On the Stability of Stochastic Jump Kinetics

Engblom, Stefan

doi:10.4236/am.2014.519300

Cited by 30 publications

(23 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This approach has been specialized to the case of stochastic reaction networks in [32,51]. Additional results on the stability of reaction networks have also been provided in [3,22,53]. We have the following general result that we specialize to our setup [49]:…”

Section: Ergodicity Analysis Of Stochastic Reaction Networkmentioning

confidence: 99%

Ergodicity analysis and antithetic integral control of a class of stochastic reaction networks with delays

Briat

Khammash

2018

Preprint

View full text Add to dashboard Cite

Delays are important phenomena arising in a wide variety of real world systems, including biological ones, because of diffusion/propagation effects or as simplifying modeling elements. We propose here to consider delayed stochastic reaction networks, a class of networks that has been relatively few studied until now. The difficulty in analyzing them resides in the fact that their state-space is infinite-dimensional. We demonstrate here that by restricting the delays to be phase-type distributed, one can represent the associated delayed reaction network as a reaction network with finitedimensional state-space. This can be achieved by suitably adding chemical species and reactions to the delay-free network following a simple algorithm which is fully characterized. Since phase-type distributions are dense in the set of probability distributions, they can approximate any distribution arbitrarily closely and this makes their consideration only a bit restrictive. As the state-space remains finite-dimensional, usual tools developed for non-delayed reaction network directly apply. In particular, we prove, for unimolecular mass-action reaction networks, that the delayed stochastic reaction network is ergodic if and only if the delay-free network is ergodic as well. Bimolecular reactions are more difficult to consider but slightly stronger analogous results are nevertheless obtained. These results demonstrate that delays have little to no harm to the ergodicity property of reaction networks as long as the delays are phase-type distributed, and this holds regardless the complexity of their distribution. We also prove that the presence of those delays adds convolution terms in the moment equation but does not change the value of the stationary means compared to the delay-free case. The covariance, however, is influenced by the presence of the delays. Finally, the control of a certain class of delayed stochastic reaction network using a delayed antithetic integral controller is considered. It is proven that this controller achieves its goal provided that the delay-free network satisfy the conditions of ergodicity and output-controllability. IntroductionDelays are omnipresent physical phenomena induced by memory, propagation or transport effects [12,24,30,42,50,52]. They naturally arise in population dynamics [28], ecology [29], epidemiology [19], biology [11,20,21,25,39] and engineering [12,30,43,52]. It is commonly understood that delays have, in general, detrimental effects in engineering as they may lead to instabilities such as oscillations. While this destabilizing effect is undesirable in this setting, their role can be crucial in biology when one wants, for instance, to design oscillators [11,48,58]. In the stochastic setting, delays have indeed been shown to be helpful for generating oscillations [11], but also to accelerate signaling [40] and to be responsible for an increase in intrinsic variability [55]. Delays can be easily incorporated in the dynamics of a deterministic reaction network by simply substituting del...

show abstract

Section: Ergodicity Analysis Of Stochastic Reaction Networkmentioning

confidence: 99%

Ergodicity analysis and antithetic integral control of a class of stochastic reaction networks with delays

Briat

Khammash

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…For the stochastic case (2.3), and in the non-spatial setting, an analysis in the form of assumptions and various a priori bounds has been developed previously [10]. We borrow many ideas from this work in what follows.…”

Section: 2mentioning

confidence: 99%

“…normalized such that min i w i = 1. Following [10] we formulate Assumption 2.2 (Reaction regularity). For a mesh M ∈ M consisting of voxel volumes (V j ) J j=1 we assume the density dependent scaling,…”

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

Pathwise Error Bounds in Multiscale Variable Splitting Methods for Spatial Stochastic Kinetics

Chevallier¹,

Engblom²

2018

SIAM J. Numer. Anal.

Self Cite

View full text Add to dashboard Cite

Stochastic computational models in the form of pure jump processes occur frequently in the description of chemical reactive processes, of ion channel dynamics, and of the spread of infections in populations. For spatially extended models, the computational complexity can be rather high such that approximate multiscale models are attractive alternatives. Within this framework some variables are described stochastically, while others are approximated with a macroscopic point value.We devise theoretical tools for analyzing the pathwise multiscale convergence of this type of variable splitting methods, aiming specifically at spatially extended models. Notably, the conditions we develop guarantee wellposedness of the approximations without requiring explicit assumptions of a priori bounded solutions. We are also able to quantify the effect of the different sources of errors, namely the multiscale error and the splitting error, respectively, by developing suitable error bounds. Computational experiments on selected problems serve to illustrate our findings.

show abstract

“…Conditions on the set of reaction channels can be imposed to ensure uniqueness [11] and to avoid explosions in finite time [37,26,21] 2.2. The Explicit-TL Approximation.…”

mentioning

confidence: 99%

Multilevel hybrid split-step implicit tau-leap

2016

View full text Add to dashboard Cite

Abstract. In biochemically reactive systems with small copy numbers of one or more reactant molecules, the dynamics is dominated by stochastic effects. To approximate those systems, discrete state-space and stochastic simulation approaches have been shown to be more relevant than continuous state-space and deterministic ones. In systems characterized by having simultaneously fast and slow timescales, existing discrete space-state stochastic path simulation methods, such as the stochastic simulation algorithm (SSA) and the explicit tau-leap (explicit-TL ) method, can be very slow. Implicit approximations have been developed to improve numerical stability and provide efficient simulation algorithms for those systems. Here, we propose an efficient Multilevel Monte Carlo (MLMC) method in the spirit of the work by Anderson and Higham (2012). This method uses split-step implicit tau-leap (SSI-TL ) at levels where the explicit-TL method is not applicable due to numerical stability issues. We present numerical examples that illustrate the performance of the proposed method.

show abstract

On the Stability of Stochastic Jump Kinetics

Cited by 30 publications

References 32 publications

Ergodicity analysis and antithetic integral control of a class of stochastic reaction networks with delays

Ergodicity analysis and antithetic integral control of a class of stochastic reaction networks with delays

Pathwise Error Bounds in Multiscale Variable Splitting Methods for Spatial Stochastic Kinetics

Multilevel hybrid split-step implicit tau-leap

Contact Info

Product

Resources

About