Obtaining Initial Rate Data

The significance of obtaining rate data for the study of enzymes has been discussed elsewhere, and the reader is referred to the General References for additional information. Although usually relevant to the in-depth study of the mechanism of an enzyme reaction, such concerns are beyond the scope of the present discussion. Of concern in this text are the problems associated with obtaining initial rate data with the HPLC assay method. 

The experiments described in Section 4.2.1 will yield values for two parameters the amount of the enzyme required to form sufficient detectable product and the incubation time required to form this amount of product. Additional experiments now are required to refine the values of both parameters. Keep in mind that to generate the straight line needed to obtain the initial rate, 

Once a suitable concentration of enzyme has been established so that three or four samples can be analyzed, the quantitative data can be obtained. A reaction is started, the reaction mixture is sampled at intervals, chromatograms are obtained, and the amount of product formed is determined directly from the chromatogram by means of either peak height or electronic integration of the peaks. These values should be plotted as a function of reaction time. 

a second and third series of reaction mixtures should be prepared, with enzyme added at concentrations of half and twice the value used in the first. These reactions are started and sampled, chromatograms are obtained, and the data are plotted as a function of reaction time. At this early stage in the optimization of the assay, it is advisable to continue sampling one of the incubations until the rate of product formation becomes nonlinear or the amount of substrate present is exhausted. This prolonged incubation provides information about the extent of the primary reaction and also allows any secondary reactions to take place and form enough products to be detectable. 

The chromatograms used to obtain the initial rate data described in Section 4.2.2 may be examined to provide information on the fate of the substrate during the course of the incubation. For example, assuming that no other reactions involving the substrate have taken place, the amount of product that was formed should equal the amount of substrate that was lost. It is important to determine this point first. 

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Very recently, Luckner et al. (116) obtained initial rate data for the metathesis of propene using the W0r-Si02 catalyst at flow rates where mass transfer effects were found to be negligible. Their experimental data referring to measurements at 0.1 to 0.9 MNm-2 and 672 to 727 K could be correlated by Eq. (53). 

Benzyl alcohol can be produced from benzaldehyde (S) by a dehydrogenation reaction catalyzed by yeast alcohol dehydrogenase (YADH). Nikolova et al. (1995) obtained initial-rate data for this reaction using immobilized YADH immersed in iso-octane with 1% v/v water. The following data were obtained ... 

The HPLC assay method is particularly useful when it is necessary to obtain initial rate data for a study of an enzymatic activity. Optimal assay conditions for the HPLC must be established first. Usually, the optimization process involves the determination of several variables, such as the optimal substrate concentration, pH, temperature, and enzyme concentration. It is assumed that the reader is familiar with the problems associated with assay conditions such as pH, buffer, and temperature. This chapter discusses only factors that might present problems for the HPLC assay method. For additional information, see the works cited in the General References. 

Obtaining initial rate data is, of course, a first step in the kinetic analysis of an enzyme-catalyzed reaction, and the reader is referred to the General References for several reviews and monographs describing the methods for this analysis. 

To obtain initial rate data, it is necessary to have a minimum of three samples of a reaction carried out under optimized conditions. The formation of product at several enzyme and substrate concentrations will yield data for the calculation of kinetic constants. 

Sinfelt et al. (1960) obtained initial rate data using a differential reactor for the dehydrogenation of methylcyclohexane (Af) over a 0.3% Pt-Al203 catalyst in the presence of H2 to reduce coking. The reaction is ... 

Casado et al. have analyzed the error of estimating the initial rate from a tangent to the concentration-time curve at t = 0 and conclude that the error is unimportant if the extent of reaction is less than 5%. Chandler et al. ° fit the kinetic data to a polynomial in time to obtain initial rate estimates. 

In contrast to consecutive reactions, with parallel competitive reactions it is possible to measure not only the initial rate of isolated reactions, but also the initial rate of reactions in a coupled system. This makes it possible to obtain not only the form of the rate equations and the values of the adsorption coefficients, but also the values of the rate constants in two independent ways. For this reason, the study of mutual influencing of the reactions of this type is centered on the analysis of initial rate data of the single and coupled reactions, rather than on the confrontation of data on single reactions with intergal curves, as is usual with consecutive reactions. 

On the basis of regression analysis of the initial rate data obtained for both isolated reactions (Vila) and (Vllb) and for each of the three reactions proceeding in the coupled system [reactions (Vlla)-(VIIc)], of the set of twenty-five equations, the best equation was always found to be of the... 

The catalyst consists of 3-mm pellets that pack to a bulk density of 1350 kg/m and = 0.5. Mercury porosimetry has found 7 ore = 5nm. The feed mixture to a differential reactor consisted of 5mol% SO2 and 95mol% air. The following initial rate data were obtained at atmospheric pressure ... 

