Model equations for the kinetics of covalent irreversible enzyme inhibition and spontaneous reactivation: esterases and organophosphorus compounds.

Type B carboxylesterases (acetylcholinesterases, neuropathy target esterase, serine peptidases), catalyse the hydrolysis of carboxyl-ester substrates by formation of a covalent acyl-enzyme intermediate and subsequent cleavage and release of the acyl group. Organophosphorus compounds, carbamates, and others exert their mechanism of neurotoxicity by permanent covalent organophosphorylation or carbamylation at the catalytic site of carboxylesterases. Classical kinetic studies converted the exponential kinetic equation to a logarithmic equation for graphic analysis. This process, however, does not allow analysing complex situations. In this paper, kinetic model equations are reviewed and strategies developed for the following cases: (a) single enzyme, with classical linear equation; (b) multi-enzymatic system-discriminating several inhibitor-sensitive and inhibitor-resistant components; (c) 'ongoing inhibition'-high sensitive enzymes can be significantly inhibited during the substrate reaction time, the model equations need a correction; (d) spontaneous reactivation (de-phosphorylation)-one or several components can be simultaneously inhibited and spontaneously reactivated; (e) spontaneous reactivation from starting time with the enzyme being partly or totally inhibited; (f) aging-single enzyme can be inhibited, spontaneously reactivated and dealkylating reaction ('aging') simultaneously occurs; and (g) aging and spontaneous reactivation from starting time with the enzyme being partly or totally inhibited. Analysis of data using the suggested equations allows the deduction of inhibition kinetic constants and the proportions of each of the enzymatic components. Strategies for practical application of the models and for obtaining consistent kinetic parameters, using multi-steps approaches or 3D fitting, are presented.