Steady-state treatments

There are various indications that a reaction is occurring by a complex mechanism. 

An obvious piece of evidence is when the kinetic law is more complex than would be consistent with the occurrence of the reaction in a single step two examples of this have already been noted. Enzyme-catalyzed reactions provide additional examples. Thus, if an enzyme reaction involving a single substrate occurred by a simple bimolecular reaction between enzyme and substrate, the kinetics would be first-order with respect to substrate. However, the behavior as far as the substrate is concerned is rarely so simple, and it can therefore be concluded that the reaction occurs in more than one stage. 

Another indication of complexity is provided when intermediates can be detected by chemical or other means during the course of reaction. When this can be done a kinetic scheme must be developed which will account for the existence of these intermediates. Sometimes these intermediates are relatively stable substances, while in other cases they are labile substances such as atoms and free radicals. Enzyme-substrate complexes are of the latter class, in that they usually cannot be isolated and preserved and can be detected only by special methods such as spectroscopic ones. Free radicals can sometimes be observed by spectroscopic methods, and evidence for their existence may be obtained by causing them to undergo certain specific reactions which less active substances cannot bring about. 

When the nature of the reaction intermediates has been determined by methods such as those outlined above, the next step is to devise a reaction scheme which will involve these intermediates and account for the kinetic features of the reaction. 

If such a scheme fits the data satisfactorily it can be tentatively assumed that the mechanism is the correct one. It should be emphasized, however, that additional kinetic work frequently leads to the overthrow of schemes which had previously been supposed to be firmly established. 

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The quantitative description of enzyme kinetics has been developed in great detail by applying the steady-state approximation to all intermediate forms of the enzyme. Some of the kinetic schemes are extremely complex, and even with the aid of the steady-state treatment the algebraic manipulations are formidable. Kineticists have, therefore, developed ingenious schemes for writing down the steady-state rate equations directly from the kinetic scheme without carrying out the intermediate algebra." -" ... 

Of course it is also possible for a reaction system not to belong to any of these classes of approximate description.) Only in class III can equilibrium be said to be a special case of the steady-state treatment. Note that, for class III systems, the steady-state concentration of intermediate is very large,whereas for class I it is very small. Zuman and Patel have discussed the equilibrium and steady-state approximations in terms similar to the present treatment. 

Previous theoretical kinetic treatments of the formation of secondary, tertiary and higher order ions in the ionization chamber of a conventional mass spectrometer operating at high pressure, have used either a steady state treatment (2, 24) or an ion-beam approach (43). These theories are essentially phenomenological, and they make no clear assumptions about the nature of the reactive collision. The model outlined below is a microscopic one, making definite assumptions about the kinematics of the reactive collision. If the rate constants of the reactions are fixed, the nature of these assumptions definitely affects the amount of reaction occurring. 

Let us compare these results with the predictions of the theory formulated by Lampe etal. (24) in terms of a steady-state concentration of collision complexes. This is a classical macroscopic treatment insofar as it makes no assumptions about the collision dynamics, but its postulate of collision complexes implies that v8 = vp/2 for the system treated above. Thus, its predictions might be expected to coincide with those of the collision-complex model. Figure 3 shows that this is not so the points calculated from the steady-state theory (Ref. 25, Equation 10) coincide exactly with the curve for which v8 = vv. The reason for this is that the steady-state treatment assumes a constant time available for reaction irrespective oC the number of reactions occurring in any one reaction... 

The reaction is discussed in terms of the scheme (71)-(74) for oxidation by simple complexes of Fe(III) except that the one-equivalent oxidation to NHj is disregarded. A steady state treatment for [N2H3-] leads to... 

A steady-state treatment for the radical (and inclusion of similar reactions incorporating RS -> -RS ) produces the observed rate law. 

Steady-state treatment for the transients (H02- and UO2 ) leads to the observed rate law. The chain reaction is indicated by (/) strong catalysis by Cu " ions and ill) partial and complete inhibition respectively by added Cl" and Ag" ions. The inhibition by Ag" is not indefinite, however, and takes the form of an induction period, during which time metallic silver is deposited. 

