Error structure of enzyme kinetic experiments. Implications for weighting in regression analysis of experimental data

Knowledge of the error structure of a given set of experimental data is a necessary prerequisite for incisive analysis and for discrimination between alternative mathematical models of the data set. A reaction system consisting of glutathione S-transferase A (glutathione S-aryltransferase), glutathione, and 3,4-dichloro-1-nitrobenzene was investigated under steady-state conditions. It was found that the experimental error increased with initial velocity, v, and that the variance (estimated by replicates) could be described by a polynomial in v Var (v) = K0 + K1 - v + K2 - v2 or by a power function Var (v) = K0 + K1 - vK2. These equations were good approximations irrespective of whether different v values were generated by changing substrate or enzyme concentrations. The selection of these models was based mainly on experiments involving varying enzyme concentration, which, unlike v, is not considered a stochastic variable. Different models of the variance, expressed as functions of enzyme concentration, were examined by regression analysis, and the models could then be transformed to functions in which velocity is substituted for enzyme concentration owing to the proportionality between these variables. Thus, neither the absolute nor the relative error was independent of velocity, a result previously obtained for glutathione reductase in this laboratory [BioSystems 7, 101-119 (1975)]. If the experimental errors or velocities were standardized by division with their corresponding mean velocity value they showed a normal (Gaussian) distribution provided that the coefficient of variation was approximately constant for the data considered. Furthermore, it was established that the errors in the independent variables (enzyme and substrate concentrations) were small in comparison with the error in the velocity determinations. For weighting in regression analysis the inverted value of the local variance in each experimental point should be used. It was found that the assumption of proportionality between variance and valpha (where alpha is an empirically determined exponent) was a good approximation for the weighting. The value of alpha was 1.6 in the present case. The weight function was tested in the fitting of a rate equation to a kinetic-data set involving variable substrate concentrations. Recommendations are given regarding the establishment of the error structure in a general case and its application in regression analysis.