Kleene's Theorem and the Solution of Metabolic Carbon Labeling Systems

Carbon Labeling Systems (CLS) are large equation systems that describe the dynamics of labeled carbon atoms in a metabolic network. The rapid solution of these systems is the algorithmic backbone of C Metabolic Flux Analysis (MFA) which has become one of the most widely used tools in Metabolic Engineering. A new algorithm is presented for the solution of CLS which is not based on iteration schemes or numerical linear algebra methods but on path tracing of labeled particles. It is shown that the set of all paths from the system input to the internal network nodes directly gives the clue to an explicit solution of CLS. The promising potential of this new solution algorithm are outlined. 1 Metabolic Flux Analysis In recent years Metabolic Flux Analysis (MFA) by using C isotopes has become one of the most widely used tools in Metabolic Engineering [Wi01, W02]. MFA allows to determine quantitatively all fluxes in the central metabolism of a micro organism or higher cell. The metabolic flux maps resulting from this analysis serve to compare different strains of a micro organism to diagnose the effects of a genetic manipulation or even give hints to further improve the production capabilities of a given organism. MFA is based on a carbon labeling experiment where C labeled substrates are fed to the cells. The C isotopes are then distributed all over the metabolic network due to the metabolic activity. Finally, the enrichment of C isotopes in the intra-cellular metabolite pools tends to an isotopically stationary state, which means that constant fractions of unlabeled and labeled carbon atoms are encountered in all pools. In this state the isotope enrichment is measured by NMR or MS instruments [Sz98]. From this measurement data the metabolic fluxes are estimated based on a mathematical model of the carbon labeling dynamics in the system. The sole biological knowledge that is required for this procedure is the biochemical structure of the metabolic network, and the carbon atom transitions in each single reaction step. This knowledge is rather well established for central metabolism, but the method can also be used to distinguish between network variances.