Rapid-equilibrium reaction

The construction of the structural kinetic model proceeds as described in Section VIII.E. Note that in contrast to previous work [84], no simplifying assumptions were used the model is a full implementation of the model described in Refs. [113, 331]. The model consists of m = 18 metabolites and r = 20 reactions. The rank of the stoichiometric matrix is rank (N) = 16, owing to the conservation of ATP and total inorganic phosphate. The steady-state flux distribution is fully characterized by four parameters, chosen to be triosephosphate export reactions and starch synthesis. Following the models of Petterson and Ryde-Petterson [113] and Poolman et al. [124, 125, 331], 11 of the 20 reactions were modeled as rapid equilibrium reactions assuming bilinear mass-action kinetics (see Table VIII) and saturation parameters O1 1. 

This reaction is analogous to the rapid equilibrium reaction of H2 with [NiFe] hydro-genases described above ... 

The Rapid Equilibrium Reaction If the rate-determining step for the overall reaction is the interconversion step of the central complexes, then all of the binding steps can be described in terms of dissociation constants ... 

The Rapid Equilibrium Reaction If the interconversion step between the central complexes is rate-determining (ie., rapid equilibrium), then for the reaction E -E A... 

The assumption that the consumption of the intermediate in the slow step is insignificant relative to its formation and decomposition in the first step is called a pre-equilibrium condition. A pre-equilibrium arises when an intermediate is formed in a rapid equilibrium reaction prior to a slow step in the mechanism. The slowest elementary step in a sequence of reactions—in our example, the reaction between 02 and N202—is called the rate-determining step of the reaction. The rate-determining step is so much slower than the rest that it governs the rate of the overall reaction (Fig. 13.24). A rate-determining step is like a slow ferry on the route between two cities. The rate at which the traffic arrives at its destination is governed by the rate at which it is ferried across the river, because this part of the journey is much slower than any of the others. 

In the +6 oxidation state, the most important solution species are the yellow chromate ion (Cr042-) and the orange dichromate ion (Cr2072-). These ions are interconverted by the rapid equilibrium reaction... 

Establishing the inhibition patterns in an enzyme-catalyzed reaction is usually an important step in elucidating the reaction mechanism. One complication in the interpretation of such data is the possible formation of dead-end complexes (i.e., a complex of the form EAP in the above scheme). This is especially important in rapid-equilibrium reactions [ones in which all steps except the rate constants for the central isomerization step (EAB EPQ in the above example) are very large]. 

Let us, now, depart from monosubstrate reactions and turn our attention to a much more realistic case of a hyperbolic inhibition in bisubstrate reactions (Segel, 1975 Dixon Webb, 1979 f rich Allison, 2000). In the rapid equilibrium reaction (6.14), A and B are the substrates while I is a nonexclusive inhibitor ... 

One common kind of reaction involves proton transfer occurring as a rapid equilibrium preceding the rate-determining step, for example, in the reaction of an alcohol with hydrobromic acid to give an alkyl bromide ... 

Because proton-transfer reactions between oxygen atoms are usually very fast, step 3 can be assumed to be a rapid equilibrium. With the above mechanism assume4 let us examine the rate expression which would result, depending upon which of the steps is rate-determining. 

The details of proton-transfer processes can also be probed by examination of solvent isotope effects, for example, by comparing the rates of a reaction in H2O versus D2O. The solvent isotope effect can be either normal or inverse, depending on the nature of the proton-transfer process in the reaction mechanism. D3O+ is a stronger acid than H3O+. As a result, reactants in D2O solution are somewhat more extensively protonated than in H2O at identical acid concentration. A reaction that involves a rapid equilibrium protonation will proceed faster in D2O than in H2O because of the higher concentration of the protonated reactant. On the other hand, if proton transfer is part of the rate-determining step, the reaction will be faster in H2O than in D2O because of the normal primary kinetic isotope effect of the type considered in Section 4.5. 

