Double reciprocal method

In evaluation of kinetic parameters, the double reciprocal method is used for linearisation of the Michaelis-Menten equation (5.7.3). 

Molecular recognitions of amino acids. The binding constant is a measure of recognition interactions of different amino acids with metallohematoporphyrin. It can be estimated by Benesi-Hildebrand equation (the double-reciprocal method) ... 

Lineweaver-Burk plot Method of analyzing kinetic data (growth rates of enzyme catalyzed reactions) in linear form using a double reciprocal plot of rate versus substrate concentration. 

FIGURE 5.19 Method of Barlow for measurement of affinity of a partial agonist, (a) Guinea pig ileal smooth muscle contraction to histamine (filled circles) and partial histamine receptor agonist E-2-P (N,N-diethyl-2-(l-pyridyl)ethylamine (open circles). Dotted lines show equiactive concentrations of each agonist used for the double reciprocal plot shown in panel b. (b) Double reciprocal plot of equiactive concentrations of histamine (ordinates) and E-2-P (abscissae). Linear plot has a slope of 55.47 and an intercept of 1.79 x 10s. This yields a KB (1 — tp/ta) = 30.9 pM. (c) Variant of double reciprocal plot according to Equation 5.8. (d) Variant of double reciprocal plot according to Equation 5.10. Data redrawn from [10],... 

Historically, Gaddum and colleagues [3] devised a method to measure the affinity of insurmountable antagonists based on a double reciprocal linear transformation. With this method, equiactive concentrations of agonist in the absence ([A]) and presence ([A ]) of a noncompetitive antagonist ([B]) are compared in a double reciprocal plot... 

FIGURE 12.6 Measurement of full agonist affinity by the method of Furchgott. (a) Dose-response curve to oxotremorine obtained before (filled circles) and after (open circles) partial alkylation of the receptor population with controlled alkylation with phenoxybenzamine (10 jiM for 12 minutes followed by 60 minutes of wash). Real data for the curve after alkylation was compared to calculated concentrations from the fit control curve (see arrows), (b) Double reciprocal of equiactive concentrations of oxotremorine before (ordinates) and after (abscissae) alkylation according to Equation 5.12. The slope is linear with a slope of 609 and an intercept of 7.4 x 107 M-1. 

Gadduin analysis, Gadduin (method of), this method (Q. J. Exp. Physiol. 40, 49-74, 1955) compares equiactive concentrations of an agonist in the absence and presence of a concentration of noncompetitive antagonist that depresses the maximal agonist response. These are compared in a double reciprocal plot (or variant thereof) to yield the equilibrium dissociation constant of the noncompetitive antagonist-receptor complex (see Chapters 6.4 and 12.2.8). 

Gadduin equation (noncompetitive antagonism), this technique measures the affinity of a noncompetitive antagonist based on a double reciprocal plot of equiactive agonist concentrations in the absence and presence of the noncompetitive antagonist. The antagonist must depress the maximal response to the agonist for the method to be effective see Chapter 6.4. 

In each of these cases, the double-reciprocal plots according to the Lineweaver-Burk method are linear. The appearance of these plots, and the parameters obtained from them, are developed in Problem 4-15. 

For either of the ternary complex mechanisms described above, titration of one substrate at several fixed concentrations of the second substrate yields a pattern of intersecting lines when presented as a double reciprocal plot. Hence, without knowing the mechanism from prior studies, one can not distinguish between the two ternary complex mechanisms presented here on the basis of substrate titrations alone. In contrast, the data for a double-displacement reaction yields a series of parallel lines in the double reciprocal plot (Figure 2.15). Hence it is often easy to distinguish a double-displacement mechanism from a ternary complex mechanism in this way. Also it is often possible to run the first half of the reaction in the absence of the second substrate. Formation of the first product is then evidence in favor of a doubledisplacement mechanism (however, some caution must be exercised here, because other mechanistic explanations for such data can be invoked see Segel, 1975, for more information). For some double-displacement mechanisms the intermediate E-X complex is sufficiently stable to be isolated and identified by chemical and/or mass spectroscopic methods. In these favorable cases the identification of such a covalent E-X intermediate is verification of the reaction mechanism. 

