Kinetics steady-state

In the almost 20 years since this volume s predecessor appeared enzyme kinetics has come of age. Extended theory and the availability of much more sensitive measuring equipment have made possible incisive kinetic analysis of multi-substrate enzymes. One must also add, however, that the full potential of the method has been achieved in rather few cases. Much of the published information has been collected by investigators primarily interested in function rather than mechanism, and is therefore of descriptive value only. Even when a more thorough-going analysis is attempted, it is often difficult and tedious to obtain enough data to remove all ambiguity. Hence doubt and controversy regarding the mechanisms of many important enzymes remain. In the space available here it is not possible to go into much detail about individual cases. The intention, therefore, is to sketch out current approaches and problems. 

Whilst detailed spectroscopic and kinetic data for APX have been very slow to emerge, largely as a result of relatively poor yields and (in some cases) instability of the purified enzyme from the early isolations (section 11), steady state data, on the other hand, have been a fairly prominent feature of most of the early literature on APX. Values of Km for H2O2 have been reported for a number of purified and partially purified APXs and are 

These conclusions are supported by a series of related experiments on the same recombinant pea cytosolic enzyme in which the steady state oxidation of / -cresol has also been found to exhibit sigmoidal kinetics (Celik et al., 1999). In this case, the data were satisfactorily fitted to the Hill equation, Eq. (2), 

Replacement of the Glull2 residue with an alanine group (E112A variant), which would similarly he expected to destabilize the dimer interaction, provides a less conclusive and slightly puzzling scenario. Hence, in contrast to the E122K variant, E112A is not monomeric under any of the experimental conditions examined. 

Initial rate measurements, especially with alternative substrates and with a product or substrate analog as inhibitor, and measurements of the rate of isotope exchange at equilibrium, can give a great deal of information about mechanism, and in some cases allow estimates of individual velocity constants and dissociation constants. The results of such studies, which require little enzyme, are an essential basis for the proper interpretation, in relation to the overall catalytic reaction, of pre-steady-state studies and kinetic and thermodynamic studies of enzyme-coenzyme reactions in isolation. 

Phenomenological Initial Rate Equations fob Two-Substrate and Thbee-Substbate Reactions (16) 

For many simple dehydrogenases which catalyze only reversible hydrogen transfer between a coenzyme and another substrate, it has been found 

Including the coenzyme but excluding water and the hydrogen ion. Dehydrogenases with flavin prosthetic groups are not considered. 

For glyceraldehyde-3-phosphate dehydrogenase, glutamate dehydrogenase, and the NADP-linked oxidative decarboxylases, which have three substrates in one direction, initial rate measurements with a fixed concentration of any one of the three substrates also conform to Eq. (1). This again rules out any form of enzyme-substitution mechanism in which free product is formed before all the substrates have combined with the enzyme and indicates the involvement of a quaternary enzyme complex. The appropriate generalized form of Eq. (1) is 

The first investigation of the CO oxidation on supported model catalysts has been performed by the Poppa-Boudart group at the beginning of the eighties [112]. Using a Pd/Al203(l 012) model catalyst, 

The effect of the reverse spillover in the oxidation of CO on supported model catalysts has been observed by several other authors on various systems Pd/mica [133], Pd/alumina [103,131, 132, 144, 163] Pd/MgO [45, 161], Pd/silica [104] it can increase the reaction rate by a factor as large as 10. 

More recently several studies have been performed using molecular beam techniques in our group and in the Freund s group on Pd/MgO(l 00) [45, 145-148] and on the Pd/Al2O3/NiAl(110) [46, 103, 153-157]. 

In the experiments the small particles (2.8 and 6.8 nm) appear more active than the largest ones (13 nm) especially in the medium range of temperature. First we have tried to take into account for the reverse spillover of CO by using Eq. (13). The reaction probability has been fitted from the universal curve given from experiments on various Pd extended surfaces [124]  

In conclusion we have seen that by decreasing particle size the activity of the CO particles changes but the observed effects have different origins  

The three basic steps in the polymerization process can be expressed in general terms as follows  

As the iiutial decomposition is slow compared with both the rate of addition of a primary radical to a monomer and the termination reaction, it is the rate 

Combination Combination Mainly combination Mainly disproportionation Mainly disproportionation Mainly disproportionation 

The quantum yield ( ) replaces / and defines the initiator efficiency is related to the incident light intensity / , the monomer concentration, and the extinction 

When the monomer is a poor absorber of radiation, small quantities of a photosensitizer may be added to absorb the energy and then transfer this to the monomer to create an active center. In this case, [M] is replaced by the concentration of photosensitizer. 

