Michaelis substrate inhibitor

Case 2 Integrated Michaelis-Menten Equation in the Presence of Substrate Inhibitor... 

Measurement of the in vitro efficacy of compounds as substrates is usually deduced by comparison of their k JK ratios where is the first-order rate constant for product formation and is the Michaelis equilibrium constant [38]. For those compounds which are classical, reversible inhibitors, K, the dissociation (or inhibition) equilibrium constant, and (kassoc) the rate constant for enzyme inhibition, are the most commonly reported kinetic values. These values may be measured while using either a high-molecular-weight natural substrate or a low-molecular-weight synthetic substrate. For alternate-substrate inhibitors, that is, compounds which form a stable complex (an acyl-enzyme ) that dissociates to enzyme and intact inhibitor or to enzyme and an altered form of the inhibitor, the usually reported value is K, the apparent K. For compounds which irreversibly inactivate the enzyme, the kinetics are usually measured under conditions such that the initial enzyme concentration [E] is much lower than the inhibitor concentration [I] which in turn is much lower than the Ky Under these conditions the commonly reported value is obs/[I]> the apparent... 

For alternate-substrate inhibitors, the enzyme (E) and inhibitor (I) first interact to give a reversible Michaelis complex (E I). This complex rapidly progresses to a relatively stable acyl-enzyme (E I) which may slowly either revert back to the active enzyme (E) and the intact inhibitor (I) or continue on via the normal catalytic machinery to the active enzyme (E) and the modified inhibitor (I ). With dual-acting inhibitors the acyl-enzyme contains a second reactive functionality which acylates or alkylates a second amino acid residue in the enzyme active-site, while the compound is still tethered to Ser-195 [177], resulting in inactivated enzyme (E I). 

The inactivation kinetics of this type can indeed be treated as a special case of the Michaelis-Menten mechanism in which the turnover of the substrate is too slow compared with the rate (ki) of enzyme inactivation. Penicillins are typical examples of such substrate inhibitors for /3-lactamases. Certain classes of irreversible inhibitors, called suicide inhibitors, are chemically unreactive in the absence of target enzymes. When an enzyme binds the innocuous inhibitor with the same specificity as the substrate, however, the inhibitor is activated into a powerful irreversible inhibitor. 

Rubinow, S.I Lebowitz, J. L. (1970). Time-dependent michaelis-menten kinetics for an enzyme-substrate-inhibitor system. J. Am. Chem. Soc. 92, 3888-3893. 

The inactivation is normally a first-order process, provided that the inhibitor is in large excess over the enzyme and is not depleted by spontaneous or enzyme-catalyzed side-reactions. The observed rate-constant for loss of activity in the presence of inhibitor at concentration [I] follows Michaelis-Menten kinetics and is given by kj(obs) = ki(max) [I]/(Ki + [1]), where Kj is the dissociation constant of an initially formed, non-covalent, enzyme-inhibitor complex which is converted into the covalent reaction product with the rate constant kj(max). For rapidly reacting inhibitors, it may not be possible to work at inhibitor concentrations near Kj. In this case, only the second-order rate-constant kj(max)/Kj can be obtained from the experiment. Evidence for a reaction of the inhibitor at the active site can be obtained from protection experiments with substrate [S] or a reversible, competitive inhibitor [I(rev)]. In the presence of these compounds, the inactivation rate Kj(obs) should be diminished by an increase of Kj by the factor (1 + [S]/K, ) or (1 + [I(rev)]/I (rev)). From the dependence of kj(obs) on the inhibitor concentration [I] in the presence of a protecting agent, it may sometimes be possible to determine Kj for inhibitors that react too rapidly in the accessible range of concentration. ... 

In this scheme, EOH is the enzyme, IX is the inhibitor (either a carbamate or an organophosphate). EOH(IX) is analogous to the Michaelis Menton comploc seen with the substrate reaction. EOI is the acyl-enzyme intermediate for carbamates or a phosphoro-enzyme intermediate for the organophosphates. The equilibrium constant for this reaction (K ) is defined as k /k and the phosphorylation or carbamylation constant is defined as k2- In this study 42)y ANTX-A(S) was found to be more specific for AChE than BUChE. The double reciprocal and Dixon plot of the inhibition of electric eel AChE indicated that the toxin is a non-competitive inhibitor decreases, k remains unchanged) (Figure 2). 

Fig. 39.17. Schematic illustration of Michaelis-Menten kinetics in the absence of an inhibitor (solid line) and in the presence of a competitive inhibitor (dashed line), (a) Plot of initial rate (or velocity) V against amount (or concentration) of substrate X. Note that the two curves tend to the same horizontal asymptote for large values of X. (b) Lineweaver-Burk linearized plot of 1/V against l/X. Note that the two lines intersect at a common intercept on the vertical axis.

We now consider the case of a competitive inhibitor which has been added to the above reaction at the fixed concentration of 40 mM [15]. The following initial velocities of the competitively inhibited Michaelis-Menten process are observed at the same substrate concentrations as above ... 

