Multivariate moment closure techniques for stochastic kinetic models

Lakatos, Eszter; Ale, Angelique; Kirk, Paul; Stumpf, Michael

doi:10.1063/1.4929837

Cited by 76 publications

(87 citation statements)

References 57 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Information on these methods can be found in classic text books [98,110] and in a recent review [274]. Although moment equations encode the complete information about a stochastic process, they typically constitute an infinite hierarchy whose evaluation requires a truncation by some closure scheme [275][276][277][278][279][280][281][282][283]. On the other hand, the KramersMoyal and the system-size expansion approximate the master equation in terms of a Fokker-Planck equation.…”

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

confidence: 99%

See 1 more Smart Citation

Master equations and the theory of stochastic path integrals

2017

View full text Add to dashboard Cite

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...

show abstract

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

confidence: 99%

“…Since the bra n| is a left eigenvector of the number matrix N · · = ca with eigenvalue n, one may write A(n) n| = n|A(N ) for an analytic observable A. After inserting the exponential solution (283) into the averaged observable (282), an evaluation of the sums therefore results in…”

Section: Intermezzo: Alternative Derivation Based On Coherent Statesmentioning

confidence: 99%

Master equations and the theory of stochastic path integrals

2017

View full text Add to dashboard Cite

show abstract

“…Perhaps an approximate algorithm could be developed on the basis of Algorithm (1) to handle such situations. Alternatively, the tau-leaping algorithm [32], or moment expansion [33, 34], may be more appropriate for models involving these kinds of feedback interactions. Algorithm (1) could, however, be naturally extended to models involving regulatory interactions between non-coding RNAs as the simulation of that part of the model is equivalent to Gillespie’s exact algorithm.…”

Section: Discussionmentioning

confidence: 99%

Inferring extrinsic noise from single-cell gene expression data using approximate Bayesian computation

2016

View full text Add to dashboard Cite

Background: Gene expression is known to be an intrinsically stochastic process which can involve single-digit numbers of mRNA molecules in a cell at any given time. The modelling of such processes calls for the use of exact stochastic simulation methods, most notably the Gillespie algorithm. However, this stochasticity, also termed "intrinsic noise", does not account for all the variability between genetically identical cells growing in a homogeneous environment. Despite substantial experimental efforts, determining appropriate model parameters continues to be a challenge. Methods based on approximate Bayesian computation can be used to obtain posterior parameter distributions given the observed data. However, such inference procedures require large numbers of simulations of the model and exact stochastic simulation is computationally costly. In this work we focus on the specific case of trying to infer model parameters describing reaction rates and extrinsic noise on the basis of measurements of molecule numbers in individual cells at a given time point. Results: To make the problem computationally tractable we develop an exact, model-specific, stochastic simulation algorithm for the commonly used two-state model of gene expression. This algorithm relies on certain assumptions and favourable properties of the model to forgo the simulation of the whole temporal trajectory of protein numbers in the system, instead returning only the number of protein and mRNA molecules present in the system at a specified time point. The computational gain is proportional to the number of protein molecules created in the system and becomes significant for systems involving hundreds or thousands of protein molecules. Conclusions: We employ this simulation algorithm with approximate Bayesian computation to jointly infer the model's rate and noise parameters from published gene expression data. Our analysis indicates that for most genes the extrinsic contributions to noise will be small to moderate but certainly are non-negligible.

show abstract

“…Typically approximate closure schemes are employed to solve for moments in such cases. [98][99][100][101][102][103][104][105][106][107][108] Here, we use the Linear Noise Approximation (LNA) method, where assuming small fluctuations in x f (t) and x b (t) around their respective mean values x f and x b , the nonlinear term is linearized as…”

Section: Bound Tf's Degradation Titrates the Regulating Activity Of Tfmentioning

confidence: 99%

Enhancement of gene expression noise due to nonspecific transcription factor binding

Dey

Soltani

Singh

2019

Preprint

View full text Add to dashboard Cite

The genome contains several high-affinity non-functional binding sites for transcription factors (TFs) creating a hidden and unexplored layer of gene regulation. We investigate the role of such "decoy sites" in controlling noise (random fluctuations) in the level of a TF that is synthesized in stochastic bursts. Prior studies have assumed that decoy-bound TFs are protected from degradation, and in this case decoys function to buffer noise. Relaxing this assumption to consider arbitrary degradation rates for both bound/unbound TF states, we find rich noise behaviors. For low-affinity decoys, noise in the level of unbound TF always monotonically decreases to the Poisson limit with increasing decoy numbers. In contrast, for high affinity decoys, noise levels first increase with increasing decoy numbers, before decreasing back to the Poisson limit. Interestingly, while protection of bound TFs from degradation slows the time-scale of fluctuations in the unbound TF levels, decay of bounds TFs leads to faster fluctuations and smaller noise propagation to downstream target proteins. In summary, our analysis reveals stochastic dynamics emerging from nonspecific binding of TFs, and highlight the dual role of decoys as attenuators or amplifiers of gene expression noise depending on their binding affinity and stability of the bound TF.

show abstract

Multivariate moment closure techniques for stochastic kinetic models

Cited by 76 publications

References 57 publications

Master equations and the theory of stochastic path integrals

Master equations and the theory of stochastic path integrals

Inferring extrinsic noise from single-cell gene expression data using approximate Bayesian computation

Enhancement of gene expression noise due to nonspecific transcription factor binding

Contact Info

Product

Resources

About