Flux cost functions and the choice of metabolic fluxes

Metabolic fluxes in cells are governed by physical, biochemical, physiological, and economic principles. Cells may show"economical"behaviour, trading metabolic performance against the costly side-effects of high enzyme or metabolite concentrations. Some constraint-based flux prediction methods score fluxes by heuristic flux costs as proxies of enzyme investments. However, linear cost functions ignore enzyme kinetics and the tight coupling between fluxes, metabolite levels and enzyme levels. To derive more realistic cost functions, I define an apparent"enzymatic flux cost"as the minimal enzyme cost at which the fluxes can be realised in a given kinetic model, and a"kinetic flux cost", which includes metabolite cost. I discuss the mathematical properties of such flux cost functions, their usage for flux prediction, and their importance for cells' metabolic strategies. The enzymatic flux cost scales linearly with the fluxes and is a concave function on the flux polytope. The costs of two flows are usually not additive, due to an additional"compromise cost". Between flux polytopes, where fluxes change their directions, the enzymatic cost shows a jump. With strictly concave flux cost functions, cells can reduce their enzymatic cost by running different fluxes in different cell compartments or at different moments in time. The enzymactic flux cost can be translated into an approximated cell growth rate, a convex function on the flux polytope. Growth-maximising metabolic states can be predicted by Flux Cost Minimisation (FCM), a variant of FBA based on general flux cost functions. The solutions are flux distributions in corners of the flux polytope, i.e. typically elementary flux modes. Enzymatic flux costs can be linearly or nonlinearly approximated, providing model parameters for linear FBA based on kinetic parameters and extracellular concentrations, and justified by a kinetic model.