Problem 24 - Biochemistry: Concepts and Connections

Question

In kinetics experiments, the hydrolysis of the substrate sialic acid by neuraminidase appears to obey Michaelis-Menten kinetics. Neuraminidase activity is critical for viral infectivity; thus, this enzyme is the target of much work by pharmaceutical companies to develop a drug to treat influenza virus infection. The drug "Tamiflu" is a competitive inhibitor of neuraminidase. Initial rate data collected at \(\mathrm{pH}=6.15,37^{\circ} \mathrm{C}\) with \(0.021 \mu \mathrm{M}\) neuraminidase and \(25.0 \mu \mathrm{M}\) sialic acid gives a Lineweaver-Burk plot with a slope of \(51.2 \mathrm{~s}\). (a) Recall from Problem 23 that the \(k_{\text {cat }}\) for neuraminidase at \(\mathrm{pH}=6.15\), \(37{ }^{\circ} \mathrm{C}\) is \(26.8 \mathrm{~s}^{-1}\). Calculate \(K_{\mathrm{M}}\) for the hydrolysis of sialic acid. (b) When the reactions in part (a) are repeated in the presence of \(0.040 \mu \mathrm{M}\) of Tamiflu, the slope of the Lineweaver-Burk plot is \(198.8 \mathrm{~s}\). Calculate the value of \(K_{\mathrm{I}}\) for Tamiflu.

Step-by-Step Solution

Answer

(a) \(K_M = 28.8 \mu \mathrm{M}\). (b) \(K_I = 0.0139 \mu \mathrm{M}\).

Step 1: Understand the Michaelis-Menten Equation

The Michaelis-Menten equation is given by \(v = \frac{V_{max}[S]}{K_M + [S]}\), where \(v\) is the initial rate, \([S]\) is the substrate concentration, \(V_{max}\) is the maximum rate, and \(K_M\) is the Michaelis constant. In a Lineweaver-Burk plot (double reciprocal plot), the equation is transformed to \(\frac{1}{v} = \frac{K_M}{V_{max}[S]} + \frac{1}{V_{max}}\), which is a linear form similar to \(y = mx + b\). The slope \(m\) on this plot is \(\frac{K_M}{V_{max}}\).

Step 2: Calculate the Michaelis Constant \(K_M\)

Given in part (a), the slope of the Lineweaver-Burk plot is \(51.2 \mathrm{~s}\) and \(k_{cat} = 26.8 \mathrm{~s^{-1}}\). The slope \( \frac{K_M}{V_{max}} = 51.2 \). We find \(V_{max}\) using \(V_{max} = k_{cat}[E]\). Substitute \([E] = 0.021 \mu \mathrm{M}\) to find \(V_{max} = 26.8 \times 0.021\). Thus, \(V_{max} = 0.5628 \mathrm{~s^{-1}}\). Calculate \(K_M\) by rearranging the slope equation: \(K_M = \text{slope} \times V_{max} = 51.2 \times 0.5628 = 28.8 \mu \mathrm{M}\).

Step 3: Identify Competitive Inhibition Effects

In competitive inhibition, the modified slope in the Lineweaver-Burk plot is \( \frac{K_M (1 + \frac{[I]}{K_I})}{V_{max}}\), where \([I]\) is the inhibitor concentration and \(K_I\) is the inhibition constant. Part (b) provides the slope as \(198.8 \mathrm{~s}\) in presence of \(0.040 \mu \mathrm{M}\) Tamiflu. We need to identify the value of \(K_I\).

Step 4: Calculate the Inhibition Constant \(K_I\)

Using the competitive inhibition formula \(198.8 = \frac{28.8 (1 + \frac{0.040}{K_I})}{0.5628}\), solve for \(K_I\). So \(198.8 \cdot 0.5628 = 28.8 (1 + \frac{0.040}{K_I})\). This simplifies to \(111.8 = 28.8 + \frac{28.8 \times 0.040}{K_I}\). Hence, \(83 = \frac{28.8 \cdot 0.040}{K_I}\). Thus, \(K_I = \frac{28.8 \cdot 0.040}{83} = 0.0139 \mu \mathrm{M}\).

Key Concepts

Lineweaver-Burk Plot

The Lineweaver-Burk plot is a graphical representation used to analyze enzyme kinetics, specifically following Michaelis-Menten kinetics. Often referred to as a double-reciprocal plot, it is a linearization of the Michaelis-Menten equation, making it easier to determine important parameters like the Michaelis constant ( K_M ) and maximum velocity ( V_{max} ). This plot graphs the reciprocal of substrate concentration ( 1/[S] ) on the x-axis against the reciprocal of reaction velocity ( 1/v ) on the y-axis.
This transformation allows for the linear regression of data, offering more accurate measures for parameters that describe enzyme activity.
The slope of the Lineweaver-Burk plot is important, given by K_M/V_{max} , and the y-intercept provides the reciprocal of V_{max} . Additionally, the x-intercept corresponds to -1/K_M .

• Y-intercept = 1/V_{max}
• X-intercept = -1/K_M
• Slope (m) = K_M/V_{max}

Using this plot, we analyzed the effect of a competitive inhibitor, Tamiflu, on neuraminidase activity, determining how the presence of the inhibitor affects substrate affinity measured through changes in slope.

Competitive Inhibition

Competitive inhibition occurs when a molecule similar to the substrate competes with the substrate for binding to the enzyme's active site. This type of inhibition raises the apparent K_M , making it look like the enzyme has a lower affinity for the substrate, while V_{max} remains unchanged. Understanding competitive inhibition is crucial while studying drug actions, especially in pharmaceuticals targeting specific enzymes.
In our exercise, Tamiflu acts as a competitive inhibitor for the enzyme neuraminidase. In the presence of competitive inhibitors, the K_M increases as it requires more substrate to reach half of V_{max} .

• Increased K_M : Apparent K_M increases, implying decreased substrate affinity.
• Unchanged V_{max} : Maximum velocity stays the same.
• Slope changes: K_M(1 + [I]/K_I)/V_{max} represents the altered slope.

The inhibition constant ( K_I ) reveals the inhibitor's potency. A low K_I indicates a strong inhibitor since less is needed to inhibit the enzyme. The analysis of competitive inhibition helped us calculate K_I for Tamiflu, providing insights into the inhibitor's efficacy in restricting neuraminidase.

Neuraminidase Enzymatic Activity

Neuraminidase is an enzyme critical to viral infectivity, found in many pathogens, including the influenza virus. Enzymes like neuraminidase speed up the hydrolysis of sialic acid, influencing processes such as the release of viral particles from host cells, making this enzyme a prime drug target.
Studying neuraminidase activity is vital for developing antiviral drugs like Tamiflu. It targets the enzyme, reducing the virus's ability to spread. By observing its inhibition kinetics, we understand how drugs interfere with the enzyme function.
In our experiment, neuraminidase showed classical Michaelis-Menten kinetics. Its activity was affected by competitive inhibition due to Tamiflu. By determining K_M and V_{max} through the Lineweaver-Burk plot, we could see how effectively the enzyme catalyzes reactions. Analyzing these changes provides crucial information on improving drug efficacy.

• Essential for understanding drug impact.
• Study of inhibition informs drug design.
• Determining K_M and V_{max} aids in understanding enzyme behavior and drug interaction.

Overall, the kinetic analysis gives insight into how antiviral agents like Tamiflu can serve as powerful tools against enzyme-based pathogen spread.