Enzyme allocation problems in kinetic metabolic networks: Optimal solutions are elementary flux modes

- Optimization principles are useful tools to study metabolic network organization.
- A metabolic optimization problem with general kinetics is formulated and studied.
- A rigorous proof is given that optimal solutions are elementary flux modes.
- The proof uses the theory of oriented matroids.
- The result helps to understand metabolic switches and the occurrence of low-yield pathways.

The survival and proliferation of cells and organisms require a highly coordinated allocation of cellular resources to ensure the efficient synthesis of cellular components. In particular, the total enzymatic capacity for cellular metabolism is limited by finite resources that are shared between all enzymes, such as cytosolic space, energy expenditure for amino-acid synthesis, or micro-nutrients. While extensive work has been done to study constrained optimization problems based only on stoichiometric information, mathematical results that characterize the optimal flux in kinetic metabolic networks are still scarce. Here, we study constrained enzyme allocation problems with general kinetics, using the theory of oriented matroids. We give a rigorous proof for the fact that optimal solutions of the non-linear optimization problem are elementary flux modes. This finding has significant consequences for our understanding of optimality in metabolic networks as well as for the identification of metabolic switches and the computation of optimal flux distributions in kinetic metabolic networks.