Constraint-based modelling of metabolic networks often gives rise to a linear programming problem. Mathematically, there always exists a dual to any linear programming problem, with dual variables known as shadow prices. Here we analyse the structure of the dual problem and the properties of the shadow prices, for typical constraint-based metabolic models. We apply our results to a representative set of recent genome-scale metabolic reconstructions. The dual problem is found to have an elegant structure. The shadow prices are to be interpreted as yield coefficients Varma and Palsson, J. theor. Biol. v165, 477; 503 (1993)%DIFDELCMD < ]%%% . In combination with reaction fluxes , they can be used to decorate the metabolic model with a yield flux network. The yield flux network shows a kind of gauge invariance with respect to rescaling of the stoichiometric matrix, and is arguably more fundamental than either the reaction fluxes or shadow prices. 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 .
 %DIFDELCMD < ]%%% A metabolic model can be represented as bipartite graph comprising linked reaction and metabolite nodes. Here it is shown how a network of conserved fluxes can be assigned to 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 .
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Constraint-based modelling of metabolic networks often gives rise to a linear programming problem. Mathematically, there always exists a dual to any linear programming problem, with dual variables known as shadow prices. Here we analyse the structure of the dual problem and the properties of the shadow prices, for typical constraint-based metabolic models. We apply our results to a representative set of recent genome-scale metabolic reconstructions. The dual problem is found to have an elegant structure. The shadow prices are to be interpreted as yield coefficients
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Varma and Palsson, J. theor. Biol. v165, 477; 503 (1993)
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. In combination with reaction fluxes , they can be used to decorate the metabolic model with a yield flux network. The yield flux network shows a kind of gauge invariance with respect to rescaling of the stoichiometric matrix, and is arguably more fundamental than either the reaction fluxes or shadow prices. Complementary slackness implies that
A metabolic model can be represented as bipartite graph comprising linked reaction and metabolite nodes. Here it is shown how a network of conserved fluxes can be assigned to
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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
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