Institut F6d6ratif Biologie Structurale et Microbiologie, CNRS-BIP, 

31 chemin Joseph-Aiguier, B.P. 71,13402 Marseille Cedex 20, France 

1 Introduction to rate equations, first-order, second-order reactions etc. 

Nearly all steps that form parts of the mechanisms of enzyme atal5rsed reactions involve reactions of a single molecule, in which case they typically follow first-order kinetics  

In both cases v represents the rate of reaction, and a and b are the concentrations of the molecules involved, and k is a rate constant Because we shall be 

It might seem odd that the concentration of an intermediate can be treated as effectively constant during a reaction if it is absent at the beginning of the reaction, is there in the middle, and is absent again at the end. What is important for the SSA to be valid is that the absolute changes in concentration of the intermediate be small with respect to changes in the concentrations of the reactants and products. It is therefore easy to realize that if the concentration of the intermediate is always very small, the absolute changes in its concentration must also be very small. 

Many organic, organometallic, and bioorganic reactions involve intermediates that are indeed reactive and transient. Reactive intermediates such as carbocations, radicals, carban-ions, and carbenes are common to organic and bioorganic transformations, whereas coordi-natively unsaturated transition metals, and low and high oxidation state metals are common to organometallic reactions (see Chapter 12). 

Let s now consider a scenario where the same number of reactants are used as in the above example, but a different sequence is involved. Here the intermediate is formed in a second order reaction, and the intermediate converts to product in a first order reaction (Eq. 7.48). Eq. 7.49 expresses the rate of the reaction, and Eq. 7.50 expresses the SSA. Solving Eq. 7.50 for [I] leads to Eq. 7.51, which upon substitution into Eq. 7.49 gives Eq. 7.52. Eq. 7.52 has several rate constants incorporated into a product and quotient, which taken together is a constant that we call fcobs- This mechanistic scenario predicts that the reaction is first order in A and B, distinctly different than that presented in the last mechanistic scenario. This comparison reveals the power of a kinetic analysis when deciphering complex reaction mechanisms, because we are able to predict the order of the reaction with respect to different reactants for different possible mechanisms. However, this analysis also shows that we could not distinguish the mechanism of Eq. 7.48 from a simple elementary second order reaction of A and B, because both rate laws have a single rate constant, k or We cannot decipher whether a rate constant represents a single elementary step or a combination of several rate constants for individual elementary steps. 

Finally, let s increase the complexity just one step further. Consider a mechanism in which a reactive intermediate is formed in a first order reaction along with a stable product (Pi), followed by a second order reaction with a second reactant that converts the intermediate to product P2 (Eq. 7.53). We go through the same mathematical process of analyzing the rate (Eq. 7.54), applying the SSA (Eq. 7.55), and performing algebraic manipulation (Eq. 7.56), to arrive at the result (Eq. 7.57). The prediction is that the reaction is first order in A, less than first order in B, and is retarded by P] (since its concentration is only in the denominator). Such predictions are important in guiding the experiments used to test if a reaction fits this mechanistic scenario. 

Often a reactant will not appear in the final rate law for a reaction, making the reaction formally zero order in that reactant. No reaction can be zero order in all species and therefore be zero order overall. This would mean that the reaction has no concentration dependence upon any of the reactants, which is impossible. But a reaction can be zero order in a single component, and the way this can occur involves a reaction where the species displaying zero order kinetics reacts after the rate-determining step. 

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C. N. Montreiul, S. D. WiUiams, and A. A. Adanc2yk, Modeling Current Generation Catalytic Converters Eaboratoy Experiments and Kinetic Parameter Optimisation—Steady State Kinetics, SAE 920096, Society of Automotive Engineers, Warrendale, Pa., 1992. 

Kinetic studies involving enzymes can principally be classified into steady and transient state kinetics. In tlie former, tlie enzyme concentration is much lower tlian that of tlie substrate in tlie latter much higher enzyme concentration is used to allow detection of reaction intennediates. In steady state kinetics, the high efficiency of enzymes as a catalyst implies that very low concentrations are adequate to enable reactions to proceed at measurable rates (i.e., reaction times of a few seconds or more). Typical enzyme concentrations are in the range of 10 M to 10 ], while substrate concentrations usually exceed lO M. Consequently, tlie concentrations of enzyme-substrate intermediates are low witli respect to tlie total substrate (reactant) concentrations, even when tlie enzyme is fully saturated. The reaction is considered to be in a steady state after a very short induction period, which greatly simplifies the rate laws. 

