A-Optimal Experimental Designs for the Michaelis–Menten Model

Abstract—The kinetics of simple and complex kinetic reactions are generally described by exponential and rational models. Among the latter is the well-known Michaelis–Menten model of enzymatic kinetics. This study presents A-optimal designs for the Michaelis–Menten model. The method of elimination was used to construct them, allowing for the separation of the problems of determining nodes and weights, thereby extending this approach to other criteria and rational-type models. It is shown that the nodes of A-optimal designs are the roots of a fourth-degree algebraic equation with coefficients depending on three parameter sets, one of which is well known to specialists in enzymatic kinetics, while the other two may be highlighted for the first time. If the nodes of the A-optimal design are already known, the weights of the corresponding nodes can be determined analytically using the Pukelsheim formula. Using the Sturm sequence system constructed in general form for the obtained equation, the properties of its roots as well as the properties of the nodes of A-optimal designs were investigated. It was shown that for certain combinations of parameter values of the model, the degree of the corresponding algebraic equation is reduced to three. A partition of the set of parameter values of the Michaelis–Menten model into two subsets was found, for one of which the A-optimal design is uniquely defined, while for the other, it requires the selection of an optimal node from two possible options. For the points on the curve that form the common boundary of the specified subsets, the degree of the found algebraic equation is three, and the sought node of the A-optimal design is uniquely determined. To illustrate the obtained results, relevant numerical examples are provided.