Rapid Equilibrium Ordered System

Rapid Equilibrium Ordered system occurs when the substrates A and B combine with the enzyme in an ordered manner, that is, when B can bind only to the EA complex and EB complex is not formed  

In order to obtain the rapid equilibrium conditions, the rate constant has to be much higher than while the k constant can be the same size or smaller than jks-Thus, TjA is equal k /ku a true dissociation constant of the enzyme-substrate complex, Ka is zero, and Kb is equal k + k )/k. The velocity equation for this reaction is 

The separation of the variable from the constant substrate in Eq. (8.3) provides  

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A] at different constant concentrations of B, a series of straight lines will be obtained having a common intersection point. The intersection point will always be in the second quadrant. A double-reciprocal plot of 1/v vs. 1/[B] at different constant concentrations of A will consist of a series of straight lines all intersecting on the vertical axis. This observation is characteristic of the rapid equilibrium ordered system but not of the steady-state scheme. 

The Steady-State Ordered Bi Bi system reduces to a Rapid Equilibrium Ordered system when ks VJEo. In this case, Ka reduces to zero and the other kinetic constants reduce to dissociation constants. Let us write out again the definition of some kinetic constants in the Steady-State Ordered system (Eq. 9.9) ... 

A potential limitation encountered when one seeks to characterize the kinetic binding order of certain rapid equilibrium enzyme-catalyzed reactions containing specific abortive complexes. Frieden pointed out that initial rate kinetics alone were limited in the ability to distinguish a rapid equilibrium random Bi Bi mechanism from a rapid equilibrium ordered Bi Bi mechanism if the ordered mechanism could also form the EB and EP abortive complexes. Isotope exchange at equilibrium experiments would also be ineffective. However, such a dilemma would be a problem only for those rapid equilibrium enzymes having fccat values less than 30-50 sec h For those rapid equilibrium systems in which kcat is small, Frieden s dilemma necessitates the use of procedures other than standard initial rate kinetics. 

How do the reciprocal plots for a rapid equilibrium ordered bireactant system differ from those of a steady-state ordered bireactant system ... 

As a mle, an uncompetitive inhibition occurs only if there are more than one substrate or product (Huang, 1990). For example, an uncompetitive inhibition will take place in a Rapid Equilibrium Order bisubstrate reaction, when an inhibitor competes with B while A is the variable substrate. Thus, the equilibria shown below describe an ordered bisubstrate system in which an inhibitor competes with B but does not bind to free enzyme. 

In rapid equilibrium systems, the relative concentration of any particular enzyme form is given by a single denominator term in the velocity equation. For example, in the rapid equilibrium ordered bireactant system (Section 8.2), the velocity Eq. (8.2) is... 

Dead-end Inhibition in a Rapid Equilibrium Ordered Bisubstrate System... 

In Section 5.4, a case of a dead-end inhibition in a Rapid Equilibrium Ordered bisubstrate system was described. One can compare this system with the following example. 

Figure l. Rapid Equilibrium Ordered bisubstrate system. Graphical presentation of Eq. 84), with B as a constant and A as a variable substrate. 

PRODUCT INHIBITION IN A RAPID EQUILIBRIUM ORDERED BI BI SYSTEM... 

General rate equation. Let us examine the product inhibition in the Rapid Equilibrium Ordered Bi Bi system, in the presence of products P or Q. Both substrates and both products of reaction are binding in an ordered fashion. 

In rapid equilibrium systems, the Haldane relationship can be obtained directly from rate equations. In equilibrium, the rate equation for the Rapid Equilibrium Ordered Bi Bi system (Eq. (8.12)), becomes... 

Thus, for the Rapid Equilibrium Ordered Bi Bi system, the Haldane relationship is... 

However, the primary double reciprocal plots of some rapid equilibrium systems are identical. In rapid equilibrium systems, in the presence of the products of reaction, the primary reciprocal plots are very characteristic and depend on the number and type of enzyme-substrate and enzyme-product complexes that can form. Therefore, in order to distinguish between different t5q>es, one must revert to product inhibition patterns that can easily distinguish between aU types of rapid equilibrium bisubstrate systems (Plowman, 1972 Segel, 1975) (Table 2). 

Reduction of Steady-State Ordered to Rapid Equilibrium Ordered Bi Bi System... 

The Rapid Equilibrium Ordered Bi Bi system (Section 8.2) is a limiting case of the more realistic Steady-State Ordered Bi Bi system (Section 9.2). In bisubstrate mechanisms, the two approaches yield different velocity equations. As described... 

Thus, when the rate constant for the dissociation of A is greater than the maximal velocity in the forward direction (fc2 V,/Ko), the Ka term is eliminated from the denominator of the velocity equation, but the Ka term and other terms remain. Consequently, the velocity equation for the Steady-State Ordered Bi Bi system (Eq. (9.15)) reduces to the velocity equation for the Rapid Equilibrium Ordered Bi Bi system (Eq. (8.2)). 

Ordered and Random-Ordered Rapid Equilibrium Trisubstrate Systems... 

Let us examine several ordered and random-ordered rapid equilibrium Ter Ter systems, in addition to the aforementioned total rapid equilibrium random system ... 

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. 

An enzyme-catalyzed reaction involving two substrates and one product. There are two basic Bi Uni mechanisms (not considering reactions containing abortive complexes or those catagorized as Iso mechanisms). These mechanisms are the ordered Bi Uni scheme, in which the two substrates bind in a specific order, and the random Bi Uni mechanism, in which either substrate can bind first. Each of these mechanisms can be either rapid equilibrium or steady-state systems. 

Equation (6) is identical in form with Eq. (4). In fact, if 3 2, k-2, Eq. (6) reduces to Eq. (4). Although Eq. (5) is a more realistic mechanism compared with Eq. (1), especially when the rapid-equilibrium treatment is applied to the reversible reaction, the information obtainable from initial-rate studies of such unireactant system remains nevertheless the same Vi and K. This serves to justify the simplification used by the kineticist that is, the elimination of certain intermediates to maintain brevity of the rate equation (provided the mathematical form is unaltered). Thus, the forward reaction of an ordered Bi Bi mechanism is generally written as diagrammed below. 

The procedure to be described here was originally developed by Cha. The basic principle of his approach is to treat the rapid-equilibrium segment as though it were a single enzyme species at steady state with the other species. Let us consider the hybrid Rapid-Equilibrium Random-Ordered Bi Bi system ... 

In a simple rapid equilibrium system, = kp, a. first order rate... 

If the conversion of AB to EPQ is as rapid as the dissociation reactions, then steady-state assumptions must be used to derive the velocity equation. In xnultireactant systems, the rapid equilibrium and steady-state approaches do not yield the same final equation. For the ordered Bi Bi system, a steady-state derivation yields ... 

Hexokinase does not yield parallel reciprocal plots, so the Ping Pong mechanism can be discarded. However, initial velocity studies alone will noi discriminate between the rapid equilibrium random and steady-state ordered mechanisms. Both yield ihe same velocity equation and families of intersecting reciprocal plots. Other diagnostic procedures must be used (e.g., product inhibition, dead-end inhibition, equilibrium substrate binding, and isotope exchange studies). These procedures are described in detail in the author s Enzyme Kinetics behavior and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems, Wiley-Interscience (1975),... 

Particularly convenient for study by this technique is the common type of complex for which the binding constants falls in the range 10 -10 . Such complexes exist in rapid equilibrium with the uncomplexed species, the rate of complex formation in these systems being of the order of 10 -10 litres mol s" and the rate of dissociation Arp, by virtue of the relationship = k /... 

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