A multi-time-scale analysis of chemical reaction networks: II. Stochastic systems

Kan, Xingye; Lee, Chang Hyeong; Othmer, Hans G.

doi:10.1007/s00285-016-0980-x

Cited by 23 publications

(19 citation statements)

References 46 publications

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“…Other approaches due to Haseltine and Rawlings [18] and Goutsias [19] model the state of the system using extents of reaction as opposed to molecules of species. A singular perturbation theory based method has also recently been used to obtain a reduced stochastic description [20].…”

Section: Introductionmentioning

confidence: 99%

Revisiting the reduction of stochastic models of genetic feedback loops with fast promoter switching

Holehouse

Grima

2019

Preprint

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Propensity functions of the Hill-type are commonly used to model transcriptional regulation in stochastic models of gene expression. This leads to an effective reduced master equation for the mRNA and protein dynamics only. Based on deterministic considerations, it is often stated or tacitly assumed that such models are valid in the limit of rapid promoter switching. Here, starting from the chemical master equation describing promoter-protein interactions, mRNA transcription, protein translation and decay, we prove that in the limit of fast promoter switching, the probability distribution of protein numbers is different than that given by standard stochastic models with Hill-type propensities. We show that the differences are pronounced whenever the protein-DNA binding rate is much larger than the unbinding rate, a special case of fast promoter switching. Furthermore we show using both theory and simulations that use of the standard stochastic models leads to drastically incorrect predictions for the switching properties of positive feedback loops and that these differences decrease with increasing mean protein burst size. Our results confirm that commonly used stochastic models of gene regulatory networks are only accurate in a subset of the parameter space consistent with rapid promoter switching.

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Section: Introductionmentioning

confidence: 99%

Revisiting the reduction of stochastic models of genetic feedback loops with fast promoter switching

Holehouse

Grima

2019

Preprint

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“…Property 2 suggests, but does not provide a definitive proof, that the distribution of X is Poissonian if ε = 0 or δ = 0. In the non-interaction case (δ = 0), the proof is straightforward: the dynamics of X is that of a simple immigrationand-death process, which is known to generate a Poisson distribution [47][48][49]. If X is stable (ε = 0), the distribution is again Poisson, but the proof requires a more subtle reasoning based on the Chemical Reaction Network Theory.…”

Section: The Statement Of the Model And Main Resultsmentioning

confidence: 99%

Buffering gene expression noise by microRNA based feedforward regulation

Bokes

Hojcka

Singh

2018

Preprint

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Cells use various regulatory motifs, including feedforward loops, to control the intrinsic noise that arises in gene expression at low copy numbers. Here we study one such system, which is broadly inspired by the interaction between an mRNA molecule and an antagonistic mi-croRNA molecule encoded by the same gene. The two reaction species are synchronously produced, individually degraded, and the second species (microRNA) exerts an antagonistic pressure on the first species (mRNA). Using linear-noise approximation, we show that the noise in the first species, which we quantify by the Fano factor, is sub-Poissonian, and exhibits a nonmonotonic response both to the species lifetime ratio and to the strength of the antagonistic interaction. Additionally, we use the Chemical Reaction Network Theory to prove that the first species distribution is Poissonian if the first species is much more stable than the second. Finally, we identify a special parametric regime, supporting a broad range of behaviour, in which the distribution can be analytically described in terms of the confluent hypergeometric limit function. We verify our analysis against large-scale kinetic Monte Carlo simulations. Our results indicate that, subject to specific physiological constraints, optimal parameter values can be found within the mRNA-microRNA motif that can benefit the cell by lowering the gene-expression noise.

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“…where y i L α i is the forward operator of the catalysed network R α i , while L β of the uncatalysed network R β . The forward operator from (25) is singularly perturbed, and, in what follows, we apply perturbation theory to exploit this fact [23,27]. Substituting the power series expansion p(x, y, τ ) = p 0 (x, y, τ ) + ε p 1 (x, y, τ ) + .…”

Section: Stochastic Analysismentioning

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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A multi-time-scale analysis of chemical reaction networks: II. Stochastic systems

Cited by 23 publications

References 46 publications

Revisiting the reduction of stochastic models of genetic feedback loops with fast promoter switching

Revisiting the reduction of stochastic models of genetic feedback loops with fast promoter switching

Buffering gene expression noise by microRNA based feedforward regulation

Noise-induced mixing and multimodality in reaction networks

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