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

Chevallier, Augustin; Engblom, Stefan

doi:10.1137/16m1083086

Cited by 5 publications

(7 citation statements)

References 24 publications

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“…In order to present our general results, we formalize the multiscale reduction performed on the networks (2.1) and (3.2). Similar procedures can be found in many other studies [8,21]. The first step is to decompose S by writing S = (S (1) , S (2) ) with S (1) = (S 1 , .…”

Section: Robustness and Deficiencymentioning

confidence: 93%

“…The assumption that all species in the system are roughly of equal abundance breaks down in many biochemical systems of interest. This has motived the development of multiscale approximations where only a fraction of the species are taken to evolve continuously [10,8,27]. Below we present two examples of multiscale networks with very different behavior, while a more general derivation of the multiscale approximation is provided in section 4.…”

Section: Examples Of Multiscale Dynamicsmentioning

confidence: 99%

See 1 more Smart Citation

Robustness of Stochastic Chemical Reaction Networks to Extrinsic Noise: The Role of Deficiency

Levien¹,

Bressloff²

2018

Multiscale Model. Simul.

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The biochemical systems inside a living cell are able to reliably perform complex tasks while subjected to various sources of noise. In this study we consider stochastic models of biochemical networks evolving in the presence of dynamic random environments. These environments are themselves modeled as chemical reaction networks so that the full system can be viewed as a multiscale chemical reaction network. The multiscale structure arises from the fact that the environment and the internal system may operate on different timescales. While previous results in chemical reaction network theory have established that certain dynamic behavior can be ruled out when a topological parameter, known as the network deficiency, is zero, these results fail to capture the behavior that can be observed in multiscale networks. We demonstrate that the deficiency of the network has implications for how robust it is to environmental noise. We then show how our results can be used to prove that correlations in a population of chemical reaction networks in a random environment vanish given certain topological constraints.

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Section: Robustness and Deficiencymentioning

confidence: 93%

Section: Examples Of Multiscale Dynamicsmentioning

confidence: 99%

Robustness of Stochastic Chemical Reaction Networks to Extrinsic Noise: The Role of Deficiency

Levien¹,

Bressloff²

2018

Multiscale Model. Simul.

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“…The global step is achieved separately by solving the connected ODE in (3.6), and is usually quite fast. The approximation (3.5)-(3.6) can be understood as a split-step method and may also be analyzed as such [6,13]. The order of the approximation can then be expected to be 1/2 in the root mean-square sense,…”

Section: Numerical Coupling Of Firing Processesmentioning

confidence: 99%

“…The approximation (3.5)-(3.6) can be understood as a split-step method and may also be analyzed as such [6,13]. The order of the approximation can then be expected to be 1/2 in the root mean-square sense,…”

Section: Numerical Coupling Of Firing Processesmentioning

confidence: 99%

Multiscale modelling via split-step methods in neural firing

Bauer

Engblom

Mikulovic

et al. 2018

Mathematical and Computer Modelling of Dynamical Systems

Self Cite

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Neuronal models based on the Hodgkin-Huxley equation form a fundamental framework in the field of computational neuroscience. While the neuronal state is often modelled deterministically, experimental recordings show stochastic fluctuations, presumably driven by molecular noise from the underlying microphysical conditions. In turn, the firing of individual neurons gives rise to an electric field in extracellular space, also thought to affect the firing pattern of nearby neurons.We develop a multiscale model which combines a stochastic ion channel gating process taking place on the neuronal membrane, together with the propagation of an action potential along the neuronal structure. We also devise a numerical method relying on a split-step strategy which effectively couples these two processes and we experimentally test the feasibility of this approach. We finally also explain how the approach can be extended with Maxwell's equations to allow the potential to be propagated in extracellular space. ARTICLE HISTORY

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“…Therefore, different hybrid methods that couple the stochastic and deterministic modeling approaches are needed. In general, hybrid methods separate reactions and/or species into different groups of reactions and/or species, and they use the diffusion or the deterministic modeling approach to describe the dynamics of fast reactions and/or species with high copy numbers, while Markov chain representation is utilized for slow reactions and/or species with low copy numbers [6,7,8,10,13,27].…”

Section: Introductionmentioning

confidence: 99%

Hybrid master equation for jump-diffusion approximation of biomolecular reaction networks

Altintan

Koeppl

2019

Bit Numer Math

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Cellular reactions have multi-scale nature in the sense that the abundance of molecular species and the magnitude of reaction rates can vary in a wide range. This diversity leads to hybrid models that combine deterministic and stochastic modeling approaches. To reveal this multi-scale nature, we proposed jump-diffusion approximation in a previous study. The key idea behind the model was to partition reactions into fast and slow groups, and then to combine Markov chain updating scheme for the slow set with diffusion (Langevin) approach updating scheme for the fast set. Then, the state vector of the model was defined as the summation of the random time change model and the solution of the Langevin equation. In this study, we have proved that the joint probability density function of the jump-diffusion approximation over the reaction counting process satisfies the hybrid master equation, which is the summation of the chemical master equation and the Fokker-Planck equation. To solve the hybrid master equation, we propose an algorithm using the moments of reaction counters of fast reactions given the reaction counters of slow reactions. Then, we solve a constrained optimization problem for each conditional probability density at the time point of interest utilizing the maximum entropy approach. Based on the multiplication rule for joint probability density functions, we construct the solution of the hybrid master equation. To show the efficiency of the method, we implement it to a canonical model of gene regulation.

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

Cited by 5 publications

References 24 publications

Robustness of Stochastic Chemical Reaction Networks to Extrinsic Noise: The Role of Deficiency

Robustness of Stochastic Chemical Reaction Networks to Extrinsic Noise: The Role of Deficiency

Multiscale modelling via split-step methods in neural firing

Hybrid master equation for jump-diffusion approximation of biomolecular reaction networks

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