Rapid Equilibrium Approach

The most direct approach to analyze enzyme kinetics is by using rapid equilibrium assumption. For the equilibrium to exist, the conversion of the complex compound into product should be much slower than the rate of its formation, and reaction 

Typically, the enzyme exists either as a free enzyme or in a complex with the substrate. The substrate exists as a free or in a complex with the enzyme. The conservation equations of the enzyme and substrate are given by Equations 4.4 and 4.5, respectively 

As the rate of enzyme-catalyzed reactions is always proportional to product concentration, the initial (v reaction rate can be expressed as in Equation 4.7. The subsequent steps of the derivation are illustrated in Equations 4.8 to 4.10  

Supercritical Fluids Technology in Lipase Catalyzed Processes 

On the other hand, at low [ l, Equation 4.3 is reduced to Equation 4.12, suggesting first-order dependence of to [E and [5], with linear dependence on 

---

The rapid equilibrium approach for deriving rate equations (Segel, 1993) is the simplest approach available. This approach assumes that only the early components of a reaction are at equilibrium ... 

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. 

The above rate equation is in agreement with that reported by Madhav and Ching [3]. Tliis rapid equilibrium treatment is a simple approach that allows the transformations of all complexes in terms of [E, [5], Kls and Kjp, which only deal with equilibrium expressions for the binding of the substrate to the enzyme. In the absence of inhibition, the enzyme kinetics are reduced to the simplest Michaelis-Menten model, as shown in Figure 5.21. The rate equation for the Michaelis-Menten model is given in ordinary textbooks and is as follows 11... 

Rapid Equilibrium and Steady-State Approaches to Derive Equations. Ill... 

THE COMBINED EQUILIBRIUM AND STEADY-STATE TREATMENT. There are a number of reasons why a rate equation should be derived by the combined equilibrium and steady-state approach. First, the experimentally observed kinetic patterns necessitate such a treatment. For example, several enzymic reactions have been proposed to proceed by the rapid-equilibrium random mechanism in one direction, but by the ordered pathway in the other. Second, steady-state treatment of complex mechanisms often results in equations that contain many higher-order terms. It is at times necessary to simplify the equation to bring it down to a manageable size and to reveal the basic kinetic properties of the mechanism. 

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 ... 

For mechanisms involving three or more rapid-equilibrium segments, once the segments are properly represented as nodes in a scheme, the rate equation can be obtained by the usual systematic approach. For example, consider the case of one substrate—one product reaction in which a modifier M is in rapid equilibrium with E, EA, and EP. 

Waters, N.J., Jones, R., Williams, G. and Sohal, B. (2008) Validation of a rapid equilibrium dialysis approach for the... 

In theory, K (i.e., kjki) should be the same whether determined by kinetic or equilibrium approaches. In practice, however, moderate differences arise that are often attributed to technical problems associated with separating bound from free rapidly without losing a significant proportion of receptor-toxicant complex. This problem is troublesome, particularly when estimating the amount bound at early time points in association or dissociation experiments, when the amount of bound ligand is changing rapidly. Large differences between the KD as determined in saturation and kinetic experiments, however, may indicate that the reaction is more complex than a simple bimolecular reversible reaction. 

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 ... 

Unlike the steady-state system, the slope of the 1/u versus 1/[A] plot for the rapid equilibrium system goes to zero as [B] approaches infinity. (As [B] increases, the Ife/fB] term of the slope factor becomes very small.) Also, unlike the steady-state system, the plots of 1/u versus 1/[B] intersect on the vertical axis at 1/Vmax. (There is no intercept factor— the denominator [B] term is not multiplied by an [A]-containing term.)... 

The second-order approach was successfully used for Cr retention and transport predictions by Selim and Amacher (1988) and for Zn retention by Hinz et al. (1992). This model was recently modified such that the total adsorption sites Smax were not partitioned between Sc and Sk phases based on a fraction of sites/(Selim Amacher, 1997 Ma Selim, 1998). Instead it was assumed that the vacant sites are available to both types of Se and Sk. Therefore,/is no longer required and the amount of solute adsorbed on each type of sites is only determined by the rate coefficients associated with each type of sites. As a result, sites associates with equilibrium or instantaneous type reactions will compete for available sites prior to slow or kinetic type sites are filled. Perhaps such mechanism is in line with observations where rapid (equilibrium type) sorption is first encountered and followed by slow types of retention reactions. We are not aware of the use of this second-order approach to describe heavy metal retention kinetics and transport in soils. 

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 . 

Any solute that is driven by this ion gradient tends to equilibrate with the electrochemical potential difference of the ion species. If its entrance by Na" -linked cotransport is fast enough as compared to the Na leak pathway, then its distribution will approach rapidly equilibrium with the electrochemical activity ratio of Na" " long before the latter has dissipated to a major extent. At this point the distribution of the solute (e.g. sugar) is at its peak, as here the uptake curve of the organic solute and decay curve of the ion meet. Subsequently, as the Na" " gradient continues to decay, the sugar distribution will go down concomitantly. 

Comparatively, with the methylthio ligands, the reaction may occur selectively through complex 9a and the nucleophile approaches cis to the better rr-acceptor. Alternatively, the reaction proceeds selectively through complex 9b and the nucleophile approaches trans to the better rr-acceptor. Either the transition state is more distorted than represented here, or the two diastereomeric allyl complexes are in rapid equilibrium, and the reaction proceeds through the less favored, but possibly more reactive, intermediate 9bf (Scheme 5). 

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... 

---