Calculating Km: Videos & Practice Problems

The Michaelis constant, denoted as K_m, is a crucial parameter in enzyme kinetics that reflects the substrate concentration at which the reaction velocity is half of the maximum velocity (V_max). Understanding how to calculate K_m is essential for analyzing enzyme behavior and efficiency. There are two primary methods for calculating K_m, and this summary focuses on the first method, which involves algebraic rearrangement of the Michaelis-Menten equation.

The Michaelis-Menten equation is expressed as:

V = \frac{V_{max} \cdot [S]}{K_m + [S]}

where [S] is the substrate concentration. To isolate K_m, we can rearrange this equation. First, we multiply both sides by the denominator to eliminate it:

V \cdot (K_m + [S]) = V_{max} \cdot [S]

Next, we distribute V:

V \cdot K_m + V \cdot [S] = V_{max} \cdot [S]

To isolate K_m, we can rearrange the equation by moving terms involving [S] to one side:

V \cdot K_m = V_{max} \cdot [S] - V \cdot [S]

Factoring out [S] gives us:

V \cdot K_m = [S] \cdot (V_{max} - V)

Now, dividing both sides by V allows us to express K_m as:

K_m = \frac{[S] \cdot (V_{max} - V)}{V}

To further simplify, we can recognize that when the initial reaction velocity (V) is half of V_max, the equation becomes:

K_m = [S] \cdot \left(\frac{V_{max}}{\frac{1}{2} V_{max}} - 1\right)

This simplifies to:

K_m = [S] \cdot (2 - 1) = [S]

This indicates that K_m is equal to the substrate concentration at which the reaction velocity is half of V_max. Thus, K_m serves as a vital indicator of enzyme affinity for its substrate, with lower values suggesting higher affinity.

In summary, the first method for calculating the Michaelis constant involves algebraic manipulation of the Michaelis-Menten equation, leading to the conclusion that K_m is the substrate concentration at which the initial reaction velocity is half of the maximum velocity. This foundational understanding sets the stage for exploring additional methods of calculating K_m in subsequent discussions.

In enzyme kinetics, the Michaelis constant, denoted as \( K_m \), is a crucial parameter that helps characterize the behavior of enzymes. It can be calculated using different methods, one of which involves algebraically rearranging the Michaelis-Menten equation. Another approach to determine \( K_m \) is through the relationship between rate constants associated with enzyme-substrate interactions.

The \( K_m \) value can be expressed as the ratio of the sum of the dissociation rate constants to the association rate constant. Specifically, it is defined as:

\[K_m = \frac{k_{-1} + k_2}{k_1}\]

In this equation, \( k_{-1} \) represents the rate constant for the dissociation of the enzyme-substrate complex back to free enzyme and substrate, while \( k_2 \) is the rate constant for the conversion of the enzyme-substrate complex to product. The association rate constant \( k_1 \) describes the formation of the enzyme-substrate complex from free enzyme and substrate.

At the beginning of an enzyme-catalyzed reaction, the initial velocity (\( v_0 \)) is observed, and the dissociation of the enzyme-substrate complex can occur in two ways: backward via \( k_{-1} \) or forward via \( k_2 \). This dynamic illustrates the balance between the formation and dissociation of the enzyme-substrate complex.

Additionally, \( K_m \) can also be related to the concentrations of the free enzyme, free substrate, and the enzyme-substrate complex. This relationship can be expressed as:

\[K_m = \frac{[E][S]}{[ES]}\]

Here, \([E]\) and \([S]\) represent the concentrations of free enzyme and free substrate, respectively, while \([ES]\) is the concentration of the enzyme-substrate complex. This formulation emphasizes that a small \( K_m \) indicates a lower concentration of free enzyme and substrate, suggesting a higher formation of the enzyme-substrate complex.

In summary, understanding the calculation and implications of \( K_m \) is essential for analyzing enzyme efficiency and substrate affinity. A small \( K_m \) value signifies strong binding between the enzyme and substrate, leading to a greater concentration of the enzyme-substrate complex, which is vital for effective catalysis.

In enzyme kinetics, the Michaelis constant (K_m) is a crucial parameter that reflects the affinity of an enzyme for its substrate. It can be determined using the relationship between various rate constants in a simple enzyme-catalyzed reaction. When the initial reaction velocity (V_max) and substrate concentrations are not provided, K_m can still be calculated using the rate constants associated with the enzyme-substrate complex.

The K_m can be expressed as:

K_m = (k_-1 + k₂) / k₁

In this equation, k_-1 represents the rate constant for the dissociation of the enzyme-substrate complex back into free enzyme and substrate, while k₂ is the rate constant for the conversion of the enzyme-substrate complex into product. The term k₁ is the rate constant for the formation of the enzyme-substrate complex from free enzyme and substrate.

For the calculation, if we have the following values:

• k_-1 = 1 × 10³ s^-1
• k₂ = 5 × 10³ s^-1
• k₁ = 2 × 10⁸ M^-1s^-1

We can substitute these values into the K_m equation:

K_m = (1 × 10³ + 5 × 10³) / (2 × 10⁸)

Calculating the numerator gives us:

K_m = (6 × 10³) / (2 × 10⁸)

Performing the division results in:

K_m = 3 × 10^-5 M

This calculated K_m value indicates a relatively high affinity of the enzyme for its substrate, as lower K_m values suggest stronger binding. In this case, the correct answer for the K_m based on the provided options is 3 × 10^-5 M.