Lineweaver-Burk form

Usually, one plots the initial rate V against the initial amount X, which produces a hyperbolic curve, such as shown in Fig. 39.17a. The rate and amount at time 0 are larger than those at any later time. Hence, the effect of experimental error and of possible deviation from the proposed model are minimal when the initial values are used. The Michaelis-Menten equation can be linearized by taking reciprocals on both sides of eq. (39.114) (Section 8.2.13), which leads to the so-called Lineweaver-Burk form ... 

This variant can be derived from the Lineweaver-Burk form in eq. (39.115) by multiplying both sides by X. From a statistical point of view, it does not seem to have an advantage over the Lineweaver-Burk form [14]. The latter variant, however, can be more easily extended to more complex systems of substrate-enzyme reactions, as will be shown below. 

Note the close analogy with the Lineweaver-Burk form of the simple Michaelis-Menten equation. In a diagram representing MV against MX one obtains a line which has the same intercept as in the simple case. The slope, however, is larger by a factor (1 + YIK-) as shown in Fig. 39.17b. Usually, one first determines and in the absence of a competitive inhibitor (F = 0), as described above. Subsequently, one obtains A" from a new set of experiments in which the initial rate V is determined for various levels of X in the presence of a fixed amount of inhibitor Y. The slope of the new line can be obtained by means of robust regression. 

Repeat Example 10-1 using a linearized form of equation 10.3-1 that is alternative to the linearized form 10.3-2 (the Lineweaver-Burk form). Comment on any advantage(s) of one form over the other. (There is more than one possible alternative form.)... 

Determine the kinetics parameters Km and Vmax, assuming that the standard Michaelis-Menten model applies to this system, (a) by nonlinear regression, and (b) by linear regression of the Lineweaver-Burk form. 

Regulatory enzymes are usually identified by the deviation of their kinetics from Michaelis-Menten kinetics plots of velocity versus substrate concentration can be a sigmoidal curve or a modified hyperbola [Fig. 9-7(o)]. If these curves are plotted in the double-reciprocal (Lineweaver-Burk) form, nonlinear graphs are obtained [Fig. 9-7(6)]. 

Equation (10), for example, when plotted, results in a Lineweaver-Burk plot that gives a straight line with intercepts at HVmix on the ordinate and -HK, on the abscissa. For illustrative purposes, the data for Figure 8-4 are recast in Lineweaver-Burk form in Figure 8-5. Regarding inhibition, the Michaelis-Menten curve and its linearized plots are altered in a way that is described later. 

Rapid equilibrium analysis of Scheme 1, followed by casting the resulting velocity equation in Lineweaver-Burk form yields equation (1)... 

Mnad+ and the data then plotted in Lineweaver-Burk form to solve for and... 

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. ... 

The competitive and non-competitive inhibitors are easily distinguished in a Lineweaver-Burk plot. The competitive inhibitor intercepts on the Mv axis whereas the non-competitive inhibitor intercepts on the 1/5 axis. The reaction of inhibitors with substrate can be assumed as a parallel reaction while the undesired product is formed along with desired product. The reactions are shown as ... 

According to this expression, a plot of 1/v, versus l/[SJo will yield a straight line if the data follow the Michaelis-Menten mechanism. This line has a slope given by Km/Vmax, a y intercept of 1/Vmax, and an x intercept of -1 fKm. This is also illustrated in Fig. 4-7. Again, this treatment is valid when Eq. (4-107) applies whether or not the catalyst is an enzyme. The Lineweaver-Burk plot, Fig. 4-lb, is convenient for visualization but statistically unreliable for data fitting the form in Eq. (4-107) should be used for numerical analysis. 

In attempting to determine if a given set of experimental data is of the same mathematical form as equation 7.3.29, there are three routes that permit the graphical determination of the parameters Vmax and K. The most frequently used plot is known as a Lineweaver-Burk or reciprocal plot. It is based on rearrangement of equation 7.3.29 into the following form. 

The useful thing about the Lineweaver-Burk transform (or double reciprocal) is that the y intercept is related to the first-order rate constant for decomposition of the ES complex to E + P (7ccat or Vmax) and is equal to the rate observed with all of the enzyme in the ES complex. The slope, in contrast, is equal to the velocity when the predominant form of the enzyme is the free enzyme, E (free meaning unencumbered rather than cheap). 

At low concentrations of substrate ([S] < Km), the enzyme is predominantly in the E form. The competitive inhibitor can combine with E, so the presense of the inhibitor decreases the velocity when the substrate concentration is low. At low substrate concentration ([S] < Km), the velocity is just Vmay IKm. Since the inhibitor decreases the velocity and the velocity at low substrate concentration is proportional to Vmax/Km, the presence of the inhibitor affects the slopes of the Lineweaver-Burk plots the slope is just the reciprocal of Vmax/Km. Increasing the inhibitor concentration causes Km/Vmax to increase. The characteristic pattern of competitive inhibition can then be rationalized if you simply remember that a competitive inhibitor combines only with E. 

