Double reciprocal plot = Lineweaver-Burke

Bimolecular reactions of two molecules, A and B, to give two products, P and Q, are catalyzed by many enzymes. For some enzymes the substrates A and B bind into the active site in an ordered sequence while for others, bindingmay be iii a random order. The scheme shown here is described as random Bi Bi in a classification introduced by Cleland. Eighteen rate constants, some second order and some first order, describe the reversible system. Determination of these kinetic parameters is often accomplished using a series of double reciprocal plots (Lineweaver-Burk plots), such as those at the right. 

Consequently double reciprocal plots (Lineweaver-Burk) of 1/V = f (1/[S]), will be linear when [A] varied. Secondary plots of the slopes and intercepts of the plots of 1/v =f (1/[S]) against [A] will be hyperbolic. 

Figure 1. Idealized (A) velocity vs. substrate concentration and (B) double reciprocal plots (Lineweaver-Burk) for enzymes demonstrating (a) hyperbolic (classical Michaelis-Menten) behavior (b) positive cooperativity (c) negative cooperativity.

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). 

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... 

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-...

The x-coordinate axis for a graph of Cartesian coordinates [x,y or [x, f(x)] or the x-value for any [x,y] ordered pair. This corresponds to the [Substrate Concentration]-axis in v versus [S] plots or the ll[Substrate Concentra-fton]-axis in so-called double-reciprocal or Lineweaver-Burk plots. 

FIGURE 1 A double-reciprocal or Lineweaver-Burk plot. 

Linear forms for rate equations. To obtain Km and Vmax from experimental rate data, Eq. 9-15 can be transformed by algebraic rearrangement into one of several linear forms. The popular double-reciprocal or Lineweaver-Burk plot of 1/ v against 1 / [S] (Fig. 9-3) is described by Eq. 9-20. The values of Km/ Vmax and 1 / Vmax can be evaluated from the slope and intercept, respectively, of this straight line plot. 

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... 

Vmax and Km can be determined experimentally by measuring V0 at different substrate concentrations, and then plotting 1/V0 against 1/[S] in a double reciprocal or Lineweaver-Burk plot. The intercept on the y-axis is equal to 1 / Vmaxr the intercept on the x-axis is equal to -1 /Km and the slope of the line is equal to Km /Vmax. 

Figure 8.36. A Double-Reciprocal or Lineweaver-Burk Plot. A double-reciprocal plot of enzyme kinetics is generated... 

The Michaelis-Menten equation can be algebraically transformed into more useful way to plot the experimental data. Lineweaver and Burk have taken the reciprocal of both [S] and v of the Michaelis-Menten equation to give Double Reciprocal or Lineweaver-Burke Plot Need in form y = ax + b, so take reciprocals of both sides (Fig. 6.4) and have -... 

Figure 9-3 Double-reciprocal or Lineweaver-Burk plot of 1/vs 1/ [S], The intercept on the vertical axis gives and the slope gives IThe intercept on the horizontal axis equals...

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. 

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. 

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. 

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. 

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. 

A Scatchard plot of the data is shown in Figure 5.10C. For convenience, the fitted line is the regression of B/F on B (though, as noted earlier, this is statistically unsound) and provides an estimate for Bmax ( -intercept) of 0.654 fmol/mg dry wt. and an estimate for KL (-1/slope) of 132 pM. A Lineweaver-Burk (double-reciprocal) plot is provided for comparison in Figure 5.10D. Linear regression gives another estimate for Bmax (I v-intercept see Eq. (5.29)) of 0.610 fmol/mg dry wt. The estimate of KL from this plot (slope x Bmax) is 114 pM. 

An analysis of the influence of errors shows clearly that the double-reciprocal plot according to Lineweaver-Burk [32] is the least suitable. Although it is by far the most widely used plot in enzyme kinetics, it cannot be recommended, because it gives a grossly misleading impression of the experimental error for small values of v small errors in v lead to enormous errors in 1/y but for large values of v the same small errors in v lead to barely noticeable errors in 1/17 [23]. Due to the error distribution, that is much more uniform, the plot according to Hanes (Eq. (7)), is the most favored. 

More accurate means of determining Km and Vmax are offered by either the Lineweaver-Burke (double reciprocal) plot or the Eadie-Hofstee plot (Figure 2.7). 

The linear form of this equation is denoted hy the Lineweaver-Burk or double reciprocal plot, which is derived from the Michaelis-Menten and Hill equation and is denoted as ... 

Based on the Lineweaver-Burk equation, a plot of v versus 1/[S] gives a straight line on a Lineweaver-Burk or double reciprocal plot (Figure 3-4). 

Figure 11. The Lineweaver-Burk double reciprocal plots of the FT Ni.

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