Graph theory and qualitative analysis of reaction networks

Abstract

Different types of macroscopic reaction kinetics can be derived from microscopic molecular interactions, with the law of mass action being the most widely used one in standard situations. After such a modeling step, where primarily the types of reactions are identified, it becomes a problem to analyse qualitative properties of complete regulatory networks. This problem has to be tackled, because chemical reaction networks play a part in some of the most fundamental cellular processes such as cell metabolism and regulation of cell signalling processes. This paper discusses how reaction networks can be described and analysed by graph theoretic means. Graph theory is a useful analysis tool for complex reaction networks, in situations where there is parameter uncertainty or modeling information is incomplete. Graphs are very robust tools, in the sense that whole classes of network topologies will show similar behaviour, independently of precise information that is available about the reaction constants. Nevertheless, one still has to take care to incorporate sufficient dynamical information in the network structure, in order to obtain meaningful results.

Multi-scale genetic dynamic modelling I : an algorithm to compute generators

“…We can study the types of bifurcations occurring when system parameters are varied (see Domijan and Kirkilionis 2009;Elowitz and Leibler 2000;Gonze 2010;Guantes and Poyatos 2006;Kirkilionis 2010). As an additional aspect we might like to achieve a structural equivalence between the dynamical system and data, in terms of system components (variables) and their links, for example expressed in terms of the interaction matrix (see Domijan and Kirkilionis 2008;Kirkilionis and Sbano 2010).…”

“…where l l K l = 0 for all l. The decoupling of Markov chains in this fashion is described in more detail in Kirkilionis (2008 andKirkilionis andSbano (2010), and it has been widely used in Janus (2009). Definitions and properties for the tensor product can be found in Merris (1997).…”

“…Remark 1 It is important to observe that the averaged dynamics (10) may have nonlinear terms just stemming from the coupling of molecular concentrations with the Markov Chain (MC) on S. In fact there are systems where each vector field X (s) (x) is linear in x but nevertheless the vector field (10) is nonlinear due to the terms l s (x). In Kirkilionis (2008, 2008), it is shown that for example Hill type kinetics can be constructed through a suitable averaged dynamics approach.…”

A Calculus of Chemical Systems

“…As is well known, reactions can also be naturally represented by Petri nets, particularly P/T nets, and, indeed there is little difference between P/T nets and sets of reactions. See [4,14,25] for surveys of the now extensive applications of Petri nets to pathway simulation; see [35,16] for an early chemical reaction formalism using a bipartite graph structure equivalent to P/T nets; and see [8] for a survey of the application of graphical ideas to chemical reaction systems, including two bipartite graph formalisms [17,5].…”

Network models in the study of metabolism

“…Nonetheless, this perspective has not been able to resolve questions inherent to variations in time, is ambiguous in its representation for detailed discussions of structure and is limited as regards of the interpretation and comparative analysis of experimental metabolic data associated with other biological networks (For example: transcriptomic networks or networks of protein interactions) (Gross, 2005), due to the hypergraphic nature of metabolism. Then it is important to incorporate information relevant to flows within the network or changes of the metabolic mass over time, for a complete analysis of metabolic systems (Steuer, 2007;Domijan and Kirkilionis, 2008) due to the static nature of the topological models. We continue in this review with the conceptual models that incorporate the kinetic parameters of metabolism.…”