A Simple Rank Test to Distinguish Extreme Pathways from Elementary Modes in Metabolic Networks

Background: Metabolic pathway analysis is a powerful tool to study the metabolic structure of a cellular metabolism that comprises an intricate network for transforming metabolites through enzyme-catalyzed reactions. The approach is based on convex analysis to solve a homogeneous system of linear equations and inequality constraints derived from the steady state operation of mass conservation of metabolites. The solutions constitute the admissible flux space known as the convex polyhedral cone. Elementary Mode and Extreme Pathway Analysis are two closely related techniques that have been developed to identify pathways spanning the admissible flux space. Both elementary modes and extreme pathways are genetically independent pathways that can support steady state operation of cellular metabolism. However, the set of extreme pathways is often a subset of elementary modes, and under certain conditions only extreme pathways are the generating edges of the polyhedral cone. Because the two techniques are closely related, it is important to develop a theoretical framework to distinguish extreme pathways from elementary modes. Results: We have found a simple algebraic test to distinguish extreme pathways from elementary modes which requires only the stoichiometry matrix. The method has been tested with published metabolic networks that have been characterized with Elementary Mode Analysis and Extreme Pathway Analysis. The identity and number of elementary modes are not altered in networks subjected to splitting every reversible reaction into two different irreversible reactions, other than the spurious futile cycles involving the new reactions themselves. However, the set of extreme pathways depends strongly on the specific treatment of the reversible reactions of the network. The application of this algebraic test for efficient computation of elementary modes in very large networks is discussed. Conclusions: Elementary modes are the complete set of genetically independent pathways of a cellular metabolism that supports steady state operation. With the simple algebraic test, we can easily identify whether a given pathway is an elementary mode or an extreme pathway before computing the complete set of pathways. This test provides a convenient way to analyze and interpret network topology with Metabolic Pathway Analysis. The algebraic test is also useful for improving the efficiency of computing elementary modes in very large metabolic networks.