Duality, Thermodynamics, and the Linear Programming Problem in Constraint-Based Models of Metabolism

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 minimize free energy consumption given a free energy drain corresponding to growth. The interpretation is distinct from conventional nonequilibrium thermodynamics, although it does satisfy a minimum entropy production principle. It can be used to motivate extensions of constraint-based modeling, for example, to microbial ecosystems.

Figure 1

Schematic metabolic network for a prokaryote like E. coli, showing intracellular metabolites (open circles), extracellular metabolites (hatched circles), internal and exchange reactions (arrows), and a growth reaction (dashed arrow).

Figure 2

Chemical potential analogues (shadow prices) plotted as a function of metabolite molecular weight for the intracellular metabolites with ${\mu }_i>0$ in a genome-scale constraint-based model of E. coli [17]. A normalizing factor is included to make the ${\mu }_i$ dimensionless.