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This article describes an integrated rate equation for the time course of covalent enzyme inhibition under the conditions where the substrate concentration is significantly lower than the corresponding Michaelis constant, for example, in the Omnia assays of epidermal growth factor receptor (EGFR) kinase. The newly described method is applicable to experimental conditions where the enzyme concentration is significantly lower than the dissociation constant of the initially formed reversible enzyme-inhibitor complex (no "tight binding"). A detailed comparison with the traditionally used rate equation for covalent inhibition is presented. The two methods produce approximately identical values of the first-order inactivation rate constant (kinact). However, the inhibition constant (Ki), and therefore also the second-order inactivation rate constant kinact/Ki, is underestimated by the traditional method by up to an order of magnitude.

Optimal experimental designs for the dose-response screening of enzyme inhibitors were studied within the framework of the Box-Lucas theory. If the enzyme concentration E is considered as a fixed constant, an exact two-point D-optimal design consists of a pair of inhibitor concentrations equal to I(1)=0 and I(2)=E+K, where K is the apparent inhibition constant. If the enzyme concentration is treated as an adjustable parameter, an empirical three-point D-optimal design consists of three inhibitor concentrations equal to I(1)=0, I(2)=E+3K, and I(3)=0.7E. These results were applied to design optimized, irregularly spaced concentration series for routine inhibitor screening. A heuristic Monte Carlo simulation study confirmed that the optimized dilution series is significantly more efficient than the classic series characterized by a constant dilution ratio. An online calculator to create optimized dilution series is freely available at http://www.biokin.com/design/.

A novel rate equation to characterize the dose-response behavior of a moderately potent ("classical") enzyme inhibitor contaminated with a very potent ("tight-binding") impurity is derived. Mathematical properties of the new rate equation show that, for such contaminated materials, experimentally observed I(50) values are ambiguous. The four-parameter logistic equation, conventionally used to determine I(50) values, cannot be used to detect the presence of tight-binding impurities in inhibitor samples. In contrast, fitting the newly derived rate equation to inhibitor dose- response curves can, in favorable cases, reveal whether the unknown material is chemically homogeneous or whether it is contaminated with a tight-binding impurity. The limitations of our method with respect to the detectable range of inhibition constants (both classical and tight-binding) were examined by using Monte-Carlo simulations. To test the new analytical procedure experimentally, we added a small amount (0.02 mole%) of a tight-binding impurity (K(i)=0.065 nM) to an otherwise weak inhibitor of human mast-cell tryptase (K(i)=50.4 microM). The resulting material was treated as "unknown." Our kinetic equation predicts that such adulterated material should show I(50)=0.40 microM, which was identical to the experimentally observed value. The best-fit value of the apparent inhibition constants for the tight-binding inhibitor was K(i)=(0.107+/-0.035)nM, close to the true value of 0.065 nM.

Determination of tight-binding inhibition constants by nonlinear least-squares regression requires sufficiently good initial estimates of the best-fit values. Normally an initial estimate of the inhibition constant must be provided by the investigator. This paper describes an automatic procedure for the estimation of tight-binding inhibition constants directly from dose-response data. Because the procedure does not require human intervention, it was incorporated into an algorithm for high-throughput screening of enzyme inhibitors. A suitable computer program is available electronically (http://www.biokin.com). Representative experimental data are shown for the inhibition of human mast-cell tryptase.

When two or more tight-binding inhibitors are present in an enzyme assay, the equation that relates the initial velocity v to the concentration of reactants cannot be written in an algebraically explicit form. Rather, for n inhibitors it is an implicit polynomial equation of degree n + 1 with respect to v. The complexity of the polynomial coefficients dramatically increases with each added inhibitor. Solving the transcendental rate equation by traditional methods of numerical mathematics has proven tedious because of the sensitivity of these methods to initial estimates and because of the existence of multiple roots. However, the equation can be rearranged into a convenient recursive form, one in which the velocity appears on both sides and the solution is found iteratively. The algebraic form of the recursive rate equation is remarkably simple and differs from the rate equation for classical rather than tight-binding inhibition only by an added term. The numerical stability and the speed of convergence were tested on the case of two competitive inhibitors. Initial estimates of velocity that spanned 12 orders of magnitude converged within five iterations. The velocities computed with the recursive method for a single tight-binding inhibitor were identical with the values predicted by the Morrison equation. The method is used to analyze experimental data for the inhibition of rat liver dihydrofolate reductase by mixtures of the anticancer drug methotrexate and its metabolic precursor form, methotrexate-α-aspartate (a prodrug)

For enzymatic progress curves conforming to the Michaelis-Menten mechanism E+Sright harpoon over left harpoonES-->E+P, the minimal fitting model cast as a system of numerically integrated differential equations is the simplified, irreversible Van Slyke-Cullen mechanism E+S-->ES-->E+P. The best-fit value of the bimolecular association rate constant is identical to the specificity constant kcat/KM. An illustrative example involves a fluorogenic continuous assay of the HIV protease, analyzed by the differential-equation oriented software package DYNAFIT [P. Kuzmic, Anal. Biochem. 237 (1996) 260].

