Concentration reduced, Michaelis-Menten

When substrates are used at low, radiotracer concentrations, the Michaelis-Menten equation defining reaction rate in terms of substrate concentration and enzyme kinetic parameters reduces to ... 

At high substrate concentrations relative to Km ([S] Km), The Michaelis-Menten equation reduces to v = Vmax, substrate concentration disappears, and the dependence of velocity on substrate concentration approaches a horizontal line. When the reaction velocity is independent of the concentration of the substrate, as it is at Vmax, it s given the name zero-order kinetics. 

Substrate-limited growth in terms of reduced availability of both the electron donor and the electron acceptor is common in wastewater of sewer systems. Based on the concept of Michaelis-Menten s kinetics for enzymatic processes, Monod (1949) formulated, in operational terms, the relationship between the actual and the maximal specific growth rate constants and the concentration of a limiting substrate [cf. Equation (2.14)] ... 

In the case of Michaelis-Menten kinetics (Eq. 13.4), Fmax y and cannot be assessed uniquely if the concentration is far below the value of X m y in that case, Eq. 13.4 reduces to Eq. 13.3, and only one parameter CL, or the ratio FmaXxy/KjUxy can be calculated uniquely. 

The reduction in concentration of reactants, enzymes, and solute molecules can provide important information about kinetic systems. For example, one can readily differentiate a first-order process from a second-order process by testing whether the period required to reduce a reactant concentration to 50% of its initial value depends on dilution. First-order processes and intramolecular processes should not exhibit any effect on rate by diluting a reactant. In terms of enzyme-catalyzed processes, the Michaelis-Menten equation requires that the initial reaction velocity depends strictly on the concentration of active catalyst. Dilution can also be used to induce dissociation of molecular complexes or to promote depolymerization of certain polymers (such as F-actin and microtubules). 

There are many examples of first-order reactions dissociation from a complex, decompositions, isomerizations, etc. The decomposition of gaseous nitrogen pentoxide (2N2O5 4NO2 + O2) was determined to be first order ( d[N205]/dt = k[N205j) as is the release of product from an enzyme-product complex (EP E -t P). In a single-substrate, enzyme-catalyzed reaction in which the substrate concentration is much less than the Michaelis constant (i.e., [S] K ) the reaction is said to be first-order since the Michaelis-Menten equation reduces to... 

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... 

In binding experiments, the affinity of magnesium ADP to native membranes and to the isolated calcium dependent ATPase was found to be considerably lower than that of magnesium ATP173. On the other hand, from the inhibition of the calcium-dependent ATPase or the activation of calcium release and ATP synthesis apparent affinities for ADP are obtained that are very similar to those of ATP (Fig. 12). The affinity of ADP for the enzyme apparently depends on its functional state. The affinity of ADP for the membranes under conditions of calcium release depends markedly on the pH of the medium. When the medium pH is reduced from 7.0 to 6.0, the affinity drops by a factor of 10. At pH 7.0 the affinity of the membrane for ADP corresponds to the affinity for ATP to the high affinity binding sites in the forward running mode of the pump. In contrast to the complex dependence of the forward reaction on the concentration of ATP, the dependence of the reverse reaction on ADP seems to follow simple Michaelis-Menten kinetics. 

Here we develop the basic logic and the algebraic steps in a modern derivation of the Michaelis-Menten equation, which includes the steady-state assumption introduced by Briggs and Haldane. The derivation starts with the two basic steps of the formation and breakdown of ES (Eqns 6-7 and 6-8). Early in the reaction, the concentration of the product, [P], is negligible, and we make the simplifying assumption that the reverse reaction, P—>S (described by k 2), can be ignored. This assumption is not critical but it simplifies our task. The overall reaction then reduces to... 

This scheme is analogous to that of the Michaelis-Menten mechanism, and the reaction should thus show saturation kinetics with increasing inhibitor concentration. The kinetics were solved in Chapter 4, equation 4.46. For the simple case of pre-equilibrium binding followed by a slow chemical step, the solution reduces to... 

A special numerical relationship arises from the Michaelis-Menten equation when the initial velocity is equal to half the maximal velocity, that, is VQ = (V2)Vmax. Equation 5.24 then reduces to Km = [S0]. This means that Km is equal to the substrate concentration in moles per liter at which the reaction velocity is half its maximum value. 

