Question

For an enzyme that displays Michaelis-Menten kinetics, what is the reaction velocity, \(\left.V \text { (as a percentage of } V_{\max }\right)\), observed at the following values? (a) \([\mathrm{S}]=K_{\mathrm{M}}\) (b) \([\mathrm{S}]=0.5 K_{\mathrm{M}}\) (c) \([\mathrm{S}]=0.1 K_{\mathrm{M}}\) (d) \([\mathrm{S}]=2 K_{\mathrm{M}}\) \((\mathrm{e})[\mathrm{S}]=10 K_{\mathrm{Y}}\)

Short answer

a) 50%, b) 33.3%, c) 9.09%, d) 66.7%, e) 90.9%

Step-by-step solution

Understand the Michaelis-Menten Equation

The Michaelis-Menten equation is given by: \[ V = V_{\max} \left( \frac{[S]}{K_M + [S]} \right) \] Here, \( V \) is the reaction velocity, \( V_{\max} \) is the maximum reaction velocity, \( [S] \) is the substrate concentration, and \( K_M \) is the Michaelis constant.

Calculate the Reaction Velocity for \( [S] = K_M \)

When \([S] = K_M\), substitute \( [S] = K_M \) into the equation: \[ V = V_{\max} \left( \frac{K_M}{K_M + K_M} \right) = V_{\max} \left( \frac{K_M}{2K_M} \right) = V_{\max} \left( \frac{1}{2} \right) \] So, \( V = 0.5 V_{\max} \). The reaction velocity is 50% of \( V_{\max} \).

Calculate the Reaction Velocity for \( [S] = 0.5 K_M \)

When \([S] = 0.5 K_M\), substitute \( [S] = 0.5 K_M \) into the equation: \[ V = V_{\max} \left( \frac{0.5 K_M}{K_M + 0.5 K_M} \right) = V_{\max} \left( \frac{0.5 K_M}{1.5 K_M} \right) = V_{\max} \left( \frac{0.5}{1.5} \right) = V_{\max} \left( \frac{1}{3} \right) \] So, \( V = 0.333 V_{\max} \). The reaction velocity is approximately 33.3% of \( V_{\max} \).

Calculate the Reaction Velocity for \( [S] = 0.1 K_M \)

When \([S] = 0.1 K_M\), substitute \( [S] = 0.1 K_M \) into the equation: \[ V = V_{\max} \left( \frac{0.1 K_M}{K_M + 0.1 K_M} \right) = V_{\max} \left( \frac{0.1 K_M}{1.1 K_M} \right) = V_{\max} \left( \frac{0.1}{1.1} \right) \] So, \( V \ approx V_{\max} \left( \frac{1}{11} \right) \). The reaction velocity is approximately 9.09% of \( V_{\max} \).

Calculate the Reaction Velocity for \( [S] = 2 K_M \)

When \([S] = 2 K_M\), substitute \( [S] = 2 K_M \) into the equation: \[ V = V_{\max} \left( \frac{2 K_M}{K_M + 2 K_M} \right) = V_{\max} \left( \frac{2 K_M}{3 K_M} \right) = V_{\max} \left( \frac{2}{3} \right) \] So, \( V \ approx 0.667 V_{\max} \). The reaction velocity is approximately 66.7% of \( V_{\max} \).

Calculate the Reaction Velocity for \( [S] = 10 K_M \)

When \([S] = 10 K_M\), substitute \( [S] = 10 K_M \) into the equation: \[ V = V_{\max} \left( \frac{10 K_M}{K_M + 10 K_M} \right) = V_{\max} \left( \frac{10 K_M}{11 K_M} \right) = V_{\max} \left( \frac{10}{11} \right) \] So, \( V \ approx 0.909 V_{\max} \). The reaction velocity is approximately 90.9% of \( V_{\max} \).

Key concepts

enzyme kinetics

Enzyme kinetics is a branch of biochemistry that studies the rates at which enzyme-catalyzed reactions occur. Enzymes are proteins that speed up chemical reactions in cells, making them essential to life. Understanding how enzymes work can help in fields like medicine, agriculture, and bioengineering.
The main goal in enzyme kinetics is to understand how different factors affect the activity of an enzyme. These factors include substrate concentration, enzyme concentration, pH, and temperature.
The Michaelis-Menten model is a widely used model for studying enzyme kinetics. It helps to describe how the reaction velocity changes with differing substrate concentrations.

reaction velocity

Reaction velocity, often referred to as the rate of reaction, quantifies how quickly a reaction proceeds. For enzyme-catalyzed reactions, it is usually measured as the amount of product formed per unit of time.
The Michaelis-Menten equation gives us a way to calculate this velocity: \( V = V_{\text{max}} \left( \frac{[S]}{K_M + [S]} \right) \). Here, \( V \) is the reaction velocity, \( V_{\text{max}} \) is the maximum velocity, \( [S] \) is the substrate concentration, and \( K_M \) is the Michaelis constant.
Reaction velocity is crucial for understanding how efficient an enzyme is under different conditions. Higher velocities indicate that the enzyme is more active, while lower velocities suggest reduced activity. By studying reaction velocity, scientists can optimize conditions for enzyme activity.

Michaelis constant

The Michaelis constant (\( K_M \)) is a key parameter in enzyme kinetics. It represents the substrate concentration at which the reaction velocity is half its maximum value (\( V_{\text{max}}/2 \)).
Mathematically, when \( [S] = K_M \, V = \frac{V_{\text{max}}}{2} \). This value helps us understand how an enzyme interacts with its substrate. A low \( K_M \) means the enzyme binds tightly to the substrate, requiring less substrate to become saturated. A high \( K_M \) implies a weaker binding affinity, needing a higher substrate concentration to achieve the same rate.
The Michaelis constant is unique for each enzyme-substrate pair and can be influenced by environmental factors like pH and temperature. Knowing the \( K_M \) helps in designing drugs and industrial processes where enzyme activity plays a crucial role.