Metabolic pathway analysis in the presence of biological constraints

Metabolic pathway analysis is a key method to study a metabolism in its steady state and the concept of elementary fluxes ( EFs ) plays a major role in the analysis of a network in terms of non-decomposable pathways. The supports of the EFs contain in particular those 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, genomescale, systems. The EFs give the possible pathways in a 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 on the mathematical structure and geometrical characterization of the solution space in the presence of such biological constraints and revisit the concept of EFMs and EFs in this framework. We show that most of the results depend only on very general properties of compatibility of constraints with the sign function: either signinvariance for regulatory constraints or sign-monotonicity (a stronger property) for thermodynamic and kinetic constraints. We show in particular that EFs for sign-monotone constraints are just those of the original EFs that satisfy the constraint and we show how to integrate their computation efficiently in the double description method, the most widely used method in the tools dedicated to EFMs computation.