Mathematical modeling of (13)C label incorporation of the TCA cycle: the concept of composite precursor function.

A novel approach for the mathematical modeling of (13)C label incorporation into amino acids via the TCA cycle that eliminates the explicit calculation of the labeling of the TCA cycle intermediates is described, resulting in one differential equation per measurable time course of labeled amino acid. The equations demonstrate that both glutamate C4 and C3 labeling depend in a predictable manner on both transmitochondrial exchange rate, V(X), and TCA cycle rate, V(TCA). For example, glutamate C4 labeling alone does not provide any information on either V(X) or V(TCA) but rather a composite "flux". Interestingly, glutamate C3 simultaneously receives label not only from pyruvate C3 but also from glutamate C4, described by composite precursor functions that depend in a probabilistic way on the ratio of V(X) to V(TCA): An initial rate of labeling of glutamate C3 (or C2) being close to zero is indicative of a high V(X)/V(TCA). The derived analytical solution of these equations shows that, when the labeling of the precursor pool pyruvate reaches steady state quickly compared with the turnover rate of the measured amino acids, instantaneous labeling can be assumed for pyruvate. The derived analytical solution has acceptable errors compared with experimental uncertainty, thus obviating precise knowledge on the labeling kinetics of the precursor. In conclusion, a substantial reformulation of the modeling of label flow via the TCA cycle turnover into the amino acids is presented in the current study. This approach allows one to determine metabolic rates by fitting explicit mathematical functions to measured time courses.