Lineweaver-Burk plot approach

While the point for [S] = 0 and v = 0 cannot be plotted, the ratio v / [S] approaches Vma JKm as v approaches zero. Notice the distribution of the points in Fig. 9-4. Substrate concentrations were chosen such that the increase in velocity from point to point is more or less constant, a desirable experimental situation. The points on the Eadie-Hofstee plot are also nearly evenly distributed, but those of the Lineweaver-Burk plot are compressed at one end. (However, if the substrate concentrations for successive points are selected in the ratios 1,1 / 3,1/ 5,1/ 7, and 1/9, the spacing will be uniform on the Lineweaver-Burk plot.) A second advantage of the Eadie-Hofstee plot is that the entire range of possible substrate concentration from near zero to infinity can be fitted onto a single plot. 

The Lineweaver-Burk plot is more often employed than the other two plots because it shows the relationship between the independent variable Cs and the dependent variable r. However, 1/r approaches infinity as Cs decreases, which gives undue weight to inaccurate measurements made at low substrate concentrations, and insufficient weight to the more accurate measurements at high substrate... 

Alternatively, you can linearize the Michaelis-Menten equation by using an Eadie-Hofstee plot [23,24]. Here the reaction velocity, v, is plotted as a function of v/[S], as shown in Eq. (2.43). This approach is more robust against error-prone data than the Lineweaver-Burk plot, because it gives equal weight to data points in any range of [S] or v. The disadvantage here is that both the ordinate and the abscissa depend on v, so any experimental error will be present in both axes. 

The Lineweaver-Burke plot once provided a convenient method for estimating V ax with a discrete interpolated data point (note that l/V is equivalent to the intercept at the ordinate), since in the direct plot (Figure 8.14), the plot of w versus [5] asymptotically approaches a maximum. As it turns out, however, can readily be estimated from the direct... 

This approach is called the Lineweaver-Burk plot. It can be seen that the error structure is not additive and there is a need to take the inverse of potentially small numbers which can introduce further errors. In general, none of the methods is very good at obtaining accurate parameter values, and nonlinear regression is better. 

There are several ways of visualizing enzyme kinetic data. One approach is to plot the reciprocal rate of conversion of substrate A, I/ca, against the reciprocal substrate concentration, 1/ca, which is known as the Lineweaver-Burk plot (Lineweaver and Burk, 1934) ... 

Employing this approach, conversion rates could be calculated. Maximum rates of conversion (Vmax) the Michaelis constants (Ku) were obtained measuring reaction rates in D2O at different substrate concentrations. The ratio of trans-sialylation as well as the Lineweaver-Burke plot are shown in Fig. 9 [47, 48]. 

With these assumptions in hand, interpretation of real assay data involves plotting a model-derived value for concentration of NAD at the enzyme surface (NAD ). The value for the Mnad+ can be fitted to allow the Lineweaver-Burk plot to intercept the x-axis at a value that yields the value of Km as determined in solution. The value for Vmax is then read as the intercept at the y-axis (Figure 12.2). This approach permits derivation of a Vmax for the electrode that is independent of the effects of mass transfer. If one further assumes that the immobilization process does not affect the turnover rate of the immobilized enzyme (relative to its activity in solution), then this value of Vmax (which represents the total activity of all bound enzyme) can also be used to estimate the amount of immobilized enzyme. This model can be particularly useful when fabricating electrodes using immobilization techniques that entrap a fraction of enzyme from bulk solutions, such as direct physical absorption or co-immobilization within gels. 

Because of the hyperbolic shape of versus [S] plots, Vmax only be determined from an extrapolation of the asymptotic approach of v to some limiting value as [S] increases indefinitely (Figure 14.7) and is derived from that value of [S] giving v= V(nax/2. However, several rearrangements of the Michaelis-Menten equation transform it into a straight-line equation. The best known of these is the Lineweaver-Burk double-reciprocal plot ... 

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. 

Another possibility is to use hyperbolic approaches, similar to those discussed in Chapter 6, like the double reciprocal plot of Lineweaver -Burk, which has the form... 

Let us consider synthetic data for Enzyme-X which is similar to the actual data for pancreafic carboxypeptidase [13,14] (note the name of an enzyme ends in -ase ). We use synthetic data, so we can insert key points into an Excel plot. It should be clear that at low substrate concentrafion, the rate increases rapidly as more substrate is added. However, in spite of the efficiency of an enzyme, there is only a small amount in solution. Thus, the rate approaches a limit as more and more substrate saturates the active sites of the enzymes in solution and eventually reaches a limit, V , as shown in Figure 8.7. Special points have been inserted into the data so that you can see the limiting rate of0.090 mM/s and see that at half that rate, 0.045 mM/s, the substrate concentrafion is 0.0065 mM which is the value of Km- The value of Km is not easily seen on the first plot but when we show the double reciprocal plot in Figure 8.8 we get a more precise value of Km value in two ways. There are really two intercepts in the double-reciprocal plot, which apparently was the innovation of Lineweaver and Burk [15] in 1934. We can use the third and fourth columns of Table 8.1 to make such aplotforour Enzyme-X data in Figure 8.8. 

---