Gaussian approximations for stochastic systems with delay: Chemical Langevin equation and application to a Brusselator system

Brett, Tobias; Galla, Tobias

doi:10.1063/1.4867786

Cited by 21 publications

(29 citation statements)

References 58 publications

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“…Our theoretical approach to analysing noise-induced cycles (which is similar to that found in [50][51][52] and detailed in the Supplement) allows us to study the amplification, synchronisation and coherence of these cycles, as discussed in the following sections. Fig.…”

Section: Resultsmentioning

confidence: 99%

“…This result can be compared with the path integral for an SDE of an appropriate form. The corresponding path integral expression for the general SDE dn α j,t dt = F α j,t (n t ) + ξ α j,t , ξ α j,t ξ α j ,t = Σ αα j,j ,t,t , (S11) is given by [50][51][52] P (n, t) = 1 N   j,α t t =0 dq α j,t dp α j,t 2π   δ 1 N n − q (t) e i j,α,t p α j,t (q α j,t +∆ −q α j,t )…”

Section: A Path-integral Approachmentioning

confidence: 99%

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Intrinsic noise, Delta-Notch signalling and delayed reactions promote sustained, coherent, synchronized oscillations in the presomitic mesoderm

Baron

Galla

2019

J. R. Soc. Interface.

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Using a stochastic individual-based modelling approach, we examine the role that Delta-Notch signalling plays in the regulation of a robust and reliable somite segmentation clock. We find that not only can Delta-Notch signalling synchronize noisy cycles of gene expression in adjacent cells in the presomitic mesoderm (as is known), but it can also amplify and increase the coherence of these cycles. We examine some of the shortcomings of deterministic approaches to modelling these cycles and demonstrate how intrinsic noise can play an active role in promoting sustained oscillations, giving rise to noise-induced quasi-cycles. Finally, we explore how translational/transcriptional delays can result in the cycles in neighbouring cells oscillating in anti-phase and we study how this effect relates to the propagation of noise-induced stochastic waves.

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Section: Resultsmentioning

confidence: 99%

Section: A Path-integral Approachmentioning

confidence: 99%

Intrinsic noise, Delta-Notch signalling and delayed reactions promote sustained, coherent, synchronized oscillations in the presomitic mesoderm

Baron

Galla

2019

J. R. Soc. Interface.

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“…These are subject to the limitations outlined above in that they do not capture the effects of stochasticity. It is only recently that analytical (and also numerical) techniques for individual-based models subject to both intrinsic noise and delay have been developed [32][33][34][35][36][37][38][39][40][41][42]. In previous work we have derived Gaussian approximations of such dynamics, and we have shown that these can capture the effects of delay reactions [40,41].…”

Section: Introductionmentioning

confidence: 99%

“…It is only recently that analytical (and also numerical) techniques for individual-based models subject to both intrinsic noise and delay have been developed [32][33][34][35][36][37][38][39][40][41][42]. In previous work we have derived Gaussian approximations of such dynamics, and we have shown that these can capture the effects of delay reactions [40,41]. This analysis is conveniently carried out in terms of generating functionals [43,44], arXiv:1505.05100v1 [cond-mat.stat-mech] 19 May 2015 and the equivalent of the linear-noise approximation (LNA) for delay systems can be derived.…”

Section: Introductionmentioning

confidence: 99%

Generating functionals and Gaussian approximations for interruptible delay reactions

Brett

Galla

2015

Phys. Rev. E

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We develop a generating functional description of the dynamics of non-Markovian individual-based systems in which delay reactions can be terminated before completion. This generalizes previous work in which a path-integral approach was applied to dynamics in which delay reactions complete with certainty. We construct a more widely applicable theory, and from it we derive Gaussian approximations of the dynamics, valid in the limit of large, but finite, population sizes. As an application of our theory we study predator-prey models with delay dynamics due to gestation or lag periods to reach the reproductive age. In particular, we focus on the effects of delay on noise-induced cycles.

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“…In order to be able to describe both aperiodic and oscillatory dynamics we account for intrinsic stochasticity that is typically associated with rate-processes 44,49 . The model is implemented using delayed Chemical Langevin equations of the form 65,66…”

Section: Stochastic Model Of Hes5 Expression Dynamicsmentioning

confidence: 99%

Quantitative single-cell live imaging links HES5 dynamics with cell-state and fate in murine neurogenesis

Manning

Biga

Boyd

et al. 2018

Preprint

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During embryogenesis cells make fate decisions within complex tissue environments. The levels and dynamics of transcription factor expression regulate these decisions.Here we use single cell live imaging of an endogenous HES5 reporter and absolute protein quantification to gain a dynamic view of neurogenesis in the embryonic mammalian spinal cord. We report that dividing neural progenitors show both aperiodic and periodic HES5 protein fluctuations. Mathematical modelling suggests that in progenitor cells the HES5 oscillator operates close to its bifurcation boundary where stochastic conversions between dynamics are possible. HES5 expression becomes more frequently periodic as cells transition to differentiation which, coupled with an overall decline in HES5 expression, creates a transient period of oscillations with higher fold expression change. This increases the decoding capacity of HES5 oscillations and correlates with interneuron versus motor neuron cell fate. Thus, HES5 undergoes complex changes in gene expression dynamics as cells differentiate.

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Gaussian approximations for stochastic systems with delay: Chemical Langevin equation and application to a Brusselator system

Cited by 21 publications

References 58 publications

Intrinsic noise, Delta-Notch signalling and delayed reactions promote sustained, coherent, synchronized oscillations in the presomitic mesoderm

Intrinsic noise, Delta-Notch signalling and delayed reactions promote sustained, coherent, synchronized oscillations in the presomitic mesoderm

Generating functionals and Gaussian approximations for interruptible delay reactions

Quantitative single-cell live imaging links HES5 dynamics with cell-state and fate in murine neurogenesis

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