# Economic principles in cell metabolism

Protein allocation | Optimal states | Enzyme cost | Flux cost | Rate and yield |

Value structure | Kinetic models | Flux analysis | Optimal adaptation | Enzyme rhythms |

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### Flux cost functions and the choice of metabolic fluxes

Metabolic fluxes in cells are governed by physical, biochemical, physiological, and economic principles. Cells are believed to trade metabolic performance against high enzyme or metabolite concentrations and their costly side-effects. To predict metabolic fluxes, variants of Flux Balance Analysis score fluxes by heuristic costs, which serve as proxies for enzyme investment. However, the linear flux cost functions ignore the complex interplay between fluxes, metabolite levels and enzyme levels. As a more realistic alternative, I define the enzymatic flux cost – the minimal enzyme cost at which the fluxes can be realised in a given kinetic model – as well as the kinetic flux cost, which additionally includes a cost for metabolites. The enzymatic flux cost scales linearly with the flux distribution and is concave on the flux polytope. When fluxes change their directions, the enzymatic cost shows a jump. When flux distributions are superimposed, the enzymatic cost is usually not additive, but contains and extra “compromise cost”. Cells can avoid this extra cost by running different flux distributions separately, in different cell compartments or at different moments in time. The enzymatic flux cost leads to an estimate of the cell growth rate, a convex function on the flux polytope. Growth-maximising metabolic fluxes can be predicted by Flux Cost Minimisation (FCM), a variant of FBA that uses general flux cost functions. The set of possible flows – a convex polytope in flux space – is defined by the stationarity constraint, a predefined flux objective, and flux bounds. Since the cost function is concave on this polytope, the optimal flux distributions are located in corners of the flux polytope, i.e. they are typically elementary flux modes. Enzymatic flux costs can be linearly or nonlinearly approximated. Linear flux cost functions for FBA are justified by kinetic models and can be obtained from kinetic parameters and extracellular concentrations.

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