Polynomial time algorithms to determine weakly reversible realizations of chemical reaction networks

Rudan, János; Szederkényi, Gábor; Hangos, Katalin M.; Péni, Tamás

doi:10.1007/s10910-014-0318-0

Cited by 12 publications

(13 citation statements)

References 20 publications

(39 reference statements)

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“…Ultimately dynamical equivalence and flux equivalence are linear feasibility problems. There exist algorithms based on linear programming that search for dynamically equivalent realizations with certain properties, e.g., complex-balanced or detailed-balanced [36]; weakly reversible or reversible [32]; with minimal deficiency [26,27]. An implementation of some of these algorithms is available as a MATLAB toolbox [35].…”

Section: Dynamical Equivalencementioning

confidence: 99%

Single-target networks

Crăciun¹,

Jin²,

Yu³

2022

DCDS-B

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Reaction networks can be regarded as finite oriented graphs embedded in Euclidean space. Single-target networks are reaction networks with an arbitrarily set of source vertices, but only one sink vertex. We completely characterize the dynamics of all mass-action systems generated by single-target networks, as follows: either (i) the system is globally stable for all choice of rate constants (in fact, is dynamically equivalent to a detailed-balanced system with a single linkage class), or (ii) the system has no positive steady states for any choice of rate constants and all trajectories must converge to the boundary of the positive orthant or to infinity. Moreover, we show that global stability occurs if and only if the target vertex of the network is in the relative interior of the convex hull of the source vertices.

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Section: Dynamical Equivalencementioning

confidence: 99%

Single-target networks

Crăciun¹,

Jin²,

Yu³

2022

DCDS-B

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“…We will explore the relationship between finding weakly reversible and complex balanced realizations in Section 6. More efficient methods to compute weakly reversible realizations are available; for example, a polynomial time algorithm can be found in [38].…”

Section: Review Of Algorithmsmentioning

confidence: 99%

Realizations of kinetic differential equations

Crăciun

Johnston

Szederkényi

et al. 2020

Mathematical Biosciences and Engineering

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The induced kinetic differential equations of a reaction network endowed with mass action type kinetics is a system of polynomial differential equations. The problem studied here is: Given a system of polynomial differential equations, is it possible to find a network which induces these equations; in other words: is it possible to find a kinetic realization of this system of differential equations? If yes, can we find a network with some chemically relevant properties (implying also important dynamic consequences), such as reversibility, weak reversibility, zero deficiency, detailed balancing, complex balancing, mass conservation, etc.? The constructive answers presented to a series of questions of the above type are useful when fitting differential equations to datasets, or when trying to find out the dynamic behavior of the solutions of differential equations. It turns out that some of these results can be applied when trying to solve seemingly unrelated mathematical problems, like the existence of positive solutions to algebraic equations.2. There is an internal requirement within this branch of science: One should like to know as much as possible about the structure of differential equations that arise from modelling a chemical system.3. Given a system of polynomial differential equations in any field of pure or applied mathematics, one may wish to have statements on stability or oscillations, similar to those offered by the Horn-Jackson Theorem [26], Zero Deficiency Theorem [21], Volpert's theorem [50], or the Global Attractor Conjecture, where several cases have been proven [3,12,23,35] 1 . Then it comes in handy to see that the system of differential equations of interest belongs to a well behaving class.4. Lastly, results of formal reaction kinetics (to use an expression introduced in [5, 6]), e.g. on the existence of stationary points, may offer alternative methods for solving problems in algebraic geometry [17,32]. For instance, one might be able to show the existence of positive roots of a polynomial if the system of polynomial equations is known to be the right hand side of the induced kinetic differential equation of a reversible or weakly reversible reaction network [7].The structure of our paper is as follows. Section 2 introduces the essential concepts of reaction networks and mass action systems. Section 3 formulates the problem we are interested in, that of realizability of kinetic differential equations. Section 4 treats two special cases: finding realizations for compartmental models (defined later) and weakly reversible networks. Section 5 focuses on the general problem of realizability. We first review existing algorithms available. Then we outline several procedures that modify a reaction network while preserving the system of differential equations, including adding and removing vertices from the reaction graph. Section 6 explores the relation between weakly reversible and complex balanced realizations. Here we work with families of symbolic kinetic differential equations. In Section 7, w...

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“…Once the set of vertices is fixed, finding realizations (satisfying certain constraints like weak reversibility, minimal deficiency, or complex-balancing) is relatively simple, and algorithms based on optimization techniques exist. For example see [18,20], or [22] for a MATLAB implementation.…”

Section: Network Structure Of Wr 0 Realizationsmentioning

confidence: 99%

Uniqueness of weakly reversible and deficiency zero realizations of dynamical systems

Crăciun¹,

Jin²,

Yu³

2020

Preprint

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In general, if a dynamical system is generated by some reaction network via mass-action kinetics, then it can also be generated by many other reaction networks. Here we show that if a dynamical system is generated by a weakly reversible network that has deficiency equal to zero, then this network must be unique. Moreover, we show that both of these hypotheses (i.e., weak reversibility and deficiency zero) are necessary for uniqueness.

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Polynomial time algorithms to determine weakly reversible realizations of chemical reaction networks

Cited by 12 publications

References 20 publications

Single-target networks

Single-target networks

Realizations of kinetic differential equations

Uniqueness of weakly reversible and deficiency zero realizations of dynamical systems

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