Metabolic pathway analysis in presence of biological constraints

Metabolic pathway analysis is a key method to study metabolism at steady state and the el- ementary fluxes (EFs) is one major concept allowing one to analyze the network in terms of non- decomposable pathways. The supports of the EFs contain in particular the supports of the elementary flux modes (EFMs), which are the support-minimal pathways, and EFs coincide with EFMs when the only flux constraints are given by the irreversibility of certain reactions. Practical use of both EFMs and EFs has been hampered by the combinatorial explosion of their number in large, genome-scale, systems. The EFs give the possible pathways at steady state but the real pathways are limited by biological constraints, such as thermodynamic or, more generally, kinetic constraints and regulatory constraints from the genetic network. We provide results about the mathematical structure and geo- metrical characterization of the solutions space in presence of such biological constraints and revisit the concept of EFMs and EFs in this framework. We show that most of those results depend only on a very general property of compatibility of the constraints with sign: either sign-based for regulatory constraints or sign-monotone (a stronger property) for thermodynamic and kinetic constraints. We show in particular that EFs for sign-monotone constraints are just those original EFs that satisfy the constraint and show how to efficiently integrate their computation in the double description method, the most widely used method in the tools dedicated to EFMs computation.