Plotting Enzyme Kinetics Data

Because the v versus [S] curve is a hyperbola, it is rather difficult to determine Vmuc and, hence, the [S] that yields 5 (i.e., Km). To facilitate the 

This plot is based on the rearrangement of the Henri-Michaelis-Menten equation into a linear y = mx + fr) form  

The concentrations of substrate chosen to generate the reciprocal plot should be in the neighborhood of the value. If the concentrations chosen are very high relative to the K value, the curve will be essentially horizontal. This will allow to be determined, but the slope of the line will be near zero. Consequently, it will be difficult to determine K accurately. If the substrate concentrations chosen are very low relative to the K value, the curve will intercept both axes too close to the origin to allow either or to be determined accurately. (At very low substrate concentrations, the 

The Lineweaver-Burk reciprocal plot is not the only linear transformation of the basic velocity (or ligand binding) equation. Indeed, under some circumstances one of the other linear plots described below may be more suitable or may yield more reliable estimates of the kinetic constants. For example, the Hanes-Woolf plot of [S]/u versus [S] may be more convenient 

The Lineweaver-Burk equation may be rearranged to yield the linear equation for the Hanes-Woolf plot  

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Pig. (first) The Lineweaver-Burk method of plotting enzyme kinetic data (second) Eadie-Hofstee method of plotting enzyme kinetic data. 

Enzyme kinetics. Data for reactions that follow the Michaelis-Menten equation are sometimes analyzed by a plot of v,/tA]o versus l/[A]o. This treatment is known as an Eadie-Hofstee plot. Following the style of Fig. 4-7b, sketch this function and label its features. 

There are several ways of visualizing enzyme kinetic data, and calculating the values of Km and vmax. Each approach has advantages and disadvantages [20,21]. One way to determine the values of Km and vmax is to plot the reciprocal of the reaction velocity, 1/v, against the reciprocal substrate concentration 1/[S], since... 

For the Michaelis-Menten equation there are algebraic transformations, in addition to the Lineweaver-Burk equation, that yield straight line plots from enzyme kinetic data. One such plot is due to Eadie and Hofstee their equation takes the following form ... 

Enzyme kinetic data of vQ at different substrate concentrations is typically presented as either of two linear plots ... 

Until relatively recently this was the only method that could be used conveniently to fit data by regression. This is the reason why so many classical approaches for evaluating biochemical data depended on linearising data, sometimes by quite complex transformations. The best known examples are the use of the Lineweaver-Burk transformation of the Michaelis-Menten model to derive enzyme kinetic data, and of the Scatchard plot to analyse ligand binding equilibria. These linearisation procedures are generally no longer recommended, or necessary. 

Figure 17.6. Eisenthal-Cornish-Bowdendirect linear plot of enzyme kinetic data fitting the Michaelis-Menten equation.

Fig. 3. p-Glucosidase inhibition shown by Lineweaver-Burk plot (reproduced from [2]). Lineweaver-Burk plot of kinetic data from peak 2 cellobiase ((3-glucosidase) at several product inhibitor levels. This is an example of noncompetitive inhibition where the product is not only completing for binding in the active site but also binding to a secondary site on the enzyme that alters the enzyme catalytic ability... 

Figure 8.15 Lineweaver-Burke plot of enzyme kinetic data in which Mv is plotted against 1/[S).

Although computer software is now readily available to fit enzyme kinetic data to the Michaelis-Menten and related equations, it can be instructive to use simple graphical methods in some cases. The most convenient of these (though not necessarily the most accurate) are based on doublereciprocal methods that convert the hyperbolic rate equations into much simpler linear forms for plotting. 

For each of the four types of inhibition of a Michaelis-Menten enzyme [competitive, Eq. (5.25) noncompetitive and mixed Eq. (5.29) and uncompetitive, Eq. (5.32)], derive the corresponding Lineweaver-Burk equations [Eqs. (5.26), and (5.30), respectively] and draw the characteristic plots that are the basis for the rapid visnal identification of which type of inhibition apphes when analyzing enzyme kinetic data. 

The steady-state kinetic treatment of random reactions is complex and gives rise to rate equations of higher order in substrate and product terms. For kinetic treatment of random reactions that display the Michaelis-Menten (i.e. hyperbolic velocity-substrate relationship) or linear (linearly transformed kinetic plots) kinetic behavior, the quasi-equilibrium assumption is commonly made to analyze enzyme kinetic data. 

Although graphical analysis is a quick and useful way to visualize enzyme kinetic data, for any definitive work, the data must be subjected to statistical analysis so that the precision of the kinetic constants can be evaluated. However, there are good reasons why plotting methods are essential. The human eye is much less easily deceived than any computer program and is capable of detecting unexpected behavior even if nothing currently available is found in the literature. 

Enzymes (see Chapter 23 on the accompanying website) show saturation kinetics. At high substrate concentrations the enzyme is said to be saturated with respect to substrate. At low concentrations of substrate the enzyme activity is first order with respect to substrate but becomes almost zero order with respect to substrate concentration at higher concentrations. Plot the following enzyme kinetic data and use a logarithmic function to plot a hyperbolic curve of substrate concentration on the x-axis and enzyme rate or activity on they-axis. Extrapolate the curve to estimate the maximum rate of enzyme activity. 

Hill plot A graphical procedure used to fit experimental enzyme kinetic data to an S-shaped or sigmoidal curve that deviates firom Michaelis-Menten kinetics. It involves plotting log V/(V-v) as a function of the substrate concentration, S, where Vis the maximum velocity, and v is the observed velocity. The straight line has a gradient that indicates the number of interacting sites on the enzyme or enzyme complex. 

The three reversible mechanisms for enzyme inhibition are distinguished by observing how changing the inhibitor s concentration affects the relationship between the rate of reaction and the concentration of substrate. As shown in figure 13.13, when kinetic data are displayed as a Lineweaver-Burk plot, it is possible to determine which mechanism is in effect. 

Lineweaver-Burk plot Method of analyzing kinetic data (growth rates of enzyme catalyzed reactions) in linear form using a double reciprocal plot of rate versus substrate concentration. 

In most kinetic investigations, one assumes the enzyme remains stable over the course of the measurement. When this is the case, corrective measures must be taken to obtain valid kinetic data. A useful test for any enzyme system is to plot enzyme activity versus time. This is readily accomplished by using a standardized assay (usually at optimal or saturating substrate concentrations) to measure the enzyme s specific activity periodically during the course of some experiment. This approach may fail to detect a reduction in activity characterized by lower affinity for substrate however, use of a subsaturating substrate concentration in a time-course study will reveal this behavior. 

The main plots used in enzyme kinetics and receptor binding studies are the Scatchard plot, the Lineweaver-Burk plot, and the linearization for estimation of the Hill coefficient. This chapter gives a short survey of these transformations of enzyme kinetics or receptor binding data. 

Decarboxylase reaction Kinetic constants The optimum pH of the decarboxylase reaction was determined with the natural substrates of both enzymes, pyruvate (PDC) and benzoylformate (BFD). Both enzymes show a pH optimum at pH 6.0-6.5 for the decarboxylation reaction [4, 5] and investigation of the kinetic parameters gave hyperbolic v/[S] plots. The kinetic constants are given in Table 2.2.3.1. The catalytic activity of both enzymes increases with the temperature up to about 60 °C. From these data activation energies of 34 kj moT (PDC) and 38 kJ mol (BFD) were calculated using the Arrhenius equation [4, 6-8]. 

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