Effects of small particle numbers on long-term behaviour in discrete biochemical systems.

MOTIVATION: The functioning of many biological processes depends on the appearance of only a small number of a single molecular species. Additionally, the observation of molecular crowding leads to the insight that even a high number of copies of species do not guarantee their interaction. How single particles contribute to stabilizing biological systems is not well understood yet. Hence, we aim at determining the influence of single molecules on the long-term behaviour of biological systems, i.e. whether they can reach a steady state. RESULTS: We provide theoretical considerations and a tool to analyse Systems Biology Markup Language models for the possibility to stabilize because of the described effects. The theory is an extension of chemical organization theory, which we called discrete chemical organization theory. Furthermore we scanned the BioModels Database for the occurrence of discrete chemical organizations. To exemplify our method, we describe an application to the Template model of the mitotic spindle assembly checkpoint mechanism. AVAILABILITY AND IMPLEMENTATION: http://www.biosys.uni-jena.de/Services.html. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Cycles and the qualitative evolution of chemical systems.

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.