Parameter inference with analytical propagators for stochastic models of autoregulated gene expression
07 Apr 2021-International Journal of Nonlinear Sciences and Numerical Simulation (Walter de Gruyter GmbH)-
TL;DR: In this article, a scale separation based analytical method is proposed to estimate propagators from asymptotic expansions for the corresponding generating functions, which can be successfully applied for Bayesian parameter inference in the nonregulated model with synthetic data.
Abstract: Stochastic gene expression in regulatory networks is conventionally modelled via the Chemical Master Equation (CME) (van Kampen 1981). As explicit solutions to the CME, in the form of so-called propagators, are not readily available, various approximations have been proposed (Zechner et al. 2013, Feigelman et al 2016, Popovic, Marr and Swain 2016). A recently developed analytical method (Veerman, Marr and Popovic 2017) is based on a scale separation that assumes significant differences in the lifetimes of mRNA and protein in the network, allowing for the efficient approximation of propagators from asymptotic expansions for the corresponding generating functions. Here, we showcase the applicability of that method to a ‘telegraph’ model for gene expression that is extended with an autoregulatory mechanism. We demonstrate that the resulting approximate propagators can be successfully applied for Bayesian parameter inference in the non-regulated model with synthetic data; moreover, we show that in the extended autoregulated model, autoactivation or autorepression may be refuted under certain assumptions on the model parameters. Our results indicate that the method showcased here may allow for successful parameter inference and model identification from longitudinal single cell data.
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TL;DR: In this paper, the authors studied large deviations in a Markovian drift-jump process that combines exponentially distributed bursts with deterministic degradation, where large deviations occur as a cumulative effect of many bursts (as in diffusion) or, if the model includes negative feedback in burst size, in a single big jump.
Abstract: Synthesis of gene products in bursts of multiple molecular copies is an important source of gene expression variability. This paper studies large deviations in a Markovian drift-jump process that combines exponentially distributed bursts with deterministic degradation. Large deviations occur as a cumulative effect of many bursts (as in diffusion) or, if the model includes negative feedback in burst size, in a single big jump. The latter possibility requires a modification in the WKB solution in the tail region. The main result of the paper is the construction, via a modified WKB scheme, of matched asymptotic approximations to the stationary distribution of the drift-jump process. The stationary distribution possesses a heavier tail than predicted by a routine application of the scheme. MSC 202092C40; 60J76, 45D05, 41A60
6 citations
TL;DR: In this article, the authors studied large deviations in a Markovian drift-jump process that combines exponentially distributed bursts with deterministic degradation, where large deviations occur as a cumulative effect of many bursts (as in diffusion) or, if the model includes negative feedback in burst size, in a single big jump.
Abstract: Synthesis of gene products in bursts of multiple molecular copies is an important source of gene expression variability. This paper studies large deviations in a Markovian drift-jump process that combines exponentially distributed bursts with deterministic degradation. Large deviations occur as a cumulative effect of many bursts (as in diffusion) or, if the model includes negative feedback in burst size, in a single big jump. The latter possibility requires a modification in the WKB solution in the tail region. The main result of the paper is the construction, via a modified WKB scheme, of matched asymptotic approximations to the stationary distribution of the drift-jump process. The stationary distribution possesses a heavier tail than predicted by a routine application of the scheme. MSC 202092C40; 60J76, 45D05, 41A60
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TL;DR: In this article, a simulation algorithm for the stochastic formulation of chemical kinetics is proposed, which uses a rigorously derived Monte Carlo procedure to numerically simulate the time evolution of a given chemical system.
Abstract: There are two formalisms for mathematically describing the time behavior of a spatially homogeneous chemical system: The deterministic approach regards the time evolution as a continuous, wholly predictable process which is governed by a set of coupled, ordinary differential equations (the “reaction-rate equations”); the stochastic approach regards the time evolution as a kind of random-walk process which is governed by a single differential-difference equation (the “master equation”). Fairly simple kinetic theory arguments show that the stochastic formulation of chemical kinetics has a firmer physical basis than the deterministic formulation, but unfortunately the stochastic master equation is often mathematically intractable. There is, however, a way to make exact numerical calculations within the framework of the stochastic formulation without having to deal with the master equation directly. It is a relatively simple digital computer algorithm which uses a rigorously derived Monte Carlo procedure to numerically simulate the time evolution of the given chemical system. Like the master equation, this “stochastic simulation algorithm” correctly accounts for the inherent fluctuations and correlations that are necessarily ignored in the deterministic formulation. In addition, unlike most procedures for numerically solving the deterministic reaction-rate equations, this algorithm never approximates infinitesimal time increments df by finite time steps At. The feasibility and utility of the simulation algorithm are demonstrated by applying it to several well-known model chemical systems, including the Lotka model, the Brusselator, and the Oregonator.
