Question

The rate of reaction can be determined by measuring the change in optical rotation of the sample as a function of time if a reactant or product is chiral. This technique is especially useful for kinetic studies of enzyme catalysis involving sugars. For example, the enzyme invertase catalyzes the hydrolysis of sucrose, an optically active sugar. The initial reaction rates as a function of sucrose concentration are as follows: $$\begin{array}{cc} \text { [Sucrose] }_{\mathbf{0}}(\mathbf{M}) & \mathbf{R}_{\mathbf{0}}\left(\mathbf{M} \mathbf{~ s}^{-\mathbf{1}}\right) \\ \hline 0.029 & 0.182 \\ 0.059 & 0.266 \\ 0.088 & 0.310 \\ 0.117 & 0.330 \\ 0.175 & 0.362 \\ 0.234 & 0.361 \end{array}$$ Use these data to determine the Michaelis constant for invertase.

Short answer

The short answer for this problem involves calculating the Michaelis-Menten constant (K_m) for invertase using the given data. First, linearize the Michaelis-Menten equation using the Lineweaver-Burk plot. Calculate the reciprocal values for initial reaction rate (1/R₀) and initial sucrose concentration (1/[S₀]). Plot these values, and find the slope and y-intercept of the resulting linear regression line. Finally, calculate K_m using the formula \(K_m = slope \times \frac{1}{y-intercept}\).

Step-by-step solution

Understand the Michaelis-Menten equation

The Michaelis-Menten equation is given by: \(R_0 = \frac{V_{max} \times [S_{0}]}{K_m + [S_{0}]}\) Here, R₀ is the initial reaction rate, [S₀] is the initial concentration of the substrate (sucrose), V_max is the maximum rate of the reaction, and K_m is the Michaelis constant. We need to fit the given data to this equation to determine K_m.

Linearize the Michaelis-Menten equation using the Lineweaver-Burk plot

A Lineweaver-Burk plot is a double reciprocal plot of the Michaelis-Menten equation, which allows us to linearize the equation and easily determine K_m and V_max. The linear form of the equation is given by: \(\frac{1}{R_0} = \frac{1}{V_{max}} + \frac{K_m}{V_{max} \times [S_{0}]}\) Here, \(\frac{1}{R_0}\) is the y-axis, and \(\frac{1}{[S_{0}]}\) is the x-axis. The slope of the line will be \(\frac{K_m}{V_{max}}\), and the y-intercept will be \(\frac{1}{V_{max}}\).

Calculate the reciprocal values for initial reaction rate and sucrose concentration

Use the given data to calculate the reciprocal values for the initial reaction rate (1/R₀) and the initial sucrose concentration (1/[S₀]). $$\begin{array}{ccc} \text { [S₀]^{-1} (M^{-1}) } & \text { R₀^{-1} (M^{-1}s) } \\\ \hline 1/0.029 & 1/0.182 \\\ 1/0.059 & 1/0.266 \\\ 1/0.088 & 1/0.310 \\\ 1/0.117 & 1/0.330 \\\ 1/0.175 & 1/0.362 \\\ 1/0.234 & 1/0.361 \end{array}$$

Plot the Lineweaver-Burk plot and find the slope and y-intercept

Plot the calculated reciprocal values of initial reaction rate (1/R₀) against the reciprocal values of initial sucrose concentration (1/[S₀]). The slope of the resulting linear regression line will be \(\frac{K_m}{V_{max}}\) and the y-intercept will be \(\frac{1}{V_{max}}\).

Calculate the Michaelis constant (K_m) and maximum rate (V_max) from the Lineweaver-Burk plot

Use the slope and y-intercept obtained in step 4 to calculate the Michaelis constant (K_m) and the maximum rate (V_max). \(\frac{K_m}{V_{max}} = slope\) \(K_m = slope \times V_{max}\) \(\frac{1}{V_{max}} = y-intercept\) \(V_{max} = \frac{1}{y-intercept}\) Now, plug the value of V_max back into the equation for K_m: \(K_m = slope \times \frac{1}{y-intercept}\) The resulting value for K_m is the Michaelis constant for invertase.

Key concepts

Understanding Enzyme Kinetics

When we talk about enzyme kinetics, we are discussing the study of how enzymes bind to substrates and turn them into products.

Enzymes are biological catalysts that increase the rate of chemical reactions without being consumed or altered in the process. Enzyme kinetics is crucial in understanding biological processes and can inform us about the mechanisms of enzyme activity, their efficiency, and how they're affected by different conditions and inhibitors.

The classic model used to describe enzyme kinetics is the Michaelis-Menten equation. This model assumes that the initial reaction speed is most rapid and then slows down as the substrate concentration decreases or as the product accumulates. It is essential to grasp that in a Michaelis-Menten scenario, one enzyme molecule can only bind to one substrate molecule at a time, leading to a saturation point known as V_max, where increasing substrate concentration does not increase the reaction rate.

Explaining Reaction Rate

The reaction rate is a measure of how quickly a reactant is converted into a product by an enzyme. In our example, we consider the initial reaction rate, denoted as R₀, which tells us how fast the reaction is proceeding right when it starts.

Reaction rates can be influenced by various factors, including enzyme concentration, substrate concentration, temperature, and pH. In the context of enzyme kinetics, examining the relationship between substrate concentration and reaction rate is fundamental. As we increase the amount of substrate, the reaction rate also increases until all enzyme molecules are busy catalyzing the reaction, and we reach V_max, the maximum rate.

Deciphering the Lineweaver-Burk Plot

A Lineweaver-Burk plot is a graphical representation that aims to linearize the relationship expressed by the Michaelis-Menten equation, simplifying the determination of two critical parameters: V_max and K_m (the Michaelis constant).

In our exercise, by plotting the reciprocal of the reaction rate (1/R₀) against the reciprocal of substrate concentration (1/[S₀]), we create a straight line which makes it easier to interpret these parameters. The plot's slope gives the ratio of K_m/V_max, while the y-intercept is equivalent to 1/V_max. This linear transformation is a widely used method to extract K_m and V_max from reaction rate data.

Invertase Enzyme Catalysis

The invertase enzyme provides a fascinating example of enzyme catalysis. It specifically catalyzes the hydrolysis of sucrose into glucose and fructose.

Enzyme catalysis involves the enzyme binding to a specific substrate—in this case, sucrose— and converting it into a product faster than would occur without the enzyme. Invertase works on sucrose by breaking the glycosidic bond between glucose and fructose.

The efficiency of invertase and its characteristics, such as the Michaelis constant (K_m), are determined by the reaction's kinetics. K_m offers a measure of the affinity of the enzyme for its substrate — a lower K_m signifies a higher affinity, meaning that invertase can function effectively even at low sucrose concentrations. This property is vital when applying invertase in various industries, such as food manufacturing, where it's used to make syrups and other sweeteners.