The standard mathematical model for stepwise "slow-binding" enzyme inhibition (E+Iright harpoon over left harpoonEIright harpoon over left harpoonEI( *)) assumes that the initial enzyme-inhibitor complex EI is always at equilibrium with the free component species E and I. This assumption implies that the dissociation rate constant (EI-->E+I) is infinitely higher than the isomerization rate constant for EI-->EI( *). This paper presents a more general mathematical treatment, under the steady state approximation rather than the usual rapid-equilibrium approximation, whereby the two rate constants for the disappearance of EI are allowed to be comparable in magnitude. Experimentally relevant illustrative examples include discrimination between a single-step and a two-step mechanism for slow-binding inhibition kinetics.

Active site titration by a reversible tight-binding inhibitor normally depends on prior knowledge of the inhibition constant. Conversely, the determination of tight-binding inhibition constants normally requires prior knowledge of the active enzyme concentration. Often, neither of these quantities is known with sufficient accuracy. This paper describes experimental conditions under which both the enzyme active site concentration and the tight-binding inhibition constant can be determined simultaneously from a single dose-response curve. Representative experimental data are shown for the inhibition of human kallikrein.

This paper describes a general numerical method for the determination of rate constants that characterize the binding of a ligand L simultaneously and competitively to two different receptor molecules, R1 and R2. The experimental data consist of changes in the concentration of one receptor (e.g., R1) monitored over time. An example problem is represented by hirudin (L) binding to thrombin (R1) and to a chemical mutant of thrombin (R2). The published experimental data [Wedemeyer et al. (1997) Anal. Biochem. 248, 130-140], previously analyzed by using an appropriate algebraic method, were reanalyzed here by numerical integration [Kuzmic (1996) Anal. Biochem. 237, 260-273]. This general numerical method offers the following advantages. (1) It provides an estimate for the lower limit on feasible values of association rate constants. (2) It is many orders of magnitude more accurate. (3) It is easily extensible to more complicated reaction mechanisms. (4) It uses a simpler formalism and it is thus more accessible to nonmathematicians. (5) A suitable computer program for the analysis of competitive binding kinetics can be obtained via the Internet (http://www.biokin.com).

A computer program with the code name DYNAFIT was developed for fitting either the initial velocities or the time course of enzyme reactions to an arbitrary molecular mechanism represented symbolically by a set of chemical equations. Seven numerical tests and five graphical tests are applied to judge the goodness of fit. Experimental data on the inhibition of the dissociative dimeric proteinase from HIV were used in four test examples. A set of initial velocities was analyzed to see if a tight-binding inhibitor could bind to the HIV proteinase monomer. Three different sets of progress curves were analyzed (i) to determine the kinetic properties of an irreversible inhibitor, (ii) to investigate the dissociation and denaturation mechanism for the protease dimer, and (iii) to investigate the inhibition mechanism for a transient inhibitor. The program is available by anonymous ftp via uwmml.pharmacy.wisc.edu and on the World Wide Web via http://uwmml.pharmacy.wisc.edu.

The intrinsic fluorescence of tyrosine increases by a factor of approximately two when the carboxy group is liberated from a peptide bond by hydrolysis. The increase in fluorescence provides a novel way to monitor the hydrolysis of native tyrosine peptides that contain only proteinogenic amino acids. Thus, for example, the hydrolysis by HIV-1 proteinase of a heptapeptide viral protein fragment gag129-135, Ser-Gln-Asn-Tyr-Pro-Ile-Val, was followed continuously at excitation and emission wavelengths 275 and 305 nm. The fluorescence increase is magnified by at least a factor of a thousand when a resonance energy quencher, such as para-nitrophenylalanine, is in the vicinity. For example, the peptide Lys-Ala-Arg-Val-Tyr-Phe(p- NO2)-Glu-Ala-Nle-NH2 [Richards et al. (1990) J. Biol. Chem. 265, 7733], widely used for spectrophotometric assays of the HIV-1 proteinase, yields a substrate:product fluorescence ratio greater than 1:1000. Tyrosine-containing substrates of pepsin and trypsin showed similar behavior. The detection limit of the present method is at least one order of magnitude lower than absorbance assays of p-nitrophenylalanine peptides.

The equilibrium constant for the binding of a spectroscopically invisible ligand to its protein receptor can be determined in a competition experiment, by using a structural analog that contains a reporter group (fluorophor). A novel mathematical treatment of the multiple equilibria allows the analysis to be performed under tight-binding conditions. The equilibrium equation for mixtures of two mutually competitive tight-binding ligands can be expressed in a recursive form, a form in which the dependent variable appears on both sides and the solution is found iteratively. The algorithm is also applicable to the special case of weak binding, where the concentration of the bound ligand can be neglected in the mass balance. The fluorescence displacement method is demonstrated on the determination cyclophilin binding to cyclosporin A (CsA), in competition with its fluorescent derivative, [D-Lys(Dns)]8-CsA.