Dynamics of simple gene-network motifs subject to extrinsic fluctuations

Roberts, Elijah; Be'er, Shay; Bohrer, Christopher H.; Sharma, Rati; Assaf, Michael

doi:10.1103/physreve.92.062717

Cited by 55 publications

(63 citation statements)

References 73 publications

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“…These processes are typically simulated using approximative techniques such as τ -leaping [78,249,[258][259][260][261][262][263][264][265][266][267]]. Another challenge is posed by the evaluation of processes with time-dependent transition rates [268][269][270].…”

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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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%

Master equations and the theory of stochastic path integrals

2017

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“…We next tested FFPilot with a relatively simple biochemical network, the self regulating gene model (SRG) 6 . SRG models expression of a single protein A.…”

Section: Self Regulating Gene Modelmentioning

confidence: 99%

“…Metastable systems, because they depend upon random fluctuations, are typically modeled using a formulation of the chemical master equation 5,6 or using stochastic differential equations 7 . In the former case, the models are often numerically studied using the stochastic simulation algorithm 8,9 (SSA) or one of its many varieties [10][11][12] .…”

Section: Introductionmentioning

confidence: 99%

Automatic error control during forward flux sampling of rare events in master equation models

Klein

Roberts

2018

Preprint

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Enhanced sampling methods, such as forward flux sampling (FFS), hold a great deal of promise for accelerating stochastic simulations of nonequilibrium biochemical systems involving rare events. However, the description of the tradeoffs between simulation efficiency and error in FFS remains incomplete. We present a novel and mathematically rigorous analysis of the errors in FFS that, for the first time, covers the contribution of every phase of the simulation. We derive a closed form expression for the optimally efficient count of samples to take in each FFS phase in terms of a fixed constraint on sampling error. We introduce a new method, forward flux pilot sampling (FFPilot), that is designed to take full advantage of our optimizing equation without prior information or assumptions about the phase weights and costs along the transition path. In simulations of both singleand multi-dimensional gene regulatory networks, FFPilot is able to completely control sampling error. Higher dimensional systems have additional sources of error and we show that this extra error can be traced to correlations between phases due to roughness on the probability landscape. Finally, we show that in sets of simulations with matched error, FFPilot is on the order of tens-to-hundreds of times faster than direct sampling, in a fashion that scales with the rarity of the events. a) Correspondence to:

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“…Even though there is no unique analytical formula for the distribution produced by our model, the parameters could be found by fitting to numerical solutions. Additionally, other known sources of noise, such as extrinsic noise, would also have to be incorporated into the model [14,52]. This study provides a step toward a more detailed biophysical model of transcription during gene expression.…”

Section: Supercoiling Build-up Generates Broad Mrna Distributionsmentioning

confidence: 99%

“…In particular, rather than being peaked at a non-zero copy number, our model predicts a large peak at zero copy number with a gradually decreasing density thereafter. Broadening of the mRNA distribution is also a characteristic of extrinsic noise and certain promoter architectures [14,52], but here we focus only on the influence of supercoiling on the distributions.…”

Section: Mrna Distributionmentioning

confidence: 99%

A Biophysical Model of Supercoiling Dependent Transcription Predicts a Structural Aspect to Gene Regulation

Bohrer

Roberts

2017

Biophysical Journal

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Background: Transcription in Escherichia coli generates positive supercoiling in the DNA, which is relieved by the enzymatic activity of gyrase. Recently published experimental evidence suggests that transcription initiation and elongation are inhibited by the buildup of positive supercoiling. It has therefore been proposed that intermittent binding of gyrase plays a role in transcriptional bursting. Considering that transcription is one of the most fundamental cellular processes, it is desirable to be able to account for the buildup and release of positive supercoiling in models of transcription. Results: Here we present a detailed biophysical model of gene expression that incorporates the effects of supercoiling due to transcription. By directly linking the amount of positive supercoiling to the rate of transcription, the model predicts that highly transcribed genes' mRNA distributions should substantially deviate from Poisson distributions, with enhanced density at low mRNA copy numbers. Additionally, the model predicts a high degree of correlation between expression levels of genes inside the same supercoiling domain. Conclusions: Our model, incorporating the supercoiling state of the gene, makes specific predictions that differ from previous models of gene expression. Genes in the same supercoiling domain influence the expression level of neighboring genes. Such structurally dependent regulation predicts correlations between genes in the same supercoiling domain. The topology of the chromosome therefore creates a higher level of gene regulation, which has broad implications for understanding the evolution and organization of bacterial genomes.

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Dynamics of simple gene-network motifs subject to extrinsic fluctuations

Cited by 55 publications

References 73 publications

Master equations and the theory of stochastic path integrals

Master equations and the theory of stochastic path integrals

Automatic error control during forward flux sampling of rare events in master equation models

A Biophysical Model of Supercoiling Dependent Transcription Predicts a Structural Aspect to Gene Regulation

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