Secondary plots

Substrate and product inhibitions analyses involved considerations of competitive, uncompetitive, non-competitive and mixed inhibition models. The kinetic studies of the enantiomeric hydrolysis reaction in the membrane reactor included inhibition effects by substrate (ibuprofen ester) and product (2-ethoxyethanol) while varying substrate concentration (5-50 mmol-I ). The initial reaction rate obtained from experimental data was used in the primary (Hanes-Woolf plot) and secondary plots (1/Vmax versus inhibitor concentration), which gave estimates of substrate inhibition (K[s) and product inhibition constants (A jp). The inhibitor constant (K[s or K[v) is a measure of enzyme-inhibitor affinity. It is the dissociation constant of the enzyme-inhibitor complex. 

Figure 2. Inhibition of eel AChE by ANTX-A(S) - the secondary plot. P, the first-order rate constant which was the rate of inhibition at that ANTX-A(S) concentration obtained from the primary plot (insert). The intercept on the 1/P axis is 1/k and the intercept on the 1/[I] axis is -1/K. Figure insert Progressive irreversible inhibition of eel AChE by ANTX-A(S). The inactivation followed first-order kinetics. ANTX-A(S) concentrations, xg/mL (A) 0.083 ( ) 0.166 (o) 0.331 ( ) 0.497 (V) 0.599 ( ) control. Each point represents the mean of 3 or 4 determinations.

These values are then plotted against 1/C . The slope of this linear secondary plot provides k, while the intercept gives the value of I /k 2 + 1/ 2 2- It follows that the two rate constants k 2 and k22 may not be derived separately from this type of experiment. The same is, of course, true for the two Michaelis constants. One has to know the value of one of them independently, or at least know that one is much larger than the other. Dealing with redox enzymes, the variations of the intercept in a series of cosubstrates of increasing reducing power may be used to solve the problem. Indeed, if for the most reducing cosubstrates, the intercept becomes independent of the cosubstrate, one is entitled to conclude that it represents the value of l/k t2. The procedure is illustrated with an experimental example in the next section. 

As in the homogeneous case, expression of the plateau current in equation (5.20) gives a simple representation of the competition between substrate and cosubstrate in the kinetic control of the enzymatic reaction. Equation (5.19) suggests the construction of primary and secondary plots allowing the derivation of the kinetic constants, as will be shown in the next section. 

FIGURE 5.26. Antigen-antibody construction of a monolayer glucose oxidase electrode with an attached ferrocenium cosuhstrate and cyclic voltammetric response in a phosphate buffer (pH 8) at 25°C and a scan rate of 0.04 V/s. a Attached ferrocene alone, h Addition of the substrate, c Primary plots, d Secondary plot. The numbers on the curves in parts h and c are the values of the substrate concentration in mM. Adapted from Figure 2 in reference 24, with permission from the American Chemical Society. 

From the intercept of the secondary plot shown in Figure 5.25c it follows that = 5 x 10-11 mol cm 2 s 1. Tg may be derived from an experi-... 

When plotted on double reciprocal axes, inhibitor data for full inhibitory mechanisms cannot be distinguished easily from those for partial inhibitory mechanisms. However, with suitable data, careful inspection of Lineweaver-Burk plots may reveal subtle differences these become clear in secondary plots (replots) of slopes or intercepts, as shown later. The use of Ks rather than Km (later) reflects the convention employed by Segel (1993) as has been discussed earlier, this dissociation constant provides a good indication of the value of Km if rapid equilibrium conditions exist. 

Rule 3. Both Rules 1 and 2 are applied if the reversible inhibitor binds to more than one enzyme form. Secondary plots can be nonlinear. See Abortive Complexes... 

This linearization of the tight-binding scheme allows the investigator the opportunity to calculate values for [Etotai] and Ki, the dissociation constant for the inhibitor. In the Henderson plot, [Itotai]/(l v/Vo) is plotted as a function of vjv where Vq is the steady-state velocity of the reaction in the absence of the inhibitor. The slope of the line is the apparent dissociation constant for the inhibitor. Secondary plots (from repeating the inhibition experiment at different substrate concentrations) will yield the Ki value. The vertical intercept is equal to [Etotai]- Hence, repeating the experiment at a different concentration of enzyme will produce a parallel line. 

Secondary plots of kinetic data are used to obtain various rate constants and other kinetic parameters such as and Vmax- To simply the analysis, one choses a algebraic transform of the rate equation that allows the observed data to be graphed in a hnear format. 

In the examples above, the secondary plots utilized the slopes or intercepts of the original plot. However, replots are secondary plots for any functional dependency using data obtained from a primary graphing procedure. Secondary replots can also be used with inhibition studies. In these cases, the slope or intercept of a double-reciprocal plot is graphed as a function of the inhibitor concentration. 

From a set of these secondary plots, V( and one of the Michaelis constants can be determined (Fig. 9-6B and C). Using two sets of secondary plots, all of the constants of Eq. 9-44 may be established. Alternatively, a computer can be used to examine all of the data at once and to obtain the best values of the parameters. The latter approach is desirable because estimates of the standard deviations of the parameters can be obtained. However, the user must take care to ensure that the experimental errors are correctly estimated and are not simply estimates of how well the computer has fitted the points on the assumption that they contain no error.23... 

