Michaelis-Menten equilibrium

Michaelis and Menten (1913) treated the special case where k2 k-i- Under this assumption, K reduces to 1/ATi and the first reaction is effectively at equilibrium. Their overall rate expression corresponds to the final form in Eq. (2.5-30). though their constant K has a different meaning. Thus, it is not possible to test the accuracy of the Michaelis-Menten equilibrium assumption by reaction rate experiments in the quasi-steady-state region. Rather, one would need additional measurements very early in the reaction to allow calculation of the rate coefficients fci, k-i. and k2-... 

This brings us to the final mechanism we need to consider for a 2-substrate reaction, namely a random-order mechanism. We have assumed that we would be alerted to the possibility of a steady-state random-order mechanism by non-linear primary or secondary plots, but it is possible to get linear kinetics with a random-order mechanism. If we make the assumption that the further reaction of the ternary complex EAB is much slower than the network of reactions connecting E to EAB via EA and EB, then there are only 4 kinetically significant complexes and their concentrations are related to one another by substrate concentrations and dissociation constants. This is the rapid-equilibrium random-order mechanism, and the assumption made is analogous to the Michaelis-Menten equilibrium assumption for a 1-substrate mechanism. 

Equation 11-15 is known as the Michaelis-Menten equation. It represents the kinetics of many simple enzyme-catalyzed reactions, which involve a single substrate. The interpretation of as an equilibrium constant is not universally valid, since the assumption that the reversible reaction as a fast equilibrium process often does not apply. 

For a Michaelis-Menten reaction, ki = 7X10VAf- sec, /f i = 1 X lOVsec, and fe = 2 X lOVsec. What are the values of Ks and Does substrate binding approach equilibrium or does it behave more like a steady-state system ... 

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... 

Repeat the derivation of the Michaelis-Menten rate law, assuming that there is a pre-equilibrium between the bound and the unbound states of the substrate. 

Michaelis—Menten mechanism A model of enzyme catalysis in which the enzyme and its substrate reach a rapid pre-equilibrium with the bound substrate-enzyme complex. 

If the enzyme charged to a batch reactor is pristine, some time will be required before equihbrium is reached. This time is usually short compared with the batch reaction time and can be ignored. Furthermore, 5o Eq is usually true so that the depletion of substrate to establish the equilibrium is negligible. This means that Michaelis-Menten kinetics can be applied throughout the reaction cycle, and that the kinetic behavior of a batch reactor will be similar to that of a packed-bed PFR, as illustrated in Example 12.4. Simply replace t with thatch to obtain the approximate result for a batch reactor. 

Here MX, Y designates an outer sphere or second sphere complex. There is every reason to suppose that formation and dissociation of MX, Y occurs at rates approaching the diffusional-control limit so that the slow conversion to MY is a negligible perturbation on the equilibrium of the first step. There is a similarity here the Langmuir, the Michaelis-Menten and the Lindemann-Hinshelwood schemes. 

The Michaelis-Menten theory assumes that k-2 is sufficiently small that the second step in the process does not affect the equilibrium formation of the ES complex [61]. At steady state the rates of formation and breakdown of ES are equal ... 

Let us consider the basic enzyme catalysis mechanism described by the Michaelis-Menten equation (Eq. 2). It includes three elementary steps, namely, the reversible formation and breakdown of the ES complex (which does not mean that it is at equilibrium) and the decomposition of the ES complex into the product and the regenerated enzyme ... 

A plot of the initial reaction rate, v, as a function of the substrate concentration [S], shows a hyperbolic relationship (Figure 4). As the [S] becomes very large and the enzyme is saturated with the substrate, the reaction rate will not increase indefinitely but, for a fixed amount of [E], it reaches a plateau at a limiting value named the maximal velocity (vmax). This behavior can be explained using the equilibrium model of Michaelis-Menten (1913) or the steady-state model of Briggs and Haldane (1926). The first one is based on the assumption that the rate of breakdown of the ES complex to yield the product is much slower that the dissociation of ES. This means that k2 tj. 

The Michaelis-Menten constant defined by Eq. 11, is the equilibrium constant for the dissociation of he ES complex and is inversely related to the affinity of the enzyme for the substrate, therefore, a low KM value reflects high affinity ... 

If [S] = Km, the Michaelis-Menten equation says that the velocity will be one-half of Vmax. (Try substituting [S] for Km in the Michaelis-Menten equation, and you too can see this directly.) It s really the relationship between Km and [S] that determines where you are along the hyperbola. Like most of the rest of biochemistry, Km is backward. The larger the Km, the weaker the interaction between the enzyme and the substrate. Km is also a collection of rate constants. It may not be equal to the true dissociation constant of the ES complex (i.e., the equilibrium constant for ES E + S). 

