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

TL;DR

This paper extends fundamental polyhedral geometry theorems to better understand elementary vectors and conformal sums, providing a rigorous mathematical foundation for analyzing metabolic pathways without cancelations.

Contribution

It generalizes theorems to polyhedral cones and polyhedra, refining the concept of elementary vectors for metabolic pathway analysis.

Findings

Elementary vectors form a unique minimal set of conformal generators.

Every flux mode can be decomposed into elementary modes without cancelations.

The work provides self-contained proofs suitable for systems biologists.

Abstract

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.