Figure 9-6 Reciprocal plots used to analyze kinetics of two-substrate enzymes. (A) Plot of 1 / against 1 / [A] for a series of different concentrations of the second substrate B. (B) A secondary plot in which the intercepts from graph A are plotted against 1/ [B], (C) Secondary plot in which the slopes from graph A have been plotted against 1 / [B]. The figures have been drawn for the case that Kmp = 10 3 M, Kun, = 2 Km, and K B = KeqAKmB (Eq. 9-46) = KmJ 200 and [A] and [B] are in emits of moles per liter. Eadie-Hofstee plots of v / [A] vs vf at constant [B] can also be used as the primary plots. The student can easily convert Eq. 9-44 to the proper form analogous to Eq. 9-21.

Second order reactions 458 Secondary kinetic isotope effect 592, 600 on fumarate hydratase 684 Secondary plots for kinetics of multisubstrate enzymes 465 Secondary structure 63... 

Secondary plot for evaluation of constants for filtrate-rate equation in Example 5. 

Under certain conditions, it is possible to determine the ATa and 2 components of ki separately. For example, if the secondary plot is not linear, this is an indication of an appreciable concentration of a Michaelis-type complex. In this case, the term must be included by combining Eq. (57.3) and Eq. (57.4) and rearranging, we have... 

The mobility of a protein of unknown mass is compared with that of a series of proteins of broadly similar shape but known mass. The mobility of each protein (known and unknown) is determined at a series of acrylamide concentrations, and for each protein a plot of log Rf vs Cf is made (called a Ferguson plot), from whose slopes values of JCR are determined (Figure 4-26 ). In practice, five or more acrylamide concentrations would be needed for reliable results. A secondary plot of the derived values of KR against Mr for the standard proteins then enables the native molecular weight of the unknown protein to be determined. 

With these provisos, buffer catalysis is generally measured from gradients of plots of observed first-order rate constants against buffer concentration the separation of the second-order buffer catalysis constant into catalytic constants for acid and basic components of the buffer is achieved by secondary plots of these second-order constants against buffer ratio. This is illustrated in Figure 1.17. 

Fig. 8.8 Polyacrylamide-gel electrophoresis of the tetrameric form (G4) of human butyrylcholinesterase in microcapillary gel tubes with different gel concentrations, T. Gel patterns for electrophoresis at 2 kbar and 20°C for (a) butyrylcholinesterase, and (b) butyrylcholinesterase in the presence of 2 m sorbitol (the gels were stained for enzyme activity according to the procedure of Karnovsky and Roots ). The numbers indicate the acrylamide concentrations T. (c) Ferguson plots constructed for butyrylcholinesterase at different pressures (o), atmospheric pressure, ( ) 0.5 kbar, ( ) 1 kbar, (A) 1.5 kbar, and ( ) 2 kbar. (d) A secondary plot of /Cr against pressure for ( ) bovine serum albumin, ( ) butyryicholinesterase, and ( ) butyrylchoiinesterase in the presence of 2 m sorbitol, (e) A secondary plot of log Vo against pressure for butyrylchoiinesterase.

Fig. 12. Secondary plots of 2-substrate initial-rate data. From a set of primary plots such as is schematically illustrated in Fig. 11, one obtains a set of slopes and intercepts. The replot (a) of slopes, S, against 1 /[B] yields estimates of 4 and and the replot (b) of intercepts, 7, gives < and 0g. If Eqn. 16 is a valid description of the kinetic behaviour over the substrate range investigated and if the experimental data are adequate, these 4 constants then allow calculation of the initial rate for any combination of substrate concentrations over the range for which Eqn. 16 is obeyed.

This brings us to the final mechanism we need to consider for a 2-substrate reaction, namely a random-order mechanism. We have assumed that we would be alerted to the possibility of a steady-state random-order mechanism by non-linear primary or secondary plots, but it is possible to get linear kinetics with a random-order mechanism. If we make the assumption that the further reaction of the ternary complex EAB is much slower than the network of reactions connecting E to EAB via EA and EB, then there are only 4 kinetically significant complexes and their concentrations are related to one another by substrate concentrations and dissociation constants. This is the rapid-equilibrium random-order mechanism, and the assumption made is analogous to the Michaelis-Menten equilibrium assumption for a 1-substrate mechanism. 

Thus, when SI is plotted as a function of [I], we obtain the results shown in Fig. 5-30. The intercept of this secondary plot on the abscissa is -K, hence Ki is estimated to be 0.5 mmol L" . The intercept on the ordinate is KJV. Since we have already estimated then K can be calculated to be 0.1 mmol L. ... 

Fig. 3.6 Secondary plots for the determination of kinetic parameters. Cl competitive inhibition NCI non-competitive inhibition UCl uncompetitive inhibition MTl mixed-type inhibition...

In the case of ordered mechanism, V is directly obtained from the 1/v versus 1/b plot. Straight lines in the 1/v versus 1/a plot will intersect at a point that is easily demonstrated to correspond to 1/Ka. However, all kinetic parameters can be obtained from secondary plots as shown in Fig. 3.91. In the case of random mechanism, as in ordered mechanism, straight lines in the 1/v versus 1/a plot will... 

Fig. 3.9 Secondary plots for the determintition of kinetic parameters in sequential mechanisms. I ordered 11 random...

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