Zn2+-catalyzed cleavage of (Zn2+ HPNPP)2 in ethanol discussed above.85 For the slower reacting MNPP, the chemical cleavage step (represented by k3 and requiring a methoxide which is probably coordinated to one or both of the metal ions in 35 2Zn(II)) is relatively slow, so that both the pre-equilibrium steps are established and typical Michaelis-Menten behavior is observed with saturation at higher [35 2Zn(II)]. On the other hand, with the far more reactive HPNPP the chemical cyclization step, /c3, is proposed to be faster than the k 2 step in the concentration range of 35 2Zn(II) used here. In this event, the observed kinetics would be linear in [35 2Zn(II)] as is the case in Fig. 20, with kohs — /c, [35 2Zn(I )]k2/(k, + k2). 

Most catalytic cycles are characterized by the fact that, prior to the rate-determining step [18], intermediates are coupled by equilibria in the catalytic cycle. For that reason Michaelis-Menten kinetics, which originally were published in the field of enzyme catalysis at the start of the last century, are of fundamental importance for homogeneous catalysis. As shown in the reaction sequence of Scheme 10.1, the active catalyst first reacts with the substrate in a pre-equilibrium to give the catalyst-substrate complex [20]. In the rate-determining step, this complex finally reacts to form the product, releasing the catalyst... 

For reversible enzymatic reactions, the Haldane relationship relates the equilibrium constant KeqsNith the kinetic parameters of a reaction. The equilibrium constant Keq for the reversible Michaelis Menten scheme shown above is given as... 

The scaled elasticities of a reversible Michaelis Menten equation with respect to its substrate and product thus consist of two additive contributions The first addend depends only on the kinetic propertiesand is confined to an absolute value smaller than unity. The second addend depends on the displacement from equilibrium only and may take an arbitrary value larger than zero. Consequently, for reactions close to thermodynamic equilibrium F Keq, the scaled elasticities become almost independent of the kinetic propertiesof the enzyme [96], In this case, predictions about network behavior can be entirely based on thermodynamic properties, which are not organism specific and often available, in conjunction with measurements of metabolite concentrations (see Section IV) to determine the displacement from equilibrium. Detailed knowledge of Michaelis Menten constants is not necessary. Along these lines, a more stringent framework to utilize constraints on the scaled elasticities (and variants thereof) as a determinant of network behavior is discussed in Section VIII.E. 

For any arbitrary metabolic network, the Jacobian matrix can be decomposed into a sum of three fundamental contributions A term M eg that relates to allosteric regulation. A term M in that relates to the kinetic properties of the network, as specified by the dissociation and Michaelis Menten parameters. And, finally, a term that relates to the displacement from thermodynamic equilibrium. We briefly evaluate each contribution separately. 

Figure 40. A simple example Cellular metabolism is modeled as a linear chain of reactions, with long range interactions mimicking the cellular environment and interactions within the metabolic network. The parameters are the number of metabolites m, the number of regulatory interactions, the probability p of positive versus negative interaction, as well as the maximal displacement ymax from equilibrium for each reaction. Each reaction is modeled as a reversible Michaelis Menten equation according to the methodology described in Section VIII.

In the absence of an enzyme, the reaction rate v is proportional to the concentration of substance A (top). The constant k is the rate constant of the uncatalyzed reaction. Like all catalysts, the enzyme E (total concentration [E]t) creates a new reaction pathway, initially, A is bound to E (partial reaction 1, left), if this reaction is in chemical equilibrium, then with the help of the law of mass action—and taking into account the fact that [E]t = [E] + [EA]—one can express the concentration [EA] of the enzyme-substrate complex as a function of [A] (left). The Michaelis constant lknow that kcat > k—in other words, enzyme-bound substrate reacts to B much faster than A alone (partial reaction 2, right), kcat. the enzyme s turnover number, corresponds to the number of substrate molecules converted by one enzyme molecule per second. Like the conversion A B, the formation of B from EA is a first-order reaction—i. e., V = k [EA] applies. When this equation is combined with the expression already derived for EA, the result is the Michaelis-Menten equation. 

STEADY STATE TREATMENT. While the Michaelis-Menten model requires the rapid equilibrium formation of ES complex prior to catalysis, there are many enzymes which do not exhibit such rate behavior. Accordingly, Briggs and Haldane considered the case where the enzyme and substrate obey the steady state assumption, which states that during the course of a reaction there will be a period over which the concentrations of various enzyme species will appear to be time-invariant ie., d[EX]/dr s 0). Such an assumption then provides that... 

THE MICHAELIS-MENTEN EQUATION AS A LIMITING CASE OF THE STEADY STATE EQUATION. To achieve a rapid equilibrium between E and EX, ki[S] and k2 must each be much greater than ks. [Note the rate constant ki is a bimolecular rate constant with units of molarity seconds, and we must use ki[S]... 

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