Direct linear plots

Cornish-Bowden, A. and Eisenthal, R. (1978). Estimation of Michaelis constant and maximum velocity from the direct linear plot. Biochim Biophys Acta 523, 268-272. 

The direct linear plot (O Figure 4-6) requires no transformation of data, and this approach is arguably the most appropriate for determining and values. Individual values of [S] are plotted as their negatives on the x-axis and measured values of v are plotted on the axis. Corresponding pairs of ( — [S], 0)... 

A direct linear plot made from seven pairs of (v, [S]) data. The dotted lines mark the lowest and highest points of intersection. Clearly, a graph showing 21 horizontal and 21 vertical dotted lines, equivalent to the number of intersections from seven data pairs (see text), would be cluttered and difficult to interpret, and these lines are not shown. Rather, the hatched lines indicate Km and values obtained from nonlinear regression of the same data not surprisingly, these lie close to the median intersection points that would be obtained from a full direct linear analysis... 

Further drawbacks associated with the direct linear plot include the fact that this analysis does not readily lend itself to standard computerized graphing methods (for example, use of GraphPad Prism), although specialized software is available (Henderson, 1993). Of course, one of the major advantages of the direct linear plot is the ability to obtain kinetic constants by eye, without the need for a computer. However, for presentation purposes, the use of graphing software is still desirable. Furthermore, any behavior more complicated than simple, single substrate kinetics - for example, turnover in the presence of an inhibitor, or multisubstrate kinetics - caimot readily be shown on a direct linear plot. This is in contrast with the flexibility afforded by nonhnear regression approaches. 

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... 

Tyrosinase, a copper-containing oxidoreductase, catalyzes the orthohydroxy-lation of monophenols and the aerobic oxidation of catechols. The enzyme activity will be assayed by monitoring the oxidation of 3,4-dihydroxyphenyl-alanine (dopa) to the red-colored dopachrome. The kinetic parameters Ku and Vmax will be evaluated using Lineweaver-Burk or direct linear plots. Inhibition of tyrosinase by thiourea and cinnamate will also be studied. Two stereoisomers, L-dopa and D-dopa, will be tested and compared as substrates. 

Eisenthal and Cornish-Bowden (direct linear) plot for an enzyme-catalyzed reaction. 

A Lineweaver-Burk plot of enzyme kinetics in the presence and absence of a noncompetitive inhibitor is shown in Figure E5.5. Umax in the presence of a noncompetitive inhibitor is decreased, but KM is unaffected. The effect of a competitive inhibitor on the direct linear plot is shown in Figure E5.6. 

Second, an enzyme assay may be used to measure the kinetic properties of an enzyme such as Ku, Vmax, and inhibition characteristics. In this situation, different experimental conditions must be used. If Ku for a substrate is desired, the assay conditions must be such that the measured initial rate is first order in substrate. To determine Ku of a substrate, constant amounts of enzyme are incubated with varying amounts of substrate. A Lineweaver-Burk plot (1/v vs. 1/[S]) or direct linear plot may be used to determine Ku and V. If a reaction involves two or more substrates, each must be evalu-... 

Whether an inhibitor acts in a competitive or noncompetitive manner is deduced from a Lineweaver-Burk or direct linear plot using varying concentrations of inhibitor and substrate. In separate assays, two substances will be added to the dopa-tyrosinase reaction mixture, and the effect on enzyme activity will be quantified. The structures of the potential inhibitors, cinnamic acid and thiourea, are shown in Figure E5.9. The inhibition assays must be done immediately following the KM studies. To measure inhibition, reaction rates both with and without inhibitor must be used and the tyrosinase activity must not be significantly different. If it is necessary to do the inhibition studies later, the Ku assay for L-dopa must be repeated with freshly prepared tyrosinase solution. 

The treatment of results will be described for L-dopa. The procedure for D-dopa is identical. Prepare a table of L-dopa concentration per assay (mmo-lar) vs. A/i/min. Convert all AA/mm units to /xmoles/min as described in part B. Prepare a Michaehs-Menten curve (/xmoles/min vs. [S]) as in Figure E5.1 and a Lineweaver-Burk plot (l//xmole/min vs. 1/[S]) as in Figure E5.2. Alternatively, you may wish to use the direct linear plot. Estimate Ku and Vmax from each graph. The intercept on the rate axis of the Lineweaver-Burk plot is equal to 1/V-. For example, if the line intersects the axis at 0.02,... 

W. Wood et al., Biochemistry, A Problems Approach, 2nd ed. (1981), Benjamin/Cum-mings (San Francisco), pp. 144-172. Enzyme kinetics with an introduction to the direct linear plot. 

D Voet, J Voet, and C Pratt, Fundamentals of Biochemistry, (1999), John Wiley Sons (New York), pp 281-347 Enzymes and kinetics C Whiteley, Biochem. Educ. 25, 144-146 (1997) Enzyme kinetics W Wood et al, Biochemistiy, A Problems Approach, 2nd ed (1981), Benjamin/Cum-mmgs (San Francisco), pp 144-172 Enzyme kinetics with an introduction to the direct linear plot... 

To determine KM values, conduct rate measurements (v) with at least four different concentrations of any given substrate, [S], and then analyze the data by any suitable kinetic plot such as the Lineweaver-Burke plot or the direct-linear plot (refer to any standard textbook of biochemistry for more information). 

Eisenthal and Cornish-Bowden (21) and Cornish-Bowden (20) have described a rather different type of enzyme kinetics plot that should be useful in analyzing affinity-labeling kinetics. The equation that forms the basis of the direct linear plot (21) is obtained from Equation 11 by rearrangement of terms to give Equation 12... 

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. 

Cornish-Bowden, A., Eisenthal, R. (1974) Statistical Considerations in the Estimation of Enzyme Kinetic Parameters by the Direct Linear Plot and other Methods, Biochem.J. 139, 721-730. 

Much of the chemistry of albumin can be understood from detailed observations of the 2,3,5-triiodobenzoic acid (TIB) complexes with albumin. The crystalline complex of TIB with HSA and ESA has been determined with resolution sufficient to position the molecule within the binding pocket unambiguously and to identify the chemistry of interaction (Fig. 15, see color insert). This ligand has a moderate and equal affinity for IIA and IIIA in both HSA and ESA. Association constants were estimated by Scatchard analysis (Scatchard, 1949) using direct linear plots (Eisenthal and Cornish-Bowden, 1974) to be 2.2 x 10 M and 8.3 X lO" M for HSA and ESA, respectively. 

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... 

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