The Lineweaver-Burk plot uses the reciprocal of the Michaelis-Menten equation in the form of the equation of a straight line, y = mx+ b, having the form shown in equation 2.3 ... 

Compare the Lineweaver-Burk linear forms resulting from the four cases in problem 10-8, together with those given by equations 10.3-2 and 10.4-11. Note which cases have the same... 

Equation (2) in form of Lineweaver-Burk plot can be written as... 

When data in the presence of an enzyme inhibitor are presented in the form of a Lineweaver-Burk plot, a series of straight lines should be obtained. The slopes of these hnes may or may not change, and the hnes may or may not intersect at a common point. The relationships between slopes, intersection points, and inhibitor mechanisms are outlined later. Further information regarding these mechanisms, including velocity equations describing data obtained in the presence of inhibitors with diverse mechanisms, can be found in (Segel, 1993). 

O Figure 4-10a shows a reaction scheme for interactions of enzyme and substrate with a full noncompetitive inhibitor. The inhibitor interacts with a site distinct from the active site, and the ESI complex is incapable of yielding product. It is thus possible, at saturating concentrations of inhibitor, to drive all enzymes to a nonproductive form, and so activity can be completely inhibited. Furthermore, the affinity of the inhibitor for the saturable allosteric inhibitory site remains independent of substrate concentration. A Lineweaver-Burk plot (O Figure 4-1 Ob) reveals a common intersection point on the 1/ [ S] axis for the data obtained at different inhibitor concentrations. It can be seen that as inhibitor concentration increases toward infinity, the slope of the Lineweaver-Burk plot increases toward infinity. Thus, a replot of the slopes versus inhibitor concentrations (O Figure 4-lOc) generates a straight line, which intersects the [i] axis at a value equal to —Ki. 

Partial uncompetitive inhibition does not resemble full uncompetitive inhibition in terms of having an ordered mechanism, but it instead represents a very specific form of partial mixed inhibition (discussed later). However, it is sometimes referred to as partial uncompetitive inhibition due to the parallel displacement of Lineweaver-Burk plots in the presence of inhibitor, and it is thus related to full uncompetitive inhibition in the same way that partial competitive inhibition is related to full competitive inhibition. 

The Lineweaver-Burk equation is a reciprocal form of the Michaelis-Menten equation. The same data graphed in this way yield a straight line as shown in Figure 1-8-6. The actual data are represented by the portion of the graph to the right of the y-axis, but the line is extrapolated into the Idt quadrant to determine its intercept with the x-axis. The intercept of the line with the x-axis gives the value of The intercept of the line with the y-axis gives the value of... 

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 ... 

D. Although it may seem from Point B in Figure 3-3 that the can be determined from this representation of the velocity data, in practice, it is more accurate to use the Lineweaver-Burk equation, a modified form of the Michaelis-Menten equation, for estimation of and (Figure 3-4). 

This form of the Michaelis-Menten equation is called the Lineweaver-Burk equation. For enzymes obeying the Michaelis-Menten relationship, a plot of 1/F0 versus 1/[S] (the double reciprocal of the V0 versus [S] plot we have been using to this point) yields a straight line (Fig. 1). This line has a slope of Km/Vmax, an intercept of 1/Fmax on the 1/F0 axis, and an intercept... 

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... 

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. AA/mm. Convert all AA/mm units to p,moles/min as desenbed in part B. Prepare a Michaelis-Menten curve (jumoles/min vs. [S]) as in Figure E5.1 and a Lineweaver-Burk plot (1/p.mole/min vs 1/[S]) as in Figure E5.2. Alternatively, you may wish to use the direct linear plot. Estimate and V max from each graph. The intercept on the rate axis of the Lineweaver-Burk plot is equal to UVm3zr For example, if the line intersects the axis at 0.02, then Vmax = 1/0.02 or 50 panoles of product formed per minute. The line intersects the 1/[S] axis at a point equivalent to — /Ku. If the intersection point on the 1/[S] axis is —0.67, then AM = -1/ -0.067 = 15 p-molar. Repeat this procedure for the data obtained for D-dopa. Compare the KM and max va ues and explain any differences. 

The value of K, is equal to the concentration of a competitive inhibitor which gives an apparent doubling of the value of Km. Graphically, a form of the Lineweaver-Burk plot"7 is used (see Section 5.4.6). ... 

The inhibition effect of poly (vinyl alcohol) on the amylose hydrolysis was investigated. Figure 7 shows Lineweaver-Burk plots of the amylose hydrolysis rates catalyzed by the random copolymer in the presence of poly (vinyl alcohol). The reaction rate is found to decrease with increasing the concentration of poly (vinyl alcohol), and all of the straight lines obtained in the plots cross with each other at a point on the ordinate. This is a feature of the competitive inhibition in the enzymatic reactions. In the present reaction system, however, it is inferred to suggest that the copolymer and poly (vinyl alcohol) molecules competitively absorb the substrate molecules. The elementary reaction can be described in the most simplified form as in Equation 3 where Z, SI, and Kj[ are inhibitor, nonproductive complex, and inhibitor constant, respectively. Then the reaction rate is expressed with Equation 4. 

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