2015/12/01 by Stefan Müller, Stefan C. Müller, Müller, Stefan +2 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #52Bxx #FOS: Biological sciences #FOS: Mathematics #Gene Regulatory Network Analysis #Metric Geometry (math.MG) #Microbial Metabolic Engineering and Bioproduction #Molecular Networks (q-bio.MN) #Peroxisome Proliferator-Activated Receptors #math.MG #msc:52Bxx #q-bio.MN
paper · pdf · doi:10.48550/arxiv.1512.00267
openalex publication_date 2015/12/01 · arxiv created 2016/04/30 · arxiv updated 2016/05/03 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
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 Carathéodory'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.