Lineweaver-Burke transformation

As we have discussed in Chapter 8, this is a typical transformably linear system. Using the well-known Lineweaver-Burk transformation. Equation 17.2 becomes... 

Engine Kinetics Results from the Lineweaver-Burk transformation. 

The useful thing about the Lineweaver-Burk transform (or double reciprocal) is that the y intercept is related to the first-order rate constant for decomposition of the ES complex to E + P (7ccat or Vmax) and is equal to the rate observed with all of the enzyme in the ES complex. The slope, in contrast, is equal to the velocity when the predominant form of the enzyme is the free enzyme, E (free meaning unencumbered rather than cheap). 

Inhibition experiments are performed by varying the concentration of substrate around the Km just as you would in an experiment to determine the Km and Vmax, except that the experiment is repeated at several different concentrations of an inhibitor. On a Lineweaver-Burk transformation (1/v vs. 1/[S]), each different inhibitor concentration will be represented as a different straight line (Fig. 8-6). The pattern that the lines make tells you the kind of inhibition. There are three possibilities (1) The inhibitor can affect only the slopes of the plot (competitive), (2) the inhibitor can affect only the y intercepts of the plot (uncompetitive), or (3) the inhibitor can affect both the slopes and the intercepts (noncompetitive). Plots are plots, and what s really important is not the pattern on a piece of paper but what the pattern tells us about the behavior of the enzyme. 

To make it easier to measure mathematically a Lineweaver-Burke transformation can be performed by taking reciprocals of both sides of the initial equation. 

GRAPHICAL REPRESENTATION. The above expression represents the equation of a hyperbola (i.e., f(x) = axl(b + x) where a and b are constants) and a plot of the initial velocity as a function of [S] will result in a rectangular hyperbola. Another way for representing the Michaelis-Menten equation is by using the doublereciprocal (or, Lineweaver-Burk ) transformation ... 

The Hanes plot also starts with the Lineweaver-Burk transformation (equation 5.15) of the Michaelis-Menten equation which in this instance is multiplied by S throughout on simplification this yields ... 

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. 

The rate of this type of inhibition is described by Equation 5 the Lineweaver-Burk transformation of this rate equation is given... 

The Lineweaver-Burk-transformation on papaverine for instance yields the calibration shown in Fig. 16. The scattering of the original data is distributed normally, according to the test designed by David 53,54a) and homogeneous variances exist, according to the test developed by Cochran 55,54 b) however, it is easy to see form Figs. 16 and 17 that this is not the case after transformation. 

Figure 8-5 Lineweaver-Burk transformation of the curve in Figure 8-4, vv ith i/v plotted on the ordinate (y-axis),and t/[S] on the abscissa (x-axis).The indicated intercepts permit calculation of Vmox and K -The units of v and [S] are those given in Figure 8-4.

Figure 4.8 Scatter plot of simulated data from a Michaelis-Menten model with Vmax — 100 and Km — 20 (top) and Lineweaver-Burke transformation of data (bottom). Stochastic variability was added by assuming normally distributed constant variability with a standard deviation of 3.

Fig. 9.16. The Lineweaver-Burk transformation (shown in blue) for the Michaelis-Menten equation converts it to a straight line of the form y = mx + b. When [S] is infinite, 1/[S] = 0, and the line crosses the ordinate (y-axis) at 1/y = W ,ax- The slope of the line is K Wmax-Where the line intersects the abscissa (x-axis), 1/[S] = -1/K .

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 -... 

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