A fundamental result in metabolic pathway analysis states that every element of the flux cone can be written as a sum of elementary modes.  The result is an immediate consequence of a theorem by Rockafellar which states that every element of a linear subspace can be written as a conformal sum (a sum without cancelations) of elementary vectors (support-minimal vectors). In this work, we extend the theorem to general polyhedral cones and polyhedra , thereby refining Minkowski's and Caratheodory 's theorems  in polyhedral geometry. In general, elementary vectors need not be support-minimal ; in fact, they are conformally non-decomposable and form a unique minimal set of conformal generators. 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. We define special cones arising from linear subspaces and nonnegativity conditions  and use them to analyze general polyhedral cones and polyhedra  .
A fundamental result in metabolic pathway analysis states that every flux mode can be decomposed into a sum of elementary modes. However, only a decomposition without cancelations is biochemically meaningful, since a reversible reaction cannot have different directions in the contributing elementary modes. This essential requirement has been largely overlooked by the metabolic pathway community. Indeed, every flux mode can be decomposed into elementary modes without cancelations. The result is an immediate consequence of a theorem by Rockafellar which states that every element of a linear subspace is a conformal sum (a sum without cancelations) of elementary vectors (support-minimal vectors). In this work, we extend the theorem , first to "subspace cones" and then to general polyhedral cones and polyhedra . Thereby, we refine Minkowski's and Carath\'eodory 's theorems , two fundamental results in polyhedral geometry. We note that, in general, elementary vectors need not be support-minimal , in fact, they are conformally non-decomposable and form a unique minimal set of conformal generators. Our treatment is mathematically rigorous, but suitable for systems biologists, since we give self-contained proofs for our results and use concepts motivated by metabolic pathway analysis. In particular, we study cones defined by linear subspaces and nonnegativity conditions - like the flux cone - and use them to analyze general polyhedral cones and polyhedra . Finally, we review applications of elementary vectors and conformal sums in metabolic pathway analysis .
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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
Our treatment is mathematically rigorous, but suitable for systems biologists, since we give self-contained
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