Linear plots Eadie-Hofstee

See Double-Reciprocal Plot Hanes Plot Direct Linear Plot Dixon Plot Dixon-Webb Plot Eadie-Hofstee Plot Substrate Concentration Range Frieden Protocol Fromm Protocol Point-of-Convergence Method Dal-ziel Phi Relationships Scatchard Plots Hill Plots... 

Most common Michaelis-Menten analytics are associated with the typical linearization plots namely Lineweaver-Burk (or double-reciprocal plot), Eadie-Hofstee " and Hanes plots. All three have in common that they are using the experimental differential values of either product formation or substrate uptake. Very often these values cannot be determined directly and must be differentiated from the corresponding concentration-time values numerically. This can naturally cause large errors, in particular when the measured data are very noisy and/or just a very low number of data points exist. Nevertheless, the Hanes plot is the most favoured one due to its much more uniform error distribution (see eqn (4.6) with v = d[P]/dt). 

Historically, data have been transformed to facilitate plotting on linear plots such as Lineweaver-Burk (1/y versus 1/[S]), Hanes-Woolf ([S]/y versus [S]), or Eadie-Hofstee (v/[S] versus y). However, with the present availability of affordable nonlinear regression and graphing software packages such as GraphPad Prism,... 

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 ). 

Evaluate the Michaelis-Menten kinetic parameters by employing (a) the Langmuir plot, (b) the Lineweaver-Burk plot, (c) the Eadie-Hofstee plot, and (d) non-linear regression procedure. 

There are well-established methods for obtaining the type of inhibition and the value of the inhibition constants from initial-rate kinetics, often from linearized plots such as lineweaver-Burk, Eadie-Hofstee, or Hanes. As these procedures are covered very well by a range of basic textbooks on biochemistry and kinetics (see the list of Suggested Further Reading ) we will not repeat these procedures here. Instead, we will discuss the situation in which an enzyme reaction is followed over more than just the initial range of conversion. Towards this end, the rate equation,... 

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 linear response range of the glucose sensors can be estimated from a Michaelis-Menten analysis of the glucose calibration curves. The apparent Michaelis-Menten constant KMapp can be determined from the electrochemical Eadie-Hofstee form of the Michaelis-Menten equation, i = i - KMapp(i/C), where i is the steady-state current, i is the maximum current, and C is the glucose concentration. A plot of i versus i/C (an electrochemical Eadie-Hofstee plot) produces a straight line, and provides both KMapp (-slope) and i (y-intercept). The apparent Michaelis-Menten constant characterizes the enzyme electrode, not the enzyme itself. It provides a measure of the substrate concentration range over which the electrode response is approximately linear. A summary of the KMapp values obtained from this analysis is shown in Table I. 

This effect can be reduced if affinity-labeling kinetic data are analyzed by other types of linear plots used by enzyme kineticists. Two which appear to be useful improvements over the Kitz-Wilson plot are analogs of the Eadie-Hofstee and the Eisenthal-Comish-Bowden plots (20,21). 

Kl. Figure 3 compares Kitz-Wilson plots obtained plus and minus L. Of course the effects of L on affinity-labeling kinetics could also be analyzed quantitatively by Eadie-Hofstee or direct linear-type plots. 

Comparing Equation 29 with Equation 9 shows that the two expressions for kobe differ only in the constants in the numerators of the right-hand sides. Both mechanisms predict first-order kinetics for the loss of site activity and identical dependence of the observed first-order rate constant, kobBy on the [R]. The similarity of Equations 9 and 29 demonstrates that the documentation of saturation kinetics as evidenced by linear Kitz-Wilson or Eadie-Hofstee plots or by the critria of the direct linear plot does not prove that true affinity labeling is involved necessarily in a site-inactivating reaction. 

Hie parameters i>max and Km of the equation can be determined from the slope and intercept of a linear plot of v l against [S]-/ ( Lineweaver-Burk plot ) or from slope and intercept of a linear plot of v against v/[S] ( Eadie-Hofstee plot ). 

Competitive inhibitors do not change the value of Vmax> which is reached when sufficiently high concentrations of the substrate are present so as to completely displace the inhibitor. However, the affinity of the substrate for the enzyme appears to be decreased in the presence of a competitive inhibitor. This happens because the free enzyme E is not only in equilibrium with the enzyme-substrate complex E. S, but also with the enzyme-inhibitor complex E. L Competitive inhibitors increase the apparent of the substrate by a factor of (1 + The evaluation of the kinetics is again greatly facilitated by the conversion of Equation 17.15 into a linear form using Line-weaver-Burk, Eadie-Hofstee, or Hanes-Woolf plots, as shown in Fig. 17.7. 

Eadie-Hofstee method plot v versus v/[S], slope = —Km, y intercept is Umax. Values obtained by linear regression are Km = 0.62 mill and Umax = 0.74 pmol/ min, with R = —0.996, using the data shown below ... 

Another method to obtain estimates for Km and is the rearrangement of the Michaelis-Menten equation to a linear form. The estimation for the initial velocities, Vo, from progress curves is not a particularly reliable method. A better way to estimate Vn is by the integrated Michaelis-Menten equation (Cornish-Bowden, 1975). Nevertheless, the graphical methods are popular among enzymolo-gists. The three most common linear transformations of the Michaelis-Menten equation are the Lineweaver-Burk plot of 1/Vo vs. 1/[S] (sometimes called the double-reciprocal plot), the Eadie-Hofstee plot, i.e. v vs. vo/[S], and the Hanes plot, i.e., [SJ/vo vs. [S] (Fig. 9.3). 

