Inhibitor concentration, inhibited-rate

Again the dependence on [S] is unaffected however, the limiting velocity at high substrate concentration is now a function of inhibitor concentration. The rate laws (5.55) and (5.56) are kinetically distinguishable and permit differentiation between these possible modes of inhibition. Another important, distinguishable process is noncompetitive inhibition (Problems 5.13 and 5.14). 

Substrate and product inhibitions analyses involved considerations of competitive, uncompetitive, non-competitive and mixed inhibition models. The kinetic studies of the enantiomeric hydrolysis reaction in the membrane reactor included inhibition effects by substrate (ibuprofen ester) and product (2-ethoxyethanol) while varying substrate concentration (5-50 mmol-I ). The initial reaction rate obtained from experimental data was used in the primary (Hanes-Woolf plot) and secondary plots (1/Vmax versus inhibitor concentration), which gave estimates of substrate inhibition (K[s) and product inhibition constants (A jp). The inhibitor constant (K[s or K[v) is a measure of enzyme-inhibitor affinity. It is the dissociation constant of the enzyme-inhibitor complex. 

To distinguish between simple, reversible slow binding (scheme B) and an enzyme isomerization mechanism (scheme C), one can examine the dependence of kobs on inhibitor concentration. If the slow onset of inhibition merely reflects inherently slow binding and/or dissociation, then the term kobs in Equations (6.1) and (6.2) will depend only on the association and dissociation rate constants k3 and k4 as follows ... 

This is a linear equation, and we can thus expect kobs to track linearly with inhibitor concentration for an inhibitor conforming to the mechanism of scheme B. As illustrated in Figure 6.4, a replot of kobs as a function of [/] will yield a straight line with slope equal to k3 and y-intercept equal to k4. It should be noted that in such an experiment the measured value of k3 is an apparent value as this association rate constant may be affected by the concentration of substrate used in the experiment, depending on the inhibition modality of the compound (vide infra). Hence the apparent value of Ki (Kfw) for an inhibitor of this type can be calculated from the ratio of... 

For example, experimental data might reveal that a novel enzyme inhibitor causes a concentration-dependent increase in Km, with no effect on and with Lineweaver-Burk plots indicative of competitive inhibition. Flowever, even at very high inhibitor concentrations and very low substrate concentrations, it is observed that the degree of inhibition levels off when some 60% of activity still remains. Furthermore, it has been confirmed that only one enzyme is present, and all appropriate blank rates have been accounted for. It is clear that full competitive inhibition cannot account for such observations because complete inhibition can be attained at infinitely high concentrations of a full competitive inhibitor. Thus, it is likely that the inhibitor binds to the enzyme at an allosteric site. 

Full and partial noncompetitive inhibitory mechanisms, (a) Reaction scheme for full noncompetitive inhibition indicates binding of substrate and inhibitor to two mutually exclusive sites. The presence of inhibitor prevents release of product, (b) Lineweaver-Burk plot for full noncompetitive inhibition reveals a common intercept with the 1/[S] axis and an increase in slope to infinity at infinitely high inhibitor concentrations. In this example, K =3 IulM. (c) Replot of Lineweaver-Burk slopes from (b) is linear, confirming a full inhibitory mechanism, (d) Reaction scheme for partial noncompetitive inhibition indicates binding of substrate and inhibitor to two mutually exclusive sites. The presence of inhibitor alters (reduces) the rate of release of product by a factor p. (e) Lineweaver-Burk plot for partial noncompetitive inhibition reveals a common intercept with the 1/[5] axis and an increase in slope to a finite value at infinitely high inhibitor concentrations. In this example, /Cj= 3 iulM and P = 0.5. (f) Replot of Lineweaver-Burk slopes from (e) is hyperbolic, confirming a partial inhibitory mechanism... 

In partial (hyperbohc) mixed inhibition (O Figure 4-12d), binding of inhibitor to a site distinct from the active site results in altered affinity of enzyme for substrate (by a factor, ot) as well as a change (by a factor, /i) in the rate at which product can be released from ESI. The effects of a partial mixed inhibitor on a Lineweaver-Burk plot depend upon the actual values, and on the relative values, of ot and fl. Once again, inhibitor plots can intersect the control plot above or below, but not on, the oeaxis, and to the left or to the right of, but not on, the y-axis. Because Vmax cannot be driven to zero, a maximum Lineweaver-Burk slope is reached at infinitely high inhibitor concentrations beyond which no further increase occurs. 

A dimensionless quantity used to assess the extent of inhibition by a particular compound at a specific concentration on the initial rate of an enzyme-catalyzed reaction. The degree of inhibition, symbolized by ei, is equal to (vo Vi)/Vo where Vo in the reaction rate in the absence of the inhibitor and Vi is the rate in the presence of the inhibitor. Whenever ej values are reported, the inhibitor concentration and initial rate conditions have to be provided as well. The degree of inhibition is a useful parameter in the early stages of an investigation. However, it does not address the type of inhibition nor provide much information on the dissociation constant for the inhibitor. See Inhibition (as well as specific type of inhibition)... 

