Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPositive Equilibria of Hill-Type Kinetic Systems
149
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
We present conditions which guarantee a parametrization of the set of positive equilibria of a generalized mass-action system. Our main results state that (i) if the underlying generalized chemical reaction network has an effective deficiency of zero, then the set of positive equilibria coincides with the parametrized set of complex-balanced equilibria and (ii) if the network is weakly reversible and has a kinetic deficiency of zero, then the equilibrium set is nonempty and has a positive, typically rational, parametrization. Via the method of network translation, we apply our results to classical mass-action systems studied in the biochemical literature, including the EnvZ-OmpR and shuttled WNT signaling pathways. A parametrization of the set of positive equilibria of a (generalized) mass-action system is often a prerequisite for the study of multistationarity and allows an easy check for the occurrence of absolute concentration robustness (ACR), as we demonstrate for the EnvZ-OmpR pathway.
The leading-order approximation to a Filippov system $f$ about a generic boundary equilibrium $x^*$ is a system $F$ that is affine one side of the boundary and constant on the other side. We prove $x^*$ is exponentially stable for $f$ if and only if it is exponentially stable for $F$ when the constant component of $F$ is not tangent to the boundary. We then show exponential stability and asymptotic stability are in fact equivalent for $F$. We also show exponential stability is preserved under small perturbations to the pieces of $F$. Such results are well known for homogeneous systems. To prove the results here additional techniques are required because the two components of $F$ have different degrees of homogeneity. The primary function of the results is to reduce the problem of the stability of $x^*$ from the general Filippov system $f$ to the simpler system $F$. Yet in general this problem remains difficult. We provide a four-dimensional example of $F$ for which orbits appear to converge to $x^*$ in a chaotic fashion. By utilising the presence of both homogeneity and sliding motion the dynamics of $F$ can in this case be reduced to the combination of a one-dimensional return map and a scalar function.
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.
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).
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
