Initial rates, measuring

Show how a rate constant can be determined from an initial rate measurement, if the reaction order is known. 

This simple example of a non-catalytic reaction demonstrates how a reaction rate law may be comprehensively defined in two substrates by just two reaction progress experiments employing two different values of excess [e]. A classical kinetics approach using initial rate measurements would require perhaps a dozen separate initial rate or pseudo-zero-order experiments to obtain the same information. 

Initial Rate Measurements. Another differential method useful in the determination of reaction rate expressions is the initial rate approach. It involves a series of rate measurements at different initial reactant concentrations but restricted to very small conversions of the limiting reagent (5 to 10% or less). This technique differs from those discussed previ-... 

Convince yourself that (a) this equation reduces to (1.104) when initial-rate measurements, [P] = 0, are used and (b) the same form of equation results when a number of intermediates arises as in... 

Then, k equals the observed reaction rate divided by the initial reactant concentration i.e., k = Vinitiai/[Ainitiai])-This method is most useful when one has an assay method that is sufficiently sensitive to ensure that only a small fraction, say 3-5%, of the reactant is depleted during the rate measurements. Typically, this is satisfactorily achieved with a UV-visible spectrophotometer, a fluorescence spectrometer, or a radioactively labeled reactant. The initial rate method is extremely convenient, and the preponderance of enzyme rate data has been obtained by initial rate measurements. Finally, one should note that the initial rate method can yield erroneous results if the initial reactant concentration is in doubt. This is not true for the plots of ln ([Ao] - [At]/([Ao] -[Aoo]) versus reaction time because one is considering the fraction of reactant A remaining. 

The initial rate assumption is one of the most powerful and widely used assumptions in the kinetic characterization of enzyme action. The proper choice of reaction conditions that satisfy the initial rate assumption is itself a challenge, but once conditions are established for initial rate measurements, the kinetic treatment of an enzyme s rate behavior becomes much more tractablek In reporting initial rate data, investigators would be well advised to provide the following information ... 

Provide a reference or direct experimental proof that the conditions chosen do provide initial rate measurements. At a minimum, the percentage of substrate consumed during the course of an initial rate determination should be specified. One should also show that under initial rate conditions a doubling of enzyme concentration should exactly produce a doubling in the observed initial rate. Likewise, if an auxiliary enzyme assay is used to monitor the primary enzyme s activity the observed rate at low or high substrate concentration should not depend on the concentration of additional enzyme(s), substrate(s), or factors used for the coupled assay. 2. Describe all assay conditions (eg., concentrations of substrates, products, inhibitors, and/or activators enzyme concentration temperature pH and buffer composition ... 

A parameter that depends on the concentration of another substrate in a multisubstrate reaction or on one or more cofactors or substances that influence reaction rate. It is an approximation of the true Michaelis constant. 2. A parameter obtained under conditions that do not rigorously satisfy the requirements of initial rate measurements. 

In cases where an acceptable kinetic mechanism can be established, it may be possible to obtain expressions, such as those in Sect. 2, which predict concentration changes with time when the values for the rate coefficients are known. However, the use of these expressions to evaluate rate coefficients from experimental data is not always straightforward, particularly with coupled reaction systems where a key reactant participants in a reversible step. Initial rate measurements are often of insufficient accuracy and, with very complex sj stems, it becomes necessary to obtain a great deal of data from experiments in which initial concentrations can be varied. 

One possible problem peculiar to a quantitative study of the inhibition of oxidation of aromatic hydrocarbons by zinc dialkyl dithiophos-phates is that peroxide decomposition could yield a phenol during the initial-rate measurement. Rate curves for the zinc diisopropyl dithio-phosphate-inhibited oxidation of cumene are shown in Figure 7. In the initial presence of hydroperoxide the uninhibited rate is never reached, and the reaction soon exhibits autoinhibition, presumably caused by the... 

Computer programs for ping-pong bi-bi kinetics which use ee-values measured at several degrees of conversion are available (Anthonsen, Hoff and Anthonsen, 1996 Rakels, Straathof and Heijnen, 1993). If both enantiomers are available in pure forms, it is also possible to determine E and K, from initial rate measurements. 