In the presence of sucrose alone as the single substrate, initial reaction rates follow Michaelis-Menten kinetics up to 200 mM sucrose concentration, but the enzyme is inhibited by higher concentrations of substrate.30 The inhibitor constant for sucrose is 730 mM. This inhibition can be overcome by the addition of acceptors.31,32 The enzyme activity is significantly enhanced, and stabilized, by the presence of dextran, and by calcium ions. 

The kinetics of the general enzyme-catalyzed reaction (equation 10.1-1) may be simple or complex, depending upon the enzyme and substrate concentrations, the presence/absence of inhibitors and/or cofactors, and upon temperature, shear, ionic strength, and pH. The simplest form of the rate law for enzyme reactions was proposed by Henri (1902), and a mechanism was proposed by Michaelis and Menten (1913), which was later extended by Briggs and Haldane (1925). The mechanism is usually referred to as the Michaelis-Menten mechanism or model. It is a two-step mechanism, the first step being a rapid, reversible formation of an enzyme-substrate complex, ES, followed by a slow, rate-determining decomposition step to form the product and reproduce the enzyme ... 

FIGURE 6.51 Determination of Vmax and Km for ATP. A Michaelis-Menten plot of PKA phosphorylation of labeled peptide substrate was used to determine Vmax and Km in the presence and absence of three concentrations of inhibitor H-89. 

If a drug is a substrate of an enzyme, it will also be a competitive inhibitor of that enzyme, but it may be a competitive inhibitor without being a substrate. This is because the rate of product formation is determined by k3 of the Michaelis-Menten equation while the rate of ES substrate dissociation and degree of enzyme inhibition is determined by the ratio of A/a i as discussed above. If A 3 is very small it will not be experimentally measurable however, the enzyme will still be bound and occupied as determined by ki/k. ... 

The inhibition can be interpreted as an increase of the Michaelis constant KM. In the case of uncompetitive inhibition (Fig. 9B), the binding of the substrate to the enzyme is not affected. However, the [ES] complex becomes inactive upon binding of the inhibitor Using Kj —> oo, the corresponding rate equation is... 

The overall influence of ATP on the rate V (ATP) is measured by a saturation parameter C (—oo, 1]. Note that, when using Eq. (139) as an explicit rate equation, the saturation parameter implicitly specifies a minimal Hill coefficient min > C necessary to allow for the reverse transformation of the parameters. The interval 6 [0,1] corresponds to conventional Michaelis Menten kinetics. For = 0, ATP has no net influence on the reactions, either due to complete saturation of a Michaelis Menten term or, equivalently, due to an exact compensation of the activation by ATP as a substrate by its simultaneous effect as an inhibitor. For < 0, the inhibition by ATP supersedes the activation of the reaction by its substrate ATP. 

The dependence (1 TP of v, on ATP is modeled as in the previous section, using an interval C [—00,1] that reflects the dual role of the cofactor ATP as substrate and as inhibitor of the reaction. All other reactions are assumed to follow Michaelis Menten kinetics with ()rs E [0, 1], No further assumption about the detailed functional form of the rate equations is necessary. Given the stoichiometry, the metabolic state and the matrix of saturation parameter, the structural kinetic model is fully defined. An explicit implementation of the model is provided in Ref. [84],... 

Figure 8.5 The kinetic effects of a competitive inhibitor. The effects of a compet- itive inhibitor can be reversed by high concentrations of substrate and the increased amount required to achieve maximum velocity results in an increased value for the Michaelis constant.

Another type of inhibitor combines with the enzyme at a site which is often different from the substrate-binding site and as a result will inhibit the formation of the product by the breakdown of the normal enzyme-substrate complex. Such non-competitive inhibition is not reversed by the addition of excess substrate and generally the inhibitor shows no structural similarity to the substrate. Kinetic studies reveal a reduced value for the maximum activity of the enzyme but an unaltered value for the Michaelis constant (Figure 8.7). There are many examples of non-competitive inhibitors, many of which are regarded as poisons because of the crucial role of the inhibited enzyme. Cyanide ions, for instance, inhibit any enzyme in which either an iron or copper ion is part of the active site or prosthetic group, e.g. cytochrome c oxidase (EC 1.9.3.1). 

Under many circumstances, the behavior of a simple unireactant enzyme system cannot be described by the Michaelis-Menten equation, although a v versus [S] plot is still hyperbolic and can be described by a modified version of the equation. For example, as will be discussed later, when enzyme activity is measured in the presence of a competitive inhibitor, hyperbolic curve fitting with the Michaelis-Menten equation yields a perfectly acceptable hyperbola, but with a value for Km which is apparently different from that in the control curve O Figure 4-7). Of course, neither the affinity of the substrate for the active site nor the turnover number for that substrate is actually altered by the presence of a competitive... 

