<i>P</i>Matrix Properties, Injectivity, and Stability in Chemical Reaction Systems

Banaji, Murad; Donnell, Pete; Baigent, Stephen

doi:10.1137/060673412

Cited by 73 publications

(115 citation statements)

References 16 publications

(38 reference statements)

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“…The implications C1 ⇒ C2, C3 ⇒ C4, C5 ⇒ C6, C7 ⇒ C8 and C9 ⇒ C10 follow by definition, while the other implications are results in [5,4,3], or are immediate consequences of these results. Note that C5 ⇔ C9 and C6 ⇒ C10 are just immediate corollaries of C1 ⇔ C3 and C2 ⇒ C4 respectively.…”

Section: Some Useful Results On Matrix Sets and Dsr Graphsmentioning

confidence: 99%

“…Continuing the theme of Remark 2.8, if Hopf bifurcation is ruled out for a chemical system with Jacobian matrices J because −J [2] is a P 0 -matrix for all J ∈ J , then it is in fact ruled out even if arbitrary degradation or outflow reactions are added. (Analogous statements follow replacing "−J [2] " with "−J" and "Hopf bifurcation" with "saddle-node bifurcation", but this is well-studied [3,5] and not the theme of this paper. )…”

Section: Hopf Bifurcationsmentioning

confidence: 99%

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Combinatorial approaches to Hopf bifurcations in systems of interacting elements

Angeli

Banaji

Pantea

2014

Communications in Mathematical Sciences

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Abstract. We describe combinatorial approaches to the question of whether families of real matrices admit pairs of nonreal eigenvalues passing through the imaginary axis. When the matrices arise as Jacobian matrices in the study of dynamical systems, these conditions provide necessary conditions for Hopf bifurcations to occur in parameterised families of such systems. The techniques depend on the spectral properties of additive compound matrices: in particular, we associate with a product of matrices a signed, labelled digraph termed a DSR [2] graph, which encodes information about the second additive compound of this product. A condition on the cycle structure of this digraph is shown to rule out the possibility of nonreal eigenvalues with positive real part. The techniques developed are applied to systems of interacting elements termed "interaction networks", of which networks of chemical reactions are a special case.

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Section: Some Useful Results On Matrix Sets and Dsr Graphsmentioning

confidence: 99%

Section: Hopf Bifurcationsmentioning

confidence: 99%

Combinatorial approaches to Hopf bifurcations in systems of interacting elements

Angeli

Banaji

Pantea

2014

Communications in Mathematical Sciences

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“…General electron transfer networks in the absence of a potential were analysed in [5] and found to have simple behaviour. Application of the theory presented in [13] should allow determination of when these networks give rise to P (−) Jacobians when interacting with a membrane potential.…”

Section: Discussionmentioning

confidence: 99%

“…It was shown in [13] that the structures of S and V along with the nonnegativity of P and F imply that J is a so called P (−) matrix (see Appendix A for the definition) 2 . This result is independent of n, the length of the chain.…”

Section: The Jacobianmentioning

confidence: 99%

Stability in generic mitochondrial models

2008

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In this paper, we use a variety of mathematical techniques to explore existence, local stability, and global stability of equilibria in abstract models of mitochondrial metabolism. The class of models constructed is defined by the biological description of the system, with minimal mathematical assumptions. The key features are an electron transport chain coupled to a process of charge translocation across a membrane. In the absence of charge translocation these models have previously been shown to behave in a very simple manner with a single, globally stable equilibrium. We show that with charge translocation the conclusion about a unique equilibrium remains true, but local and global stability do not necessarily follow. In sufficiently low dimensions -i.e. for short electron transport chains -it is possible to make claims about local and global stability of the equilibrium. On the other hand, for longer chains, these general claims are no longer valid. Some particular conditions which ensure stability of the equilibrium for chains of arbitrary length are presented.

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“…These systems are called NAC (Non AutoCatalytic) by Kaltenbach, but the hypothesis does not match the stronger and more usual definition of NAC systems (see for instance [4]) that forbids the production of some species to depend on its concentration. It rather matches the definition of monotonicity [9,10] and allows the restriction of the sign of an element in the above sum to the different components of Λ.…”

Section: Directed Species-reaction Graphmentioning

confidence: 99%

A Stronger Necessary Condition for the Multistationarity of Chemical Reaction Networks

Soliman

2013

Bull Math Biol

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To cite this version:Sylvain Soliman. A stronger necessary condition for the multistationarity of chemical reaction networks. Bulletin of Mathematical Biology, Springer Verlag, 2013, 75 (11) Abstract Biochemical reaction networks grow bigger and bigger, fed by the high-throughput data provided by biologists and bred in open repositories of models allowing merging and evolution. Nevertheless, since the available data is still very far from permitting the identification of the increasing number of kinetic parameters of such models, the necessity of structural analyses for describing the dynamics of chemical networks appears stronger every day.Using the structural information, notably from the stoichiometric matrix, of a biochemical reaction system, we state a more strict version of the famous Thomas' necessary condition for multistationarity. In particular, the obvious cases where Thomas' condition was trivially satisfied, mutual inhibition due to a multimolecular reaction and mutual activation due to a reversible reaction, can now easily be ruled out.This more strict condition shall not be seen as some version of Thomas' circuit functionality for the continuous case but rather as related and complementary to the whole domain of the structural analysis of (bio)chemical reaction systems, as pioneered by the Chemical Reaction Network Theory.

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PMatrix Properties, Injectivity, and Stability in Chemical Reaction Systems

Cited by 73 publications

References 16 publications

Combinatorial approaches to Hopf bifurcations in systems of interacting elements

Combinatorial approaches to Hopf bifurcations in systems of interacting elements

Stability in generic mitochondrial models

A Stronger Necessary Condition for the Multistationarity of Chemical Reaction Networks

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