The rate equation to be used for kinetic analysis of enzyme depletion is that for simple noncompetitive inhibition. If the Henderson equation or similar types are not employed, keep in mind that the inhibitor concentration [I] is the free inhibitor concentration. Determination of Ki may not be feasible if the rate assay is insensitive and requires an enzyme concentration much greater than K[. Alternatively, Ki may be obtained by measuring the on-off rate constants of the E l complex, provided the rate constants for any conformation change steps involved are also known. 

Equation 3-138 shows the rate of inhibited polymerization to be dependent on the first power of the initiation rate. Further, Rp is inversely dependent on the inhibitor concentration. The induction period observed for inhibited polymerization is directly proportional to the inhibitor concentration. 

A careful consideration of Eq. 3-140b shows that for a strong retarder (z 1), the polymerization rate will be negligible until the inhibitor concentration is markedly reduced. When the inhibitor concentration becomes sufficiently low, propagation can become competitive with the inhibition reaction. This is more readily seen by considering the equation... 

If the effect of an inhibitor on an enzyme is to be investigated, the Dixon plot is recommended. To obtain data for the Dixon plot, estimate the reaction rate at constant substrate concentration and vary the inhibitor concentration [I]. At competitive inhibition, all the obtained straight lines coincide at a point with the coordinates X = -Ki, y = 1/Vmax> and at non-competitive inhibition all the straight lines have the same intercept on abscissa at x = -Ki. At... 

For experimental reasons (discussed below) the relative rate of decrease should be smaller than 0.01 sec."1. Since kA is of the order of 104 liter per mole per sec., [AH] must be of the order of 10 6 mole per liter. This low concentration of inhibitor will only have an appreciable effect on the rate of autoxidation if the rate of initiation—and consequently the rate of oxidation—is low. To measure the decrease in rate during the short-lived non-steady state, one must be able to determine these low velocities within short periods of time. From the usual inhibition formulas one can compute, for instance, that in order to obtain a ratio of original to inhibited rate of about 5 with [AH] = 10 6 mole per... 

A substrate L-benzoyl arginine /)-nitroanilide hydrochloride was hydrolyzed by trypsin, with inhibitor concentrations of 0, 0.3, and 0.6 mmol 1 T The hydrolysis rates obtained are listed in Table 3.3 [5]. Determine the inhibition mechanism and the kinetic parameters (A", and /Cj) of this enzyme... 

The relative rates of inhibition as a function of pH at low inhibitor concentrations ([I] < K ) follow a bell-shaped curve, with the same pKa values as found for the pH dependence of kcJKu for the hydrolysis of substrates. 

Many inhibitors with very low dissociation constants appear to have a slow onset of inhibition when they are added to a reaction mixture of enzyme and substrate. This was once interpreted as the inhibitors having to induce a slow conformational change in the enzyme from a weak binding to a tight binding state. But in most cases, the slow binding is an inevitable consequence of the low concentrations of inhibitor used to determine its Ki. For example, consider the inhibition of trypsin by the basic pancreatic trypsin inhibitor. Kx is 6 X 10-14 M and the association rate constant is 1.1 X 106 s-1 M-1 (Table 4.1). To determine the value of Ki, inhibitor concentrations should be in the range of K1, where the observed first-order rate constant for association is (6 X Q U M) X (1.1 X 106 s-1 M-1) that is, 6.6 X 10-8 s 1. The half-life is (0.6931/6.6) X 108 s, which is more than 17 weeks. 

Determination of an inhibition constant X, is often attempted by a short-cut procedure instead of the correct method, which entails the measurement of several v-[S] plots at different concentrations [I]. This short-cut procedure calls for measuring the reaction rate v in the absence of the inhibitor, subsequent addition of different concentrations [I] of inhibitor at constant substrate concentration and determination of IC50, in other words, the inhibitor concentration at which the initial reaction rate is decreased to 50% of the rate at [I] = 0. Such a short-cut procedure can be misleading, however, as will be demonstrated in the following paragraphs. 

It should be noted that the mechanism depicted in Scheme 1 is the simplest that is consistent with mechanism-based inhibition. The mechanism for a given inhibitor and enzyme may be considerably more complex due to (a) multiple intermediates [e.g., MIC formation often involves four or more intermediates (29)], (b) detectable metabolite that may be produced from more than one intermediate, and (c) the fact that enzyme-inhibitor complex may produce a metabolite that is mechanistically unrelated to the inactivation pathway. Events such as these will necessitate alternate definitions for Z inact, Kh and r in terms of the microrate constants of the appropriate model. The hyperbolic relationship between rate of inactivation and inhibitor concentration will, however, remain, unless nonhyperbolic kinetics characterize this interaction. Silverman discussed this possibility from the perspective of an allosteric interaction between inhibitor and enzyme (5). Nonhyperbolic kinetics has been observed for the interaction of several drugs with members of the CYPs (30). 

---