Flux networks in metabolic graphs

arXiv:0808.0321v2 [q-bio.MN] 28 Sep 2009 Flux networks in metabolic graphs∗ Patrick B. Warren, Silvo M. Duarte Queiros, and Janette L. Jones Unilever R&D Port Sunlight, Bebington, Wirral, CH63 3JW, UK. A metabolic model can be represented as bipartite graph comprising linked reaction and metabo- lite 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 con- structions 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 other organisms. PACS numbers: 87.16.Yc, 87.18.Vf ABBREVIATIONS CBM: constraint-based modelling. CS: complementary slackness (a property of LP solu- tion pairs at optimality). GAM: growth-associated maintenance (in relation to ATP consumption). gDW: gram dry weight (referring to biomass). LP: linear programming. NGAM: non-growth-associated maintenance (in rela- tion to ATP consumption). A metabolic network comprises a list of biochemical reactions and their associated metabolites [1]. As such, a convenient representation is in terms of a bipartite graph containing reaction nodes and metabolite nodes, with edges between nodes indicating that a given metabolite is involved in a given reaction [2]. A schematic example is shown in Fig. 1. The metabolic network can be modelled by chemical rate equations, giving the rate of change of the metabolite concentrations in terms of the fluxes, or velocities, of the associated reactions. It is widely ac- cepted though that the metabolism comes to a steady state very quickly, so that the metabolite concentrations are unchanging in time. This means that a flux bal- ance condition holds, and the set of reaction fluxes (the ‘fluxome’) can, essentially, be regarded as the metabolic phenotype. Determination of the fluxome is therefore the focus of considerable theoretical [1], and experimental ef- fort [3, 4]. The global properties of such flux sets have been investigated [5]. When the network is represented as a bipartite graph, the fluxome is traditionally associated with the reaction nodes. Here we show how fluxes can be associated with the edges of such a bipartite graph, by combining the reaction fluxes with any metabolite property that is con- ∗Published in Phys. Biol. 6, 046006 (2009). served in the majority of reactions, such as molecular weight. Moreover, assuming the flux balance condition, such an edge-associated flux network is conserved at all the reaction and metabolite nodes apart from a hand- ful of sources and sinks. Thus the edge-associated fluxes resemble, for example, electric currents in a network of resistors [6]. This observation may help with the visual- isation of the flow of material in these complex reaction networks. If the flux-balance condition does not hold (for example away from steady state), then the edge- associated flux network can still be constructed provided a set of reaction fluxes is available. In such a case though, the edge fluxes are not in general conserved at the reac- tion nodes. Finally in the case where the set of reaction fluxes arises from the solution to a linear optimisation or linear programming (LP) problem, such as commonly encountered in constraint-based modelling (CBM), then an edge-associated yield flux network can be constructed. CBM is now a well-established approach for calculat- ing candidate sets of reaction fluxes [1]. It has been ap- plied to micro-organisms from all three domains of life [7, 8, 9, 10], and recently extended to encompass hu- man metabolism [11]. For the growth of micro-organisms a commonly used paradigm has emerged in which the metabolic network is augmented with a biomass reaction consuming the end-points of metabolism in the appropri- ate ratios, and with exchange reactions to represent the uptake of substrates and the discharge of metabolic by- products. Maximising the flux through the biomass re- action amounts to maximising the specific growth rate of the micro-organism. This approach has been highly suc- cessful at predicting the behaviour of micro-organisms [4, 12, 13], and has also been applied to problems in metabolic engineering [14, 15]. As already indicated CBM typically leads to an LP problem for the set of reaction fluxes. Mathematically, every LP problem has an associated dual [16, 17]. The dual variables are known as shadow prices reflecting an economic interpretation of th

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