On three genetic repressilator topologies

Dukarić, Maša; Errami, Hassan; Jerala, Roman; Lebar, Tina; Romanovski, Valery G.; Tóth, János; Weber, Andreas

doi:10.1007/s11144-018-1519-5

Cited by 9 publications

(4 citation statements)

References 49 publications

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“…In this example we were lucky to receive a canonical realization with so many nice properties. The reader can easily verify that this is the case with the second three dimensional repressilator model and with the six dimensional one of [17]: the canonic realization is again reversible thus assures the existence of a positive stationary point.…”

Section: Consider the Polynomial Equationẋmentioning

confidence: 81%

“…with b, g, s > 0 (a transform of the first repressilator model in [17]). The canonical representation of this equation is shown in Figure 1.…”

Section: Consider the Polynomial Equationmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Consider the Polynomial Equationẋmentioning

confidence: 81%

“…with b, g, s > 0 (a transform of the first repressilator model in [17]). The canonical representation of this equation is shown in Figure 1.…”

Section: Consider the Polynomial Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Realizations of kinetic differential equations

Crăciun

Johnston

Szederkényi

et al. 2020

Mathematical Biosciences and Engineering

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“…A very recent paper [6] by Valenzuela et al is similar to ours, but they use the normal form theory, which is computationally not very efficient. (A few of our papers on models with non mass action-rational-kinetics are [7,8]. )…”

Section: Introductionmentioning

confidence: 99%

Two Nested Limit Cycles in Two-Species Reactions

2020

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We search for limit cycles in the dynamical model of two-species chemical reactions that contain seven reaction rate coefficients as parameters and at least one third-order reaction step, that is, the induced kinetic differential equation of the reaction is a planar cubic differential system. Symbolic calculations were carried out using the Mathematica computer algebra system, and it was also used for the numerical verifications to show the following facts: the kinetic differential equations of these reactions each have two limit cycles surrounding the stationary point of focus type in the positive quadrant. In the case of Model 1, the outer limit cycle is stable and the inner one is unstable, which appears in a supercritical Hopf bifurcation. Moreover, the oscillations in a neighborhood of the outer limit cycle are slow-fast oscillations. In the case of Model 2, the outer limit cycle is unstable and the inner one is stable. With another set of parameters, the outer limit cycle can be made stable and the inner one unstable.

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