Michaelis-Menten relation

Comparison of the ordinary Michaelis-Menten relation with (5.108) shows that the inhibitor did not influence specific growth rate, vmgx, but the Michaelis-Menten constant was affected by the inhibitor and resulted in a constant, known as the apparent Michaelis constant. 

Uptake of NH4+ and NO3 into the root are described by Michaelis-Menten relations with the parameter values discussed in Section 6.3. 

Although the responses depicted in curves A, B, and C of Figure 2-15 approximate the shape of a simple Michaelis-Menten relation (transformed to a logarithmic plot), some clinical responses do not. Extremely steep dose-response curves (eg, curve D) may have important clinical consequences if the upper portion of the curve represents an undesirable extent of response (eg, coma caused by a sedative-hypnotic). Steep dose-response curves in patients can result from cooperative interactions of several different actions of a drug (eg, effects on brain, heart, and peripheral vessels, all contributing to lowering of blood pressure). 

The effect of combining the deactivation model with the simple catalytic sequence of the Michaelis-Menten relation is shown below ... 

A low Km value reflects high affinity. At substrate concentrations S < < Km the reaction rate is directly proportional to the substrate concentration (first order reaction) at high substrate concentration (S >> Km) the reaction is zero order and is no longer dependent on the substrate concentration but only on the enzyme activity. To calculate Km and Umax as well as inhibitor constants it is advantageous to transform the Michaelis-Menten relation so as to obtain linear relationships between S and vq that can be evaluated graphically. An example is the Line-weaver-Burk equation, containing the reciprocal values of vo and S ... 

This is referred to as the Michaelis-Menten relation and Km = k- /k the Michaelis constant, an estimate of the affinity of the substrate for the enzyme. When [S] Km,... 

The Michaelis - Menten relation is also obeyed in the processes involving the immobilized ADH (K =44.2 mM). The diffusion effect does not explain in a satisfactory way the increase of Kj, so that conformational modifications of the active center must be considered to affect the enzyme kinetic behaviour. It is known that the ADH molecule consists of 4 protomers bound through a Zn atom to 4 molecules of coenzyme In the coupling processes the carrier - enzyme chemical bond is formed at one protomer only, and the quaternary structure is destroyed. In fact, at lower coupling pH values (pH=8) the ADH activity is strongly diminished probably due to the structure splitting in non-bound protomers, the ADH being active as a complete structural composition only. 

For the sake of completeness, let us start by recalling some basic equations governing enzyme kinetics. A simple mechanism consistent with experimentally observed kinetic data is the Michaelis-Menten relation ° ... 

Figure 11.1 A plot of the reaction rate as a function of the substrate concentration for an enzyme catalyzed reaction. Vmax is the maximal velocity. The Michaelis constant. Km, is the substrate concentration at half Vmax- The rate v is related to the substrate concentration, [S], by the Michaelis-Menten equation ...

A special case for reduced bioavailabilty results from first-pass extraction that sometimes might be subjected to saturable Michaelis-Menten absorption kinetics. The lower the hepatic drug clearance is (Clhep) in relation to liver blood flow (Ql), or the faster the drug absorption rate constant (Ka), and the higher the dose (D) are, the more bioavailable is the drug (F). 

The rates of many catalyzed reactions depend upon substrate concentrations, as shown in Fig. 4-7. The rate at high substrate concentrations is zeroth-order with respect to [S], falling until it shows a first-order dependence in the limit of low [S], This pattern is that of a rectangular hyperbola, defined by an empirical relation known as the Michaelis-Menten equation. 

The Michaelis-Menten rate equation for enzyme reactions is typically written as the rate of formation of product (Eq. 19a). This equation implies that 1/Rate (where rate is the rate of formation of product) depends linearly on the inverse of the substrate concentration [S]. This relation allows KM to be determined. Derive this equation and sketch 1/Rate against 1/[S]. Label the axes, the y-intercept, and the slope with their corresponding functions. 

Coe and Bessell and coworkers studied the metabolic fates of 2-deoxy-2-fluoro-D-glucose (2DFG) and related compounds by using yeast hexokinase (as a model for mammalian hexokinase), and determined the kinetic constants K and V ) of the Michaelis-Menten equation D-glucose 0.17 (K in mAf)> 1 00 (relative value, D-glucose taken as 1) 2DG 0.59 0.11, 0.85 2DFG 0.19 0.03, 0.50 2-deoxy-2-fluoro-D-mannose (2DFM) 0.41 0.05, 0.85 2-deoxy-2,2-difluoro-D-nraZ>//Jo-hexose... 

