Kinetics double reciprocal plot

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. 

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 ... 

Figure 3. Double reciprocal plots of the kinetics data obtained for the decay of the transients seen for the short wavelength flash photolysis (A rr > 315 nm) of THF solutions of Ru3(CO) 2 in the presence of various ligands L (from reference 5).

The second intermediate s identity has been debated since the mid-1980s. In 1984, Liu and Tomioka suggested that it was a carbene-alkenc complex (CAC).17 Similar complexes had been previously postulated to rationalize the negative activation energies observed in certain carbene-alkene addition reactions.11,30 A second intermediate is not limited to the CAC, however. In fact any other intermediate, in addition to the carbene, will satisfy the kinetic observations i.e., that a correlation of addn/rearr vs. [alkene] is curved, whereas the double reciprocal plot is linear.31 Proposed second intermediates include the CAC,17 an excited carbene,31 a diazo compound,23 or an excited diazirine.22,26 We will consider the last three proposals collectively below as rearrangements in the excited state (RIES). 

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. 

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 =... 

A double-reciprocal plot of 1/v vx. 1/[A] at varying constant concentrations of B will be a series of straight lines having a common intersection point. A doublereciprocal plot of 1/v vx. 1/[B] at varying constant concentrations of A will also be a series of straight lines having a common intersection point. Secondary replots of the initial-rate data will yield values for the kinetic parameters. The overall steady-state rate expression is... 

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... 

Rapid Equilibrium Case. In the absence of significant amounts of product (i.e., initial rate conditions thus, [P] 0), the rate expression for the rapid equilibrium random Bi Uni mechanism is v = Uniax[A][B]/(i iai b + i b[A] + i a[B] + [A][B]) where is the dissociation constant for the EA complex, and T b are the dissociation constants for the EAB complex with regard to ligands A and B, respectively, and Umax = 9[Etotai] where kg is the forward unimolecular rate constant for the conversion of EAB to EP. Double-reciprocal plots (1/v v. 1/[A] at different constant concentrations of B and 1/v v. 1/[B] at different constant concentrations of A) will be intersecting lines. Slope and intercept replots will provide values for the kinetic parameters. 

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. 

DOUBLE-RECIPROCAL PLOT ENZYME KINETIC EQUATIONS (1. The... 

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. 

The first intensive investigation of the kinetics of step 2 was carried out by Herries et al. (496) in a study of C > p hydrolysis. The data gave linear double reciprocal plots and maximum velocities and Michaelis constants were measured as a function of pH. Similar studies on U > p have been carried out by others (497, 498), but these did not agree well with each other or with the later work of del Rosario and Hammes (499). In one case no indication was given of substrate purity and 1/15 M sulfate was employed (497). In the other product contamination was clearly a problem (498). 

Figure 1 Time- and concentration-dependent inactivation of the catalytic activity of P450 2B6 by bergamottin. Inactivation of the EFC O-deethylation activity of P450 2B6 in the reconstituted system incubated with 0.6 ( ), 1 (o), 2 ( ), 3 (O), 5 ( ), and 10 ( ) pM bergamottin. Aliquots were removed at the indicated time and assayed for residual activity. The insets show the double reciprocal plots of the initial rates of inactivation as a function of the bergamottin concentrations. The kinetic constants Kh inact, and f1/2 were determined from this plot. The data shown represent the average of three experiments that did not differ by more than 10%. Source From Ref. 72.

The inactivation of bovine Fi -ATPase by FSBA exhibited biphasic kinetics. A double reciprocal plot of the inactivation rate for the fast phase against FSBA concentration gave a curved line, whereas the same type of plot for the slow phase yielded a straight line, giving a Ka value of 0.23 mM. The slow phase was diminished when the enzyme was inactivated in the presence of 0.2 M phosphate. On complete inactivation, about three moles of FSBA bound to one mole of bovine Fi -ATPase. Regardless of the presence of phosphate, His-427 of the (8-subunit was dominantly modified at pH 6, whereas Tyr-368 of the /8-subunit was dominantly modified at pH 8. At pH 7, the two residues were modified in a similar ratio. 

Note The kinetic constant for activation with each cation and the catalytic parameters for S-o-nitrophenyl-L-cysteine (SOPC) (0.6 mM) were determined from double reciprocal plots at 25° C and pH 8.0. 

B that is the double reciprocal plots for a ping-pong kinetic mechanism are parallel lines—the slopes are not changed by altering the concentration of the second substrate. 

Figure 3-16. Relationship between the external solute concentration (c°) and the rate of influx (J n) for active uptake according to Michaelis-Menten kinetics, as given by Equation 3.28 (a) linear plot and (b) double-reciprocal plot.

---