Research Article: Cycles and the Qualitative Evolution of Chemical Systems

Date Published: October 11, 2012

Publisher: Public Library of Science

Author(s): Peter Kreyssig, Gabi Escuela, Bryan Reynaert, Tomas Veloz, Bashar Ibrahim, Peter Dittrich, Claude Prigent.


Cycles are abundant in most kinds of networks, especially in biological ones. Here, we investigate their role in the evolution of a chemical reaction system from one self-sustaining composition of molecular species to another and their influence on the stability of these compositions. While it is accepted that, from a topological standpoint, they enhance network robustness, the consequence of cycles to the dynamics are not well understood. In a former study, we developed a necessary criterion for the existence of a fixed point, which is purely based on topological properties of the network. The structures of interest we identified were a generalization of closed autocatalytic sets, called chemical organizations. Here, we show that the existence of these chemical organizations and therefore steady states is linked to the existence of cycles. Importantly, we provide a criterion for a qualitative transition, namely a transition from one self-sustaining set of molecular species to another via the introduction of a cycle. Because results purely based on topology do not yield sufficient conditions for dynamic properties, e.g. stability, other tools must be employed, such as analysis via ordinary differential equations. Hence, we study a special case, namely a particular type of reflexive autocatalytic network. Applications for this can be found in nature, and we give a detailed account of the mitotic spindle assembly and spindle position checkpoints. From our analysis, we conclude that the positive feedback provided by these networks’ cycles ensures the existence of a stable positive fixed point. Additionally, we use a genome-scale network model of the Escherichia coli sugar metabolism to illustrate our findings. In summary, our results suggest that the qualitative evolution of chemical systems requires the addition and elimination of cycles.

Partial Text

Many mechanisms are characterized by the presence of cycles, especially those that play central roles in biological systems. Specific functions such as regulation, memory and differentiation have been associated to cycles [1]–[4]. Furthermore, systems containing cycles exhibit robustness to environmental changes, which is a key feature for the evolution of biochemical networks [5], [6]. Cycles vary in appearance from simple feedback loops to coupled ones [7] or large cycles, these include some signaling cascades [8], and the Krebs and Calvin cycles [9], [10]. Frequently, simple cycles are considered as network motifs [11] and therefore, they are analyzed as isolated modules neglecting the role of the molecular environment. In contrast, it has been proposed that the topological structure of a subgraph alone cannot determine its effects over the whole network [7], [12]. Moreover, some authors have found interesting results on feedback loops and have concluded these as prerequisites for multistability in gene regulation and mixed networks [1], [2], as well as in metabolic networks [4], [13]. However, the existence of cycles as well as their necessity and contribution to stability in networks remains largely elusive. Additionally the question of how to analyze large systems, in which classical approaches like differential equations fail, is open.

We shortly summarize the needed definitions and results from COT in informal terms, still providing a mathematically precise version in the Appendix. A tool for the computation of chemical organizations is freely available on our website

Our definition of cycle can be explained using the usual definition for cycles in directed graphs. Given a reaction network we can deduce a directed graph in the following way. The nodes of that graph are the species of the reaction network, and there is an edge from species to species if there is a reaction with on its left hand side and on its right hand side. We say that is directly-causally connected to . The cycles found in this directed graph are the ones we identify as cycles of the reaction network in our terminology (see Figure 1). This definition captures a variety of specific forms of cycles, yet yielding a structural result about reaction networks.

The occurrence of specific motifs and their functionality in networks have been studied largely in biological, ecological and social systems [11]. Moreover, there are many works that present results, linking reaction networks with cyclic structures and the capacity to generate stationary states [1], [2], [4]. These studies either focus on isolated specific motifs or on the effect of feedback loops on a system’s behavior. Here however, we focus on the role of cycles in systems that undergo a change in the composition of molecular species. In Lemma 1 we showed that one possibility to achieve a qualitative transition, namely from a closed (not necessarily self-maintaining) set to an organization, is to add a cycle.