On the role of frustration in excitable systems

Kaluza, Pablo; Meyer‐Ortmanns, Hildegard

doi:10.1063/1.3491342

Cited by 26 publications

(40 citation statements)

References 15 publications

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“…However, as we have pointed out, these three distinct regimes are only obtained in our realization if the binding and unbinding rates of genes are fast as compared to the other inherent time scales, here the decay rates of the fast (A) and the slow (B) proteins. For this case we derived deterministic equations equivalent to the former ones of [15]. As soon as the binding/unbinding rates are no longer small, but of the same order as the decay time of either proteins, the averaging procedure for deriving a deterministic limit has to be changed; the proteins see no longer average values of the gene states, but tend to follow the distinct states, unless their own production is too slow to reach the appropriate state "in time".…”

Section: Discussionmentioning

confidence: 99%

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Caveats in modeling a common motif in genetic circuits

et al. 2013

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Abstract. From a coarse-grained perspective the motif of a self-activating species, activating a second species which acts as its own repressor, is widely found in biological systems, in particular in genetic systems with inherent oscillatory behavior. Here we consider a specific realization of this motif as a genetic circuit, in which genes are described as directly producing proteins, leaving out the intermediate step of mRNA production. We focus on the effect that inherent time scales on the underlying finegrained scale can have on the bifurcation patterns on a coarser scale in time. Time scales are set by the binding and unbinding rates of the transcription factors to the promoter regions of the genes. Depending on the ratio of these rates to the decay times of the proteins, the appropriate averaging procedure for obtaining a coarse-grained description changes and leads to sets of deterministic equations, which differ in their bifurcation structure. In particular the desired intermediate range of regular limit cycles fades away when the binding rates of genes are of the same order or less than the decay time of at least one of the proteins. Our analysis illustrates that the common topology of the widely found motif alone does not necessarily imply universal features in the dynamics.

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Section: Discussionmentioning

confidence: 99%

“…(20) and (21) for the time evolution of N B to obtain 3.1.1. Comparison with the deterministic description of a bistable frustrated unit Let us first briefly compare the equations (24,25) with the deterministic equations formerly used to describe the bistable frustrated unit in [8,15] …”

Section: Fast Genesmentioning

confidence: 99%

Caveats in modeling a common motif in genetic circuits

et al. 2013

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“…The different fixed-point and oscillatory regimes of such an individual unit are separated by subcritical Hopf bifurcations (for a definition see for example Ref. [10]) with corresponding hysteresis effects [5].…”

Section: The Modelmentioning

confidence: 99%

“…As it was shown in [5], this motif shows three regimes, characterized by different attractors, when a single bifurcation parameter is monotonically increased: excitable behavior, where the system approaches a fixed point (if the perturbation from the fixed point exceeds a certain threshold, the system makes a long excursion in phase space before it returns to the fixed point), limit-cycle behavior, and again excitable behavior. The individual dynamics resembles FitzHugh-Nagumo units [6,7].…”

Section: Introductionmentioning

confidence: 99%

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Networks of coupled circuits: From a versatile toggle switch to collective coherent behavior

Labavić

Meyer‐Ortmanns

2014

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We study the versatile performance of networks of coupled circuits. Each of these circuits is composed of a positive and a negative feedback loop in a motif that is frequently found in genetic and neural networks. When two of these circuits are coupled with mutual repression, the system can function as a toggle switch. The variety of its states can be controlled by two parameters as we demonstrate by a detailed bifurcation analysis. In the bistable regimes switches between the coexisting attractors can be induced by noise. When we couple larger sets of these units, we numerically observe collective coherent modes of individual fixed-point and limit-cycle behavior. It is there the monotonic change of a single bifurcation parameter that allows to control the onset and arrest of the synchronized oscillations. This mechanism may play a role in biological applications, in particular in connection with the segmentation clock. While tuning the bifurcation parameter, also a variety of transient patterns emerges upon approaching the stationary states, in particular a selforganized pacemaker in a completely uniformly equipped ensemble, so that the symmetry breaking happens dynamically.From the physics' point of view, it is of generic interest how complex the dynamics of individual building blocks should be assumed in order to reproduce the versatile collective behavior of genetic or neural networks when these blocks are coupled. As a basic unit of our network of coupled circuits we consider the motif of a self-activating species A that activates its own repressor, a second species B. The species stand for the concentrations of genes, proteins or cells. An individual unit behaves like an excitable or oscillatory element, depending on the choice of parameters. So the individual dynamics is more complex than that of phase oscillators or repressilators, but as we later shall see, it allows for a simple control when these units are coupled. When two of these units mutually repress each other, we observe already a proliferation of possible attractors, differing by their amplitude and periods of oscillations, their pattern of synchronized phases, or their fixed point values. Such a system may act like a synthetic toggle switch whose multistable states can be addressed in a * email: d.labavic@jaocbs-university. The bifurcation parameter is related to the production rate of species A. This behavior allows for a simple control of the onset and arrest of oscillations via tuning this single parameter through two bifurcations. It is possible without any finetuning or external interference. Such a mechanism seems to be simple enough to be realized in biological applications like the segmentation clock. Moreover we observe an interesting phenomenon of dynamical symmetry breaking: a transient, self-organized pacemaker emerges in our uniformly equipped ensemble of oscillators. It emits target waves for some thousands of time units before the stationary state is reached. It is then instructive to perform a detailed bifurcation anal...

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Functional Complexity Based on Topology

Meyer‐Ortmanns

2013

Advances in Network Complexity

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On the role of frustration in excitable systems

Cited by 26 publications

References 15 publications

Caveats in modeling a common motif in genetic circuits

Caveats in modeling a common motif in genetic circuits

Networks of coupled circuits: From a versatile toggle switch to collective coherent behavior

Functional Complexity Based on Topology

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