Mathematical descriptions of the steady-state flux cone of a metabolic network

The set of all possible flux distributions over a metabolic network at steady state defines a polyhedral cone, the steady-state flux cone. As any polyhedral cone, this cone may be described mathematically in two different ways: either by an inner description using sets of generating vectors, or an outer description based on systems of linear equations and inequalities. Some of the most popular concepts in metabolic pathway analysis such as elementary flux modes or extreme pathways correspond to special inner descriptions of the cone. An alternative approach is the use of an outer description based on sets of non-negativity constraints (or irreversible reactions), resulting in so-called minimal metabolic behaviors and the reversible metabolic space. The goal of this talk is to overview these different approaches, and to discuss their relationship. In particular, we will propose a general framework to show how different inner descriptions may be obtained from the outer one, and discuss the possible impact on the description size.