Reciprocal plots

In Figure 2, a double-reciprocal plot is shown Figure 1 is a nonlinear plot of as a function of [S]. It can be seen how the least accurately measured data at low [S] make the deterrnination of the slope in the double-reciprocal plot difficult. The kinetic parameters obtained in this example by making linear regression on the double-reciprocal data ate =1.15 and = 0.25 (arbitrary units). The same kinetic parameters obtained by software using nonlinear regression are = 1.00 and = 0.20 (arbitrary units). 

Equation 1-111 is known as the Lineweaver-Burk or reciprocal plot. If the data fit this model, a plot of l/V versus 1/Cg will he linear with a slope K /V x intercept l/V x-... 

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. 

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 ... 

FIGURE 14.9 The Lineweaver-Burk double-reciprocal plot, depicting extrapolations that allow the determination of the and 31-intercepts and slope. 

Reactions that fit this model are called ping-pong or double-displacement reactions. Two distinctive features of this mechanism are the obligatory formation of a modified enzyme intermediate, E, and the pattern of parallel lines obtained in double-reciprocal plots (Figure 14.19). 

FIGURE 14.18 Single-displacement bisubstrate mechanism. Double-reciprocal plots of the rates observed with different fixed concentrations of one substrate (B here) are graphed versus a series of concentrations of A. Note that, in these Lineweaver-Burk plots for singledisplacement bisubstrate mechanisms, the lines intersect to the left of the 1/v axis. 

It can be seen from Equation 5.7 that a more accurate estimate of the affinity will be obtained with partial agonists of low efficacy (i.e., as u v Vu o). Double reciprocal plots are known to produce... 

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... 

Therefore, a double reciprocal plot of equiactive agonist concentrations in the presence (1/[A ] as abscissae) and absence (1 /[A] as ordinates) of antagonist should yield a straight line. The equilibrium dissociation constant of the antagonist is calculated by... 

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. 

Show that the reaction follows first-order kinetics when [A]0 s> [S]o, and show that lAi > varies linearly with l/[A]. The same pattern is seen when maleic anhydride (B) is used instead of A, except that the line in the double reciprocal plot is parallel to the abscissa. It intersects the ordinate at the same point as the y-intercept for A. Why are the slopes for A and B different, and their intercepts the same ... 

Rate law and reaction scheme. Interpret quantitatively the data21 presented in the accompanying two graphs in terms of either or both of the sequences that might be considered for experiments in which [Ph2C2] and [CO] [Co2(CO)8]o- The reciprocal of k varies linearly with the reciprocal of the diphenyl acetylene concentration at constant [CO]. The slopes of these double reciprocal plots are directly proportional to [CO]. 

Double Reciprocal Plots Facilitate the Evaluation of Inhibitors... 

Double reciprocal plots distinguish between competitive and noncompetitive inhibitors and simpbfy evaluation of inhibition constants Aj. v, is determined at several substrate concentrations both in the presence and in the absence of inhibitor. For classic competitive inhibition, the lines that connect the experimental data points meet at they axis (Figure 8-9). Since they intercept is equal to IIV, this pattern indicates that wben 1/[S] approaches 0, Vj is independent of the presence of inhibitor. Note, however, that the intercept on the X axis does vary with inhibitor concentration—and that since is smaller than HK, (the apparent... 

For simple noncompetitive inhibition, E and EI possess identical affinity for substrate, and the EIS complex generates product at a negligible rate (Figure 8-10). More complex noncompetitive inhibition occurs when binding of the inhibitor does affect the apparent affinity of the enzyme for substrate, causing the tines to intercept in either the third or fourth quadrants of a double reciprocal plot (not shown). 

Figure 3. Plot of V against total enzyme [ET] showing the irreversible inhibition of el tric eel acetylcholinesterase (AChE) by ANTX-A(S). The enzymes were incubated with 0.32 fig/mL ANTX-A(S) for 1.0 min and acetylthiocholine (final concentrations 2.5, 4.7, 6.3, and 7.8 X 10 M) was added. V was determined from the double reciprocal plots (not shown). Key (o) control ( ) ANTX-A(S). (Reproduced with permission from Ref. 42. Copyright 1987 Pergamon Press)...

Stem juice of Dieffenbachia maculata contains an inhibitor of fungal polygalacturonase. The inhibitor is non dializable and heat stable. The double reciprocal plot indicates that the inhibitor causes a mixed type of inhibition. The paper also describes the distribution of the inhibitor in different varieties of Dieffenbachia, and some of its properties. 

Figure 2.11 Double reciprocal plot of initial velocity as a function of substrate concentration. Data from Figure 2.9 are plotted here in double reciprocal format.

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. 

Table 3.1 Diagnostic signatures of reversible inhibition modabties in double reciprocal plots...

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. 

Figure 7.6 Double reciprocal plot for a tight binding competitive enzyme inhibitor, demonstrating the curvature of such plots. The dashed lines represent an attempt to fit the data at lower substrate concentrations to linear equations. This highlights how double reciprocal plots for tight binding inhibitors can be misleading, especially when data are collected only over a limited range of substrate concentrations.

This double-reciprocal plot is based on another rearrangement of Eq. (5.3) ... 

A plot of 1 IB vs. 1/[L] will give a straight line providing that Eq. (5.3) applies when 1 IB = 0, then 1/[L] = -1 IKl, and when 1/[L] = 0, then 1 IB = 1 /) max. A Lineweaver-Burk plot is shown in Figure 5.10, where it may be compared with the Scatchard plot of the same data. The double-reciprocal plot spreads the data very poorly and is inferior to the Scatchard plot. 

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