Graphical Kinetic Plots

FIGURE 4.2 Lineweaver-Burk (double-reciprocal) plot of a reaction that follows Michaelis-Menten kinetics. Kinetic parameters are as defined previously. 

Another type of graphical plot used to analyze kinetic data is the Hanes-Woolf plot (Fig. 4.3). In this case, [S]/v is plotted along the y-axis and [S] is plotted along the x-axis. For this plot, the x-intercept = — the y-intercept = m/Finax nd the slope of the best-fit line estimates l/Vmax- The Hanes-Woolf 

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The inhibition analyses were examined differently for free lipase in a batch and immobilised lipase in membrane reactor system. Figure 5.14 shows the kinetics plot for substrate inhibition of the free lipase in the batch system, where [5] is the concentration of (S)-ibuprofen ester in isooctane, and v0 is the initial reaction rate for (S)-ester conversion. The data for immobilised lipase are shown in Figure 5.15 that is, the kinetics plot for substrate inhibition for immobilised lipase in the EMR system. The Hanes-Woolf plots in both systems show similar trends for substrate inhibition. The graphical presentation of rate curves for immobilised lipase shows higher values compared with free enzymes. The value for the... 

Imagine that one wanted to demonstrate the adherence of the kinetic data to the two equations referred to in Problem 1.3. One might do this graphically by plotting the equation of a... 

This same [e] experimental protocol leads to a graphical overlay plot that yields valuable kinetic information if the two experiments described in Table 50.1 are plotted together as reaction rate vs. [2], the two curves will fall on top of one another ( overlay ) over the range of [2] common to both only if the rate is not significantly influenced by changes in the overall catalyst concentration within the cycle, including catalyst activation, deactivation or product inhibition. Overlay in same excess plots, therefore, may be used to confirm catalyst robustness or identify problems such as catalyst deactivation or product inhibition. 

The data for the reactions of potassium cyanide with benzyl halides at 85 C and 25 C are summarized in Tables I-III and graphical representations of these data are shown in Figures 1-3. The reactions carried out at 85 C were followed to 70% completion, while those at 25 C were followed to 50% completion. In general, excellent first-order kinetic plots were obtained. Each point on the graphs represents an average of at least three kinetic determinations. It is interesting to note that in the absence of added water (solid-liquid phase transfer catalysis), the rates of benzyl halide disappearance were more accurately described by zero-order kinetics. 

This transformation suffers from a number of disadvantages. The data are reciprocals of measurements, and small experimental errors can lead to large errors in the graphically determined values of K, , especially at low substrate concentrations. Departures from linearity are also less obvious than on other kinetic plots such as the Eadie-Hofstee and Hanes plots (see reference 7 ). 

Graphical kinetic methods Methods of determining reaction rates from plots of the concentration of a reactant or a product as a function of time. 

It may be evaluated from any portion of the curve, and is proportional to the amount of enzyme. Graphically, a plot of log substrate concentration versus time gives a straight line with first-order reactions, and the negative slope of the line is proportional to the reaction constant. Occasionally it is not possible to use conditions that produce zero- or first-order rates. The rate may be a function of more than one substrate (second or higher order) or it may be influenced by the accumulation of products. The products may act as inhibitors or they may react back to yield the original substrate or other products. An obvious complication is the existence of additional reactions which may involve either substrates or products. When higher order reactions or other complications obscure rate measurements, various empirical devices are employed to establish an assay. The most familiar of these are measurement of initial rates that may tend to approach zero-order kinetics and therefore be... 

After that, the user has to decide how the results should be visualized. It is possible to print the answer in the form of individual values of the desired function, an array, etc. However, the most visual output form is the graphical one. The plotting of the results is provided by the command odeplot from the graphical library plots. Figures 3.11 and 3.12 show a solution of the differential equation set, which describes the kinetics of the first-order reversible reaction B with arbitrary rate constant values. 

This chapter on Rate of Crystallization of Polymers is incomplete for several reasons. Sometimes original data were not available. Where the deconvolution of complex graphical kinetic scenarios was unnecessarily tedious and resulted in questionable accuracy, these data were omitted. However, results are listed for rate plots that could be deciphered and digitized satisfactorily. Original data provided directly by authors are included and gratefully acknowledged by the writer. 

