Double-reciprocal

A plot of 1/v versus 1/[S], which is called a double reciprocal, or Lineweaver-Burk plot, is a straight line with a slope of Km/Vmax) a y-intercept of 1/Vmax> and an x-intercept of-l/Km (Figure 13.11). 

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). 

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. 

The Michaelis-Menten equation is, like Eq. (3-146), a rectangular hyperbola, and it can be cast into three linear plotting forms. The double-reciprocal form, Eq. (3-152), is called the Lineweaver-Burk plot in enzyme kinetics. ... 

Equation (3-150) and other equations of the same functional form can be placed in three linear plotting forms, of which the double-reciprocal form is shown as Eq. (3-152). Find the other two linear plotting forms. 

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. 

Write the Lineweaver-Burk (double-reciprocal) equivalent of this equation, and from it calculate algebraic expressions for (a) the slope (b) the y-intercepts and (c) the horizontal and vertical coor-... 

This is referred to as a double reciprocal or lineweaver Burk plot. From this linear plot, Kd = slope/intercept and the 1 /intercept = Bmax. Finally, a linear plot can be achieved with... 

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],... 

Equating stimuli from these equations and grouping terms leads to the following linear double reciprocal equation ... 

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... 

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

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]. 

Equation (35) is the equation for a straight line, 7 = ax + b, wherey= l/v[andx= 1/[S]. Aplot of 1/v as j/as a function of 1/[S] as x therefore gives a straight line whose jy intercept is 1/l iax and whose slope is KJV. Such a plot is called a double reciprocal or Lineweaver-Burk plot (Figure 8-5). Setting thej/ term of equation (36) equal to zero and solving for x reveals that the x intercept is — IK. ... 

Figure 8-5. Double reciprocal or Lineweaver-Burk plot of 1/V versus 1/[S] used to evaluate and 1/max-...

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). 

In this scheme, EOH is the enzyme, IX is the inhibitor (either a carbamate or an organophosphate). EOH(IX) is analogous to the Michaelis Menton comploc seen with the substrate reaction. EOI is the acyl-enzyme intermediate for carbamates or a phosphoro-enzyme intermediate for the organophosphates. The equilibrium constant for this reaction (K ) is defined as k /k and the phosphorylation or carbamylation constant is defined as k2- In this study 42)y ANTX-A(S) was found to be more specific for AChE than BUChE. The double reciprocal and Dixon plot of the inhibition of electric eel AChE indicated that the toxin is a non-competitive inhibitor decreases, k remains unchanged) (Figure 2). 

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. 

---