Photoinitiation is an excellent method for studying the pre- and posteffects of free radical polymerization, and from the ratio of the specific rate constant (kx) in non-steady-state conditions, together with steady-state kinetics, the absolute values of propagation (kp) and termination (k,) rate constants for radical polymerization can be obtained. 

In the case of reagents with a low content of inhibitors a steady-state polymerization rate may be set up. Steady-state kinetics are also observed... 

Steady-state kinetics. Consider the interchange of 02 and CO coordinated to myoglobin, an iron porphyrin represented as PFe ... 

Steady-state kinetics. The nickel complex Ni2(C5H5)2(CO)2 reacts with diphenylacety-Iene (dpa) to add dpa and displace both CO molecules. Assume the following mechanism ... 

Steady-state kinetics. The reaction of methylthiamine (MT+) in the presence of a large excess of SO3 and of 4-thiopyridone (= ArS-) is believed to follow the mechanism shown here,15 in which A" and B are steady-state intermediates. Derive the steady-state rate law. 

Steady-state kinetics. The cycloaddition reaction between the singlet ground state of 2-isopropylidene cyclopentane-1,3-diyl ( = S ) with acrylonitrile (A) is believed to occur by way of a biradical intermediate (BR),17... 

The reader can show that, with the steady-state approximation for [Tl2+], this scheme agrees with Eq. (6-14), with the constants k = k i and k = k j/k g. Of course, as is usual with steady-state kinetics, only the ratio of the rate constants for the intermediate can be determined. Subsequent to this work, however, Tl2+ has been generated by pulse radiolysis (Chapter 11), and direct determinations of k- and k g have been made.5... 

Figure 8.78. Steady-state kinetics of C2H4 oxidation on Pt/Ce02 as a function of catalyst potential, UWR, and oxygen partial pressure. T=500°C, Pc2h4=1-5 kPa.71 Reprinted by permission of The Electrochemical Society.

Despite these problems, the potential of research with conformationally restricted substrate analogs appears to be great. As yet, the use of these analogs with tools other than steady-state kinetics has been little explored. 

Fig. 9. The MoFe protein cycle of molybdenum nitrogenase. This cycle depicts a plausible sequence of events in the reduction of N2 to 2NH3 + H2. The scheme is based on well-characterized model chemistry (15, 105) and on the pre-steady-state kinetics of product formation by nitrogenase (102). The enzymic process has not been chsiracter-ized beyond M5 because the chemicals used to quench the reactions hydrolyze metal nitrides. As in Fig. 8, M represents an aji half of the MoFe protein. Subscripts 0-7 indicate the number of electrons trsmsferred to M from the Fe protein via the cycle of Fig. 8.

The mechanism of the first half-reaction has been studied by a combination of reductive titrations with CO and sodium dithionite and pre-steady-state kinetic studies by rapid freeze quench EPR spectroscopy (FQ-EPR) and stopped-flow kinetics 159). These combined studies have led to the following mechanism. The resting enzyme is assumed to have a metal-bound hydroxide nucleophile. Evidence for this species is based on the similarities between the pH dependence of the EPR spectrum of Cluster C and the for the for CO, deter-... 

We have studied the steady-state kinetics and selectivity of this reaction on clean, well-characterized sinxle-crystal surfaces of silver by usinx a special apparatus which allows rapid ( 20 s) transfer between a hixh-pressure catalytic microreactor and an ultra-hixh vacuum surface analysis (AES, XPS, LEED, TDS) chamber. The results of some of our recent studies of this reaction will be reviewed. These sinxle-crystal studies have provided considerable new insixht into the reaction pathway throuxh molecularly adsorbed O2 and C2H4, the structural sensitivity of real silver catalysts, and the role of chlorine adatoms in pro-motinx catalyst selectivity via an ensemble effect. 

Fig. 8. A model of bases on steady-state and pre-steady-state kinetic data. Ps indicate the two phosphorylation sites. Cl, CII and NIII refer to domains A, B, and C, respectively.