In most clinical situations the concentration of the drug, [C], is much less than the Michaelis constant, Km, and the Michaelis-Menten equation reduces to... 

Thus, Kn, the Michaelis constant, is a dynamic or pseudo-equilibrium constant expressing the relationship between the actual steady-state concentrations, rather than the equilibrium.concentrations. If Aj, is very small compared to A-i, reduces to K. A steady-state treatment of the more realistic reaction sequence E+ S ES EP E + P yields the same final velocity equation although now Km is a more complex function, composed of the rate constants of all the steps. Thus, the physical significance of K cannot be stated with any certainty in the absence of other data concerning the relative magnitudes of the various rate constants. Nevertheless, represents a valuable constant that relates the velocity of an enzyme-catalyzed reaction to the substrate concentration. Inspection of the Henri-Michaelis-Menten equation shows that Km is equivalent to the substrate concentration that yields half-maximal velocity ... 

The constant K in the above equation no longer equals the substrate concentration that yields half-maximal velocity (except when n = 1, when the equation reduces to the Henri-Michaelis-Menten equation). 

A rigorous kinetic description of interfacial catalysis has been hampered by the ill-defined physical chemistry of the lipid—water interface (Martinek et ai, 1989). Traditional kinetic assumptions are undermined by the anisotropy and inhomogeneity of the substrate aggregate. For example, the differential partitioning of reactants (enzyme, calcium ion, substrate) and products (lysolecithins, fatty acids) between the two bulk phases prevents direct measurement of enzyme and substrate concentrations. This complicates dissection of the multiple equilibria that contribute to the observed rate constants. Only recently has it become possible to describe clearly the activity of SPLA2S in terms of traditional Michaelis— Menten kinetics. Such a description required the development of methods to reduce experimentally the number of equilibrium states available to the enzyme (Berg etai, 1991). 

The enzyme lysozyme kills certain bacteria by attacking a sugar called N-acetylglucosamine (NAG) in their cell walls. At an enzyme concentration of 2 X 10 M, the maximum rate for substrate (NAG) reaction, found at high substrate concentration, is 1 X 10 " mol L s. The rate is reduced by a factor of 2 when the substrate concentration is reduced to 6 X 10 M. Determine the Michaelis-Menten constants and kz for lysozyme. 

The Mass-Action representation is clearly a special case of the GMA representation in which all exponents are positive integers. The Michaelis-Menten representation is, in turn, a special case of the traditional Mass-Action representation in which two important restrictions have been imposed (Savageau, 1992). First, it is assumed that the mechanism is in quasi-steady state. The derivatives of the dependent state variables in the Mass-Action Formalism can then be set to zero, thereby reducing the description from differential equations to algebraic equations. Second, it is assumed that complexes do not occur between different forms of an enzyme or between different enzymes. The algebraic equations will then be linear in the concentrations of the various enzyme forms, and one can derive the rational function that is the representation of the rate law within the Michaelis-Menten Formalism. 

D23.4 Refer to eqns 23.26 and 23.27, which are the analogues of the Michaelis-Menten and Lineweaver-Burk equations (23.21 and 23,22), as well as to Figure 23.13, There are three major modes of inhibition that give rise to distinctly different kinetic behavior (Figure 23.13), In competitive inhibition the inhibitor binds only to the active site of the enzyme and thereby inhibits the attachment of the substrate. This condition corresponds to a > 1 and a = 1 (because ESI does not form). The slope of the Lineweaver-Burk plot increases by a factor of a relative to the slope for data on the uninhibited enzyme (a = a = I), The y-intercept does not change as a result of competitive inhibition, In uncompetitive inhibition, the inhibitor binds to a site of the enzyme that is removed from the active site, but only if the substrate is already present. The inhibition occurs because ESI reduces the concentration of ES, the active type of the complex, In this case a = 1 (because El does not form) and or > 1. The y-intercepl of the Lineweaver-Burk plot increases by a factor of a relative to they-intercept for data on the uninhibited enzyme, but the slope does not change. In non-competitive inhibition, the inhibitor binds to a site other than the active site, and its presence reduces the ability of the substrate to bind to the active site. Inhibition occurs at both the E and ES sites. This condition corresponds to a > I and a > I. Both the slope and y-intercept... 

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