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![Stoichiometric BCs in a toy network. The conversion diagram depicts a base network with m=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=5$$\end{document} species, n=7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=7$$\end{document} complexes, and r=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=6$$\end{document} base reactions, two of which are reversible. The network has four BCs, which will remain balanced even if one adds the dashed reaction (highlighted in green). However, once one adds the dotted reaction (highlighted in red), only two of these complexes remain balanced.](publication/369882907/figure/fig3/AS:11431281140082543@1680923724902/Stoichiometric-BCs-in-a-toy-network-The-conversion-diagram-depicts-a-base-network-with.png)
Stoichiometric BCs in a toy network. The conversion diagram depicts a base network with m=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=5$$\end{document} species, n=7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=7$$\end{document} complexes, and r=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=6$$\end{document} base reactions, two of which are reversible. The network has four BCs, which will remain balanced even if one adds the dashed reaction (highlighted in green). However, once one adds the dotted reaction (highlighted in red), only two of these complexes remain balanced.
Source publication
Balanced complexes in biochemical networks are at core of several theoretical and computational approaches that make statements about the properties of the steady states supported by the network. Recent computational approaches have employed balanced complexes to reduce metabolic networks, while ensuring preservation of particular steady-state prop...
Citations
... Here, we rely on the recently introduced classification of balanced complexes (Langary, et al., 2023), in particular the class of nonstoichiometric balanced complexes, to define the notion of effective deficiency of a network. Nonstoichiometric balanced complexes have been shown to arise as a combined result of several factors, including the algebraic and graphical structure of the network as well as operational bounds on reaction kinetics (Langary, et al., 2023). ...
... Here, we rely on the recently introduced classification of balanced complexes (Langary, et al., 2023), in particular the class of nonstoichiometric balanced complexes, to define the notion of effective deficiency of a network. Nonstoichiometric balanced complexes have been shown to arise as a combined result of several factors, including the algebraic and graphical structure of the network as well as operational bounds on reaction kinetics (Langary, et al., 2023). A remarkable feature of the effective deficiency is that, like that of the classical, so-called structural, it is defined and can be calculated for networks of arbitrary kinetics. ...
... As was shown in (Langary, et al., 2023), a complex ∈ is a BC, if and only if ...
The deficiency of a (bio)chemical reaction network can be conceptually interpreted as a measure of its ability to support exotic dynamical behavior and/or multistationarity. The classical definition of deficiency relates to the capacity of a network to permit variations of the complex formation rate vector at steady state, irrespective of the network kinetics. However, the deficiency is by definition completely insensitive to the fine details of the directionality of reactions as well as bounds on reaction fluxes. While the classical definition of deficiency can be readily applied in the analysis of unconstrained, weakly reversible networks, it only provides an upper bound in the cases where relevant constraints on reaction fluxes are imposed. Here we propose the concept of effective deficiency, which provides a more accurate assessment of the network’s capacity to permit steady state variations at the complex level for constrained networks of any reversibility patterns. The effective deficiency relies on the concept of nonstoichiometric balanced complexes, which we have already shown to be present in real-world biochemical networks operating under flux constraints. Our results demonstrate that the effective deficiency of real-world biochemical networks is smaller than the classical deficiency, indicating the effects of reaction directionality and flux bounds on the variation of the complex formation rate vector at steady state.
... Here, we rely on the recently introduced classification of balanced complexes (Langary, et al., 2021), in particular the class of nonstoichiometric balanced complexes, to define the notion of effective deficiency of a network. Nonstoichiometric balanced complexes have been shown to arise as a combined result of several factors, including the algebraic and graphical structure of the network as well as operational bounds on reaction kinetics (Langary, et al., 2021). ...
... Here, we rely on the recently introduced classification of balanced complexes (Langary, et al., 2021), in particular the class of nonstoichiometric balanced complexes, to define the notion of effective deficiency of a network. Nonstoichiometric balanced complexes have been shown to arise as a combined result of several factors, including the algebraic and graphical structure of the network as well as operational bounds on reaction kinetics (Langary, et al., 2021). A remarkable feature of the effective deficiency is that, like that of the classical, so-called structural, it is defined and can be calculated for networks of arbitrary kinetics. ...
... As was shown in (Langary, et al., 2021), a complex ∈ is a BC, if and only if ...
The deficiency of a (bio)chemical reaction network can be conceptually interpreted as a measure of its ability to support exotic dynamical behavior and/or multistationarity. The classical definition of deficiency relates to the capacity of a network to permit variations of the complex formation rate vector at steady state, irrespective of the network kinetics. However, the deficiency is by definition completely insensitive to the fine details of the directionality of reactions as well as bounds on reaction fluxes. While the classical definition of deficiency can be readily applied in the analysis of unconstrained, weakly reversible networks, it only provides an upper bound in the cases where relevant constraints on reaction fluxes are imposed. Here we propose the concept of effective deficiency, which provides a more accurate assessment of the networks capacity to permit steady state variations at the complex level for constrained networks of any reversibility patterns. The effective deficiency relies on the concept of nonstoichiometric balanced complexes, which we have already shown to be present in real-world biochemical networks operating under flux constraints. Our results demonstrate that the effective deficiency of real-world biochemical networks is smaller than the classical deficiency, indicating the effects of reaction directionality and flux bounds on the variation of the complex formation rate vector at steady state.
The deficiency of a (bio)chemical reaction network can be conceptually interpreted as a measure of its ability to support exotic dynamical behavior and/or multistationarity. The classical definition of deficiency relates to the capacity of a network to permit variations of the complex formation rate vector at steady state, irrespective of the network kinetics. However, the deficiency is by definition completely insensitive to the fine details of the directionality of reactions as well as bounds on reaction fluxes. While the classical definition of deficiency can be readily applied in the analysis of unconstrained, weakly reversible networks, it only provides an upper bound in the cases where relevant constraints on reaction fluxes are imposed. Here we propose the concept of effective deficiency, which provides a more accurate assessment of the network’s capacity to permit steady state variations at the complex level for constrained networks of any reversibility patterns. The effective deficiency relies on the concept of nonstoichiometric balanced complexes, which we have already shown to be present in real-world biochemical networks operating under flux constraints. Our results demonstrate that the effective deficiency of real-world biochemical networks is smaller than the classical deficiency, indicating the effects of reaction directionality and flux bounds on the variation of the complex formation rate vector at steady state.
