Menten with Competitive Inhibitor

The concept of the active site in a floating, mobile, water-soluble enzyme can be extended to biological receptors that are fixed in cell membranes and a similar analysis can be applied to competitors to natural substrates. This means that it is important to have an analysis procedure for competitive inhibition. Consider the Michaelis-Menten equations with an inhibitor 1  

Now we proceed with the familiar Michaelis-Menten derivation for the usual steady-state approximation  

Insert the formula for [Efree] into the steady state, so 

But we need to isolate [E S] and you should follow this step with pencO and paper  

Cancel the (1 + iTi[l]) denominator on both sides of the equation and we obtain 

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It is revealing to compare the equation for the uninhibited case. Equation (14.23) (the Michaelis-Menten equation) with Equation (14.43) for the rate of the enzymatic reaction in the presence of a fixed concentration of the competitive inhibitor, [I]... 

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. ... 

Note the close analogy with the Lineweaver-Burk form of the simple Michaelis-Menten equation. In a diagram representing MV against MX one obtains a line which has the same intercept as in the simple case. The slope, however, is larger by a factor (1 + YIK-) as shown in Fig. 39.17b. Usually, one first determines and in the absence of a competitive inhibitor (F = 0), as described above. Subsequently, one obtains A" from a new set of experiments in which the initial rate V is determined for various levels of X in the presence of a fixed amount of inhibitor Y. The slope of the new line can be obtained by means of robust regression. 

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... 

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... 

Hyperbolic curve fits to control enzymatic data and to data obtained in the presence of a competitive inhibitor. Curve fitting to the Michaelis-Menten equation results in two different values for Km- However, Km does not, in actuality, change, and the value in the presence of inhibitor (15 uiM) is an apparent value. Fitting with the correct equation, that for turnover in the presence of a competitive inhibitor ( Eq. 5), results in plots identical in appearance to those obtained with the Michaelis-Menten equation. However, nonlinear regression now reveals that Km remains constant at 5 ulM and that [l]/Ki = 2.5 with knowledge of [/], calculation of K is straightforward... 

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.

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... 

Reversible inhibition occurs rapidly in a system which is near its equilibrium point and its extent is dependent on the concentration of enzyme, inhibitor and substrate. It remains constant over the period when the initial reaction velocity studies are performed. In contrast, irreversible inhibition may increase with time. In simple single-substrate enzyme-catalysed reactions there are three main types of inhibition patterns involving reactions following the Michaelis-Menten equation competitive, uncompetitive and non-competitive inhibition. Competitive inhibition occurs when the inhibitor directly competes with the substrate in forming the enzyme complex. Uncompetitive inhibition involves the interaction of the inhibitor with only the enzyme-substrate complex, while non-competitive inhibition occurs when the inhibitor binds to either the enzyme or the enzyme-substrate complex without affecting the binding of the substrate. The kinetic modifications of the Michaelis-Menten equation associated with the various types of inhibition are shown below. The derivation of these equations is shown in Appendix S.S. 

This is in the form of the Michaelis-Menten equation with being divided by the Thus, for a simple linear non-competitive inhibitor, Km remains unchanged while that factor (1 + (//, )). mu s altered so... 

Substances that cause enzyme-catalyzed reactions to proceed more slowly are termed inhibitors, and the phenomenon is termed inhibition. When an enzyme is subject to inhibition, the reaction still may obey Michaelis-Menten kinetics but with apparent Km and Vmax values that vary with the inhibitor concentration. If the inhibitor acts only on the apparent Km, it is a competitive inhibitor if it affects only the apparent Vmax, it is a noncompetitive inhibitor and if it affects both constants, it is an uncompetitive inhibitor. 

Many substances interact with enzymes to lower their activity that is, to inhibit them. Valuable information about the mechanism of action of the inhibitor can frequently be obtained through a kinetic analysis of its effects. To illustrate, let us consider a case of competitive inhibition, in which an inhibitor molecule, I, combines only with the free enzyme, E, but cannot combine with the enzyme to which the substrate is attached, ES. Such a competitive inhibitor often has a chemical structure similar to the substrate, but is not acted on by the enzyme. For example, malonate (-OOCCH2COO-) is a competitive inhibitor of succinate (-OOCCH2CH2COO-) dehydrogenase. If we use the same approach that was used in deriving the Michaelis-Menten equation together with the additional equilibrium that defines a new constant, an inhibitor constant, A),... 

