Enzymes, inhibition, substrate Michaelis-Menten equation

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. ... 

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. 

The Michaelis-Menten equation, especially if derived with the steady-state concept as above, is a rigorous rate law which not only fits almost all one-substrate enzyme kinetics, except in the case of inhibition (see Section 5.3), but also allows identification of the kinetic constants with the elementary steps in Eq. (5.1). 

If two different substrates bind simultaneously to the active site, then the standard Michaelis-Menten equations and competitive inhibition kinetics do not apply. Instead it is necessary to base the kinetic analyses on a more complex kinetic scheme. The scheme in Figure 6 is a simplified representation of a substrate and an effector binding to an enzyme, with the assumption that product release is fast. In Figure 6, S is the substrate and B is the effector molecule. Product can be formed from both the ES and ESB complexes. If the rates of product formation are slow relative to the binding equilibrium, we can consider each substrate independently (i.e., we do not include the formation of the effector metabolites from EB and ESB in the kinetic derivations). This results in the following relatively simple equation for the velocity ... 

Enzyme kinetics and the mode of inhibition are well described by transformation of the Michaelis-Menten equation. The binding affinity of the inhibitor to the enzyme is defined as the inhibition constant Ki, whereas the affinity, with which the substrate binds, is referred to the Michaelis-Menten coefficient Km. Michaelis-Menten kinetics base on three assumptions ... 

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),... 

Although it is quite simple to set up an experiment to determine the variation of v with [5], the exact value of V,n,y is not easily determined from hyperbolic curves. Furthermore, many enzymes deviate from ideal behavior at high substrate concentrations and indeed may be inhibited by excess substrate, so the calculated value of cannot be achieved in practice. In the past it was common practice to transform die Michaelis-Menten equation (9) into one of several reciprocal forms (equations [10] and [11]), and either 1/v was plotted against 1/[S], or [S]/v was plotted against [S]. 

In textbooks dealing with enzyme kinetics, it is customary to distinguish four types of reversible inhibitions (i) competitive (ii) noncompetitive (iii) uncompetitive and, (iv) mixed inhibition. Competitive inhibition, e.g., given by the product which retains an affinity for the active site, is very common. Non-competitive inhibition, however, is very rarely encountered, if at all. Uncompetitive inhibition, i.e. where the inhibitor binds to the enzyme-substrate complex but not to the free enzyme, occurs also quite often, as does the mixed inhibition, which is a combination of competitive and uncompetitive inhibitions. The simple Michaelis-Menten equation can still be used, but with a modified Ema, or i.e. ... 

Which of these plots should be used To generally understand the behavior of enzymes, use the simple graph of initial velocity against substrate concentration. The linearized forms are useful for calculation of ATM and Fmax. The Lineweaver-Burke plot is useful for distinguishing between types of inhibition (Chapter 8). The Eadie-Hofstee plot is better than the Lineweaver-Burke plot at picking up deviations from the Michaelis-Menten equation. 

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. 

The mechanisms of enzyme inhibition fall into three main types, and they yield particular forms of modified Michaelis-Menten equations. These can be derived for single-substrate/single-product enzymic reactions using the steady-state analysis of Sec. 5.10, as follows. 

Such an effect can be useful in the case of enzymatic substrate determinations (cf. 2.6.1.3). When inhibitor activity is absent, i. e. [I] = 0, Equation 2.72 is transformed into the Michaelis-Menten equation (Equation 2.41). The Line-weaver-Burk plot (Fig. 2.30a) shows that the intercept 1 /V with the ordinate is the same in the presence and in the absence of the inhibitor, i. e. the value of V is not affected although the slopes of the lines differ. This shows that the inhibitor can be fully dislodged by the substrate from the active site of the enzyme when the substrate is present in high concentration. In other words, inhibition can be overcome at high substrate concentrations (see application in Fig. 2.49). The inhibitor constant, Ki, can be calculated from the corresponding intercepts with the abscissa in Fig. 2.30a by calculating the value of from the abscissa intercept when [I] = 0. 

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). 

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 ... 

The Michaelis-Menten kinetic scheme, which involves a single substrate and a single product, is obviously the simplest type of enzyme catalysis. Equation (1.7) may hold for many mechanisms, but the mechanisms can be different from each other and the expression of kinetic parameters may also differ. When there is a substrate inhibition or activation due to the binding of a second substrate molecule, the Michaelis-Menten equation does not hold. 

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... 

Kmi would be the standard Michaelis constant for the binding of the first substrate, if [ESS] = 0. Km2 would be the standard Michaelis constant for the binding of the second substrate, if [E] = 0 (i.e., the first binding site is saturated). In the complete equation, these constants are not true Km values, but their form (i.e., Km] = (k2 + k25)/k 2) and significance are analogous. Likewise, k25 and k35 are Vmi/Et and Vm2/Et terms when the enzyme is saturated with one and two substrate molecules, respectively. Equation (10) describes several non-Michaelis-Menten kinetic profiles. Autoactivation (sigmoidal saturation curve) occurs when k35 > k24 or Km2 < Km 1, substrate inhibition occurs when k24 > 35, and a biphasic saturation... 

Unfortunately, most enzymes do not obey simple Michaelis-Menten kinetics. Substrate and product inhibition, presence of more than one substrate and product, or coupled enzyme reactions in multi-enzyme systems require much more complicated rate equations. Gaseous or solid substrates or enzymes bound in immobilized cells need additional transport barriers to be taken into consideration. Instead of porous spherical particles, other geometries of catalyst particles can be apphed in stirred tanks, plug-flow reactors and others which need some modified treatment of diffusional restrictions and reaction technology. 

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... 

The reaction rate for the hydrolysis of starch (Eq. 2.1 in Table 3) is a Michaelis-Menten type model, which considers competitive product inhibition of glucose and substrate inhibition of starch. The hydrolysis of maltose (Eq. 2.2 in Table 3) is represented by a Michaelis-Menten type model with competitive product inhibition. These equations were tested by Lopez et al. [3] for hydrolysis of chestnut puree by an alpha and glucoamylase mixture. As the enzyme STARGEN also contains amormts of alpha and glucoamylase, it was not surprising that they (Eqs. 2.1-2.2 in Table 3) fit the hydrolysis data better than non-inhibitory Michaelis-Menten kinetics. 

---