COVALENT COMPOUNDS, METAL IONS OXIDATION-REDUCTION Steady-state treatment for L- leads to a rate law -d[02]/dr = fc2( i[Fe(acac)3])°- ... 

Now, assuming again the concepts of dynamic equilibria at the electrode, e.g., the cathode (cf., p. 104), applying a steady-state treatment we may consider the following ... 

Equations (2.10) and (2.12) are identical except for the substitution of the equilibrium dissociation constant Ks in Equation (2.10) by the kinetic constant Ku in Equation (2.12). This substitution is necessary because in the steady state treatment, rapid equilibrium assumptions no longer holds. A detailed description of the meaning of Ku, in terms of specific rate constants can be found in the texts by Copeland (2000) and Fersht (1999) and elsewhere. For our purposes it suffices to say that while Ku is not a true equilibrium constant, it can nevertheless be viewed as a measure of the relative affinity of the ES encounter complex under steady state conditions. Thus in all of the equations presented in this chapter we must substitute Ku for Ks when dealing with steady state measurements of enzyme reactions. 

Lien EA, Solheim E, Ueland PM (1991) Distribution of tamoxifen and its metabolites in rat and human tissues diming steady-state treatment. Cancer Res 51 4837-4844... 

HEINRICH, R., RAPOPORT, T.A., A linear steady-state treatment of enzymatic chains. General properties, control and effector strength, Eur. J. Biochem., 1974, 42, 89-95. 

Where competitive inhibition is observed between two solutes (i.e. binding to a single, identical carrier), it is also possible to estimate carrier concentrations using a steady-state treatment [193-195], In that case, data from the competing solutes are used to generate a sufficient number of equilibrium expressions (e.g. equations (38) and (39)) and corresponding mass balance equations (e.g. equations (40) and (41)) to resolve for the total carrier concentration. 

A steady-state treatment of the kinetic scheme yields the differential equation... 

A steady-state treatment of reactions (56) to (61) gives the approximate relationship... 

Applying the usual steady-state treatment for consecutive first-order reactions kt at 16 torr pressure over the temperature range 597-701 °C is given by 1.8 x 1011 exp(—47,000/Kr) sec Within experimental error, reactions (1) and (2) were homogeneous processes. However, both k2 and k2 were functions of the total pressure in the system. This dependence is shown in Fig. 1. The methyl zinc decomposition is apparently in its second-order region. Therefore, assuming four effective oscillators and a mean temperature of 1050 °K, = Eohs.+i nRT... 

Steady-state treatment then predicts the observed f-order kinetics and leads to... 

Their algebraic formulation is inconsistent with their reaction scheme in that they represent the propagation step as kinetically of third order they use a steady-state treatment despite the fact that the limited yields obtained in these reactions show that this is inappropriate their equation for the DP as a function of [PJ is not of a form which has a maximum. For these and other reasons their treatment is not valid. (A simple explanation of the DP maximum is proposed in the Appendix to this Chapter.)... 

On applying the steady-state treatment with respect to [R ], we get... 

The concentration of complex in such a case must be calculated by making the use of steady-state-treatment. This type of complex is known as a Vant Hoff complex. 

However, if complex is of Vant Hoff type complex, [X] is determined by steady state treatment. On applying steady state conditions with respect to X, we get... 

However, when k[ k2, the rate law becomes same as in case of Arrhenius complex. Thus, the steady state treatment is the general one, and reduces to the equilibrium treatment (Arrhenius complex) when k[ k2. 

In highly basic solution the ES form will be dominating and again the rate will be low. At some intermediate pH, when EHS is dominating, the rate will be maximum. Application of steady state treatment to this scheme of mechanism leads to a very complicated rate law and it is difficult to apply it to the experimental data. 

The application of steady state treatment to this scheme, gives the following rate equation ... 

By application of the steady-state treatment to Scheme 7, the authors calculate the general rate expression for reaction at the 3-position to produce adducts 10 (fcfast), and the rate expression for product formation (ks ow), respectively (equations 21 and 22). 

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