Metabolic isotopomer labeling systems. Part III: Path tracing

• Path tracing leads to an explicit solution theory for isotope labeling systems.
• Labeled molecules are rigorously traced through a metabolic network.
• Some algorithms can be both interpreted as path tracers and as equation solvers.
• New insights into the structure of isotopic labeling systems are obtained.
• Path tracing speeds up the computational solution of isotope labeling systems.

Isotope labeling systems (ILSs) are sets of balance equations that quantitatively describe the distribution of isotopic tracers in metabolic networks. The solution of ILSs, i.e., the calculation of isotopic labeling distributions (mostly in steady state) is the fundamental computational step of 13C metabolic flux analysis (MFA). Aiming at a deeper analytical understanding of ILSs a new approach for solving ILSs is developed. It is based on the straightforward idea of tracing labeled molecules through the metabolic network. The new approach allows to calculate the label distribution in isotopic tracer experiments in an analytical way that directly reflects the underlying network structure. The theory of path tracing is formally developed by introducing regular expressions for representing all possible paths through the labeling network. These expressions are generated by classical path tracing algorithms, e.g. by the Kleene algorithm. As a central theoretical result, a framework for proving the correctness of such path tracing algorithms in their application to ILSs is developed. Finally, by mapping path expressions to algebraic expressions, the solution of an ILS is computed. As an offspring of the developed theory, the relation between path tracing and former approaches for ILS solution is worked out and several consequences for the numerical solution and analysis of ILSs and – more general – compartmental systems used in pharmaco-kinetic modeling will be sketched.

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