Rapid-equilibrium assumption

The rapid-equilibrium treatment assumes that the reactants are part of a rapidly attained equilibrium that is always maintained during the course of the reaction, as shown by 

Substitution for [B] into Eq. (2.18) yields an equation with only [C] as the concentration variable 

This equation has the same form as Eq. (2.11) and can be integrated to give an analogous result. But, as noted in Ae preceding section, it is simpler to recognize that can be obtained directly from the coefficient of [C] as 

The expression for k may be compared to that derived from the steady-state assumption under the condition that kj. The k is missing in the present example because we have assumed an irreversible model, but otherwise the steady-state and equilibrium models are the same if 1 (in which case the concentration of B is small). 

The preceding discussion can leave the incorrect imfnessimi that B is like a particularly stable intermediate on the reaction pathway from A to C. A somewhat different perspective is gained if one views B as the starting material and A as some unreactive form of B. ITiis situation produces the same rate law as Eq. (2.21). The important general lesson is that all rapid equilibria involving the reactant(s) will enter into the rate law, even if the species involved are not on the net reaction pathway. 

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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. 

Derivation of Enzymatic Rate Equation from Rapid Equilibrium Assumption... 

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. 

Using the rapid equilibrium assumption between the inhibitor and the enzyme, the expressions for the complexes are given as... 

Comparing with Eq. (44) and using the rapid equilibrium assumption with dissociation constants K, the total enzyme concentration can be written as... 

In order for an equilibrium to exist between E -E S and ES, the rate constant kp would have to be much smaller than k i However, for the majority of enzyme activities, this assumption is unlikely to hold true. Nevertheless, the rapid equilibrium approach remains a most useful tool since equations thereby derived often have the same form as those derived by more correct steady-state approaches (see later), and although steady-state analyses of very complex systems (such as those displaying cooperative behavior) are almost impossibly complicated, rapid equilibrium assumptions facilitate relatively straightforward derivations of equations in such cases. 

Another term used to indicate a rapid equilibrium assumption in a kinetic process. The most prominent example in biochemistry is the assumption that enzyme and substrate rapidly form a preequilibrium enzyme-substrate complex in the Michaelis-Menten treatment. 

RAPID EQUILIBRIUM ASSUMPTION ENZYME RATE EQUATIONS (2.)... 

COMMITMENT-TO-CATALYSIS RAPID EQUILIBRIUM ASSUMPTION KINETIC ISOTOPE EFFECT ISOTOPE TRAPPING COMPARTMENTAL ANALYSIS CATENARY MODEL... 

RAPID BUFFER EXCHANGE Rapid equilibrium assumption,... 

UNI UNI ENZYME KINETIC MECHANISM MICHAELIS-MENTEN EQUATION ISO UNI UNI MECHANISM RAPID EQUILIBRIUM ASSUMPTION Unpaired electron,... 

Rapid equilibrium assumption 467 Rapid photometric methods 468 Ras 577-579... 

Even this scheme represents a complex situation, for ES can be arrived at by alternative routes, making it impossible for an expression of the same form as the Michaelis-Menten equation to be derived using the general steady-state assumption. However, types of non-competitive inhibition consistent with the Michaelis-Menten type equation and a linear Linweaver-Burk plot can occur if the rapid-equilibrium assumption is valid (Appendix S.A3). In the simplest possible model, involving simple linear non-competitive inhibition, the substrate does not affect the inhibitor binding. Under these conditions, the reactions... 

To analyze this system we simplify the kinetic mechanism by assuming that the binding and unbinding of glucose from the transporter are rapid, with dissociation constant Kd on both sides of the membrane. This rapid-equilibrium assumption yields ... 

It would be instructive to examine the effect of the product on the initial forward velocity. For example, suppose we have a solution containing a certain concentration of S and a certain concentration of P. In the absence of an appropriate enzyme, the reaction does not occur at a measurable rate. Now we add an enzyme catalyzing the reversible reaction S P. In which direction and at what rate will the reaction progress The direction of the reaction will depend on the ratio of [P]/[S] relative to K. An equation for the net velocity can be derived quite easily from rapid equilibrium assumptions (where K g = Ks, and K f = Kv). 

Equation 75 (as all others derived from rapid equilibrium assumptions) is really an equilibrium binding equation that gives the ratio of occupied to total sites. We obtain a velocity equation by assuming that the velocity is proportional to the concentration of occupied sites. In other words, a velocity equadon is obtained when we equate Ys to v/Vm xi... 

Ishikawa, H., Maeda, T., Hikita, H and Miyatake, K. (1988) The computerised derivation of rate equations for enzyme reactions on the basis of the pseudo-steady-state assumption and the rapid-equilibrium assumption. Biochem. J. 251, 175-181. 

