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![A simple example of a closed network consisting of one reaction. a Two molecules of superoxide and two hydron atoms react to each other and produce one molecule of dioxygen and a molecule of hydrogen perixide. The reaction rate here follows the mass-action kinetics with the reaction rate constant k. b The system of ODE equations of the network. The concentrations of O2-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {O}_{2}}^{-}$$\end{document}, H+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {H}^{+}$$\end{document}, O2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {O}_{2}$$\end{document} and H2O2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {H}_{2}\hbox {O}_{2}$$\end{document} are denoted by x1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_1$$\end{document}, x2,x3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_2, x_3$$\end{document} and x4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_4$$\end{document} respectively](publication/363888332/figure/fig2/AS:11431281086777643@1664331654816/A-simple-example-of-a-closed-network-consisting-of-one-reaction-a-Two-molecules-of.png)
A simple example of a closed network consisting of one reaction. a Two molecules of superoxide and two hydron atoms react to each other and produce one molecule of dioxygen and a molecule of hydrogen perixide. The reaction rate here follows the mass-action kinetics with the reaction rate constant k. b The system of ODE equations of the network. The concentrations of O2-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {O}_{2}}^{-}$$\end{document}, H+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {H}^{+}$$\end{document}, O2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {O}_{2}$$\end{document} and H2O2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {H}_{2}\hbox {O}_{2}$$\end{document} are denoted by x1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_1$$\end{document}, x2,x3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_2, x_3$$\end{document} and x4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_4$$\end{document} respectively
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In this paper we introduce a new representation for the multistationarity region of a reaction network, using polynomial superlevel sets. The advantages of using this polynomial superlevel set representation over the already existing representations (cylindrical algebraic decompositions, numeric sampling, rectangular divisions) is discussed, and al...
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Citations
... This has been accomplished for certain biochemically significant networks, by using algebraic techniques that harness the structure and sparsity appearing in polynomial systems arising in many biochemical networks [1,3,9,23]. Another approach, due to Sadeghimanesh and England, is to approximate multistationarity regions by polynomial super-level sets [19]. A related attempt to gain a detailed understanding of such regions was pursued by Bradford et al., who used computational (symbolic and numerical) techniques [2]. ...
... (a) vectors of positive rate constants κ (such multistionarity regions were studied in [6,7]), (b) total-constant vectors c (as in [3]), and (c) pairs (κ; c) (as in [1,2,17,19]). We focus on options (a) and (c). ...
... Indeed, Σ is the region above the graph of the positive function h(κ, κ) (in Case 1) or the region between two graphs, one of which lies above the other (Cases 2 and 3). Finally, the fact that the multistationarity-allowing region equals R 2 >0 is verified easily using (19) and Proposition 4.4. ...
A multistationarity region is the part of a reaction network's parameter space that gives rise to multiple steady states. Mathematically, this region consists of the positive parameters for which a parametrized family of polynomial equations admits two or more positive roots. Much recent work has focused on analyzing multistationarity regions of biologically significant reaction networks and determining whether such regions are connected. Here we focus on the multistationarity regions of small networks, those with few species and few reactions. For two families of such networks -- those with one species and up to three reactions, and those with two species and up to two reactions -- we prove that the resulting multistationarity regions are connected. We also give an example of a network with one species and six reactions for which the multistationarity region is disconnected. Our proofs rely on the formula for the discriminant of a trinomial, a classification of small multistationary networks, and a recent result of Feliu and Telek that partially generalizes Descartes' rule of signs.
