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Siegel, David; Maclean, D J

doi:10.1023/a:1019183206064

Cited by 55 publications

(8 citation statements)

References 6 publications

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“…Then it is easy to see from Eq. (3) thatThe space of concentrationshas been referred to as the positive reaction simplex in Horn and Jackson (1972) and the positive stoichiometric compatibility class (corresponding to ) in Feinberg (1995), Anderson (2011) and Siegel and MacLean (2000). In this paper we will use the terminology positive stoichiometric compatibility class to refer to .…”

Section: Stoichiometry Positive Compatibility Class and Conservationmentioning

confidence: 99%

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Global stability of a class of futile cycles

Rao

2016

J. Math. Biol.

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In this paper, we prove the global asymptotic stability of a class of mass action futile cycle networks which includes a model of processive multisite phosphorylation networks. The proof consists of two parts. In the first part, we prove that there is a unique equilibrium in every positive compatibility class. In the second part, we make use of a piecewise linear in rates Lyapunov function in order to prove the global asymptotic stability of the unique equilibrium corresponding to a given initial concentration vector. The main novelty of the paper is the use of a simple algebraic approach based on the intermediate value property of continuous functions in order to prove the uniqueness of equilibrium in every positive compatibility class.

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Section: Stoichiometry Positive Compatibility Class and Conservationmentioning

confidence: 99%

“…This method has been used in Siegel and MacLean (2000) in order to prove the nonexistence of boundary equilibria in enzyme kinetic reaction networks that are similar to but simpler than the network (1). We briefly explain the concept of reachability as in Vol’pert and Hudjaev (1985).…”

Section: Uniqueness Of Equilibriummentioning

confidence: 99%

Global stability of a class of futile cycles

Rao

2016

J. Math. Biol.

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“…The global attractor conjecture is the oldest and best known of these conjectures, and has resisted efforts for a proof for over four decades, but proofs of many special cases have been obtained during this time, for example in , . The conjecture originated from the 1972 breakthrough work by Horn and Jackson, and was formulated in its current form by Horn in 1974 .…”

Section: Persistence and Global Stabilitymentioning

confidence: 99%

Mathematical Analysis of Chemical Reaction Systems

Crăciun

2018

Israel Journal of Chemistry

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The use of mathematical methods for the analysis of chemical reaction systems has a very long history, and involves many types of models: deterministic versus stochastic, continuous versus discrete, and homogeneous versus spatially distributed. Here we focus on mathematical models based on deterministic mass‐action kinetics. These models are systems of coupled nonlinear differential equations on the positive orthant. We explain how mathematical properties of the solutions of mass‐action systems are strongly related to key properties of the networks of chemical reactions that generate them, such as specific versions of reversibility and feedback interactions.

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“…The purpose of this paper is to generalize some results presented in [2,4] so that we can use these in the system with time-delay. Time delayed system is defined on the functional space C([−τ, 0]; R n ≥0 ), firstly we get the ω-limit set theorem [26]in this functional space, namely, each trajectory in complex balanced system with time delay trends to either the unique positive constant equilibrium or the constant boundary equilibriums. Then we give the definition of "face" and "facet" in this system and prove the delayed form of theorems in [2,4]: For a complex balanced system with arbitrary time delay, if Z W is empty or Z W is a facet(dim Z W = dim S − 1) or a vertex(dim Z W = 0) of stoichiometric subspace S for all semilocking set W, this system is persistent.…”

Section: Introductionmentioning

confidence: 99%

Persistence of Delayed Complex Balanced Chemical Reaction Networks

Zhang¹,

Gao²

2019

Preprint

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Time delay is a very common phenomenon in real biological or industrial systems. This paper studies the persistence of complex balanced mass action systems with any constant time delays(DeMAS). We give several sufficient conditions for DeMASs to be persistent based on the dimension of intersection of stoichiometric compatibility class and R n ≥0 . These results are analogous to the theorems which have proven by Anderson in [2] and [4] for the non-delayed mass action system. Besides, persistence implies global asymptotic stability for complex balanced DeMASs which means that all trajectories starting at a positive concentration trend to the unique positive constant equilibrium in its stoichiometric compatibility class. From this paper, we can learn further about the long-term property of the delayed complex balanced system.

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Cited by 55 publications

References 6 publications

Global stability of a class of futile cycles

Global stability of a class of futile cycles

Mathematical Analysis of Chemical Reaction Systems

Persistence of Delayed Complex Balanced Chemical Reaction Networks

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