Nonlinear inhibition

Consider the standard Uni Uni mechanism (E + A EX E + P). A noncompetitive inhibitor, I, can bind reversibly to either the free enzyme (E) to form an El complex (having a dissociation constant K s), or to the central complex (EX) to form the EXl ternary complex (having a dissociation constant Xu). Both the slope and vertical intercept of the standard double-reciprocal plot (1/v vx. 1/[A]) are affected by the presence of the inhibitor. If the secondary replots of the slopes and the intercepts (thus, slopes or vertical intercepts vx [I]) are linear (See Nonlinear Inhibition), then the values of those dissociation constants can be obtained from these replots. If Kis = Xu, then a plot of 1/v vx 1/[A] at different constant concentrations of the inhibitor will have a common intersection point on the horizontal axis (if not. See Mixed-Type Inhibition). Note that the above analysis assumes that the inhibitor binds in a rapid equilibrium fashion. If steady-state binding conditions are present, then nonlinearity may occur, depending on the magnitude of the [I] and [A] terms in the rate expression. See also Mixed Type Inhibition... 

This type of inhibition differs from that exhibited by classical competitive inhibitors, because the substrate can still bind to the El complex and the EIS complex can go on to form product (albeit at a slower rate) without the inhibitor being released from the binding site. While standard double-reciprocal plots of partial competitive inhibitors will be linear (except for some steady-state, i.e., non-rapid-equilibrium, cases), secondary slope replots will be nonlinear. See Nonlinear Inhibition... 

NSPORT metal ion EXCHANGE TIMES MICROENVIRONMENT MULTIPLE ATTACK UCLEOPHILIC SUBST TUTION REACTION NONLINEAR INHIBITION RANDOM SCISSIOI VTE MFC MYOGLOBIN OXYGENATION NUCLEATION ONE DIMENSIONA... 

Indirect response models require differential equations and numerical integration algorithms to describe the nonlinear inhibition or stimulation. Partially integrated solutions for these models have been developed (50, 51), which allow qualitative examination of the relationships between response... 

Further division of reversible inhibitors is made according to their influence on the form of rate equations thus, we can make a difference between the linear and a nonlinear inhibition. In Chapter 5, we shall describe the linear inhibition and in Chapter 6 the main forms of the nonlinear inhibitioa Thus, we could distinguish tfie various types of inhibition stiU further by referring to competitive inhibition as linear, hyperbolic, or paraboUc inhibition. Fven more complex forms are possible (Cleland, 1970). 

In Chapter 5, we have examined the properties of several simple linear types of inhibition systems. All these types have, in common, a dead-end El or a nonproductive EAI complex, or both. In order to introduce more complex cases of inhibition, we shall start with an analysis of nonlinear inhibition with less-demanding monosubstrate reactions (Cleland, 1970, 1977 Fromm, 1975). 

Figure l. Nonlinear inhibition in a mooosubatrate reaction. Graphical presentation of Eq. (6.3). 

An infinitely high A will drive aU the enzyme to a mixture of EA and EAI, and at infinitely high I all the enzyme will be converted to El and EAI. Because EAI can form the product, the velocity cannot be driven to zero by increasing I (Fig. 1). This clearly shows that the nonlinear inhibition is not a complete inhibition, as the velocity does not approach zero even at higher concentrations of an inhibitor. 

Thus, it is clear that the primary double reciprocal plots of i/Dq versus 1/A represent a valuable general diagnostic tool for the estimation of the type of nonlinear inhibition mechanisms. 

Similarly as for the nonlinear inhibition with a single substrate and a single inhibitor molecule (Section 6.1), the double reciprocal plots of i/vo versus 1/A or i/u versus 1/B, are a family of straight lines with a single intersection point to the left of the vertical axis. Also, the slope and intercept replots are hyperbolic. 

Kinetically, the nonessential activation can be treated in the same manner as a general nonlinear inhibition however, this time the changes are in the opposite direction wherever we have had an inhibition, now we have an... 

Compare Fig. 1 for the nonlinear inhibition of Chapter 6, and notice that the double reciprocal plot in both figures is a family of straight lines with a common intersection point this intersection point has different coordinates in activation and inhibition systems. 

Similarly to nonlinear inhibition, at infinitely high X, Eq. (7.2) reduces to... 

In this case, since both functions are nonlinear, it is possible to determine the kinetic constants a and fi by the application of the differential method, similarly as described for the nonlinear inhibition in Chapter 6 (Section 6.2) (Fig. 3). 

Nonlinear inhibition (Chapter 6). In nonlinear inhibition, the replots of slope or intercept function from the primary plot of i/Uo versus 1/A plot, at different constant concentrations of I, are not linear. The replots may be curved parabolically or hyperbolically, according to equations ... 

Hirst JD. Nonlinear quantitative structure-activity relationship for the inhibition of dihydrofolate reductase by pyrimidines. J Med Chem 1996 39(18) 3526-32. 

The IC50 can thus be accurately determined by fitting the concentration-response data to Equation (5.1) through nonlinear curve-fitting methods. Some investigators prefer to plot data in terms of % inhibition rather than fractional activity. Using the mass-balance relationships discussed above, we can easily recast Equation (5.1) as follows ... 

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... 

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... 

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. 

The MO concentrations versus time profiles were fitted to second order polynomial equations and the parameters estimated by nonlinear regression analysis. The initial rates of reactions were obtained by taking the derivative at t=0. The reaction is first order with respect to hydrogen pressure changing to zero order dependence above about 3.45 MPa hydrogen pressure. This was attributed to saturation of the catalyst sites. Experiments were conducted in which HPLC grade MIBK was added to the initial reactant mixture, there was no evidence of product inhibition. 

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