chapter 3 - Bentham Science

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108 Applications of Spreadsheets in Education The Amazing Power of a Simple Tool Sinex and Gage Fig. (6.1): Tabs on Excelet. handled by the “just add data” aspect of the Excelet, where all graphs and calculations are already set up. Students need to know the basics of mathematical modeling (e.g., regression, goodness-offit, and so on) as described in [6]. 6.2 The Kinetics of Enzyme Reactions Excelet To get the most out of this discussion, the reader needs to open the interactive Excel spreadsheet and accompanying activity. First we review some basic information from chemical kinetics and link to other Excelets that review the basic concepts even further. Understanding this information is required if students are going to grasp enzyme kinetics. We are investigating the classic reaction with the rate constants as given by S+E k1 −−−−⇀ ↽−−−− k2 ES kcat −−−−−→ E+ P (6.1) where S is the substrate, E is the enzyme, ES is the enzyme-substrate complex, and P is the product. The calculations are similar to concepts outlined in [7] and can demonstrate the validity of the steady state approximation. The similarity of enzyme kinetics to homogeneous catalysis is shown in [8], wherein some elucidating points on the mechanisms occurring in enzyme kinetics are illustrated. This Excelet uses manipulatable variables that feed into the Michaelis-Menten equation to calculate the initial velocity of the reaction, v0, on the kinetics plot tab according to where v0 = vmax(S) KM+(S) vmax = kcat(E)0 and KM = k2+ kcat The kinetics plot tab illustrates a variety of enzymes with substrates. We are trying to get students to see how these substrate/enzyme variables influence the graph. How vmax and KM are determined from the graph is also illustrated. Students then explore the behavior of the concentrations of the substrate, enzyme, enzymesubstrate complex, and the product over time on the S, E, & P over time tab by changing the various rate constants (k1, k2, and kcat) and initial concentrations of substrate, (S)0, and enzyme, (E)0. This part of the Excelet is a system’s dynamic model that uses difference equations to calculate concentrations over time. The actual mathematical equations (differential equations converted to k1 (6.2) (6.3)
Enzyme Kinetics for Novice Learners Applications of Spreadsheets in Education The Amazing Power of a Simple Tool 109 difference equations or see [7]) are given on this tab of the Excelet if you scroll down below the graph, a sample of which is shown in Fig. (6.2). Fig. (6.2): Concentration over time plot. A similar plot is employed on the Initial Rate S tab that shows how the initial rate, which is a common measurement in enzyme experiments, is determined from the graph. 6.3 Transformed Data If the enzyme kinetics data follows the form of the Michaelis-Menten equation then mathematically it is a rectangular hyperbola and can be fit by non-linear regression techniques or the data can be transformed to a linear plot and fit using linear regression. We look at the common transformations (Table 6.1) used by biochemists [9] to fit the data to find the various parameters: vmax, KM, and kcat (see transformed data tab). Historically, the data were transformed to produce linear plots, which in the pre-technology age was the only way to analyze the data (using rulers and graph paper). Analysis improved with the addition of linear regression, especially employing computational technology. A big advantage of linear plots is the ability to spot behavior that is non-linear. Many errors that occur when using the transformed data are pointed out in [10]. The three plots corresponding to the transformations in Table 6.1 are shown in Fig. (6.3) with the lowest concentration value as a green point. One might notice how the points are not regularly spaced, especially on the Lineweaver-Burk and Eadie-Hofstee plots. We explore these three methods simultaneously and their sensitivity to random and systematic error, both constant and proportional. If students need an introduction to investigating error, see [11, 12].
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