Metabolic fluxes in cells are often assumed to reflect an economical use of enzymes. To apply this principle of "minimal enzyme cost" to metabolic modelling and to study its biological consequences, I consider two types of computational models: models in which the metabolic fluxes are predefined and metabolite concentrations are optimised in order to minimise enzyme cost; and models in which the fluxes are also optimised. In the first case, optimal enzyme and metabolite levels can be computed by solving a convex optimization problem. The formalism highlights the close relationship between enzyme demand and reaction thermodynamics. In the second case, we obtain a nonlinear version of flux balance analysis, a method commonly used in flux prediction. The optimal fluxes can be determined by screening all elementary flux modes, a well-defined set of maximally sparse flux distributions. Altogether, we obtain a tractable method for predicting optimal fluxes, metabolite levels, and enzyme levels in models of central metabolism.