Figure 13.3a shows a plot of the rate of packaging as a function of the ATP concentration under a constant force of 5pN. This data is well described by the characteristic Michaelis-Menten behavior with a Pmax 100 bp s and a Km 30 gM. Interestingly, the fit, done to a Michaelis-Menten-Hill equation reveals a Hill coefficient n I, indicating that the binding of the ATP to the motor is not cooperative. These same studies revealed that ADP is a competitive inhibitor of the motor and that phosphate release should be a nearly irreversible step [55], as its concentration in solution can be varied three orders of magnitude without affecting the rate of the motor. 

Unlike the major cellulase components, beta-glucosidase is a soluble enzyme acting on a soluble substrate. Beta-glucosidase is characterized by classical Michaelis-Menten kinetics, with glucose acting as a competitive inhibitor [32] ... 

The term should be used for enzymes that display Michaelis-Menten kinetics. Thus, it is not used with allosteric enzymes. Technically, competitive and noncompetitive inhibition are also terms that are restricted to Michaelis-Menten enzymes, although the concepts are applicable to any enzyme. An inhibitor that binds to an allosteric enzyme at the same site as the substrate is similar to a classical competitive inhibitor. One that binds at a different site is similar to a noncompetitive inhibitor, but the equations and the graphs characteristic of competitive and noncompetitive inhibition don t work the same way with an allosteric enzyme. 

Most of the compoimds that have been foimd to be good synthetase inhibitors are structurally related to the nicotinamide moiety of the enzyme substrate, NAD. Additionally, these synthetase inhibitors have been found to act competitively with respect to NAD. Rearrangement of the Michaelis-Menten equation for a competitive inhibitor yields the following ... 

More detailed studies of the oxidation of propylene [380, 381] and 1-octene [382] have shown that the actual rate equation is of the Michaelis-Menten type with the co-product alcohol being the competitive inhibitor. The fact that alcohols inhibit the molybdenum catalyzed epoxidation of olefins is well known [377, 378, 383]. In addition, it has been noted that induction periods are observed [378, 382] during which time the active catalyst, a molybdenum(VI) species [384, 379] is formed. 

The Michaelis-Menten model was fitted to the experimental data using standard nonlinear regression techniques to obtain estimates of and K (Fig. 4.1). Best-fit values of and K of corresponding standard errors of the estimates plus the number of values used in the calculation of the standard error, and of the goodness-of-fit statistic are reported in Table 4.3. These results suggest that succinate is a competitive inhibitor of fumarase. This prediction is based on the observed apparent increase in Ks in the absence of changes in Vmax (see Table 4.1). At this point, however, the experimenter cannot state with any certainty whether the observed apparent increase in Ks is a tme effect of the inhibitor or merely an act of chance. A proper statistical analysis has to be carried out. For the comparison of two values, a two-tailed t-test is appropriate. When more than two values are compared, a one-way analysis of variance (ANOVA),... 

Consider an enzyme-catalyzed reaction that follows Michaelis-Menten kinetics with Km = 3.0 mmol dm". What concentration of a competitive inhibitor characterized by Ki = 20 pmol dm" will reduce the rate of formation of product by 50 per cent when the substrate concentration is held at 0.10 mmol dm" ... 

With the clear way non-competitive inhibition mimics the kinetics observed with the Michaelis-Menten equation, the manner in which competitive inhibitors affect enzyme activity becomes obscure. This can be demonstrated through the same rearrangement of the inhibitory term which directly affects the substrate affinity (Equations 15-19). 

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... 

Even this scheme represents a complex situation, for ES can be arrived at by alternative routes, making it impossible for an expression of the same form as the Michaelis-Menten equation to be derived using the general steady-state assumption. However, types of non-competitive inhibition consistent with the Michaelis-Menten type equation and a linear Linweaver-Burk plot can occur if the rapid-equilibrium assumption is valid (Appendix S.A3). In the simplest possible model, involving simple linear non-competitive inhibition, the substrate does not affect the inhibitor binding. Under these conditions, the reactions... 

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