Highly accurate computation of dynamic sensitivities in metabolic reaction systems by a Taylor series method

We have previously developed the software for calculation of dynamic sensitivities, SoftCADS, in which one can calculate dynamic sensitivities with high accuracy by just setting the differential equations for metabolite concentrations. However, SoftCADS did not always provide calculated values with the machine accuracy of a computer, although a Taylor series method was employed to numerically solve the differential equations. This is because numerical derivatives calculated from an approximate formula were directly used in the derivation of the differential equations for sensitivities from those for metabolite concentrations. The present work therefore attempts to further enhance the performance of SoftCADS, including not only the accuracies of the calculated values but also the calculation time. To overcome the problem, the approximate formula is expanded into a Taylor series in time and the first-term value of the series is replaced by the exact coefficient on the second term of the flux function expanded into a Taylor series in an independent or dependent variable. The result reveals that this replacement certainly provides not only numerical derivatives but also dynamic sensitivities with superhigh accuracies comparable to the machine accuracy, regardless of the degree of stiffness of the differential equations. Moreover, a comparison indicates that the improved SoftCADS shortens the calculation time of the dynamic sensitivities without reducing their accuracies, even when the simplest approximate derivative formula is used.