Résumé. 2014 Nous présentons un modèle décrivant l'élasticité d'un réseau de chaînes rigides. La variation du module élastique avec la fraction volumique de

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It is shown that the dual to the linear programming problem that arises in constraint-based models of metabolism can be given a thermodynamic interpretation in which the shadow prices are chemical potential analogues, and the objective is to minimise free energy consumption given a free energy drain corresponding to growth. The interpretation is distinct from conventional non-equilibrium thermodynamics, although it does satisfy a minimum entropy production principle. It can be used to motivate extensions of constraint-based modelling, for example to microbial ecosystems.PACS numbers: 05.70. Ln, In biology, the metabolism of an organism provides energy and raw materials for maintenance and growth. As such, an interesting and important question concerns the application of thermodynamics to metabolic reaction networks [1,2,3,4]. For example, Prigogine and Wiame suggested a long time ago that an organism's metabolism might be governed by a minimum entropy production (MEP) principle [5]. From the physical point of view, a metabolic reaction network is an excellent example of a system in a non-equilibrium steady state, since one can usually assume that the metabolite concentrations are unchanging after a short transient relaxation period. The appropriate generalisation of thermodynamics and statistical mechanics to non-equilibrium steady-states is a large field [6], which continues to attract attention to this present day [7]. In this Letter, we show that a novel thermodynamic interpretation can be given to the dual linear programming problem which arises in constraintbased models of metabolism. The resulting interpretation is rigorously defined, and uniquely determined by the mathematics. It is closely analogous to, but distinctly different from, conventional non-equilibrium thermodynamics. We also show that it satisfies an MEP principle similar to that proposed by Prigogine and Wiame.Constraint-based modelling (CBM) of metabolic networks has been pioneered by Palsson and co-workers [8]. In a typical application, described in more detail below, the steady-state assumption is combined with a target function to make a linear optimisation or linear programming (LP) problem. The LP variables are the fluxes through the various reactions that comprise the network, and the LP constraints arise from basic considerations of stoichiometry and from the reversibility or otherwise of the reactions. The LP objective function is biologically motivated, for example a 'growth' reaction is commonly inserted, and the target is to maximise flux through this reaction to correspond to maximal growth rate. CBM has been applied to microorganisms from all three domains of life [9,10,11], and has been remarkably successful in predicting phenotypic behaviour [12,13,14].Mathematically, every LP problem has a unique dual [15]. It was in determining the dual to the CBM LP problem that we noticed a striking analogy to non-equilibrium thermodynamics. Let us start therefore with a general discussion of LP duality, before specialising to the case of ...

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A metabolic model can be represented as a 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 visualization 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 other organisms.