Before revision:  As an application, every flux mode can be decomposed into elementary modes without cancelations. Only such a decomposition is biochemically meaningful, in the sense that a reversible reaction cannot have different directions in the contributing elementary modes. We give elementary proofs for our mathematical results , in particular, we do not assume previous knowledge of polyhedral geometry.
After revision:  Our treatment is mathematically rigorous, but suitable for systems biologists, since we give self-contained proofs for our mathematical results , in particular, we do not assume previous knowledge of polyhedral geometry.