Distribution Approximations for the Chemical Master Equation: Comparison of the Method of Moments and the System Size Expansion

Andreychenko, Alexander; Bortolussi, Luca; Grima, Ramon; Thomas, Philipp; Wolf, Verena

doi:10.1007/978-3-319-45833-5_2

Cited by 25 publications

(30 citation statements)

References 49 publications

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“…We observe that the approximations accurately capture the skewed distribution, with increasing accuracy at higher order. The figure is taken from [137].…”

Section: Construction Of Distributions From Momentsmentioning

confidence: 99%

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Approximation and inference methods for stochastic biochemical kinetics—a tutorial review

Schnoerr

Sanguinetti

Grima

2017

J. Phys. A: Math. Theor.

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320

413

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Stochastic fluctuations of molecule numbers are ubiquitous in biological systems. Important examples include gene expression and enzymatic processes in living cells. Such systems are typically modelled as chemical reaction networks whose dynamics are governed by the Chemical Master Equation. Despite its simple structure, no analytic solutions to the Chemical Master Equation are known for most systems. Moreover, stochastic simulations are computationally expensive, making systematic analysis and statistical inference a challenging task. Consequently, significant effort has been spent in recent decades on the development of efficient approximation and inference methods. This article gives an introduction to basic modelling concepts as well as an overview of state of the art methods. First, we motivate and introduce deterministic and stochastic methods for modelling chemical networks, and give an overview of simulation and exact solution methods. Next, we discuss several approximation methods, including the chemical Langevin equation, the system size expansion, moment closure approximations, time-scale separation approximations and hybrid methods. We discuss their various properties and review recent advances and remaining challenges for these methods. We present a comparison of several of these methods by means of a numerical case study and highlight some of their respective advantages and disadvantages. Finally, we discuss the problem of inference from experimental data in the Bayesian framework and review recent methods developed the literature. In summary, this review gives a self-contained introduction to modelling, approximations and inference methods for stochastic chemical kinetics.

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“…We observe that the approximations accurately capture the skewed distribution, with increasing accuracy at higher order. The figure is taken from [137].…”

Section: Construction Of Distributions From Momentsmentioning

confidence: 99%

“…numerical methods. The maximum entropy method for the construction of the marginal distributions of single species has recently been used in combination with moment closure methods and the system size expansion in [136,137]. Figure 7 shows the result for one example system.…”

Section: Construction Of Distributions From Momentsmentioning

confidence: 99%

Approximation and inference methods for stochastic biochemical kinetics—a tutorial review

Schnoerr

Sanguinetti

Grima

2017

J. Phys. A: Math. Theor.

Self Cite

320

413

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“…The results revealed that the implications of a partially finite state space had not been completely understood in earlier studies and that the number of classical moment equations that are really needed is smaller than what was previously claimed. 11,12,35 The results of this paper can also be seen as the basis for a hybrid approach for the analysis of stochastic models of biochemical reaction networks. The binary variable representation of the network, and the derivation of moment equations from it, automatically keeps the full probability distribution over the finite part of the state space while resorting to low-order moments over the infinite part.…”

Section: Discussionmentioning

confidence: 93%

Minimal moment equations for stochastic models of biochemical reaction networks with partially finite state space

Ruess¹

2015

The Journal of Chemical Physics

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Many stochastic models of biochemical reaction networks contain some chemical species for which the number of molecules that are present in the system can only be finite (for instance due to conservation laws), but also other species that can be present in arbitrarily large amounts. The prime example of such networks are models of gene expression, which typically contain a small and finite number of possible states for the promoter but an infinite number of possible states for the amount of mRNA and protein. One of the main approaches to analyze such models is through the use of equations for the time evolution of moments of the chemical species. Recently, a new approach based on conditional moments of the species with infinite state space given all the different possible states of the finite species has been proposed. It was argued that this approach allows one to capture more details about the full underlying probability distribution with a smaller number of equations. Here, I show that the result that less moments provide more information can only stem from an unnecessarily complicated description of the system in the classical formulation. The foundation of this argument will be the derivation of moment equations that describe the complete probability distribution over the finite state space but only low-order moments over the infinite state space. I will show that the number of equations that is needed is always less than what was previously claimed and always less than the number of conditional moment equations up to the same order. To support these arguments, a symbolic algorithm is provided that can be used to derive minimal systems of unconditional moment equations for models with partially finite state space.

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“…The dynamics of these processes is therefore well described by stochastic Markov processes in continuous time with discrete state space [15,22,42]. While few-component or linear-kinetics systems [16] allow for exact analysis, in more complex system one often uses approximative methods [12], such as moment closure [4], linear-noise approximation [3,9], hybrid formulations [25,26,33], and multi-scale techniques [38,39].…”

Section: Introductionmentioning

confidence: 99%

Stationary distributions and metastable behaviour for self-regulating proteins with general lifetime distributions

Çelik

Bokes

Singh

2020

Preprint

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Regulatory molecules such as transcription factors are often present at relatively small copy numbers in living cells. The copy number of a particular molecule fluctuates in time due to the random occurrence of production and degradation reactions. Here we consider a stochastic model for a self-regulating transcription factor whose lifespan (or time till degradation) follows a general distribution modelled as per a multidimensional phase-type process. We show that at steady state the protein copy-number distribution is the same as in a one-dimensional model with exponentially distributed lifetimes. This invariance result holds only if molecules are produced one at a time: we provide explicit counterexamples in the bursty production regime. Additionally, we consider the case of a bistable genetic switch constituted by a positively autoregulating transcription factor. The switch alternately resides in states of up-and downregulation and generates bimodal protein distributions. In the context of our invariance result, we investigate how the choice of lifetime distribution affects the rates of metastable transitions between the two modes of the distribution. The phase-type model, being non-linear and multi-dimensional whilst possessing an explicit stationary distribution, provides a valuable test example for exploring dynamics in complex biological systems.

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Distribution Approximations for the Chemical Master Equation: Comparison of the Method of Moments and the System Size Expansion

Cited by 25 publications

References 49 publications

Approximation and inference methods for stochastic biochemical kinetics—a tutorial review

Approximation and inference methods for stochastic biochemical kinetics—a tutorial review

Minimal moment equations for stochastic models of biochemical reaction networks with partially finite state space

Stationary distributions and metastable behaviour for self-regulating proteins with general lifetime distributions

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