Progress curve

Case E, where the three rate constants are comparable, does not feature A2 s> A3. That is, the progress curve is not a single exponential. None of the approximations is very good, although fcjmp is the best. Case F, characterized by a value of k 1 that is much smaller than kss or 2, is similar. The fit for k is quite poor that for ktmp is the best, but it is hardly adequate. 

Figure 2. Progress curve for the incorporation of label from UDP-Gal.

The sum of the individual turnover numbers for each oligomer also shows that the rate of hydrolysis of the oligomers with DP4 and DPS is much slower than of those with higher DP which is reflected in the product progression curves in Fig. 1. 

E I is a kinetic chimera Kj and kt are the constants characterizing the inactivation process kt is the first-order rate constant for inactivation at infinite inhibitor concentration and K, is the counterpart of the Michaelis constant. The k,/K, ratio is an index of the inhibitory potency. The parameters K, and k, are determined by analyzing the data obtained by using the incubation method or the progress curve method. In the incubation method, the pseudo-first-order constants /cobs are determined from the slopes of the semilogarithmic plots of remaining enzyme activity... 

The ratio kJK, is obtained as /cobs/[I] at low inhibitor concentrations. With efficient inhibitors, parameters Kr and kt can be obtained using the progress curve method in which the enzyme substrate competes with the inhibitor as described for example in Ref. 21. 

Figure 2.6 (A) Typical enzyme product progress curve for a reaction going to near completion.

In Chapter 2 we described the typical product progress curve for a well-behaved enzyme and introduced the concept of initial velocity. In assays designed to quantify the ability of a test compound to inhibit the target enzyme, it is critical to restrict... 

Figure 4.3 Product progress curves for an enzyme-catalyzed reaction in the absence (closed circles) and presence open circles) of an inhibitor at a concentration that reduces the reaction rate by 50%. Inset The initial velocity phase of these progress curves.

Figure 4.4 Calculated % inhibition as a function of reaction time from die progress curves shown in Figure 4.3. Note that as die reaction continues past die initial velocity phase (shown in die inset), the apparent % inhibition is dramatically diminished.

Figure 4.5 Example of a reaction progress curve obtained by discontinuous measurement of 33P incorportation into a peptide substrate of a kinase. Each data point represents a measurement made at a discrete time point after initiation of the reaction with y-33P-ATP.

In this chapter we have reviewed some of the basic biochemical considerations that must be taken into account in the design of assays for HTS purposes. We saw that activity measurements must be made during the initial velocity phase of the reaction progress curve to ensure the best chances of observing inhibition by library... 

If the inhibition is found to be rapidly reversible, we must next determine if the approach to equilibrium for the enzyme-inhibitor complex is also rapid. As described in Chapter 4, some inhibitors bind slowly to their target enzymes, on a time scale that is long in comparision to the time scale of the reaction velocity measurement. The effect of such slow binding inhibition is to convert the linear progress curve seen in the absence of inhibitor to a curvilinear function (Figure 5.10). When nonlinear progress curves are observed in the presence of inhibitor, the analysis of... 

Figure 5.10 Progress curves for an enzyme in the absence (open circles) and presence (closed circles) of an slow-binding inhibitor. See Chapter 6 for more details on this form of inhibition.

If the inhibitor is found to bind rapidly (linear progress curves) and dissociate rapidly (rapid recovery of activity upon dilution) from its target enzyme, then one can proceed to analyze its inhibition modality and affinity by classical methods. The modes of reversible inhibition of enzymes were described in Chapter 3. In the next section of this chapter we will describe convenient methods for determining reversible inhibition modality of lead compounds and lead analogues during compound optimization (i.e., SAR) studies. 

The hallmark of slow binding inhibition is that the degree of inhibition at a fixed concentration of compound will vary over time, as equilibrium is slowly established between the free and enzyme-bound forms of the compound. Often the establishment of enzyme-inhibitor equilibrium is manifested over the time course of the enzyme activity assay, and this leads to a curvature of the reaction progress curve over a time scale where the uninhibited reaction progress curve is linear. We saw... 

