Efficient computation of transient solutions of the chemical master equation based on uniformization and quasi-Monte Carlo

Hellander, Andreas

doi:10.1063/1.2897976

Cited by 16 publications

(22 citation statements)

References 28 publications

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“…Such a decomposition is called a uniformization or randomization and was first proposed by Jensen [224]. The series (16) can be evaluated via numerically stable algorithms and truncation errors can be bounded [225,226]. Nevertheless, the uniformization method requires the computation of the powers of a matrix having as many rows and columns as the system has states.…”

Section: Analytical and Numerical Methods For The Solution Of Master mentioning

confidence: 99%

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Master equations and the theory of stochastic path integrals

2017

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Abstract. This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of masters equations most often rely on lownoise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a "generating functional", which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a "forward" and a "backward" path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon expanding the forward and the backward path integrals around stationary paths, we then discuss and extend a recent method for the computation of rare event probabilities. Besides, we also derive path integral representations for processes with continuous state spaces whose forward and backward master equations admit Kramers-Moyal expansions. A truncation of the backward expansion at the level of a diffusion approximation recovers a classic path integral representation of the (backward) Fokker-Planck equation. One can rewrite this path integral in terms of an Onsager-Machlup function and, for purely diffusive Brownian motion, it simplifies to the path integral of Wiener. To make this review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory and do not assume any knowledge of the latter. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit. arXiv:1609.02849v2 [cond-mat...

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Section: Analytical and Numerical Methods For The Solution Of Master mentioning

confidence: 99%

“…Consequently, a numerical implementation of the method is only feasible for sufficiently small state spaces. Further information on the method and on its improvements can be found in [224][225][226][227][228][229].…”

Section: Analytical and Numerical Methods For The Solution Of Master mentioning

confidence: 99%

Master equations and the theory of stochastic path integrals

2017

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“…Numerical solution algorithms for the CME are usually based on matrix descriptions of the discrete-state Markov process [47]; anyway, these methods are computationally expensive and not always feasible, especially for systems consisting of many molecular species, for which the number of reachable states is huge or even (countably) infinite. Several analytical solution algorithms for the CME exist, for instance those based on uniformization methods [48]–[50], finite state projection algorithms [51], [52] or the sliding window method [53]; other methods were also introduced for special reaction systems characterized by particular initial conditions (see, e.g., [54] and references therein). A different strategy to solve the CME consists in generating trajectories of the underlying Markov process.…”

Section: Methodsmentioning

confidence: 99%

cuTauLeaping: A GPU-Powered Tau-Leaping Stochastic Simulator for Massive Parallel Analyses of Biological Systems

et al. 2014

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Tau-leaping is a stochastic simulation algorithm that efficiently reconstructs the temporal evolution of biological systems, modeled according to the stochastic formulation of chemical kinetics. The analysis of dynamical properties of these systems in physiological and perturbed conditions usually requires the execution of a large number of simulations, leading to high computational costs. Since each simulation can be executed independently from the others, a massive parallelization of tau-leaping can bring to relevant reductions of the overall running time. The emerging field of General Purpose Graphic Processing Units (GPGPU) provides power-efficient high-performance computing at a relatively low cost. In this work we introduce cuTauLeaping, a stochastic simulator of biological systems that makes use of GPGPU computing to execute multiple parallel tau-leaping simulations, by fully exploiting the Nvidia's Fermi GPU architecture. We show how a considerable computational speedup is achieved on GPU by partitioning the execution of tau-leaping into multiple separated phases, and we describe how to avoid some implementation pitfalls related to the scarcity of memory resources on the GPU streaming multiprocessors. Our results show that cuTauLeaping largely outperforms the CPU-based tau-leaping implementation when the number of parallel simulations increases, with a break-even directly depending on the size of the biological system and on the complexity of its emergent dynamics. In particular, cuTauLeaping is exploited to investigate the probability distribution of bistable states in the Schlögl model, and to carry out a bidimensional parameter sweep analysis to study the oscillatory regimes in the Ras/cAMP/PKA pathway in S. cerevisiae.

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“…We discretize the system using adaptive uniformization, which has been introduced by van Moorsel [41] as a variant of standard uniformization [31,34,44,15,35]. Numerical methods based on uniformization have the advantage that they are numerically stable and often more efficient than other methods [37].…”

Section: Numerical Reachability Analysismentioning

confidence: 99%

Approximation of Event Probabilities in Noisy Cellular Processes

Didier

Henzinger

Mateescu

et al. 2009

Computational Methods in Systems Biology

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Abstract. Molecular noise, which arises from the randomness of the discrete events in the cell, significantly influences fundamental biological processes. Discrete-state continuous-time stochastic models (CTMC) can be used to describe such effects, but the calculation of the probabilities of certain events is computationally expensive. We present a comparison of two analysis approaches for CTMC. On one hand, we estimate the probabilities of interest using repeated Gillespie simulation and determine the statistical accuracy that we obtain. On the other hand, we apply a numerical reachability analysis that approximates the probability distributions of the system at several time instances. We use examples of cellular processes to demonstrate the superiority of the reachability analysis if accurate results are required.

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Efficient computation of transient solutions of the chemical master equation based on uniformization and quasi-Monte Carlo

Cited by 16 publications

References 28 publications

Master equations and the theory of stochastic path integrals

Master equations and the theory of stochastic path integrals

cuTauLeaping: A GPU-Powered Tau-Leaping Stochastic Simulator for Massive Parallel Analyses of Biological Systems

Approximation of Event Probabilities in Noisy Cellular Processes

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