10,275 citations
TL;DR: Using a quantitative model, the first genome-scale prediction of synthesis rates of mRNAs and proteins is obtained and it is found that the cellular abundance of proteins is predominantly controlled at the level of translation.
Abstract: Gene expression is a multistep process that involves the transcription, translation and turnover of messenger RNAs and proteins. Although it is one of the most fundamental processes of life, the entire cascade has never been quantified on a genome-wide scale. Here we simultaneously measured absolute mRNA and protein abundance and turnover by parallel metabolic pulse labelling for more than 5,000 genes in mammalian cells. Whereas mRNA and protein levels correlated better than previously thought, corresponding half-lives showed no correlation. Using a quantitative model we have obtained the first genome-scale prediction of synthesis rates of mRNAs and proteins. We find that the cellular abundance of proteins is predominantly controlled at the level of translation. Genes with similar combinations of mRNA and protein stability shared functional properties, indicating that half-lives evolved under energetic and dynamic constraints. Quantitative information about all stages of gene expression provides a rich resource and helps to provide a greater understanding of the underlying design principles.
5,635 citations
TL;DR: This work constructed strains of Escherichia coli that enable detection of noise and discrimination between the two mechanisms by which it is generated and reveals how low intracellular copy numbers of molecules can fundamentally limit the precision of gene regulation.
Abstract: Clonal populations of cells exhibit substantial phenotypic variation. Such heterogeneity can be essential for many biological processes and is conjectured to arise from stochasticity, or noise, in gene expression. We constructed strains of Escherichia coli that enable detection of noise and discrimination between the two mechanisms by which it is generated. Both stochasticity inherent in the biochemical process of gene expression (intrinsic noise) and fluctuations in other cellular components (extrinsic noise) contribute substantially to overall variation. Transcription rate, regulatory dynamics, and genetic factors control the amplitude of noise. These results establish a quantitative foundation for modeling noise in genetic networks and reveal how low intracellular copy numbers of molecules can fundamentally limit the precision of gene regulation.
5,209 citations
TL;DR: The results demonstrate that gene expression in mammalian cells is subject to large, intrinsically random fluctuations and raise questions about how cells are able to function in the face of such noise.
Abstract: Individual cells in genetically homogeneous populations have been found to express different numbers of molecules of specific proteins. We investigated the origins of these variations in mammalian cells by counting individual molecules of mRNA produced from a reporter gene that was stably integrated into the cell's genome. We found that there are massive variations in the number of mRNA molecules present in each cell. These variations occur because mRNAs are synthesized in short but intense bursts of transcription beginning when the gene transitions from an inactive to an active state and ending when they transition back to the inactive state. We show that these transitions are intrinsically random and not due to global, extrinsic factors such as the levels of transcriptional activators. Moreover, the gene activation causes burst-like expression of all genes within a wider genomic locus. We further found that bursts are also exhibited in the synthesis of natural genes. The bursts of mRNA expression can be buffered at the protein level by slow protein degradation rates. A stochastic model of gene activation and inactivation was developed to explain the statistical properties of the bursts. The model showed that increasing the level of transcription factors increases the average size of the bursts rather than their frequency. These results demonstrate that gene expression in mammalian cells is subject to large, intrinsically random fluctuations and raise questions about how cells are able to function in the face of such noise.
1,728 citations
TL;DR: This work established various gene trap cell lines and transgenic cell lines expressing a short-lived luciferase protein from an unstable mRNA, and recorded bioluminescence in real time in single cells, demonstrating that bursting kinetics are highly gene-specific.
Abstract: In prokaryotes and eukaryotes, most genes appear to be transcribed during short periods called transcriptional bursts, interspersed by silent intervals. We describe how such bursts generate gene-specific temporal patterns of messenger RNA (mRNA) synthesis in mammalian cells. To monitor transcription at high temporal resolution, we established various gene trap cell lines and transgenic cell lines expressing a short-lived luciferase protein from an unstable mRNA, and recorded bioluminescence in real time in single cells. Mathematical modeling identified gene-specific on- and off-switching rates in transcriptional activity and mean numbers of mRNAs produced during the bursts. Transcriptional kinetics were markedly altered by cis-regulatory DNA elements. Our analysis demonstrated that bursting kinetics are highly gene-specific, reflecting refractory periods during which genes stay inactive for a certain time before switching on again.
891 citations