On Reduced Models for the Chemical Master Equation

Jahnke, Tobias

doi:10.1137/110821500

Cited by 66 publications

(72 citation statements)

References 32 publications

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“…Since it can be shown with standard arguments that p(t, n, m) ≥ 0 (cf., e.g., Section 2.4 in [28]), this shows that p [M ] (t, ·, ·) is a probability distribution at any time t ≥ 0.…”

Section: Markov Process and Chemical Master Equationmentioning

confidence: 79%

Error Bound for Piecewise Deterministic Processes Modeling Stochastic Reaction Systems

Jahnke¹,

Kreim²

2012

Multiscale Model. Simul.

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Biological processes involving the random interaction of d species with integer particle numbers are often modeled by a Markov jump process on N d 0 . Realizations of this process can, in principle, be generated with the classical stochastic simulation algorithm proposed in [19], but for very reactive systems this method is usually inefficient. Hybrid models based on piecewise deterministic processes offer an attractive alternative which can decrease the simulation time considerably in applications where species with rather low particle numbers interact with very abundant species. We investigate the convergence of the hybrid model to the original one for a class of reaction systems with two distinct scales. Our main result is an error bound which states that, under suitable assumptions, the hybrid model approximates the marginal distribution of the discrete species and the conditional moments of the continuous species up to an error of O`M −1´w here M is the scaling parameter of the partial thermodynamic limit.

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“…Since it can be shown with standard arguments that p(t, n, m) ≥ 0 (cf., e.g., Section 2.4 in [28]), this shows that p [M ] (t, ·, ·) is a probability distribution at any time t ≥ 0.…”

Section: Markov Process and Chemical Master Equationmentioning

confidence: 79%

Error Bound for Piecewise Deterministic Processes Modeling Stochastic Reaction Systems

Jahnke¹,

Kreim²

2012

Multiscale Model. Simul.

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“…From a hybrid systems perspective, an appealing approach is to group the interacting species into highly and lowly abundant species and to model them using continuous deterministic and discrete stochastic dynamics, respectively. Methods to analyze such hybrid models have been developed recently (Jahnke, 2011), but their use for parameter inference or optimal control has so far not been documented.…”

Section: Discussionmentioning

confidence: 99%

Bayesian inference for stochastic individual-based models of ecological systems: a pest control simulation study

Parise

Lygeros

Ruess

2015

Front. Environ. Sci.

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Mathematical models are of fundamental importance in the understanding of complex population dynamics. For instance, they can be used to predict the population evolution starting from different initial conditions or to test how a system responds to external perturbations. For this analysis to be meaningful in real applications, however, it is of paramount importance to choose an appropriate model structure and to infer the model parameters from measured data. While many parameter inference methods are available for models based on deterministic ordinary differential equations, the same does not hold for more detailed individual-based models. Here we consider, in particular, stochastic models in which the time evolution of the species abundances is described by a continuous-time Markov chain. These models are governed by a master equation that is typically difficult to solve. Consequently, traditional inference methods that rely on iterative evaluation of parameter likelihoods are computationally intractable. The aim of this paper is to present recent advances in parameter inference for continuous-time Markov chain models, based on a moment closure approximation of the parameter likelihood, and to investigate how these results can help in understanding, and ultimately controlling, complex systems in ecology. Specifically, we illustrate through an agricultural pest case study how parameters of a stochastic individual-based model can be identified from measured data and how the resulting model can be used to solve an optimal control problem in a stochastic setting. In particular, we show how the matter of determining the optimal combination of two different pest control methods can be formulated as a chance constrained optimization problem where the control action is modeled as a state reset, leading to a hybrid system formulation.

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“…1 However, computing the time evolution of the probability distribution for such models is typically challenging because the CME cannot be solved exactly, 2,3 except in some special cases. [4][5][6] An approach that has recently gained popularity is based on deriving systems of differential equations from the CME that only describe the time evolution of momentsup to some desired order L-of the underlying probability distribution.…”

Section: Introductionmentioning

confidence: 99%

Minimal moment equations for stochastic models of biochemical reaction networks with partially finite state space

Ruess¹

2015

The Journal of Chemical Physics

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Many stochastic models of biochemical reaction networks contain some chemical species for which the number of molecules that are present in the system can only be finite (for instance due to conservation laws), but also other species that can be present in arbitrarily large amounts. The prime example of such networks are models of gene expression, which typically contain a small and finite number of possible states for the promoter but an infinite number of possible states for the amount of mRNA and protein. One of the main approaches to analyze such models is through the use of equations for the time evolution of moments of the chemical species. Recently, a new approach based on conditional moments of the species with infinite state space given all the different possible states of the finite species has been proposed. It was argued that this approach allows one to capture more details about the full underlying probability distribution with a smaller number of equations. Here, I show that the result that less moments provide more information can only stem from an unnecessarily complicated description of the system in the classical formulation. The foundation of this argument will be the derivation of moment equations that describe the complete probability distribution over the finite state space but only low-order moments over the infinite state space. I will show that the number of equations that is needed is always less than what was previously claimed and always less than the number of conditional moment equations up to the same order. To support these arguments, a symbolic algorithm is provided that can be used to derive minimal systems of unconditional moment equations for models with partially finite state space.

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On Reduced Models for the Chemical Master Equation

Cited by 66 publications

References 32 publications

Error Bound for Piecewise Deterministic Processes Modeling Stochastic Reaction Systems

Error Bound for Piecewise Deterministic Processes Modeling Stochastic Reaction Systems

Bayesian inference for stochastic individual-based models of ecological systems: a pest control simulation study

Minimal moment equations for stochastic models of biochemical reaction networks with partially finite state space

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