If the rate constant k2 is much smaller than the rate constant k., of the enzyme, the substrate and the enzyme-substrate complex are in equilibrium, which is not disturbed by the decomposition of ES into E and P ( rapid equilibrium-assumption ). Based on this assumption, Michaelis and Menten derived the following rate equation (Eq. (17)) ... 

The Michaelis-Menten kinetics, represented by Eq. (18), may be extended to more complicated reactions by looking at the structure of the adsorption term. This procedure shown below is valid as long as the rapid equilibrium assumption is made. This is not valid in all cases. 

Although the derivation of reaction rates using the steady state assumption is more exact, often the rapid equilibrium assumption is used because it allows a simple derivation of the rate equation from the relevant enzyme-substrate complexes (see below) and allows fitting of the kinetic data. The following explanations are based on the rapid equilibrium assumption, and therefore all following constants K are used as dissociation constants with the component dissociating from the enzyme as the subscript, e.g. Ka, Kb, and the component remaining at the enzyme as second subscript (e.g. Kf, see below). 

In the following sections the extension of Eq. (18) to more complex reaction schemes is described. Again the rapid equilibrium assumption is used to show how more complex rate equations are derived from simple Michaelis-Menten kinetics. Attention is focused on some typical rate equations that are useful to describe enzyme kinetics with respect to a desired process optimization. The whole complexity of enzyme kinetics is of importance for a basic understanding of the enzyme mechanism, but it is not necessary for the fitting of kinetic data and the calculation of reactor performance. 

Because of the principle of microscopic reversibility each molecular process (in contrast to a macroscopic process) may occur in both forward and backward directions. As a consequence the end product P of an enzymatic conversion can act as a competitive inhibitor of the enzyme or, depending on the thermodynamic equilibrium, be transformed to the substrate S. If the interconversion of the ES to the EP complex is the rate-determining step the rapid equilibrium assumption is valid and the rate equation can be derived easily. 

For a practicable approach, the rapid equilibrium assumption is applied and the structure of kinetic models of two substrate reactions is demonstrated for the case of a random bi-uni reaction . 

In the example involving the binding of S3P and glyphosate, it was relatively easy to resolve the two-step reaction kinetics because of the separate concentration dependence for each step the first step depended on S3P and the second on glyphosate. Moreover, substrate trapping studies established that S3P dissociates at a very fast rate (i), justifying the rapid equilibrium assumption. With the... 

The rate equations for fully random and ordered mechanisms for three-substrate reactions are shown in Table II and can only be briefly discussed here. For the random mechanism, the rate equation derived by the rapid equilibrium assumption 43) contains all the terms of Eq. (2), and from experimental values for the eight kinetic coefficients for the reaction in each direction the dissociation constants for all the complexes may be calculated (c/. 43). 

A. Quasi-equilibrium assumption also known as the rapid-equilibrium assumption in which an equilibrium condition exists between the enzyme E, its substrate (A) and the enzyme-substrate complex (EA), i.e. ... 

In addition, we have admitted that El complex may have different affinities for the substrate than the free enzyme, and that EA complex may have different affinities for the inhibitor than the free enzyme. A rate equation for this general case may be derived from the rapid equilibrium assumptions ... 

The interaction factors a, p, and y represent, respectively, the effect of A on the binding of B, the effect of I on the binding of B, and the effect of I on the binding of A. The factor d represents the effect of I on the catalytic activity of the EABI complex. The derivation of rate equation for reaction (6.14) would be extremely laborious from the steady-state assumptions. Therefore, the general velocity equation is derived from Ae rapid equilibrium assumptions ... 

The rate equation for this case, derived from rapid equilibrium assumptions, is ... 

London and Steck (1969) have developed a general model, based on rapid equilibrium assumptions, for a monosubstrate enzyme that combines with substrate, activator, and a substrate-activator complex. The kinetic model for this type of activation is rather complex (Reaction (7.7)). 

The metal activator (X) not only combines with a free enzyme to form an enz5mie-activator complex (XE), but also combines with EA to form E(AX), with an EX complex to form XEX, and with E(AX) complex to form XE(AX). If the activation is nonessential, both E(AX) and XE(AX) are catalytically active. Since AX (Mg ATP ) is a tme substrate of enzyme, EA and XEA are inactive. The general velocity equation may be derived from the rapid equilibrium assumptions, in the following form ... 

In this section, we shall reviewthe rate equations forthe majortypes of trisubstrate mechanisms, written in the absence of products (Cleland, 1963 Plowman, 1972 Fromm, 1975,1979). All trisubstrate mechanisms in the rapid equilibrium category are relatively rare and the steady-state mechanisms are more common. However, the derivation of rate equations for rapid equilibrium mechanisms, in the absence of products, is less demanding, as it requires only the rapid equilibrium assumptions and, therefore, the resulting rate equations are relatively simple. 

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