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A chemical reaction network (CRN) is composed of reactions that can be seen as interactions among entities called species, which exist within the system. Endowed with kinetics, CRN has a corresponding set of ordinary differential equations (ODEs). In Chemical Reaction Network Theory, we are interested with connections between the structure of the CRN and qualitative properties of the corresponding ODEs. One of the results in Decomposition Theory of CRNs is that the intersection of the sets of positive steady states of the subsystems is equal to the set of positive steady states of the whole system, if the decomposition is independent. Hence, computational approach using independent decompositions can be used as an efficient tool in studying large systems. In this work, we provide a necessary and sufficient condition for the existence of a nontrivial independent decomposition of a CRN, which leads to a novel step-by-step method to obtain such decomposition, if it exists. We also illustrate these results using real-life examples. In particular, we show that a CRN of a popular model of anaerobic yeast fermentation pathway has a nontrivial independent decomposition, while a particular biological system, which is a metabolic network with one positive feedforward and a negative feedback has none. Finally, we analyze properties of steady states of reaction networks of specific influenza virus models.
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This paper presents novel decomposition classes of chemical reaction networks (CRNs) derived from S-system kinetics. Based on the network decomposition theory initiated by Feinberg in 1987, we introduce the concept of incidence independent decompositions and develop the theory of $mathscr{C}$- and $mathscr{C}^*$- decompositions which partition the set of complexes and the set of nonzero complexes respectively, including their structure theorems in terms of linkage classes. Analogous to Feinbergs independent decomposition, we demonstrate the important relationship between sets of complex balance equilibria for an incidence independent decomposition of weakly reversible subnetworks for any kinetics. We show that the $mathscr{C}^*$-decompositions are also incidence independent. We also introduce in this paper a new realization for an S-system that is analyzed using a newly defined class of species coverable CRNs. This led to the extension of the deficiency formula and characterization of fundamental decompositions of species decomposable reaction networks.
Many biochemical and industrial applications involve complicated networks of simultaneously occurring chemical reactions. Under the assumption of mass action kinetics, the dynamics of these chemical reaction networks are governed by systems of polynomial ordinary differential equations. The steady states of these mass action systems have been analysed via a variety of techniques, including elementary flux mode analysis, algebraic techniques (e.g. Groebner bases), and deficiency theory. In this paper, we present a novel method for characterizing the steady states of mass action systems. Our method explicitly links a networks capacity to permit a particular class of steady states, called toric steady states, to topological properties of a related network called a translated chemical reaction network. These networks share their reaction stoichiometries with their source network but are permitted to have different complex stoichiometries and different network topologies. We apply the results to examples drawn from the biochemical literature.
We study how the properties of allowing multiple positive nondegenerate equilibria (MPNE) and multiple positive linearly stable equilibria (MPSE) are inherited in chemical reaction networks (CRNs). Specifically, when is it that we can deduce that a CRN admits MPNE or MPSE based on analysis of its subnetworks? Using basic techniques from analysis we are able to identify a number of situations where MPNE and MPSE are inherited as we build up a network. Some of these modifications are known while others are new, but all results are proved using the same basic framework, which we believe will yield further results. The results are presented primarily for mass action kinetics, although with natural, and in some cases immediate, generalisation to other classes of kinetics.
We are concerned with polynomial ordinary differential systems that arise from modelling chemical reaction networks. For such systems, which may be of high dimension and may depend on many parameters, it is frequently of interest to obtain a reduction of dimension in certain parameter ranges. Singular perturbation theory, as initiated by Tikhonov and Fenichel, provides a path toward such reductions. In the present paper we discuss parameter values that lead to singular perturbation reductions (so-called Tikhonov-Fenichel parameter values, or TFPVs). An algorithmic approach is known, but it is feasible for small dimensions only. Here we characterize conditions for classes of reaction networks for which TFPVs arise by turning off reactions (by setting rate parameters to zero), or by removing certain species (which relates to the classical quasi-steady state approach to model reduction). In particular, we obtain definitive results for the class of complex balanced reaction networks (of deficiency zero) and first order reaction networks.
A decomposition of a chemical reaction network (CRN) is produced by partitioning its set of reactions. The partition induces networks, called subnetworks, that are smaller than the given CRN which, at this point, can be called parent network. A complex is called a common complex if it occurs in at least two subnetworks in a decomposition. A decomposition is said to be incidence independent if the image of the incidence map of the parent network is the direct sum of the images of the subnetworks incidence maps. It has been recently discovered that the complex balanced equilibria of the parent network and its subnetworks are fundamentally connected in an incidence independent decomposition. In this paper, we utilized the set of common complexes and a developed criterion to investigate decompositions incidence independence properties. A framework was also developed to analyze decomposition classes with similar structure and incidence independence properties. We identified decomposition classes that can be characterized by their sets of common complexes and studied their incidence independence. Some of these decomposition classes occur in some biological and chemical models. Finally, a sufficient condition was obtained for the complex balancing of some power law kinetic (PLK) systems with incidence independent and complex balanced decompositions. This condition led to a generalization of the Defficiency Zero Theorem for some PLK systems.
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