Fast Adaptive Uniformization of the Chemical Master Equation

Didier, Frédéric; Henzinger, Thomas A.; Mateescu, Maria; Wolf, Verena

doi:10.1109/hibi.2009.23

Cited by 33 publications

(52 citation statements)

References 41 publications

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“…The efficiency of these methods has been greatly advanced in the last several years [5][6][7][8][9][10][11][12][13][14][15][16] . However numerical simulations are naturally limited to a specific choice of parameters, and changing the parameters requires a completely new calculation.…”

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confidence: 99%

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Analytic Methods for Modeling Stochastic Regulatory Networks

Walczak

Mugler

Wiggins

2012

Methods in Molecular Biology

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The past decade has seen a revived interest in the unavoidable or intrinsic noise in biochemical and genetic networks arising from the finite copy number of the participating species. That is, rather than modeling regulatory networks in terms of the deterministic dynamics of concentrations, we model the dynamics of the probability of a given copy number of the reactants in single cells. Most of the modeling activity of the last decade has centered on stochastic simulation of individual realizations, i.e., Monte-Carlo methods for generating stochastic time series. Here we review the mathematical description in terms of probability distributions, introducing the relevant derivations and illustrating several cases for which analytic progress can be made either instead of or before turning to numerical computation. Contents

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confidence: 99%

“…Naturally, we are not able to cover all the developments in the field, but we hope to give the reader a useful starting point. For concreteness we also limit our discussion to the case of small gene networks and do not discuss approximations used to describe larger networks, which is currently an active area of research in many communities [10,11,15,16,28,29].…”

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confidence: 99%

Analytic Methods for Modeling Stochastic Regulatory Networks

Walczak

Mugler

Wiggins

2012

Methods in Molecular Biology

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“…If the variances of the state variables remain small, however, one can exploit that only a tractable number of states have "significant" probability, that is, only relatively few states have a probability that is greater than a small threshold. Here, we present a method based on our previous work [8] for efficiently approximating the solution of Eq. (5).…”

Section: Dynamical State Space Truncationmentioning

confidence: 99%

“…(5). Many transient solution approaches can be applied for this purpose (see, for instance, [8]). Here, we use an approximation based on numerical integration of Eq.…”

Section: Dynamical State Space Truncationmentioning

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Efficient Calculation of Rare Event Probabilities in Markovian Queueing Networks

Mikeev¹,

Sandmann²,

Wolf³

2011

Proceedings of the 5th International ICST Conference on Performance Evaluation Methodologies and Tools

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We address the computation of rare event probabilities in Markovian queueing networks with huge or possibly even infinite state spaces. For this purpose, we incorporate ideas from importance sampling simulations into a non-simulative numerical method that approximates transient probabilities based on a dynamical truncation of the state space. A change of measure technique is applied in order to accomplish a guided state space exploration. Numerical results for three different example networks demonstrate the efficiency and accuracy of our method.

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Bounding the equilibrium distribution of Markov population models

Dayar

Hermanns

Spieler

et al. 2011

Numerical Linear Algebra App

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SUMMARYWe propose a bounding technique for the equilibrium probability distribution of continuous-time Markov chains with population structure and infinite state space. We use Lyapunov functions to determine a finite set of states that contains most of the equilibrium probability mass. Then we apply a refinement scheme based on stochastic complementation to derive lower and upper bounds on the equilibrium probability for each state within that set. To show the usefulness of our approach, we present experimental results for several examples from biology.

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Fast Adaptive Uniformization of the Chemical Master Equation

Abstract: Abstract-WithinIn this paper we present an on-the-fly variant of AU, where we improve the original algorithm for AU at the cost of a small approximation error. By means of several examples, we show that our approach is particularly well-suited for biochemical reaction networks.

Cited by 33 publications

References 41 publications

Analytic Methods for Modeling Stochastic Regulatory Networks

Analytic Methods for Modeling Stochastic Regulatory Networks

Efficient Calculation of Rare Event Probabilities in Markovian Queueing Networks

Bounding the equilibrium distribution of Markov population models

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