Test Models for Statistical Inference: Two-Dimensional Reaction Systems Displaying Limit Cycle Bifurcations and Bistability

Plesa, Tomislav; Vejchodský, Tomáš; Erban, Radek

doi:10.1007/978-3-319-62627-7_1

Cited by 15 publications

(40 citation statements)

References 42 publications

(167 reference statements)

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“…It was established in [ 20 ] that, for particular choices of the rate coefficients, the deterministic model of reaction network (3.1), given as equation (S34) in the electronic supplementary material, exhibits exotic dynamics: it undergoes a homoclinic bifurcation, and displays a bistability involving a limit cycle and an equilibrium point. On the other hand, it is demonstrated in [ 21 ] that the stochastic model of (3.1) is not necessarily sensitive to the deterministic bifurcation, and may effectively behave in a monostable manner. The latter point is demonstrated in figure 2 c , where we show in red numerically approximated x 1 -solutions of equation (S34) from the electronic supplementary material, one initiated in the region of attraction of the equilibrium point, while the other of the limit cycle.…”

Section: A Two-species Exotic Systemmentioning

confidence: 99%

“…The algorithm may play a significant role in the biochemical network synthesis, allowing for a deterministic–stochastic hybrid approach. More precisely, when constructing abstract and physical networks, one may use the deterministic model to guide the construction [ 20 , 21 ], and then apply the algorithm to favourably modify the intrinsic noise in the stochastic model, while preserving the desired deterministic dynamics. The algorithm may also be used to adjust the intrinsic noise to favourably interact with environment-induced effects (e.g.…”

Section: Introductionmentioning

confidence: 99%

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Noise control for molecular computing

Plesa

Zygalakis

Anderson

et al. 2018

J. R. Soc. Interface.

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Synthetic biology is a growing interdisciplinary field, with far-reaching applications, which aims to design biochemical systems that behave in a desired manner. With the advancement in nucleic-acid-based technology in general, and strand-displacement DNA computing in particular, a large class of abstract biochemical networks may be physically realized using nucleic acids. Methods for systematic design of the abstract systems with prescribed behaviours have been predominantly developed at the (less-detailed) deterministic level. However, stochastic effects, neglected at the deterministic level, are increasingly found to play an important role in biochemistry. In such circumstances, methods for controlling the intrinsic noise in the system are necessary for a successful network design at the (more-detailed) stochastic level. To bridge the gap, the noise-control algorithm for designing biochemical networks is developed in this paper. The algorithm structurally modifies any given reaction network under mass-action kinetics, in such a way that (i) controllable state-dependent noise is introduced into the stochastic dynamics, while (ii) the deterministic dynamics are preserved. The capabilities of the algorithm are demonstrated on a production–decay reaction system, and on an exotic system displaying bistability. For the production–decay system, it is shown that the algorithm may be used to redesign the network to achieve noise-induced multistability. For the exotic system, the algorithm is used to redesign the network to control the stochastic switching, and achieve noise-induced oscillations.

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Section: A Two-species Exotic Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Noise control for molecular computing

Plesa

Zygalakis

Anderson

et al. 2018

J. R. Soc. Interface.

Self Cite

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“…where y * (γ) is the unique normalized equilibrium obtained by setting the deterministic conservation constant to M = 1 in (20). Substituting (31) and (33) into (29), one finally obtains p 0 (x) = y∈π N n N ! (y * (γ)) y y!…”

Section: Stochastic Analysismentioning

confidence: 99%

“…In this case, in contrast to Section 5.1, the auxiliary PMFs are not Poissonians (more generally, Theorem 2.2 is not applicable). The resulting fast-slow network displays an arbitrary number of noisy limit cycles (known as stochastic multicyclicity [29]), and may illustrate the kind of stochastic dynamics arising when a gene produces a protein whose concentration oscillates in time.…”

Section: Stochastic Multicyclicitymentioning

confidence: 99%

Noise-induced mixing and multimodality in reaction networks

2018

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We analyze a class of chemical reaction networks under mass-action kinetics and involving multiple time-scales, whose deterministic and stochastic models display qualitative differences. The networks are inspired by gene-regulatory networks, and consist of a slow-subnetwork, describing conversions among the different gene states, and fast-subnetworks, describing biochemical interactions involving the gene products. We show that the long-term dynamics of such networks can consist of a unique attractor at the deterministic level (unistability), while the long-term probability distribution at the stochastic level may display multiple maxima (multimodality). The dynamical differences stem from a novel phenomenon we call noise-induced mixing, whereby the probability distribution of the gene products is a linear combination of the probability distributions of the fast-subnetworks which are 'mixed' by the slow-subnetworks. The results are applied in the context of systems biology, where noise-induced mixing is shown to play a biochemically important role, producing phenomena such as stochastic multimodality and oscillations.In this section, chemical reaction networks are defined [14,17,18,19], which are used to model the biochemical processes considered in this paper, together with their deterministic and stochastic dynamical models. We begin with some notation.Definition 2.1 Set R is the space of real numbers, R ≥ the space of nonnegative real numbers, and R > the space of positive real numbers. Similarly, Z is the space of integer numbers, Z ≥ the space of nonnegative integer numbers, and Z > the space of positive integer numbers. Euclidean vectors are denoted in boldface, x = (x 1 , x 2 , . . . , x m ) ∈ R m . The support of x is defined by supp(x) = {i ∈ {1, 2, . . . , m}|x i = 0}. Given a finite set S, we denote its cardinality by |S|.Definition 2.2 A chemical reaction network is a triple {S, C, R}, where (i) S = {S 1 , S 2 , . . . , S m } is the set of species of the network.

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“…In 13 , we studied the system (1.2) with all parameters not zeros and proved the uniqueness of limit cycle. There are many articles in the field of limit cycles and homoclinic orbits for example see 20,21,22 .…”

Section: R I mentioning

confidence: 99%