# Half-time analysis of the integrated Michaelis equation. Simulation and use of the half-time plot and its direct linear variant in the analysis of some α-chymotrypsin-, papain- and fumarase-catalysed reactions

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## Abstract

Substitution of half-time parameters in the integrated form of the Michaelis–Menten equation for any enzyme-catalysed reaction yields an equation that gives a linear relationship between the half-time of the reaction and the substrate concentration at that point of the reaction. The logarithmic term of the integrated equation becomes a constant as a result of the substitution, which means that the use of the half-time plot of the equation requires calculation only of half-time and substrate-concentration values at various stages of the reaction. The half-time method is both simple and exact, being analogous to an [S0]/vi against [S0] plot. A direct linear form of the half-time plot has been devised that allows very simple estimation of Michaelis parameters and/or initial velocities from progress-curve data. This method involves no approximation and is statistically valid. Simulation studies have shown that linear-regression analysis of half-time plots provides unbiased estimates of the Michaelis parameters. Simulation of the effect of error in estimation of the product concentration at infinite time [P∞] reveals that this is always a cause for concern, such errors being magnified approximately an order of magnitude in the estimate of the Michaelis constant. Both the half-time plot and the direct linear form have been applied to the analysis of a variety of experimental data. The method has been shown to produce excellent results provided certain simple rules are followed regarding criteria of experimental design. A set of rules has been formulated that, if followed, allows progress-curve data to be acquired and analysed in a reliable fashion. It is apparent that the use of modern spectrophotometers in carefully designed experiments allows the collection of data characterized by low noise and accurate [P∞] estimates. [P∞] values have been found, in the present work, to be precise to within ±0.2% and noise levels have always been below 0.1% (signal-to-noise ratio≃1000). As a result of the considerations above, it is concluded that there is little to be feared with regard to the analysis of enzyme kinetics using complete progress curves, despite the generally lukewarm recommendations to be found in the literature. The saving in time, materials and experimental effort amply justify analysis of enzyme kinetics by progress-curve methods. Half-time plots linear to ≥90% of reaction have been obtained for some α-chymotrypsin-, papain- and fumarase-catalysed reactions.