Numerical Approximation of Rare Event Probabilities in Biochemically Reacting Systems

Mikeev, Linar; Sandmann, Werner; Wolf, Verena

doi:10.1007/978-3-642-40708-6_2

Cited by 8 publications

(11 citation statements)

References 34 publications

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“…For other queries, alternative variance reduction techniques such as control variates are available [5]. Apart from sampling-based approaches, dynamic finite-state projections have been employed by Mikeev et al [34], but are lacking automated truncation schemes.…”

Section: Related Workmentioning

confidence: 99%

Analysis of Markov Jump Processes under Terminal Constraints

Backenköhler

Bortolussi

Großmann

et al. 2021

Tools and Algorithms for the Construction and Analysis of Systems

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Many probabilistic inference problems such as stochastic filtering or the computation of rare event probabilities require model analysis under initial and terminal constraints. We propose a solution to this bridging problem for the widely used class of population-structured Markov jump processes. The method is based on a state-space lumping scheme that aggregates states in a grid structure. The resulting approximate bridging distribution is used to iteratively refine relevant and truncate irrelevant parts of the state-space. This way, the algorithm learns a well-justified finite-state projection yielding guaranteed lower bounds for the system behavior under endpoint constraints. We demonstrate the method’s applicability to a wide range of problems such as Bayesian inference and the analysis of rare events.

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Section: Related Workmentioning

confidence: 99%

Analysis of Markov Jump Processes under Terminal Constraints

Backenköhler

Bortolussi

Großmann

et al. 2021

Tools and Algorithms for the Construction and Analysis of Systems

Self Cite

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“…For the expansion of state-space, Finite State Projection ( ) [20] and Sliding Windows ( ) [17] are used to find the domain. Whereas, methods like Krylov subspace [13] and Runge Kutta [21] are popularly used for approximation (of the series) of the CME Eq. (5).…”

Section: Initial Value Problemmentioning

confidence: 99%

“…In every expansion step, the domain is validated by Eq. (21) and new states are added in as long as:…”

Section: (C) Cease Of Criterion After Updatingmentioning

confidence: 99%

Novel domain expansion methods to improve the computational efficiency of the solution of Chemical Master Equation for large biological networks.

Kosarwal¹,

Kulasiri²,

Samarasinghe³

2020

Preprint

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Background: Numerical solutions of the chemical master equation (CME) are important to understand the stochasticity of biochemical systems. However, solving CMEs is a formidable task due to the nonlinear nature of the reactions and size of the networks that result in different realisations and, most importantly, the exponential growth of the size of the state-space with respect to the number of different species in the system. When the size of the biochemical system is very large in terms of the number of variables, the solution to the CME becomes intractable. Therefore, we introduce the intelligent state projection (𝐼𝑆𝑃) method to use in the stochastic analysis of these systems. For any biochemical reaction network, it is important to capture more than one moment to describe the dynamic behaviour of the system. 𝐼𝑆𝑃 is based on a state-space search and the data structure standards of artificial intelligence (𝐴𝐼) to explore and update the states of a biochemical system. To support the expansion in 𝐼𝑆𝑃, we also develop a Bayesian likelihood node projection (𝐵𝐿𝑁𝑃) function to predict the likelihood of the states. Results: To show the acceptability and effectiveness of our method, we apply the 𝐼𝑆𝑃 to several biological models previously discussed in the literature. According to the results of our computational experiments, we show that 𝐼𝑆𝑃 is effective in terms of speed and accuracy of the expansion, accuracy of the solution, and provides a better understanding of the state-space of the system in terms of blueprint patterns. Conclusions: The 𝐼𝑆𝑃 is the de-novo method to address the accuracy as well as the performance problems for the solution of the CME. It systematically expands the projection space based on predefined inputs, which are useful in providing accuracy in the approximation and an exact analytical solution at the time of interest. The 𝐼𝑆𝑃 was more effective in terms of predicting the behaviour of the state-space of the system and in performance management, which is a vital step towards modelling large biochemical systems.

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“…To address the state explosion problem, some approaches attempt to truncate the state space. For instance, [32] presents a method for selectively exploring states involving rare events; however, this technique requires the modification of parameters in the system to help guide this exploration. Other approaches attempt to dynamically explore the state space and continually add states until the resulting state space satisfies a desired level of precision [5,33,34].…”

Section: Related Workmentioning

confidence: 99%

Approximation Techniques for Stochastic Analysis of Biological Systems

Neupane

Zhang

Madsen

et al. 2019

Computational Biology

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There has been an increasing demand for formal methods in the design process of safety-critical synthetic genetic circuits. Probabilistic model checking techniques have demonstrated significant potential in analyzing the intrinsic probabilistic behaviors of complex genetic circuit designs. However, its inability to scale limits its applicability in practice. This chapter addresses the scalability problem by presenting a state-space approximation method to remove unlikely states resulting in a reduced, finite state representation of the infinite-state continuous-time Markov chain that is amenable to probabilistic model checking. The proposed method is evaluated on a design of a genetic toggle switch. Comparisons with another state-of-art tool demonstrates both accuracy and efficiency of the presented method.

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Numerical Approximation of Rare Event Probabilities in Biochemically Reacting Systems

Cited by 8 publications

References 34 publications

Analysis of Markov Jump Processes under Terminal Constraints

Analysis of Markov Jump Processes under Terminal Constraints

Novel domain expansion methods to improve the computational efficiency of the solution of Chemical Master Equation for large biological networks.

Approximation Techniques for Stochastic Analysis of Biological Systems

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