Deduction of intracellular sub-systems from a topological description of the network{ Torbjo ¨ rn E. M. Nordling,* a Noriko Hiroi, bcdef Akira Funahashi bcdg and Hiroaki Kitano bcgh DOI: 10.1039/b702142a Non-linear behaviour of biochemical networks, such as intracellular gene, protein or metabolic networks, is commonly represented using graphs of the underlying topology. Nodes represent abundance of molecules and edges interactions between pairs of molecules. These graphs are linear and thus based on an implicit linearization of the kinetic reactions in one or several dynamic modes of the total system. It is common to use data from different sources—experiments conducted under different conditions or even on different species—meaning that the graph will be a superposition of linearizations made in many different modes. The mixing of different modes makes it hard to identify functional modules, that is sub- systems that carry out a specific biological function, since the graph will contain many interactions that do not naturally occur at the same time. The ability to establish a boundary between the sub-system and its environment is critical in the definition of a module, contrary to a motif in which only internal interactions count. Identification of functional modules should therefore be done on graphs depicting the mode in which their function is carried out, i.e. graphs that only contain edges representing interactions active in the specific mode. In general, when an interaction between two molecules is established, one should always state the mode of the system in which it is active. Modelling of intracellular networks of molecules Today the wealth of bio-molecular infor- mation is used to compose mainly two types of models: network graphs and detailed mechanistic models. Recent reviews of modelling of intracellular networks 1–4 and graph-based analysis. 5 Both types of models are widely used and intensively explored. Commonly, how- ever, only one or the other is used in the same work, and that is perhaps why some aspects of their relation to one- another hardly is discussed. We will here illustrate how different dynamic modes of a network can be observed as different topologies, for example, two molecules that interact in one mode do not interact in another. This has severe implications on model- ling networks as graphs, which we will discuss. Our illustration is based on two network examples: a detailed mechanistic model consisting of non-linear ordinary differential equations (ODEs), and a graph describing the network of interac- tions in a mammalian fibroblast cell. The non-linear ODE model Eissing and co-workers 6 have composed a model of receptor induced apoptosis (Fig. 1). The extended version of this model consists of eight non-linear ODEs deduced from the law of mass action. It includes eight state variables: pro- caspase 8, activated caspase 8, pro- caspase 3, activated caspase 3, Inhibitor of Apoptosis Protein (IAP), IAP bound activated caspase 3, Caspase 8- and 10-Associated RING Proteins (CARP) which essentially is equivalent to the Bifunctional Apoptosis Regulator (BAR) (used in Fig. 1 A), and CARP bound activated caspase 8. This model describes one functional module respon- sible for caspase induced apoptosis. It works as a bistable switch creating two distinct modes: proliferation and apoptosis, which can be represented by the corresponding stable steady-states. The fibroblast network As a part of a pilot study of modelling of large-scale intracellular networks, we constructed a directed graph describing a mammalian fibroblast (Fig. 2), which consists of 1447 reactions, involving some 1343 different molecules. 7 The model includes the cell cycle, several Integrin signals, EGF, HMMR, midkine (MDK) and TGF-b signals, the Ras, Raf and MAP kinase pathway, several pathways involving Cyclic AMP, Cyclic GMP, PIP 2 and Ca 2+ , Purine, Pyrimidine and Deoxyribonucleotide synthesis, transcription and translation of key proteins and the apoptosis pathway. We thus attempted to include the complete cell cycle with all major signal trans- duction pathways and everything that a Automatic Control - School of Electrical Engineering, Royal Institute of Technology (KTH), Sweden. E-mail: tn@kth.se; http://www.ee.kht.se/ytn/ b JST/ERATO SORST Kitano Symbiotic Systems Project, Japan c School of Fundamental Science and Technology, Keio University, Japan d School of Medicine, Keio University, Japan e EMBL-European Bioinformatics Institute (EBI), UK. E-mail: noriko.hiroi@ebi.ac.uk f Graduate School of Medicine and Faculty of Medicine, The University of Tokyo, Japan g The Systems Biology Institute, Japan h Sony Computer Science Laboratories, Inc., Japan { Electronic supplementary information (ESI) available: Reactions.sif file used to create Fig. 2. See DOI: 10.1039/b702142a OPINION www.rsc.org/molecularbiosystems | Molecular BioSystems This journal is ß The Royal Society of Chemistry 2007 Mol. BioSyst., 2007, 3, 523–529 | 523