Double reciprocal plots nonlinear

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). 

A procedure used to assist in identifying sequential mechanisms when the double-reciprocal plots exhibit parallel lines ". In some cases, bireactant mechanism can have various collections of rate constants that result in so-called parallel line kinetics, even though the mechanism is not ping pong. However, if the concentrations of A and B are kept in constant ratio with respect to each other, a sequential mechanism in a 1/v v. 1/[A] plot would be nonlinear (since in the denominator the last term of the double-reciprocal form of the rate expression contains [A] for example, for the steady-state ordered Bi Bi reaction scheme in which [B] = a[A], the double-reciprocal rate expression becomes 1/v =... 

Depressed catalytic activity occurring when an inhibitor binds more than once to a single enzyme form (or forms). While standard double-reciprocal plots are usually linear, secondary replots of the data (i.e., plots of slopes and/or intercepts vx. [I], the concentration of the inhibitor) will be nonlinear depending on the relative magnitude of the [I], [If,. .., and [If terms in the rate expression. 

Consider the standard Uni Uni mechanism (E + A EX E + P). A noncompetitive inhibitor, I, can bind reversibly to either the free enzyme (E) to form an El complex (having a dissociation constant K s), or to the central complex (EX) to form the EXl ternary complex (having a dissociation constant Xu). Both the slope and vertical intercept of the standard double-reciprocal plot (1/v vx. 1/[A]) are affected by the presence of the inhibitor. If the secondary replots of the slopes and the intercepts (thus, slopes or vertical intercepts vx [I]) are linear (See Nonlinear Inhibition), then the values of those dissociation constants can be obtained from these replots. If Kis = Xu, then a plot of 1/v vx 1/[A] at different constant concentrations of the inhibitor will have a common intersection point on the horizontal axis (if not. See Mixed-Type Inhibition). Note that the above analysis assumes that the inhibitor binds in a rapid equilibrium fashion. If steady-state binding conditions are present, then nonlinearity may occur, depending on the magnitude of the [I] and [A] terms in the rate expression. See also Mixed Type Inhibition... 

This term usually applies to reversible inhibition of an enzyme-catalyzed reaction in which nonlinearity is detected (a) in a double-reciprocal plot (i.e., 1/v versus 1/ [S]) in the presence of different, constant concentrations of inhibitor or (b) in replots of slope or intercept values obtained from primary plots of 1/v versus 1/[S]). Nonline-... 

This type of inhibition differs from that exhibited by classical competitive inhibitors, because the substrate can still bind to the El complex and the EIS complex can go on to form product (albeit at a slower rate) without the inhibitor being released from the binding site. While standard double-reciprocal plots of partial competitive inhibitors will be linear (except for some steady-state, i.e., non-rapid-equilibrium, cases), secondary slope replots will be nonlinear. See Nonlinear Inhibition... 

For non-rapid-equilibrium cases (i.e., steady-state cases) the enzyme rate expression is much more complex, containing terms with [A] and with [I]. Depending on the relative magnitude of those terms in the initial rate expression, there may be nonlinearity in the standard double-reciprocal plot. In such cases, computer-based numerical analysis may be the only means for obtaining estimates of the magnitude of the kinetic parameters involving the partial inhibition. See Competitive Inhibition... 

Except for very simple systems, initial rate experiments of enzyme-catalyzed reactions are typically run in which the initial velocity is measured at a number of substrate concentrations while keeping all of the other components of the reaction mixture constant. The set of experiments is run again a number of times (typically, at least five) in which the concentration of one of those other components of the reaction mixture has been changed. When the initial rate data is plotted in a linear format (for example, in a double-reciprocal plot, 1/v vx. 1/[S]), a series of lines are obtained, each associated with a different concentration of the other component (for example, another substrate in a multisubstrate reaction, one of the products, an inhibitor or other effector, etc.). The slopes of each of these lines are replotted as a function of the concentration of the other component (e.g., slope vx. [other substrate] in a multisubstrate reaction slope vx. 1/[inhibitor] in an inhibition study etc.). Similar replots may be made with the vertical intercepts of the primary plots. The new slopes, vertical intercepts, and horizontal intercepts of these replots can provide estimates of the kinetic parameters for the system under study. In addition, linearity (or lack of) is a good check on whether the experimental protocols have valid steady-state conditions. Nonlinearity in replot data can often indicate cooperative events, slow binding steps, multiple binding, etc. 

