Before revision:  Complementary slackness implies that the yield flux network also satisfies a conservation law, which can be used to explain why the shadow prices are strongly correlated with measures of molecular complexity such as molecular weight and atom count. For the genome-scale models , the shadow prices have a broad distribution and the overall pattern reflects the URLanisation of the metabolism .
After revision:  Complementary slackness implies that the edges of such a graph by combining the reaction fluxes with a conserved metabolite property such as molecular weight. A similar flux network can be constructed by combining the primal and dual solutions to the linear programming problem that typically arises in constraint-based modelling. Such constructions may help with the visualisation of flux distributions in complex metabolic networks. The analysis also explains the strong correlation observed between metabolite shadow prices (the dual linear programming variables) and conserved metabolite properties. The methods were applied to recent metabolic models for Escherichia coli, Saccharomyces cerevisiae, and Methanosarcina barkeri. Detailed results are reported for E. coli; similar results were found for the URLanisms .