Atom Transition Networks and Isotope Labeling Patterns in Large Chemical Reaction Networks

Isotope tracing experiments are an important source of information in particular for the analysis of complex metabolic networks. Their evaluation, however, is a notoriously difficult and time-consuming task. Here, we derive a simple and complete solution for an important special case, namely the specific labeling of a single atom in a single feed compound. Starting from the atom-to-atom map of a reaction we first derive a transition matrix that exactly describes the propagation of the label from reactants to products, taking into account the relevant symmetries. These are subsequently combined into a global atom transition network. Assuming a steady-state flux F through the chemical reaction network we derive a system of affine differential equations describing how the reaction network is flooded by labeled atoms from an external reservoir. This leads to a unique asymptotically stable steady state distribution of labels that can be computed by solving a non-singular system of linear equations. Linear combinations of these single-atom labeling patterns also solve the problem of computing the enrichment of multiple, simultaneous labels, albeit without providing information on correlations between distinct labels.