Enzyme kinetics Lineweaver-Burke transformation

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. / Enzyme Kinetics Results from the Lineweaver-Burk transformation.

The hyperbohc saturation function of the form ax i [x + b) often arises in biophysical and biochemical appHcations. It is more obvious that this represents a hyperbola, if it is written in a double-reciprocal form, as in a Lineweaver-Burk transformation of the MichaeHs-Menten enzyme kinetics that employs this type of function. Other contexts where this function appears are monomolecular photochemical kinetics and visual physiology. In the present context of action spectroscopy, x would stand for the fluence or, in some cases, the fluence rate. When x = b, the function is at the half-maximum level that is often chosen to be the criterion response. So, when one performs least-squares fits using such a function, the parameter b is the estimate of the fluence needed for the criterion response, and the effectiveness (action spectrum ordinate) is just h -... 

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. 

Graphical transformation of the representation of enzyme kinetics is useful as the value of V max is impossible to obtain directly from practical measurements. A series of graphical transformations/linearisations may be used to overcome this problem. Lineweaver and Burk (see reference(,7)) simply inverted the Michaelis-Menten equation (equation 5.10). Thus ... 

Figure 22 Examples of enzyme kinetic plots used for determination of Km and Vmax for a normal and an allosteric enzyme Direct plot [(substrate) vs. initial rate of product formation] and various transformations of the direct plot (i.e., Eadie-Hofstee, Lineweaver-Burk, and/or Hill plots) are depicted for an enzyme exhibiting traditional Michaelis-Menten kinetics (coumarin 7-hydroxylation by CYP2A6) and one exhibiting allosteric substrate activation (testosterone 6(3-hydroxylation by CYP3A4/5). The latter exhibits an S-shaped direct plot and a hook -shaped Eadie-Hofstee plot such plots are frequently observed with CYP3A4 substrates. Km and Vmax are Michaelis-Menten kinetic constants for enzymes. K is a constant that incorporates the interaction with the two (or more) binding sites but that is not equal to the substrate concentration that results in half-maximal velocity, and the symbol n (the Hill coefficient) theoretically refers to the number of binding sites. See the sec. III.C.3 for additional details.

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 ... 

To make an appropriate assessment of the pattern of inhibition, one need only compare the pattern of reaction velocity versus [S] observed relative to the pattern predicted from an application of the hyperbolic kinetics model. This requires making an estimate of V ax and from the data available. Transforming the original data to a Lineweaver-Burke plot (despite the aforementioned limitations) indicates that only four data points (at low [S]) can be used to estimate Vmax and Km (as 3.58 units and 0.48 mM, respectively. Fig. 14.10). The predicted (uninhibited) behavior of the enzyme activity can now be calculated by applying the rectangular hyperbola [Eq. (14.5)] (yielding the upper curve in Fig. 14.11), and it becomes clear that inhibition was obvious at [S] <1 mM. The degree of inhibition is expressed appropriately as the difference between observed and predicted activity at any [S] value, if one makes interpretations within the context of the Michaelis-Menten model. 

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