A mathematical approach to emergent properties of metabolic networks: Partial coupling relations, hyperarcs and flux ratios Sayed-Amir Marashi a,n , Mojtaba Tefagh b,n a Department of Biotechnology, College of Science, University of Tehran, Tehran, Iran b Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran HIGHLIGHTS  We introduce some of the emergent properties of metabolic networks.  These properties generally emerge because of hyperarcs and irreversible reactions.  Graph-based models are not suitable for the analysis of such networks. article info Article history: Received 15 October 2013 Received in revised form 7 March 2014 Accepted 14 April 2014 Available online 19 April 2014 Keywords: Emergence Flux coupling analysis FCA Hypergraph Flow networks abstract Emergent properties in systems biology are those which arise only when the biological system passes a certain level of complexity. In this study, we introduce some of the emergent properties which appear in the constraint-based analysis of metabolic networks. These properties generally appear as a result of existence of hfdeyperarcs and irreversible reactions in networks. Here, we present examples of metabolic networks in which there exist at least two reactions whose fluxes cannot be written as products and/or ratios of the stoichiometric coefficients of the network. We show that any such network contains at least one hyperarc. Additionally, we prove that partial coupling cannot appear in consistent metabolic networks with less than four reactions, or with less than three irreversible reactions, or without hyperarc(s). & 2014 Elsevier Ltd. All rights reserved. 1. Introduction 1.1. Emergent properties in systems biology Aristotle (384-322 BC) was probably the first person to describe the concept of emergence in his famous statement: “the whole is more than the sum of its parts” (Mazzocchi, 2008). It is now widely accepted that biological systems cannot be fully understood merely on a molecular level (Powell and Dupre, 2009). Emergent properties in systems biology are those which arise only when the system passes a certain level of complexity (Ferrell, 2009). Example 1. As a concrete example of emergence, we show how the number of components in a gene regulatory network can determine its oscillatory behavior. In a gene regulatory network, a negative feedback loop is a feedback loop which includes an odd number of inhibitory interactions. In Fig. 1 two examples of negative feedback loops are presented. If there is no time delay, it can be shown that a network with two components may only show damped (and not sustained) oscillation. In contrast, a net- work with three genes can function as a sustained oscillator (Ferrell et al., 2011). Therefore, oscillatory behavior emerges at the level of a three gene system (Ferrell, 2009). Typically, emergence appears as a result of non-linear interac- tions among the components of system. Therefore, reductionist and deterministic attempts fail to explain it (Powell and Dupre, 2009; Mazzocchi, 2011). Emergent properties are commonly observed in systems biol- ogy. There are several instances of emergent properties reported in signalling networks (Bhalla and Iyengar, 1999; Papin and Palsson, 2004; Liu et al., 2013; Appleton and Luttrell, 2013), gene regulatory networks (Barberis et al., 2007; Ferrell et al., 2011; Torres-Sosa et al., 2012), immune systems (De Boer and Perelson, 1991; Nardai et al., 2006), metabolic systems (Boogerd et al., 2005; Palumbo et al., 2007), and even cell-to-cell communication networks in bacteria (Wintermute and Silver, 2010). Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/yjtbi Journal of Theoretical Biology http://dx.doi.org/10.1016/j.jtbi.2014.04.011 0022-5193/& 2014 Elsevier Ltd. All rights reserved. n Corresponding authors. E-mail addresses: marashi@ut.ac.ir (S.-A. Marashi), mtefagh@dena.sharif.ir (M. Tefagh). Journal of Theoretical Biology 355 (2014) 185–193