Summing over trajectories of stochastic dynamics with multiplicative noise

Tang, Ying; Yuan, Ruoshi; Ao, Ping

doi:10.1063/1.4890968

Cited by 30 publications

(32 citation statements)

References 51 publications

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“…These corrections are vanishingly small in the small noise limit which we are going to take later. 42,54 The terms A s i and D s i stand for drift and diffusion coefficients corresponding to gene state s i at time t i . We may formally take the limit of ∆t i → 0 and write the transition probabilities between protein numbers in a path integral form,…”

Section: A Schematic Discussion Of Trajectory Statistics In Gene Netmentioning

confidence: 99%

Dichotomous noise models of gene switches

Potoyan

Wolynes

2015

The Journal of Chemical Physics

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Molecular noise in gene regulatory networks has two intrinsic components, one part being due to fluctuations caused by the birth and death of protein or mRNA molecules which are often present in small numbers and the other part arising from gene state switching, a single molecule event. Stochastic dynamics of gene regulatory circuits appears to be largely responsible for bifurcations into a set of multi-attractor states that encode different cell phenotypes. The interplay of dichotomous single molecule gene noise with the nonlinear architecture of genetic networks generates rich and complex phenomena. In this paper, we elaborate on an approximate framework that leads to simple hybrid multi-scale schemes well suited for the quantitative exploration of the steady state properties of large-scale cellular genetic circuits. Through a path sum based analysis of trajectory statistics, we elucidate the connection of these hybrid schemes to the underlying master equation and provide a rigorous justification for using dichotomous noise based models to study genetic networks. Numerical simulations of circuit models reveal that the contribution of the genetic noise of single molecule origin to the total noise is significant for a wide range of kinetic regimes. C 2015 AIP Publishing LLC. [http://dx

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Section: A Schematic Discussion Of Trajectory Statistics In Gene Netmentioning

confidence: 99%

Dichotomous noise models of gene switches

Potoyan

Wolynes

2015

The Journal of Chemical Physics

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“…It is often assumed that the continuous-time chain rule (7) can be applied when manipulating the action (see for instance [30]) or that the formulae (16) (ii) similarly, that non-linear changes of variables are allowed in the action but are also wrong if one applies the chain rule (7).…”

Section: Stochastic Calculus In the Path Integral Actionmentioning

confidence: 99%

Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager–Machlup approach

Cugliandolo¹,

Lecomte²

2017

J. Phys. A: Math. Theor.

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The definition and manipulation of Langevin equations with multiplicative white noise require special care (one has to specify the time discretisation and a stochastic chain rule has to be used to perform changes of variables). While discretisation-scheme transformations and non-linear changes of variable can be safely performed on the Langevin equation, these same transformations lead to inconsistencies in its path-integral representation. We identify their origin and we show how to extend the well-known Itō prescription (dB 2 = dt) in a way that defines a modified stochastic calculus to be used inside the path-integral representation of the process, in its Onsager-Machlup form.

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“…The Stratonovich version of the action with κ = 1 2 is, for example, employed in Seifert's review on stochastic thermodynamics [237]. For a recent, more thorough discussion of the above discretization schemes, as well as of conflicting approaches, we refer the reader to [398] (the above action is discussed in the appendices).…”

Section: Alternative Discretization Schemesmentioning

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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Summing over trajectories of stochastic dynamics with multiplicative noise

Cited by 30 publications

References 51 publications

Dichotomous noise models of gene switches

Dichotomous noise models of gene switches

Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager–Machlup approach

Master equations and the theory of stochastic path integrals

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