Analysis of Direct, Median-Based Estimators for Enzyme Parameters in Two Substrate Reactions

Abstract

The common Michaelis-Menten equation in enzyme kinetics has the form y=Vmaxx/(KM+x) that describes a rectangular hyperbola. Non-linear least squares regression may be used to find the kinetic parameters (Vmax and KM) but if the data contain a large number of  bad observations (‘outliers’) non-parametric alternatives are generally more precise. The direct linear plot (Eisenthal R, Cornish-Bowden A, Biochem J 1974;139:715-20) is one of the robust statistical methods used when the distribution of errors in the experiments is not normal, as it seems often to be the case. For models of enzyme kinetics that involve two substrates the equation is more complex and it is analysed by keeping the concentration of a substrate constant while varying the concentration of the second substrate. Apparent values of Vmax and KM at different concentration of one substrate are thus obtained and are used in a second step to get the final estimates of all the kinetic parameters (indirect method). Direct fitting of all the data simultaneously by solving sets of non-linear equations numerically gives better estimates and a closer fit than indirect methods (Nimmo IA, Atkins GL, Biochem J 1976;157:489-92).
A robust, direct method for estimating enzyme kinetic parameters, derived from the direct linear plot, is presented. It was used with the equation for a two-substrate reaction that follows michaelian kinetics and it depends on solving sets of linear systems of equations. The medians of the solutions obtained are used to calculate the final estimates. The systems of equations are constructed by combining all the experimental values obtained with different concentrations of the two substrates. The method performed well under the conditions of simulated experiments with a variety of error structures. It is shown that this method gives more precise parameter estimates than do some usual indirect (or replot) methods. It worked well with data that contained a high content of outliers. The algorithm is easy to implement as a computer program but a large number of data points require a long computational time. For five or six concentrations in each substrate the necessary time is reasonably short.