In conclusion, the steady-state kinetics of mannitol phosphorylation catalyzed by II can be explained within the model shown in Fig. 8 which was based upon different types of experiments. Does this mean that the mechanisms of the R. sphaeroides II " and the E. coli II are different Probably not. First of all, kinetically the two models are only different in that the 11 " model is an extreme case of the II model. The reorientation of the binding site upon phosphorylation of the enzyme is infinitely fast and complete in the former model, whereas competition between the rate of reorientation of the site and the rate of substrate binding to the site gives rise to the two pathways in the latter model. The experimental set-up may not have been adequate to detect the second pathway in case of II " . The important differences between the two models are at the level of the molecular mechanisms. In the II " model, the orientation of the binding site is directly linked to the state of phosphorylation of the enzyme, whereas in the II" model, the state of phosphorylation of the enzyme modulates the activation energy of the isomerization of the binding site between the two sides of the membrane. Steady-state kinetics by itself can never exclusively discriminate between these different models at the molecular level since a condition may be proposed where these different models show similar kinetics. The II model is based upon many different types of data discussed in this chapter and the steady-state kinetics is shown to be merely consistent with the model. Therefore, the II model is more likely to be representative for the mechanisms of E-IIs. 

The Kmax (and K, see below) constants determined from steady-state kinetic measurements are thus seen to be complex constants containing two or more of the individual rate constants illustrated in Fig. 2. 

Steady state and non steady state kinetic measurements suggest that methane carbon dioxide reforming proceeds in sequential steps combining dissociation and surface reaction of methane and CO2 During admission of pulses of methane on the supported Pt catalysts and on the oxide supports, methane decomposes into hydrogen and surface carbon The amount of CH, converted per pulse decreases drastically after the third pulse (this corresponds to about 2-3 molecules of CH< converted per Pt atom) indicating that the reaction stops when Pt is covered with (reactive) carbon CO2 is also concluded to dissociate under reaction conditions generating CO and adsorbed... 

Ogura, Y. and Yamazaki, I. (1983). Steady-state kinetics of the catalase reaction in the presence of cyanide. J. Biochem. (Tokyo) 94, 403-408. 

The calculated conversions presented in Table VIII used Eq. (57). They are quite remarkable. They reproduce experimental trends of lower conversion and higher peak bed temperature as the S02 content in the feed increases. Bunimovich et al. (1995) compared simulated and experimental conversion and peak bed temperature data for full-scale commercial plants and large-scale pilot plants using the model given in Table IX and the steady-state kinetic model [Eq. (57)]. Although the time-average plant performance was predicted closely, limiting cycle period predicted by the... 

Model dynamics were forced to steady state by setting derivatives for the melt complexes in Eq. (61) to zero (Bunimovich et al., 1995). This should make the model behave as though the steady-state kinetic model... 

Fig. 17. Comparison of the variation of the time-average S02 conversion and the maximum bed temperature predicted for stationary cycling condition by an unsteady-state and a steady-state kinetic model for a packed-bed S02 converter operating with periodic flow reversal...

Factors Affecting the Steady State Kinetic Constants... 

Figure 2.8 Relationship between steady state kinetic constants and specific portions of the enzyme reaction pathway.

For our purposes the most important factor that can impact the individual steady state kinetic constants is the presence of an inhibitor. We will see in Chapter 3 how specific modes of inhibitor interactions with target enzymes can be diagnosed by the effects that the inhibitors have on the three steady state kinetic constants. 

An inhibitor that binds exclusively to the free enzyme (i.e., for which a = °°) is said to be competitive because the binding of the inhibitor and the substrate to the enzyme are mutually exclusive hence these inhibitors compete with the substrate for the pool of free enzyme molecules. Referring back to the relationships between the steady state kinetic constants and the steps in catalysis (Figure 2.8), one would expect inhibitors that conform to this mechanism to affect the apparent value of KM (which relates to formation of the enzyme-substrate complex) and VmJKM, but not the value of Vmax (which relates to the chemical steps subsequent to ES complex formation). The presence of a competitive inhibitor thus influences the steady state velocity equation as described by Equation (3.1) ... 

Table 3.3 Effects of inhibitors of different modalities on the apparent values of steady state kinetic constants and on specific steps in catalysis... 

In this chapter we described the thermodynamics of enzyme-inhibitor interactions and defined three potential modes of reversible binding of inhibitors to enzyme molecules. Competitive inhibitors bind to the free enzyme form in direct competition with substrate molecules. Noncompetitive inhibitors bind to both the free enzyme and to the ES complex or subsequent enzyme forms that are populated during catalysis. Uncompetitive inhibitors bind exclusively to the ES complex or to subsequent enzyme forms. We saw that one can distinguish among these inhibition modes by their effects on the apparent values of the steady state kinetic parameters Umax, Km, and VmdX/KM. We further saw that for bisubstrate reactions, the inhibition modality depends on the reaction mechanism used by the enzyme. Finally, we described how one may use the dissociation constant for inhibition (Kh o.K or both) to best evaluate the relative affinity of different inhibitors for ones target enzyme, and thus drive compound optimization through medicinal chemistry efforts. 

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