Mechanism III cannot be distinguished from the first two on the basis of kinetics alone, because the reactive species shown is in rapid equilibrium with the anion and therefore equivalent to it in terms of reaction kinetics. 

Carbonyl compounds are in a rapid equilibrium with called keto-enol tautomerism. Although enol tautomers to only a small extent at equilibrium and can t usually be they nevertheless contain a highly nucleophilic double electrophiles. For example, aldehydes and ketones are at the a position by reaction with Cl2, Br2, or I2 in Alpha bromination of carboxylic acids can be similarly... 

Enzyme reaction kinetics were modelled on the basis of rapid equilibrium assumption. Rapid equilibrium condition (also known as quasi-equilibrium) assumes that only the early components of the reaction are at equilibrium.8-10 In rapid equilibrium conditions, the enzyme (E), substrate (S) and enzyme-substrate (ES), the central complex equilibrate rapidly compared with the dissociation rate of ES into E and product (P ). The combined inhibition effects by 2-ethoxyethanol as a non-competitive inhibitor and (S)-ibuprofen ester as an uncompetitive inhibition resulted in an overall mechanism, shown in Figure 5.20. 

The number of components in a system can change with experimental conditions, and one must exercise care in defining the system. For example, a mixture of H2, 02, and H20(g) at low temperature is a three-component mixture.11 However, heating the mixture to a high temperature causes the three species to be in rapid equilibrium through the reaction... 

Since a first-order rate constant does not depend on [A]o, one need not know either the initial concentration or the exact instant at which the reaction began. This characteristic should not be used to rationalize experimentation on impure materials. These features do allow, however, a procedure in which measurements of slower reactions are not taken until the sample has reached temperature equilibrium with the thermostating bath. The first sample is simply designated as t = 0. Likewise, for rapidly decaying reaction transients, knowing the true zero time is immaterial. 

Prior equilibrium. Consider the net reaction between certain metal halide and alkyl cobalt complexes, RCo + MX = Co+ + RM + X". There is a rapid equilibrium ... 

The rate law that we have derived is not the same as the experimental one. We have stressed that a reaction mechanism is plausible only if its predictions are in line with experimental results so should we discard our proposal Before doing so, it is always wise to explore whether under certain conditions the predictions do in fact agree with experimental data. In this case, if the rate of step 2 is very slow relative to the rapid equilibrium in step 1—so that N202] 2[N202][02],... 

In either neat dioxane or THF, carbene-ether ylides are observed as a broad IR absorption band between 1560 and 1610 cm , distinct from the IR bands of the free carbenes. With discrete spectroscopic signatures for the free carbene and its corresponding ether ylides, TRIR spectroscopy was used to confirm that the effects described above with dilute ether in Freon-113 were due to specific solvation of the carbene (Scheme 4.6, Reaction 2) rather than a pre-equilibration with the coordinating solvent (Scheme 4.6, Reaction 3) or reactivity of the ylide itself (Scheme 6, Reaction 4). In Freon-113 containing 0.095M THF simultaneous TRIR observation of both the free carbene (x = ca. 500 ns) and the carbene-THF ylide (x = ca. 5ps) was possible7 The observation that lifetimes of these species were observed to be so different conclusively demonstrates that the free carbene and the carbene-THF ylide are not in rapid equilibrium and that Reaction 3 of Scheme 4.6 is not operative. By examining the kinetics of the carbene 34 at 1635 cm directly in Freon-113 with small amounts of added dioxane, it was observed that the rate of reaction with TME was reduced, consistent with Reaction 2 (and not Reaction 4) of Scheme 4.6. 

The discussion above of enzyme reactions treated the formation of the initial ES complex as an isolated equilibrium that is followed by slower chemical steps of catalysis. This rapid equilibrium model was first proposed by Henri (1903) and independently by Michaelis and Menten (1913). However, in most laboratory studies of enzyme reactions the rapid equilibrium model does not hold instead, enzyme... 

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. 

Figure 2.12 Reaction pathway for a bi-bi rapid equilibrium, random sequential ternary complex reaction mechanism.

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