A highly accurate numerical method for calculating apparent kinetic parameters of immobilized enzyme reactions: 1. Theory

To establish a highly reliable numerical method for calculation of the values of apparent kinetic parameters in various types of immobilized enzyme reactions, the performance of the shooting method combined with a variable- order, variable-step Taylor-series method has been discussed in a series of two papers. In this first paper, every formulae that are necessary to execute the numerical calculation are given and the procedure and technique for the calculation are described. As an example, the effectiveness factor is calculated for the Thiele modulus from 1 to 700 and the calculated values are found to have accuracies that are almost equal to the machine accuracy of the personal computer used. In the Taylor-series method, the step sizes are efficiently estimated from the ratio of the appropriate two of the first few terms in each Taylor-series solution to the relevant simultaneous differential equations, which contributes greatly to reduction in both the loss-of-significance error and the execution time.