Stochastic approach to chemical kinetics

McQuarrie, Donald A.

doi:10.2307/3212214

Cited by 717 publications

(141 citation statements)

References 111 publications

(102 reference statements)

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“…at least when the corresponding transition rate describes a simple chemical reaction [122][123][124][125][126][127][128]327]. The generating function effectively replaces the discrete variable n by the continuous variable q.…”

Section: The Probability Generating Functionmentioning

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: The Probability Generating Functionmentioning

confidence: 99%

Master equations and the theory of stochastic path integrals

2017

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“…However, this deterministic framework fails within single cells where many species occur at very low molecular counts and change by discrete integer amounts whenever a reaction occurs inside the cell. The time evolution of such low-copy biochemical species is more accurately represented by a stochastic formulation of chemical kinetics which treats x(t) as a stochastic process (McQuarrie 1967).…”

Section: (A) Stochastic Formulation Of Chemical Kineticsmentioning

confidence: 99%

Stochastic hybrid systems for studying biochemical processes

Singh

Hespanha

2010

Phil. Trans. R. Soc. A.

126

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Many protein and mRNA species occur at low molecular counts within cells, and hence are subject to large stochastic fluctuations in copy numbers over time. Development of computationally tractable frameworks for modelling stochastic fluctuations in population counts is essential to understand how noise at the cellular level affects biological function and phenotype. We show that stochastic hybrid systems (SHSs) provide a convenient framework for modelling the time evolution of population counts of different chemical species involved in a set of biochemical reactions. We illustrate recently developed techniques that allow fast computations of the statistical moments of the population count, without having to run computationally expensive Monte Carlo simulations of the biochemical reactions. Finally, we review different examples from the literature that illustrate the benefits of using SHSs for modelling biochemical processes.

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“…(1) is the so-called CME. [11][12][13] The CME is also generally derived using the Markov property, by writing the ChapmanKolmogorov equation, an identity that must be obeyed by the transition probability of any Markov process.…”

Section: Stochastic Modeling Of Biochemical Reactionsmentioning

confidence: 99%

“…9 These behaviors can only be captured and thoroughly understood in the context of stochastic computational models. The current stochastic representation of biological systems adopts the formalism of the chemical master equation (CME), [11][12][13] an N dimensional differential-difference equation (N is the number of model species), or its accompanying numerical stochastic simulation algorithm (SSA). 14 However, for many realistic biological models, the CME is computationally intractable and the SSA is too numerically inefficient.…”

Section: Background and Motivationmentioning

confidence: 99%

A data-integrated method for analyzing stochastic biochemical networks

Chevalier

El-Samad

2011

The Journal of Chemical Physics

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Variability and fluctuations among genetically identical cells under uniform experimental conditions stem from the stochastic nature of biochemical reactions. Understanding network function for endogenous biological systems or designing robust synthetic genetic circuits requires accounting for and analyzing this variability. Stochasticity in biological networks is usually represented using a continuous-time discrete-state Markov formalism, where the chemical master equation (CME) and its kinetic Monte Carlo equivalent, the stochastic simulation algorithm (SSA), are used. These two representations are computationally intractable for many realistic biological problems. Fitting parameters in the context of these stochastic models is particularly challenging and has not been accomplished for any but very simple systems. In this work, we propose that moment equations derived from the CME, when treated appropriately in terms of higher order moment contributions, represent a computationally efficient framework for estimating the kinetic rate constants of stochastic network models and subsequent analysis of their dynamics. To do so, we present a practical data-derived moment closure method for these equations. In contrast to previous work, this method does not rely on any assumptions about the shape of the stochastic distributions or a functional relationship among their moments. We use this method to analyze a stochastic model of a biological oscillator and demonstrate its accuracy through excellent agreement with CME/SSA calculations. By coupling this moment-closure method with a parameter search procedure, we further demonstrate how a model's kinetic parameters can be iteratively determined in order to fit measured distribution data.

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Stochastic approach to chemical kinetics

Cited by 717 publications

References 111 publications

Master equations and the theory of stochastic path integrals

Master equations and the theory of stochastic path integrals

Stochastic hybrid systems for studying biochemical processes

A data-integrated method for analyzing stochastic biochemical networks

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