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A fundamental result in metabolic pathway analysis states that every fluxmode can be decomposed into a sum of elementary modes. However, only adecomposition without cancelations is biochemically meaningful, since areversible reaction cannot have different directions in the contributingelementary modes. This essential requirement has been largely overlooked by themetabolic pathway community.  Indeed, every flux mode can be decomposed into elementary modes withoutcancelations. The result is an immediate consequence of a theorem byRockafellar which states that every element of a linear subspace is a conformalsum (a sum without cancelations) of elementary vectors (support-minimalvectors). In this work, we extend the theorem, first to "subspace cones" andthen to general polyhedral cones and polyhedra. Thereby, we refine Minkowski'sand Carath\'eodory's theorems, two fundamental results in polyhedral geometry.We note that, in general, elementary vectors need not be support-minimal, infact, they are conformally non-decomposable and form a unique minimal set ofconformal generators.  Our treatment is mathematically rigorous, but suitable for systemsbiologists, since we give self-contained proofs for our results and useconcepts motivated by metabolic pathway analysis. In particular, we study conesdefined 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 inmetabolic pathway analysis.

Elementary vectors and conformal sums in polyhedral geometry and their relevance for metabolic pathway analysis