A decomposition approach for computing elementary flux modes in genome-scale metabolic networks

Abstract

The appearance of high-throughput experimental techniques to measure biological data in recent decades gives birth to Systems Biology which studies the emergent properties of biological systems by mathematical modelling. The most ubiquitous structure in biological systems is the network structure. Among different biological networks, a particular important one is the metabolic network consisting of all the biochemical reactions and compounds in a cell. Reconstructed from the whole genome of a cell, the so-called genome-scale metabolic network successfully describes the cellular metabolism. A fundamental computational framework applied to metabolic networks is the flux balance analysis (FBA) derived from the steady-state assumption. In FBA, the metabolic flux distribution, which is the vector containing all reaction rates in a metabolic network, can be obtained from solving a simple linear program given the stoichiometric information of reactions and a biological objective for optimization. Metabolic pathway analysis (MPA) is a computational technique relevant to FBA to analyze metabolic pathways in metabolic networks. The first mathematically defined metabolic pathway, elementary flux mode (EFM), has theoretical as well as practical importance. One significant role of EFMs is that every flux distribution can be decomposed into a set of EFMs and a number of methods to study flux distributions originate from it. Yet finding such decompositions requires the complete set of EFMs, which is intractable in genome-scale metabolic networks due to combinatorial explosion. In this research, we propose an algorithm to decompose flux distributions into EFMs in genome-scale networks. It is an iterative scheme of a mixed integer linear program. The algorithm is also able to approximate the EFM of largest contribution to an objective reaction in a flux distribution. Complimentary to existing methods, our algorithm is capable of finding EFMs of flux distributions with complex structures, closer to the realistic case in which a cell is subject to various constraints. Our algorithm is first applied to study the growth of Escherichia coli (E. coli) under simple growth condition and we find that the employment of different EFMs is highly dynamic and sensitive to growth condition in order to achieve an optimal state of metabolism. This suggests a possible reason for the enormous redundancy of EFMs consuming the same set of uptake substrates and producing the same set of metabolites. A case of growth of E. coli in the Lysogeny broth (LB) medium in which the situation is complicated by the presence of various carbon sources is simulated and studied via our algorithm. Essential metabolites and their syntheses are located. Information on the contribution of each carbon source not obvious from the apparent flux distribution is also revealed. Finally, we apply our algorithm to analyze a real experimental flux distribution in mouse cardiomyocyte. Results consistent with literature are obtained. Interestingly, a mode of oxidative phosphorylation uncoupled from adenosine triphosphate (ATP) synthesis is discovered and this is not obvious from the flux distribution. In conclusion, the algorithm can facilitate MPA in genome-scale metabolic networks. It provides an analytic method that prepares for the future breakthrough in experimental techniques to measure in vivo fluxes in a huge scale. One of the future directions is the improvement, refinement and further applications of the algorithm. Another possibility is the development of a more general algorithm to decompose a flux distribution into a set of EFMs with respect to a given optimization objective in a genome-scale metabolic network. Also, in the future, by further case studies and evaluations of different schemes for decomposition, a well-structured methodology may be established to analyze flux distributions in different situations as thorough as possible by their decompositions into EFMs.