Subnetwork analysis for multistationarity in mass action kinetics

Flockerzi, Dietrich; Conradi, Carsten

doi:10.1088/1742-6596/138/1/012006

Cited by 12 publications

(13 citation statements)

References 25 publications

(48 reference statements)

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“…where the rate constants ofĨ k andÃ k :=Ĩ aĨk are defined byk h1(i) = k i isÑ is proper and (19) ifÑ is strongly resolvably improper, and Ψ K (x) has entries [Ψ K (x)] h2(j) = Ψ j (x) for j ∈ CR K . (Notice that, for proper translations, CR K = CR and h 2 is bijective so that this coincides with the definition of Ψ K (x) given in the proof of Lemma 4.1.…”

Section: Connection With Toric Steady Statesmentioning

confidence: 99%

Translated Chemical Reaction Networks

Johnston

2014

Bull Math Biol

175

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Many biochemical and industrial applications involve complicated networks of simultaneously occurring chemical reactions. Under the assumption of mass action kinetics, the dynamics of these chemical reaction networks are governed by systems of polynomial ordinary differential equations. The steady states of these mass action systems have been analysed via a variety of techniques, including elementary flux mode analysis, algebraic techniques (e.g. Groebner bases), and deficiency theory. In this paper, we present a novel method for characterizing the steady states of mass action systems. Our method explicitly links a network's capacity to permit a particular class of steady states, called toric steady states, to topological properties of a related network called a translated chemical reaction network. These networks share their reaction stoichiometries with their source network but are permitted to have different complex stoichiometries and different network topologies. We apply the results to examples drawn from the biochemical literature

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Section: Connection With Toric Steady Statesmentioning

confidence: 99%

Translated Chemical Reaction Networks

Johnston

2014

Bull Math Biol

175

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“…We say that the network has the capacity for multistationarity if there exists a choice of reaction rate constants κ such that there are two or more steady states in one stoichiometric compatibility class for some initial state x 0 , that is, for an appropriate choice of total conservation constants. Starting with [7,8], several articles studied the capacity for multistationarity from the structure of the digraph [1,11,12,13,20,23,24]. Once the capacity for multistationarity is determined, the following difficult question is to find values of multistationary parameters as exhaustively and explicitly as possible.…”

Section: Introductionmentioning

confidence: 99%

Lower bounds for positive roots and regions of multistationarity in chemical reaction networks

2020

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Given a real sparse polynomial system, we present a general framework to find explicit coefficients for which the system has more than one positive solution, based on the recent article by Bihan, Santos and Spaenlehauer [2]. We apply this approach to find explicit reaction rate constants and total conservation constants in biochemical reaction networks for which the associated dynamical system is multistationary.

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“…Thus one can use Theorem 1 (and all results derived from it in the following sections) to establish multistationarity with respect to arbitrary linear subspaces (see e.g. Flockerzi and Conradi 2008).…”

Section: Incorporating the Linear Constraints Z T A = Z T Bmentioning

confidence: 99%

Multistationarity in mass action networks with applications to ERK activation

Conradi

Flockerzi

2011

J. Math. Biol.

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Ordinary Differential Equations (ODEs) are an important tool in many areas of Quantitative Biology. For many ODE systems multistationarity (i.e. the existence of at least two positive steady states) is a desired feature. In general establishing multistationarity is a difficult task as realistic biological models are large in terms of states and (unknown) parameters and in most cases poorly parameterized (because of noisy measurement data of few components, a very small number of data points and only a limited number of repetitions). For mass action networks establishing multistationarity hence is equivalent to establishing the existence of at least two positive solutions of a large polynomial system with unknown coefficients. For mass action networks with certain structural properties, expressed in terms of the stoichiometric matrix and the reaction rate-exponent matrix, we present necessary and sufficient conditions for multistationarity that take the form of linear inequality systems. Solutions of these inequality systems define pairs of steady states and parameter values. We also present a sufficient condition to identify networks where the aforementioned conditions hold. To show the applicability of our results we analyse an ODE system that is defined by the mass action network describing the extracellular signal-regulated kinase (ERK) cascade (i.e. ERK-activation).

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Subnetwork analysis for multistationarity in mass action kinetics

Cited by 12 publications

References 25 publications

Translated Chemical Reaction Networks

Translated Chemical Reaction Networks

Lower bounds for positive roots and regions of multistationarity in chemical reaction networks

Multistationarity in mass action networks with applications to ERK activation

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