For compounds that conform to the mechanism of scheme C, an alternative method for defining inhibition modality is to measure progress curves (or preincubation effects vide supra) at varying inhibitor and substrate concentrations, and to then construct a double reciprocal plot of 1/v, as a function of l/[.Sj. Using the analysis methods and equations described in Chapter 3, one can then determine the modality of inhibition for the inhibitor encounter complex. Similarly, for inhibitors that conform to the mechanism of scheme B, a double reciprocal plot analysis of l/vs as a function of 1/[S] can be used to define inhibition modality. 

There are three basic methods for carrying out alternative substrate inhibition studies. In the first, the investigator seeks to observe numerical changes in the coefficients of the double-reciprocal form of the enzyme rate expression in the presence and absence of the alternative substrate. For some mechanisms, only certain coefficients will be altered. This method requires extremely accurate estimates of the magnitudes of the coefficients and should always be supplemented with other kinetic probes . 

This method is widely used because it provides hnear transformation of the hyperbolic function describing the rate saturation process. Double-reciprocal plots can be reasonably accurate if rate data can be obtained over a reasonable range of saturation, say from 0.3 E ax to 0.8 E ax. 

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... 

A ratio used to assess the degree of cooperativity exhibited by an enzyme. It is equal to the true Ymax value (typically extrapolated from the high-substrate-concen-tration end of a double-reciprocal plot) divided by the apparent Emax value obtained from extrapolating the asymptote in the low-substrate-concentration portion of the double-reciprocal plot. For a noncooperative system, Ry will equal one positively cooperative systems will have values greater than one and negatively cooperative systems will have values less than one . This method requires good estimates of the asymptotes. 

The calculated binding constants assuming a 1 1 interaction are listed in Table 3. There is a clear difference between the plotting methods. Only by using the x-reciprocal plot does it become clear that there seem to be higher order equilibria between the compounds. The nonlinear regression leads to similar results as with the y-reciprocal fit. The double reciprocal... 

Amphetamine Me-/3 Graphical methods Double reciprocal A-reciprocal y-reciprocal Nonlinear regression 67 correlation between log P and KB... 

Double reciprocal x-reciprocal Nonlinear regression Graphical method Graphical method Nonlinear regression... 

It is very useful to transform the Michaelis-Menten equation into a linear form for analyzing data graphically and detecting deviations from the ideal behavior. One of the best known methods is the double-reciprocal or Lineweaver-Burk plot. Inverting both sides of equation 3.1 and substituting equation 3.2 gives the Lineweaver-Burk plot 4... 

Repeat Question 11, but graph the data as a linear, double reciprocal plot in the spirit of the Lineweaver-Burk equation (see Chapter 4). Plot l/ATm vs. 1 /(N/nt) and perform a linear regression to determine the best-fit line (Equation 4.a). The x-intercept corresponds to the KD of the DNA-netropsin complex. The KD value from this method should be more accurate than the estimation in Question 11. 

These hyperbolic equations are analogous to the Michaelis-Menten equation. Nonlinear regression is preferable to the method proposed in the 1960s by Kitz and Wilson, which necessitates a double-reciprocal linear transformation of the data (analogous to a Lineweaver-Burk plot) that can bias the estimates of /clnact and A). 

Application of a least-squares method to the linearized plots (e.g., Scatchard and Hames) is not reasonable for analysis of drug-protein binding or other similar cases (e.g., adsorption) to obtain the parameters because the experimental errors are not parallel to the y-axis. In other words, because the original data have been transformed into the linear form, the experimental errors appear on both axes (i.e., independent and dependent variables). The errors are parallel to the y-axis at low levels of saturation and to the x-axis at high levels of saturation. The use of a double reciprocal plot to determine the binding parameters is recommended because the experimental errors are parallel to the y-axis. The best approach to this type of experimental data is to carry out nonlinear regression analysis on the original equation and untransformed data. 

A mathematical model has also been proposed for evaluating cellulase preparations. Sattler et al.209 describe a relationship between hydrolysis extent, reaction time, and enzyme concentration. This procedure permits the effectiveness of different enzymes and of different pretreatment methods to be ranked. This method examines cellulose hydrolysis data collected from hyperbolic functions of substrate concentration versus cellulase enzyme concentration at various timed incubations. The model is based on a double reciprocal plot of the relationship... 

Fig. 13.4. DGBP competitively inhibits GGDPS with respect to FPP. In vitro GGDPS assays were performed utilizing various concentrations of FPP substrate or DGBP using an established method [110]. A double reciprocal plot liV vs. lA) is shown.

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