Michaelis-Menten rate expression

Although equation 7.3.28 and, in particular, equation 7.3.29 are known as Michaelis-Menten rate expressions, these individuals used a somewhat different approach to arrive at this mathematical form for an enzymatic rate expression (35). 

Often the key entity one is interested in obtaining in modeling enzyme kinetics is the analytical expression for the turnover flux in quasi-steady state. Equations (4.12) and (4.38) are examples. These expressions are sometimes called Michaelis-Menten rate laws. Such expressions can be used in simulation of cellular biochemical systems, as is the subject of Chapters 5, 6, and 7 of this book. However, one must keep in mind that, as we have seen, these rates represent approximations that result from simplifications of the kinetic mechanisms. We typically use the approximate Michaelis-Menten-type flux expressions rather than the full system of equations in simulations for several reasons. First, often the quasi-steady rate constants (such as Ks and K in Equation (4.38)) are available from experimental data while the mass-action rate constants (k+i, k-i, etc.) are not. In fact, it is possible for different enzymes with different detailed mechanisms to yield the same Michaelis-Menten rate expression, as we shall see below. Second, in metabolic reaction networks (for example), reactions operate near steady state in vivo. Kinetic transitions from one in vivo steady state to another may not involve the sort of extreme shifts in enzyme binding that have been illustrated in Figure 4.7. Therefore the quasi-steady approximation (or equivalently the approximation of rapid enzyme turnover) tends to be reasonable for the simulation of in vivo systems. 

The time-dependent transformation of N compounds by microbial oxidation and reduction reactions can also be described by Michaelis-Menten rate expression... 

Kinetic experiments with synthetic iron oxyhydroxides have shown that the initial microbial reduction rate increases with increasing initial ferric iron concentration up to a given maximum reduction rate (Bonneville et al. 2004). This observation was explained by a saturation of active membrane sites with Fe(III) centers. The respective reaction was best described with a Michaelis-Menten rate expression with the maximum reduction rate per cell positively correlating with the solubility of the iron oxyhydroxides (Bonneville et al. 2004). Kinetic studies involving iron are not only inherently important to describe reaction pathways and to derive rate constants, which can be used in models. Kinetic studies also increasingly focus on iron isotopic fractionation to better understand the iron isotopic composition of ancient sediments, which may assist in the reconstruction of paleo-environments. Importantly, iron isotope fractionation occurs in abiotic and biotic processes the degree of isotopic fractionation depends on individual reaction rates and the environmental conditions, e.g. whether reactions take place within an open or closed system (Johnson et al. 2004). 

For practical purposes, it is convenient to rearrange the Michaelis-Menten rate expression to Equation 4-11 ... 

J. Kumar and S. Nath [Chem. Eng. Sci., 52, 3455-3462 (1997)] simulated the behavior of a CSTR in which a liquid-phase autocatalytic reaction and a reaction that obeys a Michaelis-Menten rate expression are occurring simultaneously. 

The Michaelis-Menten rate expression for a monomolecular enzyme-catalyzed reaction is very similar to the Langmuir kinetic expression that we discussed previously for heterogeneous catalyzed systems. The Michaelis-Menten relationship is readily deduced for a simple model where the enzyme molecule (E) equilibrates with substrate molecule (S) to form the enzyme substrate complex (ES). The enzyme-substrate complex then reacts to the product molecule (P) and regenerates the active site of the enzyme (E) in what is considered to be the rate-limiting step of the proposed scheme ... 

The Michaelis-Menten rate expression as written in biochemistry is... 

Develop a suitable rate expression using the Michaelis-Menten rate equation and the quasi-steady-state approximations for the intermediate complexes formed. 

To obtain an expression for the Michaelis Menten rate equation, the dissociation of the product from the complex needs to be evaluated. Using a... 

Figure 3. Michaelis-Menten rate law expressed in Cartesian coordinates. The maximum velocity of the reaction V is the asymptotic value of the rate v at high concentrations of the substrate S. The parameter /C is given by the value of S that yields half the maximum velocity, or V /2.

Figure 4. Michaelis-Menten rate law expressed in double-reciprocal coordinates. In this plot, which is attributed to Lineweaver-Burk, 1 /V is given by the intercept on the //vaxis. The parameter K can be obtained from the slope of the straight line or the intercept on the negative i/S axis.

Derive the rate expression for an enzyme-catalyzed unimolecular (single substrate) reaction, such as that shown in steps 9.1 and 9.2, assuming that the decomposition of the reactive intermediate to give the product is reversible, rather than irreversible as indicated in step 9.2. Can the initial rate in the forward direction and the initial rate in the reverse direction be expressed in the form of a Michaelis-Menten rate equation If so, how If not, why ... 

Fractional orders sometimes are observed when power-law rate equations are used in place of more fundamental forms, for example, Langmuir-Hinshelwood or Michaelis-Menten kinetic expressions. Consider the rate equation... 