Figure 6.1 Typical progress curve for an enzyme reaction in the presence of a slow binding inhibitor. The initial (v,) and steady state (vs) velocities are defined by the slope values in the early and late stages of the progress curve, respectively, as indicated by the dashed lines.

Fitting of a progress curve, such as that shown in Figure 6.1 to either Equation (6.1) or (6.2) allows one to obtain an estimate of kohs, vi( and vs at a specific concentration of compound. 

Figure 6.4 (A) Progress curves for an enzymatic reaction in the presence of increasing concentra-...

Figure 6.5 Concentratioin esponse plot of inhibition by a slow binding inhibitor that conforms to scheme B of Figure 6.3. The progress curves of Figure 6.4A were fitted to Equation (6.1). The values of vs thus obtained were used together with die velocity of the uninhibited reaction (v0) to calculate the fractional activity (vs/v0) at each inhibitor concentration. The value of Kf9 is then obtained as the midpoint (i.e., die IC50) of die isotherm curve, by fitting die data as described by Equation (6.8).

Thus, if the reaction progress curve can be followed for a long enough time, under conditions where the unihibited enzyme remains stable, one may be able to measure a small, but nonzero, value for vs. Combining this value with y and obs would allow one to determine k6 from Equation (6.12). 

For compounds that conform to the mechanism of scheme C, an alternative method for defining inhibition modality is to measure progress curves (or preincubation effects vide supra) at varying inhibitor and substrate concentrations, and to then construct a double reciprocal plot of 1/v, as a function of l/[.Sj. Using the analysis methods and equations described in Chapter 3, one can then determine the modality of inhibition for the inhibitor encounter complex. Similarly, for inhibitors that conform to the mechanism of scheme B, a double reciprocal plot analysis of l/vs as a function of 1/[S] can be used to define inhibition modality. 

We have already used the interactions of methotrexate with dihydrofolate reductase (DHFR) several times within this text to illustrate some key aspects of enzyme inhibition. The reader will recall that methotrexate binds to both the free enzyme and the enzyme-NADPH binary complex but displays much greater affinity for the latter species. The time dependence of methotrexate binding to bacterial DHFR was studied by Williams et al. (1979) under conditions of saturating [NADPH], In the presence of varying concentrations of methotrexate, the progress curves for DHFR activity became progressively more nonlinear (Figure 6.14). The value of kobs from... 

Figure 6.14 Progress curves for the enzymatic reaction of dihydrofolate reductase in the presence of the indicated concentrations of methotrexate.

Addition of the L-732,531 FKBP binary complex to a calcineurin activity assay resulted in increasingly nonlinear progress curves with increasing binary complex concentration. The htting of the data to Equation (6.3) revealed an inhibitor concentration effect on v-, as well as on vs and obs, consistent with a two-step mechanism of inhibition as in scheme C of Figure 6.3. Salowe and Hermes analyzed the concentration-response effects of the binary complex on v, and determined an IC50 of 0.90 pM that, after correction for I.S I/A (assuming competitive inhibition), yielded a A) value for the inhibitor encounter complex of 625 nM. 

These practical approaches are by no means mutually exclusive, and attempts should be made to combine as many of these as possible to improve ones ability to experimentally measure the K-pp of tight binding inhibitors. Thus one should always work at the lowest enzyme concentration possible, and drive the substrate concentration as high as possible, when dealing with competitive inhibitors. A long preincubation step should be used before activity measurements, or the progress curves should be fitted to Equation (6.2) so that accurate determinations of the steady state velocity at each inhibitor concentration can be obtained. Finally, the concentration-response data should be fitted to Morrison s quadratic equation to obtain good estimates of the value of Arfpp. 

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