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A mathematical treatment for the general case of enzyme inactivation by an inhibitor that breaks down in solution in a first-order reaction is presented. Cathepsin D was inactivated by fluorescein isothiocyanate with a K(i) of 4.47mum. Kinetic constants were also determined for the inactivation of cathepsin D by 1,1-bis(diazoacetyl)-2-phenylethane, and the inactivation of pepsin C by diazoacetyl-dl-norleucine methyl ester.

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A mathematical analysis of biphasic protein modification reactions is presented, and it is shown that, in addition to the protein species modification reactions, one more time-dependent step must be postulated to exist in the reaction process. This step involves the interconversion of the different protein species, such as binding of ligand with protein, or the change in the isomerization state of the protein. The kinetic description of the reaction process is effected through a second order homogeneous linear differential equation, with time as the independent variable, and unmodified protein concentration as the dependent variable. A simple procedure of graphical analysis of the experimental data is described, and it is shown that, by a process of elimination, the nature of the protein species interconversion time-dependent step may be recognized, and also the dependence of the protein species inactivation rate constants on various parameters in the preparation may be evaluated. The method is illustrated by the detailed analysis of one example from the literature, the inactivation of phosphorylase b by 5,5-dithiobis (2-nitrobenzoic acid).

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A kinetic analysis is presented of reactions of protein modification, and/or of modification-induced enzyme inactivation, which can formally be described by a single exponential function, or by a summation of two exponential functions, of reaction time plus a constant term. The reaction schemes compatible with the kinetic formalism of these cases are given, and a simple kinetic criterion is described whereby the identification of one of these cases, strong negative protein modification co-operativity, may be carried out. The treatment outlined in this paper is applied to a case from the literature, the inactivation of glyceraldehyde-3-phosphate dehydrogenase by butane-2,3-dione [Asriyants, Benkevich & Nagradova (1983) Biokhimiya (Engl. Transl.) 48, 164-171].