When asked to determine a rate law and rate constant, we must determine the order of the reaction. The rate law for this reaction may contain the concentrations of NO2, F2, and NO2 F raised to powers x, y, and z that must be determined Rate = k [N02] [F2] [NO2 F] Because the rate law contains more than one species, we need to use either isolation or initial rates to determine the orders of reaction. In the experiments whose data are shown, initial rate data are obtained for various combinations of initial concentrations. We apply the ratios of these initial rates to evaluate the orders. 

Mardaleishvilli, Sin-Chou, and Smorodin-skaya [Kinetics and Catalysis, 8 (664), 1967] have studied the catalytic decomposition of ammonia on quartz. The following initial rate data were obtained by these investigators at 951 C... 

For the solid catalyzed reaction, 2A B+C, initial rate data were obtained as tabulated, starting with pure A. Consider these three reaction mechanisms,... 

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. 

Figure 8.9 shows that the concentration of intermediate in reversible series reactions need not pass through a maximum, while Fig. 8.10 shows that a product may pass through a maximum concentration typical of an intermediate in the irreversible series reaction however, the reactions may be of a different kind. A comparison of these figures shows that many of the curves are similar in shape, making it difficult to select a mechanism of reaction by experiment, especially if the kinetic data are somewhat scattered. Probably the best clue to distinguishing between parallel and series reactions is to examine initial rate data—data obtained for very small conversion of reactant. For series reactions the time-concentration curve for S has a zero initial slope, whereas for parallel reactions this is not so. 

Aside from the time required to prepare reagents, the least amount of time is required per lipase assay by the spectrophotometric method, and the greatest amount of time is required per assay for the titrimetric method. Although all assays are described as requiring up to 30 min for the reaction mixture to be subsampled, time savings can be realized by subsampling more frequently over a shorter period of time, as long as one obtains valid initial rate data. Thus, for all assays, the time involved to run the lipase reaction can be normalized to be the same at -10 to 15 min. The difference in time requirements for the protocols becomes embedded in sample workup procedures. 

From a series of batch runs with a constant enzyme concentration, the following initial rate data were obtained as a function of initial substrate concentration. 

The classical procedures used by the chemist or engineer to obtain polymerization rate data have usually involved dilatometry, sealed ampoules, or samples withdrawn from model reactors—batch, tubular, and CSTR s alone or in various combinations. These rate data, together with data on molecular weight can be used to obtain the chain initiation constant and certain ratios such as kp2/kt and ktr/kp. Some basic relationships are shown in Figure 5. To determine individual rate constants such as kp and kt, other techniques are needed. For example, by periodic photochemical initiation it is possible to obtain kp/kt. If the ratio kp2/kt (discussed above) is also known, kp and kt can each be calculated. Typical techniques are described by Flory (20). 

In the first approach, we examine the rate progress curves at various substrate concentrations, and use linear regression to evaluate initial rates. These initial rates are then fitted to the Michaelis-Menten equation (Eqn. 9.14) (Exercise 3 Michaelis-Menten kinetics I). This method has the advantage of being simple and robust. It has the disadvantage that the choice of data points used to obtain initial rates is often arbitrary, and also that the progress curves at low substrate concentrations show marked curvature because of substrate depletion. 

Butler obtained the initial rate data given in the following table in a manner analogous to that illustrated in Example 3.5.1. Show that a reaction rate expression that follows Rules III and IV can be used to describe these data. 

For the generic reaction A=>B the following three reaction rate expressions were proposed to correlate the initial rate data obtained. Describe how a differential tubular reactor could be used to discriminate among these models ... 

If initial rate data are obtained, and if there is no B in the feed stream, then the concentration of B at low conversion is small. Thus, at these conditions the rate expressions are ... 

For elucidation of mechanisms, rate data at very low conversions—that is, initial rates—may be highly desirable. They can be obtained more easily from a batch reactor than from a continuous stirred-tank or plug-flow tubular reactor. The reason is that for both of the latter, very high flow rates would be required, or the tubular reactor would have to be equipped with a sampling port near its inlet, a mechanical complication apt to perturb the flow pattern. On the other hand, if the reaction can be confined to a very small flow reactor, a differential reactor operated once-through may be an even better choice for obtaining initial rates (see also Sections 3.1.3 and 3.1.4 farther below). 

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.

Selection of Equations and Evaluation of Constants. Several techniques making use of initial rates have been employed in the interpretation of experimental data to select the proper equation for a given reaction. To use this method, initial rate data have to be obtained at several different pressures. If Eq. (24) is rewritten for the initial rate (zero conversion),... 

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