

Using the Eadie-Hofstee plot, determine the vertical intercept (\( V_{\text{max}} \)) and the slope (\( -K_m \)).

Discuss Discrepancy

Compare the \( V_{max} \) and \( K_m \) values obtained by inspection (Step 1) and by the Eadie-Hofstee method (Step 4). Explain any discrepancies.

Determine Fit to Michaelis-Menten Kinetics

Assess if the data fit the Michaelis-Menten kinetics based on the values obtained and their graphical representation.

Suggest Alternative Rate Expression

If the data do not fit Michaelis-Menten kinetics, suggest an alternative rate expression based on the graphical representation and calculated values.

Key concepts

Enzyme Kinetics

Enzyme kinetics is the study of how fast enzymatic reactions occur. It helps us understand how enzymes work and the factors influencing their efficiency. When studying enzyme kinetics, scientists look at reaction rates, substrate concentrations, and how these quantities change. This knowledge allows us to determine important constants like the maximum reaction rate (\( V_{\max} \)) and the Michaelis constant (\( K_m \)). These constants are crucial for understanding enzyme behavior and designing experiments.

Michaelis-Menten Equation

The Michaelis-Menten equation is a core concept in enzyme kinetics. It describes how the reaction rate (\( v \)) depends on the substrate concentration (\( [S] \)). The equation is written as: \[ v = \frac{V_{\max} [S]}{K_m + [S]} \] Here, \( V_{\max} \) is the maximum rate of the reaction, and \( K_m \) is the substrate concentration at half of \( V_{\max} \). This equation assumes that enzyme-substrate complex formation is a reversible process and that the breakdown of this complex to release the product is the rate-limiting step. The Michaelis-Menten equation is pivotal for enzyme kinetics analysis, as it allows us to understand how changes in substrate concentration affect the reaction rate.

Substrate Concentration

Substrate concentration (\( [S] \)) is a key determinant in enzymatic reactions. It refers to the amount of substrate present in the reaction mixture. By varying \( [S] \), we can observe how it influences the initial reaction rate. Typically, at low substrate concentrations, the reaction rate increases rapidly as \( [S] \) rises. However, at higher substrate concentrations, the reaction rate approaches a maximum (\( V_{\max} \)), where increases in \( [S] \) have little effect. This phenomenon is explained by the Michaelis-Menten model, where enzyme molecules become saturated with substrate, and additional substrate cannot further increase the rate.

Initial Reaction Rate

The initial reaction rate (\( v_0 \)) is the rate of reaction observed at the very beginning, typically measured when the substrate is first added. It is crucial because it reflects the enzyme's activity without any complications from product accumulation or enzyme deactivation. The initial reaction rate is used to derive key kinetic parameters such as \( V_{\max} \) and \( K_m \). By plotting the initial reaction rate against substrate concentration, scientists can analyze the enzyme's catalytic efficiency and affinity for the substrate.

Vmax and Km Determination

Determining \( V_{\max} \) and \( K_m \) is essential for characterizing enzyme kinetics. \( V_{\max} \) represents the maximum rate achieved by the enzyme when the substrate concentration is high enough to saturate the enzyme. \( K_m \) is the substrate concentration at which the reaction rate is half of \( V_{\max} \).
\ To find these values, scientists often use a method called the Eadie-Hofstee plot. This involves plotting \( v \) (initial reaction rate) on the y-axis against \( \frac{v}{[S]} \) on the x-axis. The vertical intercept of this plot gives \( V_{\max} \), while the slope corresponds to \( -K_m \). This graphical method helps identify these crucial constants and understand their role in enzyme kinetics.