G.L. Schott, "Chain Branching and Initiation Rates Measured by Spatially Integrated Light Emission During Reflected Shock-Wave Ignition , Ibid, pp 569-78... 

Detailed account of kinetics and inhibition of pancreatic lipase action on emulsified triacylglycerol, including special treatment of initial rate measurements and pKa of liberated fatty acids. 

An important inhibitor of alkaline phosphatase is Pi, normally a very effective competitor with an affinity comparable to that for good substrates (107, 117). In consequence the kinetics of hydrolysis are often approximately first order whatever the substrate concentration, and for this reason initial rate measurements should be limited to 10% hydrolysis. With poor substrates, e.g., phosphocreatine (113) or o-carboxyphenyl phosphate (176), additional care is required. Arsenate is an even more powerful competitive inhibitor (101a, 118) and can prevent the incorporation of Pi by intestinal phosphatase at pH 5 (169). Phosphonates have been reported as weak inhibitors of intestinal phosphatase (120), and one has been used recently as a chromophoric probe for E. coli phosphatase (177). 

A large-scale enzymic synthesis of the trisaccharide a-D-Neu5Ac-(2—>3)-j8-D-Gal-( 1 — 3)-d-G1cNAc (65), the tumor-associated carbohydrate antigen CA 50, has been achieved.113 This is a further illustration of the cross reactivity and efficiency of STB, This sialyltransferase, like STA, was partially purified from porcine liver, according to a modification of Conradt s procedure. 114 The initial rate measured for j8-d-Gal-( 1 - 3)-d-G1cNAc at saturating concentration was 18% of the one measured for the real substrate / -d-... 

Deactivation of the enzyme catalyst often goes unnoticed, especially if the biocatalyst is fairly stable and/or the time of observation is short, such as during an initial rate measurement. If we analyze the short-time behavior of Eq. (17.23) we can replace the exponential by a linear term deviating from unity. At short times, exp (-kd obs t) 1 - kdobs t, and Eq. (17.24) is recovered. 

Initial rate measurements of the system trimethylantimony vs antimony trichloride indicate the following first-step reaction to be first order in each component ... 

Even though the loss of L2(H20)Rh002+ was monitored under both sets of conditions, the stoichiometric factors obtained are different for the two types of measurements. The rate constant provided the information on the disappearance of HN02, whereas the initial rates measured the disappearance of L2(H20)Rh002 +. 

Combining the effects due to substrate binding and transition state stabilization, this parameter is useful for assessing altered substrate specificity. Differences in log(kcJKm) provide an accurate measure of the lowering of the transition state activation energy (AG T ). The kcat, Km, and keJKm values were determined from the initial rate measurements and the results are listed in Table 3 and graphically represented in Figure 4. 

The kinetic study by Hinshelwood and Green cited in Problem 7.8 also included initial rate measurements over a range of partial pressures. ... 

Another technique of the differentiation method is the initial rate measurement. A series of experiments are carried out for different initial concentrations over a short time period (5 to 10% or less conversion). This approach is different from the experimental run discussed in Figure 5.7. Each rate measurement requires a new experiment with a different initial concentration. The initial rate of the reaction is determined from the curve of the concentration vs. time, as shown Figure 5.9a. The log of the initial rate is then plotted against the log of the initial concentration (Figure 5.9b). If the order of the reaction calculated from the concentration-time curve is different from the one determined by initial rate experiments, interference by the reaction products is expected, leading to complex reaction kinetics. 

FIGURE 5.9 Method of initial rate measurement for determining the order of reaction. 