Lineweaver-Burk plot a graphical means for evaluating enzyme kinetics, (p. 638)... 

The following graphical patterns obtained from kinetic experiments have several possible interpretations depending on the nature of the experiment and the variables being plotted. Give at least two possibilities for each. 

The plotting of Dixon plot and its slope re-plot (see 5.9.5.9) is a commonly used graphical method for verification of kinetics mechanisms in a particular enzymatic reaction.9 The proposed kinetic mechanism for the system is valid if the experimental data fit the rate equation given by (5.9.4.4). In this attempt, different sets of experimental data for kinetic resolution of racemic ibuprofen ester by immobilised lipase in EMR were fitted into the rate equation of (5.7.5.6). The Dixon plot is presented in Figure 5.22. 

The temporal reaction heat flow data may be graphically manipulated to reveal the overall second order dependence in a quantitative manner. Reaction heat flow is converted to reaction rate using eq. (1), and the concentration of the limiting substrate 5 may be calculated according to eq. (3). From these calculations we may constract the plot in Figure 50.2b of reaction rate vs. [5]. The reaction is known to be first order in both [5] and [6] these plots reveal the curvature typical of overall second order kinetics. 

In experimental kinetics studies one measures (directly, or indirectly) the concentration of one or more of the reactant and/or product species as a function of time. If these concentrations are plotted against time, smooth curves should be obtained. For constant volume systems the reaction rate may be obtained by graphical or numerical differentiation of the data. For variable volume systems, additional numerical manipulations are necessary, but the process of determining the reaction rate still involves differentiation of some form of the data. For example,... 

Equation 11.6 shows that Km corresponds to the substrate concentration at which the rate has fallen to half the maximum. The kinetic parameters Vmax and Vmax/KM can be obtained by least squares regression of v vs. [S] plots or graphically by linearizing Equation 11.6. 

A linear graphical method for analyzing the initial rate kinetics of enzyme-catalyzed reactions. In the Hanes plot, [A]/v is plotted as a function of [A], where v is the initial rate and [A] is the substrate concentration ". ... 

The proper evaluation and assessment of the calculated or graphically determined values of the kinetic parameters requires the application of statistical analysis . This is also true when looking for possible patterns in the various plots (e.g., parallel lines V5-. intersecting lines). When reporting kinetic values, the error limits should always be provided. Programs are available that statistically evaluates kinetic data. See Statistics A Primer)... 

If a noncompetitive or an uncompetitive inhibitor were present with the substrate at constant ratio, then graphical analysis would suggest that the phenomenon of substrate inhibition is present. If an investigator analyzed the apparent substrate inhibition via a Marmasse plot, wrong estimates of both the K a and K s values would be reported and the investigator would be mislead with respect to the kinetic mechanism. If partial inhibitors or alternative substrates are present in constant ratio, depending on the relative sizes of the Ymax and values,... 

Pourbaix diagrams (Pourbaix, 1963) indicate graphically the conditions of redox potential (Eh) and pH under which different types of corrosion behaviour may be expected. These plots of potential vs. pH indicate the phase and species in equilibrium with iron under various conditions (see Chap. 8). The solid phases indicated are those that are thermodynamically the most stable owing to kinetic factors other phases may be present during the initial stages of the corrosion process. What the different regions show, however, are the predominant oxidation states to be expected. 

Thus, by the Eineweaver-Burk plot the kinetic parameters K, Ky and can be graphically evaluated, as shown in Figure 3.8. 

The results can be presented graphically, as in Fig. 19.14. The plot shows the kinetically dominant type of nucleation as a function of grain size (via Rb), AQB, and yaa/ 0lP. By setting the nucleation rate, J, at a fixed value, a curve such as abode can be plotted to indicate, for given value of L/S, the dominant modes of nucleation at the designated nucleation rate at various values of yao / fat3. 

Figure 6.2 Graphical method of analyzing the kinetics of a reaction obeying equation 6.16. The logarithm of [B] is plotted against time. The rate constant for the slower process is obtained from the slope of the linear region after the faster process has died out. The rate constant for the faster process is obtained by plotting the logarithm of A (the difference between the value of [B] at a particular time and the value of [B] extrapolated back from the linear portion of the plot) against time for the earlier points. The rate constants for this example are 20 and 2 s 1, respectively.

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