Sato et al. (1991) expanded their earlier PBPK model to account for differences in body weight, body fat content, and sex and applied it to predicting the effect of these factors on trichloroethylene metabolism and excretion. Their model consisted of seven compartments (lung, vessel rich tissue, vessel poor tissue, muscle, fat tissue, gastrointestinal system, and hepatic system) and made various assumptions about the metabolic pathways considered. First-order Michaelis-Menten kinetics were assumed for simplicity, and the first metabolic product was assumed to be chloral hydrate, which was then converted to TCA and trichloroethanol. Further assumptions were that metabolism was limited to the hepatic compartment and that tissue and organ volumes were related to body weight. The metabolic parameters, (the scaling constant for the maximum rate of metabolism) and (the Michaelis constant), were those determined for trichloroethylene in a study by Koizumi (1989) and are presented in Table 2-3. 

Let us consider the determination of two parameters, the maximum reaction rate (rITOIX) and the saturation constant (Km) in an enzyme-catalyzed reaction following Michaelis-Menten kinetics. The Michaelis-Menten kinetic rate equation relates the reaction rate (r) to the substrate concentrations (S) by... 

Due to the Michaelis-Menten metabolism of phenytoin, alterations in its protein binding will result in increased severity of dose-related adverse effects. In patients with suspected changes in protein binding, it is useful to measure unbound phenytoin concentrations. 

When valproate protein binding is altered, the risk for severe dose-related adverse effects is much less compared to phenytoin. Michaelis-Menten metabolism is not a factor with valproate, so hepatic enzymes are able to efficiently metabolize the additional unbound portion. 

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. 

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 ... 

This relation is the broadly known Michaelis-Menten equation. The effect of substrate concentration ni on the rate predicted by this equation follows a characteristic pattern. Where substrate concentration is considerably smaller than the half saturation constant (ni <reactive intermediate EA depends on the availability of the substrate A. In this case, (mA + K A ) and reaction rate r+ given by 17.18 is proportional to mA. For the opposite case, (mA K ), little free enzyme E is available to complex with A. Now, (mA + mA and reaction... 

Probably the most important variable to consider in defining optimal conditions or standard conditions is the initial substrate concentration. Most enzymes show a hyperbolic curve as relation between initial reaction velocity and substrate concentration, well known now as the Michaelis-Menten curve. With increasing substrate concentration (S) the velocity (o) rises asymptotically to a maximum value (V) (Fig. 3), according to the expression ... 

Combining the flux given by equation (63) with a Michaelis-Menten-like expression for Ju, one recovers the modified Best equation (17), where the bioconversion capacity parameter b is now related to the total concentration of free and labile species of M ... 

Characteristically, within certain concentration limits, if a chemical is absorbed by passive diffusion, then the concentration of toxicant in the gut and the rate of absorption are linearly related. However, if absorption is mediated by active transport, the relationship between concentration and rate of absorption conforms to Michaelis-Menten kinetics and a Lineweaver-Burk plot (i.e., reciprocal of rate of absorption plotted against reciprocal of concentration), which graphs as a straight line. 

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... 

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. 

Velocity data may be plotted in any one of a number of ways to illustrate the relation between v and [S], and plotting procedures are detailed later in this chapter. However, the Michaelis-Menten equation is an equation for a rectangular hyperbola, and plotting v versus [S] yields a hyperbolic curve, in the absence of cooperative or other unusual behaviors (O Figure 4-3). 

Transport of NO3 formed from NH4+ towards the root and its consumption in denitrification and uptake by the root. Denitrification is described with Michaelis-Menten kinetics with an inhibition function related to [O2]. 

A normalization parameter used in treating ligand binding equilibria to convert two extensive variables, dissociation sud substrate concentration, into a parameter whose value is related to the fractional saturation of ligand binding sites. For the simple Michaelis-Menten treatment, v = + i m/[S], if R is the reduced... 

At this point, it is instructive to notice that the numerical example given above for the minimal Michaelis-Menten scheme will probably be very relevant to the situation for the majority of the enzymes considered so far. In consequence, it appears that the effects of organic solvents on the enantioselectivity are not restricted to their relative effects on the ground state system and the transition state. Instead, substantial contributions from the diffusional process parameters have to be taken into account as well. Since these contributions are probably better described by the Einstein-Smoluchovski relation,... 

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