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... 

Almost every reaction scheme that gives rise to Michaelis-Menten kinetics will proceed at a rate directly proportional to [E]j. It is customary to express Emax as... 

This is the Michaelis-Menten expression (which dates back to 1913) for the rate of an enzymatic reaction. 

Compare the Michaelis-Menten expression for the rate of an enzyme-catalyzed reaction with the Langmuir-Hinshelwood expression for the same reaction on a metal surface. Are the two expressions equivalent ... 

The reaction rate for this enzyme kinetics example is expressed by the Michaelis-Menten equation and with product inhibition. 

Pyruvate kinase (PK) is one of the three postulated rate-controlling enzymes of glycolysis. The high-energy phosphate of phosphoenolpyruvate is transferred to ADP by this enzyme, which requires for its activity both monovalent and divalent cations. Enolpyruvate formed in this reaction is converted spontaneously to the keto form of pyruvate with the synthesis of one ATP molecule. PK has four isozymes in mammals M, M2, L, and R. The M2 type, which is considered to be the prototype, is the only form detected in early fetal tissues and is expressed in many adult tissues. This form is progressively replaced by the M( type in the skeletal muscle, heart, and brain by the L type in the liver and by the R type in red blood cells during development or differentiation (M26). The M, and M2 isozymes display Michaelis-Menten kinetics with respect to phosphoenolpyruvate. The Mj isozyme is not affected by fructose-1,6-diphosphate (F-1,6-DP) and the M2 is al-losterically activated by this compound. Type L and R exhibit cooperatively in... 

Each enzyme has a working name, a specific name in relation to the enzyme action and a code of four numbers the first indicates the type of catalysed reaction the second and third, the sub- and sub-subclass of reaction and the fourth indentifies the enzyme [18]. In all relevant studies, it is necessary to state the source of the enzyme, the physical state of drying (lyophilized or air-dried), the purity and the catalytic activity. The main parameter, from an analytical viewpoint is the catalytic activity which is expressed in the enzyme Unit (U) or in katal. One U corresponds to the amount of enzyme that catalyzes the conversion of one micromole of substrate per minute whereas one katal (SI unit) is the amount of enzyme that converts 1 mole of substrate per second. The activity of the enzyme toward a specific reaction is evaluated by the rate of the catalytic reaction using the Michaelis-Menten equation V0 = Vmax[S]/([S] + kM) where V0 is the initial rate of the reaction, defined as the activity Vmax is the maximum rate, [S] the concentration of substrate and KM the Michaelis constant which give the relative enzyme-substrate affinity. 

Figure 3. Schematic view of the substrate uptake rate versus concentration relationship as described by the whole-cell Michaelis-Menten kinetics. Q is the substrate uptake rate, <2max the biologically determined maximum uptake rate per biomass, c the substrate concentration, and Kj the whole-cell Michaelis constant, i.e. the concentration resulting in 2max/2 (mass of substrate per volume). At c

Aiming at a computer-based description of cellular metabolism, we briefly summarize some characteristic rate equations associated with competitive and allosteric regulation. Starting with irreversible Michaelis Menten kinetics, the most common types of feedback inhibition are depicted in Fig. 9. Allowing all possible associations between the enzyme and the inhibitor shown in Fig. 9, the total enzyme concentration Er can be expressed as... 

Sometimes, however, initial velocities are needed. If measurements of the rate u, are made at the time t, simple division of Michaelis-Menten expressions yields for the initial velocity Oo... 

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. 

It was Henri who first proposed that enzyme catalysis depended on the formation of a transient complex of enzyme and substrate, followed by the breakdown i.e., chemical conversion) of bound substrate into product. Nonetheless, credit for derivation of the rate expression for the initial rate phase of one-substrate enzyme-catalyzed reactions is given to Michaelis and Menten. Both treatments gave the same general result ... 

Symbol for maximal velocity of an enzyme-catalyzed reaction, usually expressed as the molarity change in product per unit time (usually, second). Fmax must not be confused with or specific activity the former has dimensions of time, and the latter is usually expressed as micromol product per unit time per milligram of protein. See Michaelis-Menten Equation Enzyme Rate Equations (1. The Basics)... 

Referring to reactions in which the reaction velocity is independent of the reactant under consideration. For example, for the reaction A + B C, if the empirical rate expression is v = A [B], the reaction is first order with respect to B but zero order with respect to A. See Chemical Kinetics Rate Saturation Michaelis-Menten Equation... 

The order of reaction is not always integral and sometimes the rate cannot be expressed in a simple form. For example, some enzyme-catalysed reactions obey Michaelis—Menten kinetics and the rate is given by... 

We wdl encounter similar rate expressions of this type when we consider surface or enzyme-catalyzed reactions in Chapter 7. These rate expressions are called Langmuir-Hinshelwood and Michaelis-Menten kinetics, respectively. 

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