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 compl
ex 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.
This paper studies chemical kinetic systems which decompose into weakly reversible complex factorizable (CF) systems. Among power law kinetic systems, CF systems (denoted as PL-RDK systems) are those where branching reactions of a reactant complex ha
ve identical rows in the kinetic order matrix. Mass action and generalized mass action systems (GMAS) are well-known examples. Schmitzs global carbon cycle model is a previously studied non-complex factorizable (NF) power law system (denoted as PL-NDK). We derive novel conditions for the existence of weakly reversible CF-decompositions and present an algorithm for verifying these conditions. We discuss methods for identifying independent decompositions, i.e., those where the stoichiometric subspaces of the subnetworks form a direct sum, as such decompositions relate positive equilibria sets of the subnetworks to that of the whole network. We then use the results to determine the positive equilibria sets of PL-NDK systems which admit an independent weakly reversible decomposition into PL-RDK systems of PLP type, i.e., the positive equilibria are log-parametrized, which is a broad generalization of a Deficiency Zero Theorem of Fortun et al. (2019).
A complex balanced kinetic system is absolutely complex balanced (ACB) if every positive equilibrium is complex balanced. Two results on absolute complex balancing were foundational for modern chemical reaction network theory (CRNT): in 1972, M. Fein
berg proved that any deficiency zero complex balanced system is absolutely complex balanced. In the same year, F. Horn and R. Jackson showed that the (full) converse of the result is not true: any complex balanced mass action system, regardless of its deficiency, is absolutely complex balanced. In this paper, we revive the study of ACB systems first by providing a partial converse to Feinbergs Theorem. In the spirit of Horn and Jacksons result, we then describe several methods for constructing new classes of ACB systems with positive deficiency and present classes of power law kinetic systems for each method. Furthermore, we illustrate the usefulness of the ACB property for obtaining new results on absolute concentration robustness (ACR) in a species, a concept introduced for mass action systems by Shinar and Feinberg in 2010, for a class of power law systems. Finally, we motivate the study of ACB in poly-PL systems, i.e. sums of power law systems, and indicate initial results.
There have been recent theoretic results that provide sufficient conditions for the existence of a species displaying absolute concentration robustness (ACR) in a power law kinetic (PLK) system. One such result involves the detection of ACR among net
works of high deficiency by considering a lower deficiency subnetwork with ACR as a local property. In turn, this smaller subnetwork serves as a building block for the larger ACR-possessing network. Here, with this theorem as foundation, we construct an algorithm that systematically checks ACR in a PLK system. By slightly modifying the algorithm, we also provide a procedure that identifies balanced concentration robustness (BCR), a weaker form of concentration robustness than ACR, in a PLK system.
This work introduces a novel approach to study properties of positive equilibria of a chemical reaction network $mathscr{N}$ endowed with Hill-type kinetics $K$, called a Hill-type kinetic (HTK) system $left(mathscr{N},Kright)$, including their multi
plicity and concentration robustness in a species. We associate a unique positive linear combination of power-law kinetic systems called poly-PL kinetic (PYK) system $left( {mathscr{N},{K_text{PY}}} right)$ to the given HTK system. The associated system has the key property that its equilibria sets coincide with those of the Hill-type system, i.e., ${E_ + }left( {mathscr{N},K} right) = {E_ + }left( {mathscr{N},{K_text{PY}}} right)$ and ${Z_ + }left( {mathscr{N},K} right) = {Z_ + }left( {mathscr{N},{K_text{PY}}} right)$. This allows us to identify two novel subsets of the Hill-type kinetics, called PL-equilibrated and PL-complex balanced kinetics, to which recent results on absolute concentration robustness (ACR) of species and complex balancing at positive equilibria of power-law (PL) kinetic systems can be applied. Our main results also include the Shinar-Feinberg ACR Theorem for PL-equilibrated HT-RDK systems (i.e., subset of complex factorizable HTK systems), which establishes a foundation for the analysis of ACR in HTK systems, and the extension of the results of Muller and Regensburger on generalized mass action systems to PL-complex balanced HT-RDK systems. In addition, we derive the theory of balanced concentration robustness (BCR) in an analogous manner to ACR for PL-equilibrated systems. Finally, we provide further extensions of our results to a more general class of kinetics, which includes quotients of poly-PL functions.
Absolute concentration robustness (ACR) is a condition wherein a species in a chemical kinetic system possesses the same value for any positive steady state the network may admit regardless of initial conditions. Thus far, results on ACR center on ch
emical kinetic systems with deficiency one. In this contribution, we use the idea of dynamic equivalence of chemical reaction networks to derive novel results that guarantee ACR for some classes of power law kinetic systems with deficiency zero. Furthermore, using network decomposition, we identify ACR in higher deficiency networks (i.e. deficiency $geq$ 2) by considering the presence of a low deficiency subnetwork with ACR. Network decomposition also enabled us to recognize and define a weaker form of concentration robustness than ACR, which we named as `balanced concentration robustness. Finally, we also discuss and emphasize our view of ACR as a primarily kinetic character rather than a condition that arises from structural sources.
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 decomposit
ions 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.
Magombedze and Mulder in 2013 studied the gene regulatory system of Mycobacterium Tuberculosis (Mtb) by partitioning this into three subsystems based on putative gene function and role in dormancy/latency development. Each subsystem, in the form of S
-system, is represented by an embedded chemical reaction network (CRN), defined by a species subset and a reaction subset induced by the set of digraph vertices of the subsystem. For the embedded networks of S-system, we showed interesting structural properties and proved that all S-system CRNs (with at least two species) are discordant. Analyzing the subsystems as subnetworks, where arcs between vertices belonging to different subsystems are retained, we formed a digraph homomorphism from the corresponding subnetworks to the embedded networks. Lastly, we explored the modularity concept of CRN in the context of digraph.
The fundamental decomposition of a chemical reaction network (also called its $mathscr{F}$-decomposition) is the set of subnetworks generated by the partition of its set of reactions into the fundamental classes introduced by Ji and Feinberg in 2011
as the basis of their higher deficiency algorithm for mass action systems. The first part of this paper studies the properties of the $mathscr{F}$-decomposition, in particular, its independence (i.e., the networks stoichiometric subspace is the direct sum of the subnetworks stoichiometric subspaces) and its incidence-independence (i.e., the image of the networks incidence map is the direct sum of the incidence maps images of the subnetworks). We derive necessary and sufficient conditions for these properties and identify network classes where the $mathscr{F}$-decomposition coincides with other known decompositions. The second part of the paper applies the above-mentioned results to improve the Multistationarity Algorithm for power-law kinetic systems (MSA), a general computational approach that we introduced in previous work. We show that for systems with non-reactant determined interactions but with an independent $mathscr{F}$-decomposition, the transformation to a dynamically equivalent system with reactant-determined interactions -- required in the original MSA -- is not necessary. We illustrate this improvement with the subnetwork of Schmitzs carbon cycle model recently analyzed by Fortun et al.
