Abstract
Functional modules of metabolic networks are essential for understanding the metabolism of an organism as a whole. With the vast amount of experimental data and the construction of complex and large-scale, often genome-wide, models, the computer-aided identification of functional modules becomes more and more important. Since steady states play a key role in biology, many methods have been developed in that context, for example, elementary flux modes, extreme pathways, transition invariants and place invariants. Metabolic networks can be studied also from the point of view of graph theory, and algorithms for graph decomposition have been applied for the identification of functional modules. A prominent and currently intensively discussed field of methods in graph theory addresses the Q-modularity. In this paper, we recall known concepts of module detection based on the steady-state assumption, focusing on transition-invariants (elementary modes) and their computation as minimal solutions of systems of Diophantine equations. We present the Fourier-Motzkin algorithm in detail. Afterwards, we introduce the Q-modularity as an example for a useful non-steady-state method and its application to metabolic networks. To illustrate and discuss the concepts of invariants and Q-modularity, we apply a part of the central carbon metabolism in potato tubers (Solanum tuberosum) as running example. The intention of the paper is to give a compact presentation of known steady-state concepts from a graph-theoretical viewpoint in the context of network decomposition and reduction and to introduce the application of Q-modularity to metabolic Petri net models.
Highlights
The knowledge of biochemical networks, in particular of metabolic networks, increases daily with the capabilities of new upcoming high-throughput technologies to measure all the participating molecules and the relations between them
We report two types of approaches for functional module detection: those that are based on the steady-state assumption and those that are based on graph-theoretical methods without a steady-state consideration
We describe the computation of t-invariants (EM), which can be further decomposed by several approaches into disjunctive or overlapping subnetworks
Summary
The knowledge of biochemical networks, in particular of metabolic networks, increases daily with the capabilities of new upcoming high-throughput technologies to measure all the participating molecules and the relations between them This enables us to construct large and complex models for many pathways of different species. In most cases, quantitative data in sufficient amounts and of high quality are rare and only available for rather small metabolic systems This situation motivated the development of qualitative methods, which enable us to analyze statements on functional behavior and dynamic properties of the system without any knowledge of the kinetic parameters. Afterwards, we continue with a recapitulation of steady-state network decomposition methods and their application to metabolic systems, including a brief consideration of network representation as hypergraphs and bipartite graphs, the definition of Petri nets and a detailed explanation of the Fourier-Motzkin algorithm for invariant computation.
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