When appreciable cation-independent transferase activities were found (F3, HIO), double reciprocal plots of enzyme activity against concentration of added Mg + were nonlinear, but became linear when only the stimulated enzyme fractions were plotted. The apparent values so obtained were independent of the concentration of UDP-glucuronic acid, and vice versa (HIO), suggesting that Km for Mg + represents the dissociation constant of an Mg +-enzyme complex (D5). 

Fig. 2. The double-reciprocal plot of /v as a function of 1 / [5]. Two determinations of v for each [5] are shown. l/vmax = 0.87 and —1/Km = —4.05 (arbitrary units). Figure 1 is a nonlinear regression plot of the same data where umax = 1.00 and Km = 0.20 (arbitrary units).

Two other examples of sigmoidal reactions that are made linear by an activator include a report by Johnson et al. (31), who showed that pregnenolone has a nonlinear double-reciprocal plot that was made linear by the presence of 5 pM 7,8-benzoflavone, and Ueng et al. (23), who showed that aflatoxin B1 has sigmoidal saturation curve that is made more hyperbolic by 7,8-benzoflavone. As with the effect of quinine on carbamazepine metabolism, 7,8-benzoflavone is an activator at low aflatoxin B1 concentrations and an inhibitor at high aflatoxin B1 concentrations. 

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. 

Fitting the data directly to either Equation 5.14 or Equation 5.15 eliminates bias in the data imposed by reciprocal linear curve fitting. Figure 5.21 shows the use of nonlinear curve fitting to measure the affinity of the a-adrenoceptor agonist oxymetazoline in rat anococcygeus muscle after alkylation of a portion of the receptors with phenoxybenzamine. This data shows how all three curves can be used for a better estimate of the affinity with nonlinear curve fitting, a technique not possible with the double reciprocal plot approach where only two dose-response curves can be used. The use of three curves increases the power of the analysis... 

The nonlinear 2 2 function kinetics should be differentiated from other nonlinear kinetics such as allosteric/cooperative kinetics (Bardsley and Waight, 1978) and the formation of the abortive substrate complex (Dalziel and Dickinson, 1966). The cooperative kinetics (of the double reciprocal plots) can either concave up (positive cooperativity) or... 

Figure 4 shows the survey of different t)q)es of nonlinear hyperbolic inhibition mechanisms and their characteristics. A basic property of aU nonlinear mechanisms, shown in Fig. 4, is that the double reciprocal plot of i/uq versus /A, in the presence of different constant concentrations of an inhibitor is a family of straight lines with a common intersection point. This common intersection point is found either in the I, in, or in the IV quadrant, depending on the mechanism only in Case 5 (hyperbolic uncompetitive type), the double reciprocal plot is a family of parallel straight lines without a common intersection point. 

Thus, it is clear that the primary double reciprocal plots of i/Dq versus 1/A represent a valuable general diagnostic tool for the estimation of the type of nonlinear inhibition mechanisms. 

Similarly as for the nonlinear inhibition with a single substrate and a single inhibitor molecule (Section 6.1), the double reciprocal plots of i/vo versus 1/A or i/u versus 1/B, are a family of straight lines with a single intersection point to the left of the vertical axis. Also, the slope and intercept replots are hyperbolic. 

Compare Fig. 1 for the nonlinear inhibition of Chapter 6, and notice that the double reciprocal plot in both figures is a family of straight lines with a common intersection point this intersection point has different coordinates in activation and inhibition systems. 

In the literature, it is sometimes argued that linearisation of nonlinear functions in the era of personal computers is obsolete. Moreover linearisation of Michaelis—Menten kinetics through a double reciprocal plot transforms the region of zero order to a single point, which is not favorable for parameter estimation. 

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... 

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). 

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)]. 

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