The behavior of the reaction rate as a function of temperature dispels any notion that the reaction is simple. Figure 3 shows that there is a maximum in the first-order rate constant-temperature curve at approximately 80 °C. At such a low temperature, the rate-temperature maximum cannot be explained by depolymerization, nor can it be explained by deactivation of the catalyst as a result of more rapid polymer accumulation on the catalyst at higher temperatures since the maximum is obtained for initial rates measured as a function of temperature. Theoretical considerations predict that a maximum in the rate-temperature curve may be expected from the Langmuir-Hinshelwood model for polymerization on solid surfaces but not from the Rideal model (5). The rate of reaction for the Langmuir-Hinshelwood model is given by ... 

In order to provide a better estimate of the enantioselectivity of the catalyst, we prepared an authentic sample of (+)-chorismate by kinetic resolution of the racemate with 1F7 (37). Circular dichroism spectroscopy confirmed the identity and high optical purity of the recovered, HPLC-purified compound. Initial rate measurements with the individual isomers show that (-)-chorismate is favored over (+)-chorismate by the antibody by a factor of at least 90 to 1 at low substrate concentrations. The slight rate enhancements above background observed for the (+)-isomer may be due to general medium effects rather than interaction with a specific locus on the antibody surface. To test this possibility we are currently examining the ability of the transition state analog 3 to inhibit rearrangement of this optical isomer. 

A number of questions might be addressed in the discussion of the results. How reproducible are the initial rate measurements (Note that runs D5 and E3 are duplicates also, runs El and the standard assay for enzyme activity have identical initial concentrations.) Are the enzyme-catalyzed data compatible with the Michaelis-Menten mechanism Do the data from both runs C and D follow apparent zero-order kinetics, and how does this agree with expectations based on comparing (S) with KJ Which of the two types of analysis, Lineweaver-Burk or Eadie-Hofstee, seems to give the better results and why How does 2 agree with the estimate of the turnover number based on the specific activity Are the acid-catalyzed data consistent with the rate law given in Eq. (10) ... 

A more recent examination of the kinetics of this enzyme by initial rate measurements has included product inhibition patterns and has led to the conclusion that at least under some conditions an ordered bi-bi mechanism applies which involves a ternary complex of enzyme, NAD, and dihydrolipoamide (157). Clear spectral evidence is presented for the existence of a complex between NAD and the oxidized enzyme and this will be discussed in Section III,E. The product inhibition pattern for NAD tended toward that expected for this mechanism only at high NAD concentration. 

A kinetic study has been reported recently for the nitrosation of many symmetrical tertiary amines in aqueous acid-acetate buffers (Gowenlock et al., 1979). One experimental difficulty in tertiary amine nitrosations generally is that the reactions are much slower than for the analogous secondary amines, and are more conveniently studied at a higher temperature, typically 75°C, where the decomposition of nitrous acid is quite rapid. In this study, rate constants were obtained from the less accurate method of initial rate measurements. Nevertheless, for acidities less than pH 3.1, rate eqn (16) was established. The acid dependence is complicated by the protonation of the... 

In spite of the elimination of formic acid in a couple of steps changing the oxidation number of the rhodium metal center from -nl to -i-3 and vice versa, the reaction could take place by an alternative mechanistic pathway via cr-meta-thesis between the coordinated formate unit and the nonclassical bound hydrogen molecule [48, 49]. Initial rate measurements of a complex of the type 13 show that kinetic data are consistent with a mechanism involving a rate-limiting product formation by liberation of formic acid from an intermediate that is formed via two reversible reactions of the actual catalytically active species, first with CO2 and then with H2. The calculations provide a theoretical analysis of the full catalytic cycle of CO2 hydrogenation. From these results s-bond metathesis seems to be an alternative low-energy pathway to a classical oxidative addition/reductive elimination sequence for the reaction of the formate intermediate with dihydrogen [48 a]. 

When making initial rate measurements with coupled enzyme systems, there is often a significant lag time during which the linkage products build up to steady-state concentrations.6 Remember that some dehydrogenase reactions possess unfavorable equilibria lactate dehydrogenase, for example, which catalyzes the reaction shown in Eq. 3.12 ... 

Bioluminescence methods are distinct from absorbance and fluorescence methods because the light measured is now transient rather than steady state. Because of this, initial rate measurements yield a constant level of detected light, rather than the... 

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