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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. Feinberg 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.
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Very often, models in biology, chemistry, physics, and engineering are systems of polynomial or power-law ordinary differential equations, arising from a reaction network. Such dynamical systems can be generated by many different reaction networks. On the other hand, networks with special properties (such as reversibility or weak reversibility) are known or conjectured to give rise to dynamical systems that have special properties: existence of positive steady states, persistence, permanence, and (for well-chosen parameters) complex balancing or detailed balancing. These last two are related to thermodynamic equilibrium, and therefore the positive steady states are unique and stable. We describe a computationally efficient characterization of polynomial or power-law dynamical systems that can be obtained as complex-balanced, detailed-balanced, weakly reversible, and reversible mass-action systems.
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 multiplicity 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 chemical 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 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 have 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).
We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures. We use these general conditions to derive results for spectral properties of time-changes of unipotent flows on homogeneous spaces of semisimple groups regarding absolutely continuous spectrum as well as maximal spectral type; the time-changes of the horocycle flow are special cases of this general category of flows. In addition we use the general conditions to derive spectral results for twisted horocycle flows and to rederive certain spectral results for skew products over translations and Furstenberg transformations.
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