Chemical reaction-diffusion networks: convergence of the method of lines

Mohamed, Fatma; Pantea, Casian; Tudorascu, Adrian

doi:10.1007/s10910-017-0779-z

Cited by 7 publications

(11 citation statements)

References 39 publications

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“…Most models in the literature are of this type. (When spatial effects may not be neglected [48,49,50], partial differential equations must be considered and the connection between network structure and qualitative properties has not yet been studied to the same extent; for some recent results, see [54,55,65]. )…”

Section: Introductionmentioning

confidence: 99%

Dynamics of Posttranslational Modification Systems: Recent Progress and Future Directions

Conradi

Shiu

2018

Biophysical Journal

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Posttranslational modification of proteins is important for signal transduction, and hence significant effort has gone toward understanding how posttranslational modification networks process information. This involves, on the theory side, analyzing the dynamical systems arising from such networks. Which networks are, for instance, bistable? Which networks admit sustained oscillations? Which parameter values enable such behaviors? In this Biophysical Perspective, we highlight recent progress in this area and point out some important future directions. Along the way, we summarize several techniques for analyzing general networks, such as eliminating variables to obtain steady-state parameterizations, and harnessing results on how incorporating intermediates affects dynamics.

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Section: Introductionmentioning

confidence: 99%

Dynamics of Posttranslational Modification Systems: Recent Progress and Future Directions

Conradi

Shiu

2018

Biophysical Journal

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“…[50,54] Recent work features strong connections between reaction-diffusion equations and complex-balanced mass-action systems. [27,35,55] 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56…”

Section: Reaction-diffusion Equationsmentioning

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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“…This steady state is known to be globally stable under some additional assumptions [1,16,27,40], and is actually conjectured to be globally stable even without these assumptions [10,16]. If a reaction network is a complex-balanced system under mass-action kinetics, then other relevant models, ranging from continuous-time Markov chain models [3] to reaction-diffusion models [20,38] and delay differential equation models [35], are also stable in some sense.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Characterization of Complex-Balanced, Detailed-Balanced, and Weakly Reversible Systems

Crăciun¹,

Jin²,

Yu³

2020

SIAM J. Appl. Math.

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Very often, models in biology, chemistry, physics and engineering are systems of polynomial or power-law ordinary differential equations, arising from a reaction network. Such dynamical systems can be generated by many different reaction networks. On the other hand, networks with special properties (such as reversibility or weak reversibility) are known or conjectured to give rise to dynamical systems that have special properties: existence of positive steady states, persistence, permanence, and (for well-chosen parameters) complex balancing or detailed balancing. These last two are related to thermodynamic equilibrium, and therefore the positive steady states are unique and stable. We describe a computationally efficient characterization of polynomial or powerlaw dynamical systems that can be obtained as complex-balanced, detailed-balanced, weakly reversible, and reversible mass-action systems. redirect the reaction z → y * . Instead of the reaction z → y * with flux J z→y * , we have M reactions z → y j with fluxes J ′ z→y j = J y * →y j . Let (G ′ , J ′ ) denote this newest flux system. Recall that flux equivalence means Equation (11) holds at each vertex of G and G ′ . Here we only need to look at the vertex z to show that (G ′ , J ′ ) ∼ (G, J ). Note that y j − z = w j − w 0 . From P (G,J) (y * ) = 0, we also have M j=1 J y * →y j = J z→y * . Thus, the weighted sum of vectors coming out of z isFinally, we prove that the potentials are unchanged. Trivially P (G,J ) (y * ) = P (G ′ ,J ′ ) (y * ) = 0. Also P (G,J) (y j ) = J y * →y j = J ′ z→y j = P (G ′ ,J ′ ) (y j ) for j = 1, 2, . . . , M . Last but not least,J y * →y j = J z→y * = −P (G,J) (z).We have shown that the resulting flux system (G ′ , J ′ ) is flux equivalent to the original flux system (G, J ), and the potential at each vertex is preserved.Remark. In Lemma 4.3, the source vertex z may not be distinct from y j .

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Chemical reaction-diffusion networks: convergence of the method of lines

Cited by 7 publications

References 39 publications

Dynamics of Posttranslational Modification Systems: Recent Progress and Future Directions

Dynamics of Posttranslational Modification Systems: Recent Progress and Future Directions

Mathematical Analysis of Chemical Reaction Systems

An Efficient Characterization of Complex-Balanced, Detailed-Balanced, and Weakly Reversible Systems

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