Fig. 9.3. Determination of the parameters of the Michaelis-Menten equation and by the Lineweaver-Burk (A), Eadie-Hofstee (B), Hanes (C), and direct linear (D) plots. The error bars in A, B and C represent a variation of 5 /, of Vmi, and show the large effect small errors at low [S] may have on the estimates. Outlying lines obtained in the direct linear plot (D) are easily recognized, at least if a large fraction of the lines do converge in the same intersection.

Despite their appealing simplicity, these methods have serious limitations. The Lineweaver-Burk and Hanes plots are unreliable, e.g., the variation of the variance almost certainly results in an incorrect weighting, whereas in the Eadie-Hofstee plot Vo is present in both variables. The direct linear plot of Eisenthal and Cornish-Bowden (1974), for which the Michaelis-Menten equation is rearranged to relate to A , i.e., = Vo -f- Vo A ,/[S] is very simple but... 

Fig. 4 Mechanism of action (MOA) and inhibition studies of ML119 (compound 1) with HePTP and HePTP mutants, (a) Progress curves of HePTP (6.25 nM) activity in the presence of different doses of compound 1 (0, 0.078,0.156,0.313,0.625,1.25 /jM) and 0.3 mM OMFP in 20 mM Bis-Tris, pH 6.0,150 mM NaCI, 1 mM DH, and 0.005 % Tween-20 in 20 /jL totai assay voiume in biack 384-weii microtiter plates. No time-dependent inhibition was observed as demonstrated by the linear progress curves of the HePTP phosphatase reaction, (b) Eadie-Hofstee plot of the Michaelis-Menten kinetic study with compound I.The HePTP-catalyzed hydrolysis of OMFP was assayed at room temperature in a 60 /jL 96-well format reaction system in 50 mM Bis-Tris, pH 6.0 assay buffer containing 1.7 mM DTT, 0.005 % Tween-20, and 5 % DMSO. Recombinant HePTP (5 nM) was preincubated with various fixed concentrations of inhibitor (0,0.1,0.2,0.4,0.8,1.6 /jM) for 10 min. The reaction was initiated by addition of various concentrations of substrate (0,12.5,25,50,100,200,400 pM) to the...

Which of these plots should be used To generally understand the behavior of enzymes, use the simple graph of initial velocity against substrate concentration. The linearized forms are useful for calculation of ATM and Fmax. The Lineweaver-Burke plot is useful for distinguishing between types of inhibition (Chapter 8). The Eadie-Hofstee plot is better than the Lineweaver-Burke plot at picking up deviations from the Michaelis-Menten equation. 

Alternatively the data can be linearized in a plot of v versus v/[S], known as an Eadie-Hofstee plot as shown in Figure 8.16. 

Figure 8.28 Simulated Eadie-Hofstee plot for negative cooperatlvlty in the absence of enzyme inhibition, the plot would be linear (compare to competitive and non-competitive inhibition in Fig 8.27).

A Lineweaver-Burk plot ( ) indicates that with D-glucose as the substrate, the enzyme obeys Michaelis-Menten kinetics with a Km value of 3.2 + 0.08 mM and a Vmax of 126.0 + 0.02 micromol/mg protein/min (Figure 11). Similar results were obtained by the direct linear plot (88), Hanes and Woolf ( ) or Eadie-Hofstee plots (90). All the kinetic data reported here and subsequently, were based on the initial rates of hydrogen peroxide formation... 

FIGURE 13.2 Biochemical plots for the enz5me kinetic characterizations of biotransformation, (a) Direct concentration-rate or Michaelis-Menten plot (b), Eadie-Hofstee plot (c), double-reciprocal or Lineweaver-Burk plot. The Michaelis-Menten plot (a), typically exhibiting hyperbolic saturation, is fundamental to the demonstration of the effects of substrate concentration on the rates of metabolism, or metabolite formation. Here, the rates at 1 mM were excluded for the parameter estimation because of the potential for substrate inhibition. Eadie-Hofstee (b) and Lineweaver-Burk (c) plots are frequently used to analyze kinetic data. Eadie-Hofstee plots are preferred for determining the apparent values of and Umax- The data points in Lineweaver-Burk plots tend to be unevenly distributed and thus potentially lead to unreliable reciprocals of lower metabolic rates (1 /V) these lower rates, however, dictate the linear regression curves. In contrast, the data points in Eadie-Hofstee plot are usually homogeneously distributed, and thus tend to be more accurate. 

In the graphical analysis of initial rate data, it is pradent to use all three plots shown in Figs. 3 and 4. The direct plot of versus [A] wiU show directly the influence of substrate concentration on initial rate of reaction. The two linear plots should be used together, because the Lineweaver-Burk plot serves to visualize the influence of low concentrations whereas the Hanes plot serves to visualize the influence of high concentrations of substrates. The third plot, the Eadie-Hofstee plot, is useful in detecting exceptionally bad measurements (Section 3.11). 

The influence of external transport on kinetic parameter estimation can also be illustrated in an Eadie-Hofstee plot, as shown in Fig. 4.33 (Hartmeier, 1972 Horvath and Engasser, 1974). Significant departures from linearity, however, are observed with increasing external transport limitation >0.1), particularly when a wide range of s is examined. 

A more satisfactory distribution of error is found for S/v versus S, known as the Hanes plot, or the Hanes-Wilkinson plot (Table Fig.). A further linear transformation is represented by v versus v/S, known as the Eadie-Hofstee plot. The error increases with v/S, but since v is a component of both coordinates, the errors vary with respect to the origin rather than the axis, i.e. all error bars converge on the origin (Table Fig.). 

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