Thereafter, and V ax values for substrate turnover are determined in the absence (controls) and presence of several concentrations of the inhibitor of interest. It is recommended that substrate turnover in the presence of at least four concentrations of inhibitor are examined, at concentrations between 1/3 x IC50 and 4 x IC50. Velocity data are then plotted versus substrate concentration, yielding a control plot and plots at each of the concentrations of inhibitor assessed. Hyperbolic curves are then fitted to data with the Michaelis-Menten equation, or with whichever variation of the Michaelis-Menten equation was found to describe control enzyme behavior most appropriately (see Section 4.1.4 etseq.). In this way, a pattern of changes in Km and Vmax> or both, should become apparent with changing inhibitor concentration. 

Substrate analogs (2) have properties similar to those of one of the substrates of the target enzyme. They are bound by the enzyme, but cannot be converted further and therefore reversibly block some of the enzyme molecules present. A higher substrate concentration is therefore needed to achieve a halfmaximum rate the Michaelis constant increases (B). High concentrations of the substrate displace the inhibitor again. The maximum rate V ax is therefore not influenced by this type of inhibition. Because the substrate and the inhibitor compete with one another for the same binding site on the enzyme, this type of inhibition is referred to as competitive. Analogs of the transition state (3) usually also act competitively. 

While this model explained the action of the brain enzyme on a number of hexose substrates and nonsubstrate inhibitory analogs, the mode had its weaknesses. It assumed that the other conformations of a hexose that are in equilibrium with the active conformer act as competitive inhibitors relative to this conformer. One cannot evaluate the effect of a competitive inhibitor which is present in a constant proportion relative to the active substrate by initial velocity measurements. Moreover, the use of apparent Michaelis constants may not provide accurate estimates of affinity, which is more directly related to a dissociation constant. The chief limitation of the model, however, is that an equally great number of experimental facts can be satisfactorily explained in terms of a simpler scheme involving the binding and phosphorylation of the Cl conformer. Furthermore, one can understand more directly how the enzyme can phosphorylate glucopyranose and fructofuranose equally well. 

Figure 1. Plot of v/V ax versus the millimolar concentration of total substrate for a model enzyme displaying Michaelis-Menten kinetics with respect to its substrate MA (i.e., metal ion M complexed to otherwise inactive ligand A). The concentrations of free A and MA were calculated assuming a stability constant of 10,000 M k The Michaelis constant for MA and the inhibition constant for free A acting as a competitive inhibitor were both assumed to be 0.5 mM. The ratio v/Vmax was calculated from the Michaelis-Menten equation, taking into account the action of a competitive inhibitor (when present). The upper curve represents the case where the substrate is both A and MA. The middle curve deals with the case where MA is the substrate and where A is not inhibitory. The bottom curve describes the case where MA is the substrate and where A is inhibitory. In this example, [Mfotai = [Afotai at each concentration of A plotted on the abscissa. Note that the bottom two curves are reminiscent of allosteric enzymes, but this false cooperativity arises from changes in the fraction of total "substrate A" that has metal ion bound. For a real example of how brain hexokinase cooperatively was debunked, consult D. L. Purich H. J. Fromm (1972) Biochem. J. 130, 63.

There are quite a few enzyme reactions of aromatic substrates and inhibitors where the steric effect of aromatic substituents is separated from others with Es constant. Eq. 9 was developed for the Michaelis constant, Km, of substituted D-phenylglycines with hog kidney D-phenylglycine oxidase11). 

Also characteristic of enzymes that obey Michaelis-Menten kinetics is that suitable inhibitors can compete with the substrate for the enzyme active site, thus impeding the reaction. If the inhibitor binds reversibly to the enzyme active site, then the substrate can compete for the active site leading to competitive inhibition. To test for... 

In comparison with Equation 3.28 for the reaction without inhibition, the apparent value ofthe Michaelis constant increases by (K Ci)/Ky and hence the reaction rate decreases. At high substrate concentrations, the reaction rates approach the maximum value because large amounts of the substrate decrease the effect of the inhibitor. 

Reversible Inhibition One common type of reversible inhibition is called competitive (Fig. 6-15a). A competitive inhibitor competes with the substrate for the active site of an enzyme. While the inhibitor (I) occupies the active site it prevents binding of the substrate to the enzyme. Many competitive inhibitors are compounds that resemble the substrate and combine with the enzyme to form an El complex, but without leading to catalysis. Even fleeting combinations of this type will reduce the efficiency of the enzyme. By taking into account the molecular geometry of inhibitors that resemble the substrate, we can reach conclusions about which parts of the normal substrate bind to the enzyme. Competitive inhibition can be analyzed quantitatively by steady-state kinetics. In the presence of a competitive inhibitor, the Michaelis-Menten equation (Eqn 6-9) becomes... 

Two other types of reversible inhibition, uncompetitive and mixed, though often defined in terms of one-substrate enzymes, are in practice observed only with enzymes having two or more substrates. An uncompetitive inhibitor (Fig. 6-15b) binds at a site distinct from the substrate active site and, unlike a competitive inhibitor, binds only to the ES complex. In the presence of an uncompetitive inhibitor, the Michaelis-Menten equation is altered to... 

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