Research Article: Analysis of Metabolic Subnetworks by Flux Cone Projection

Date Published: May 29, 2012

Publisher: BioMed Central

Author(s): Sayed-Amir Marashi, Laszlo David, Alexander Bockmayr.


Analysis of elementary modes (EMs) is proven to be a powerful constraint-based method in the study of metabolic networks. However, enumeration of EMs is a hard computational task. Additionally, due to their large number, EMs cannot be simply used as an input for subsequent analysis. One possibility is to limit the analysis to a subset of interesting reactions. However, analysing an isolated subnetwork can result in finding incorrect EMs which are not part of any steady-state flux distribution of the original network. The ideal set to describe the reaction activity in a subnetwork would be the set of all EMs projected to the reactions of interest. Recently, the concept of “elementary flux patterns” (EFPs) has been proposed. Each EFP is a subset of the support (i.e., non-zero elements) of at least one EM.

We introduce the concept of ProCEMs (Projected Cone Elementary Modes). The ProCEM set can be computed by projecting the flux cone onto a lower-dimensional subspace and enumerating the extreme rays of the projected cone. In contrast to EFPs, ProCEMs are not merely a set of reactions, but projected EMs. We additionally prove that the set of EFPs is included in the set of ProCEM supports. Finally, ProCEMs and EFPs are compared for studying substructures of biological networks.

We introduce the concept of ProCEMs and recommend its use for the analysis of substructures of metabolic networks for which the set of EMs cannot be computed.

Partial Text

Metabolic pathway analysis is the study of meaningful minimal pathways or routes of connected reactions in metabolic network models [1,2]. Two closely related concepts are often used for explaining such pathways: elementary modes (EMs) [3,4] and extreme pathways (EXPAs) [5]. Mathematically speaking, EMs and EXPAs are generating sets of the flux cone [1,6]. Several approaches have been proposed for the computation of such pathways [7-14].

We consider a metabolic network N with m internal metabolites and n reactions. Formally, we describe N by its stoichiometric matrix S ∈ ℝm × n and the set of irreversible reactions Irr ⊆ {1, . . ., n}. If steady-state conditions hold, i.e., there is no net production or consumption of internal metabolites, the set of all feasible flux distributions defines a polyhedral cone

As mentioned above, the set of EMs of a genome-scale network may be large, and in general, cannot be computed with the available tools. Even if this is possible, one cannot simply extract interesting information from it. Therefore, a subset of EMs (or in case that we are interested in a subset of reactions, the set of PEMs) should be computed to reduce the running time and/or output size of the algorithm. Several approaches to this problem have been proposed in the literature. These strategies can be classified into four main categories:

In this paper, we introduce the concept of projected cone elementary modes (ProCEMs). The set of ProCEMs covers more PEMs than EFPs. Therefore, ProCEMs contain more information than EFPs. The set of ProCEMs is computable without enumerating all EMs. Is there a bigger set of vectors that covers even more PEMs and does not require full enumeration of EMs? This question is yet to be answered. One possible extension to this work is to use a more efficient implementation of polyhedral projection. With such an implementation, analysis of different subnetworks in genome-scale network models using ProCEMs is an interesting possibility for further research. For example, the ProCEMs can be used in the identification of all pathways with optimal yield [23] and in the analysis of control-effective fluxes [27].

The authors declare that they have no competing interests.

The original idea was presented by SAM, LD and AB. The mathematical results are presented by SAM, and improved by all authors. Implementation of the ProCEM method and performing the computational experiments are done by LD. The manuscript was originally drafted by SAM, and improved by all authors. The final version of the manuscript was read and approved by all authors.