Approximation scheme for master equations: Variational approach to multivariate case

Ohkubo, Jun

doi:10.1063/1.2957462

Cited by 10 publications

(12 citation statements)

References 19 publications

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“…Besides its application to networks of genetic switches [39], the variational method has been applied to signalling in enzymatic cascades in [372]. An extension of the method to multivariate processes is described in [373]. Whether or not the variational method provides useful information mainly depends on making the right ansatz for ψ L | and |ψ R .…”

Section: A Variational Methodmentioning

confidence: 99%

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: A Variational Methodmentioning

confidence: 99%

Master equations and the theory of stochastic path integrals

2017

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“…The master equation governing the stochastic equations is explained in the subsection Master equation . We use the proteomic field approximation 11 34 55 to reduce the dimensionality of the master equation so that the solution is tractable computationally. Details used for kinetic calculation are explained in the subsection Effective number of cells .…”

Section: Methodsmentioning

confidence: 99%

“…To solve the master equation, we use the proteomic field approximation 11 34 55 , which is the the Hartree-like approximation;…”

Section: Methodsmentioning

confidence: 99%

Effects of Collective Histone State Dynamics on Epigenetic Landscape and Kinetics of Cell Reprogramming

Ashwin

Sasai

2015

Sci Rep

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Cell reprogramming is a process of transitions from differentiated to pluripotent cell states via transient intermediate states. Within the epigenetic landscape framework, such a process is regarded as a sequence of transitions among basins on the landscape; therefore, theoretical construction of a model landscape which exhibits experimentally consistent dynamics can provide clues to understanding epigenetic mechanism of reprogramming. We propose a minimal gene-network model of the landscape, in which each gene is regulated by an integrated mechanism of transcription-factor binding/unbinding and the collective chemical modification of histones. We show that the slow collective variation of many histones around each gene locus alters topology of the landscape and significantly affects transition dynamics between basins. Differentiation and reprogramming follow different transition pathways on the calculated landscape, which should be verified experimentally via single-cell pursuit of the reprogramming process. Effects of modulation in collective histone state kinetics on transition dynamics and pathway are examined in search for an efficient protocol of reprogramming.

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“…Here we use it to define a flexible family of distributions for moment closure. A log-normal mixture of Poisson distributions, used within Eyink's variational framework 7 , was proposed previously and motivated using ideas from statistical physics 24 . It is however important to realize that this approach is simply moment closure using a log-normal-Poisson mixture ansatz.…”

Section: A Poisson Mixturesmentioning

confidence: 99%

A variational approach to moment-closure approximations for the kinetics of biomolecular reaction networks

Bronstein

Koeppl

2018

The Journal of Chemical Physics

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Approximate solutions of the chemical master equation and the chemical FokkerPlanck equation are an important tool in the analysis of biomolecular reaction networks. Previous studies have highlighted a number of problems with the momentclosure approach used to obtain such approximations, calling it an ad-hoc method. In this article, we give a new variational derivation of moment-closure equations which provides us with an intuitive understanding of their properties and failure modes and allows us to correct some of these problems. We use mixtures of product-Poisson distributions to obtain a flexible parametric family which solves the commonly observed problem of divergences at low system sizes. We also extend the recently introduced entropic matching approach to arbitrary ansatz distributions and Markov processes, demonstrating that it is a special case of variational moment closure. This provides us with a particularly principled approximation method. Finally, we extend the above approaches to cover the approximation of multi-time joint distributions, resulting in a viable alternative to process-level approximations which are often intractable. a)

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Approximation scheme for master equations: Variational approach to multivariate case

Cited by 10 publications

References 19 publications

Master equations and the theory of stochastic path integrals

Master equations and the theory of stochastic path integrals

Effects of Collective Histone State Dynamics on Epigenetic Landscape and Kinetics of Cell Reprogramming

A variational approach to moment-closure approximations for the kinetics of biomolecular reaction networks

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