Theoretical Basis for Dynamic Label Propagation in Stationary Metabolic Networks under Step and Periodic Inputs

The dynamics of label propagation in a stationary metabolic network during an isotope labeling experiment can provide highly valuable information on the network topology, metabolic fluxes, and on the size of metabolite pools. However, major issues, both in the experimental set-up and in the accompanying numerical methods currently limit the application of this approach. Here, we propose a method to apply novel types of label inputs, sinusoidal or more generally periodic label inputs, to address both the practical and numerical challenges of dynamic labeling experiments. By considering a simple metabolic system, i.e. a linear, non-reversible pathway of arbitrary length, we develop mathematical descriptions of label propagation for both classical and novel label inputs. Theoretical developments and computer simulations show that the application of rectangular periodic pulses has both numerical and practical advantages over other approaches. We applied the strategy to estimate fluxes in a simulated experiment performed on a complex metabolic network (the central carbon metabolism of Escherichia coli), to further demonstrate its value in conditions which are close to those in real experiments. This study provides a theoretical basis for the rational interpretation of label propagation curves in real experiments, and will help identify the strengths, pitfalls and limitations of such experiments. The cases described here can also be used as test cases for more general numerical methods aimed at identifying network topology, analyzing metabolic fluxes or measuring concentrations of metabolites.