Time-dependent propagators for stochastic models of gene expression: an analytical method.
01 Aug 2018-Journal of Mathematical Biology (Springer Berlin Heidelberg)-Vol. 77, Iss: 2, pp 261-312
TL;DR: An analytical method is proposed for the efficient approximation of propagators of stochastic models for gene expression which lends itself naturally to implementation in a Bayesian parameter inference scheme, and can be generalised systematically to related categories of stoChastic models beyond the ones considered here.
Abstract: The inherent stochasticity of gene expression in the context of regulatory networks profoundly influences the dynamics of the involved species. Mathematically speaking, the propagators which describe the evolution of such networks in time are typically defined as solutions of the corresponding chemical master equation (CME). However, it is not possible in general to obtain exact solutions to the CME in closed form, which is due largely to its high dimensionality. In the present article, we propose an analytical method for the efficient approximation of these propagators. We illustrate our method on the basis of two categories of stochastic models for gene expression that have been discussed in the literature. The requisite procedure consists of three steps: a probability-generating function is introduced which transforms the CME into (a system of) partial differential equations (PDEs); application of the method of characteristics then yields (a system of) ordinary differential equations (ODEs) which can be solved using dynamical systems techniques, giving closed-form expressions for the generating function; finally, propagator probabilities can be reconstructed numerically from these expressions via the Cauchy integral formula. The resulting 'library' of propagators lends itself naturally to implementation in a Bayesian parameter inference scheme, and can be generalised systematically to related categories of stochastic models beyond the ones considered here.
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TL;DR: The assumptions and properties of the main models in the literature are discussed, the current understanding of the relationship between them is summarized, and some of the insights gained through modeling are highlighted.
42 citations
Cites background from "Time-dependent propagators for stoc..."
...The derivation of approximate time-dependent propagators has received recent attention (65)....
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TL;DR: In this paper, the authors investigate single-cell stochastic gene expression kinetics in a minimal coupled gene circuit with positive-plus-negative feedback and derive the analytic steady-state distributions of the protein abundance for both the discrete model and its two macroscopic limits.
Abstract: Here we investigate single-cell stochastic gene expression kinetics in a minimal coupled gene circuit with positive-plus-negative feedback. A triphasic stochastic bifurcation is observed upon increasing the ratio of the positive and negative feedback strengths, which reveals a strong synergistic interaction between positive and negative feedback loops. We discover that coupled positive-plus-negative feedback amplifies gene expression mean but reduces gene expression noise over a wide range of feedback strengths when promoter switching is relatively slow, stabilizing gene expression around a relatively high level. In addition, we study two types of macroscopic limits of the discrete chemical master equation model: the Kurtz limit applies to proteins with large burst frequencies and the Levy limit applies to proteins with large burst sizes. We derive the analytic steady-state distributions of the protein abundance in a coupled gene circuit for both the discrete model and its two macroscopic limits, generalizing the results obtained by Liu et al. [Chaos 26, 043108 (2016)CHAOEH1054-150010.1063/1.4947202]. We also obtain the analytic time-dependent protein distribution for the classical Friedman-Cai-Xie random bursting model [Friedman, Cai, and Xie, Phys. Rev. Lett. 97, 168302 (2006)PRLTAO0031-900710.1103/PhysRevLett.97.168302]. Our analytic results are further applied to study the structure of gene expression noise in a coupled gene circuit, and a complete decomposition of noise in terms of five different biophysical origins is provided.
33 citations
TL;DR: It is shown that transient bimodality is a noise-induced phenomenon that occurs when the protein expression is sufficiently bursty, and a theory is used to estimate the observation time window when it is manifested.
Abstract: While the steady-state behavior of stochastic gene expression with auto-regulation has been extensively studied, its time-dependent behavior has received much less attention. Here, under the assumption of fast promoter switching, we derive and solve a reduced chemical master equation for an auto-regulatory gene circuit with translational bursting and cooperative protein-gene interactions. The analytical expression for the time-dependent probability distribution of protein numbers enables a fast exploration of large swaths of the parameter space. For a unimodal initial distribution, we identify three distinct types of stochastic dynamics: (i) the protein distribution remains unimodal at all times; (ii) the protein distribution becomes bimodal at intermediate times and then reverts back to being unimodal at long times (transient bimodality); and (iii) the protein distribution switches to being bimodal at long times. For each of these, the deterministic model predicts either monostable or bistable behavior, and hence, there exist six dynamical phases in total. We investigate the relationship of the six phases to the transcription rates, the protein binding and unbinding rates, the mean protein burst size, the degree of cooperativity, the relaxation time to the steady state, the protein mean, and the type of feedback loop (positive or negative). We show that transient bimodality is a noise-induced phenomenon that occurs when the protein expression is sufficiently bursty, and we use a theory to estimate the observation time window when it is manifested.
32 citations
TL;DR: It is proved that in the limit of fast promoter switching, the distribution of protein numbers is different than that given by standard stochastic models with Hill-type propensities, and that these differences decrease with increasing mean protein burst size.
25 citations
TL;DR: A Markov model used in the analysis of gene expression, involving the bursty production of pre-mRNA, its conversion to mature mRNA, and its consequent degradation, is explored, proving that the integration used to compute the solution of the stochastic system can be approximated by the evaluation of special functions.
Abstract: We explore a Markov model used in the analysis of gene expression, involving the bursty production of pre-mRNA, its conversion to mature mRNA, and its consequent degradation. We demonstrate that the integration used to compute the solution of the stochastic system can be approximated by the evaluation of special functions. Furthermore, the form of the special function solution generalizes to a broader class of burst distributions. In light of the broader goal of biophysical parameter inference from transcriptomics data, we apply the method to simulated data, demonstrating effective control of precision and runtime. Finally, we propose and validate a non-Bayesian approach for parameter estimation based on the characteristic function of the target joint distribution of pre-mRNA and mRNA.
20 citations
References
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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
"Time-dependent propagators for stoc..." refers methods in this paper
...…nature of stochastic models for gene expression has led to the widespread use of stochastic simulation techniques, such as Gillespie’s algorithm (Gillespie 1977), with the aim of predicting values for the associated propagators from these models; see Feigelman et al. (2016) for recent work…...
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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: Current understanding of the major factors regulating protein expression is summarized to demonstrate a substantial role for regulatory processes occurring after mRNA is made in controlling steady-state protein abundances.
Abstract: Recent advances in next-generation DNA sequencing and proteomics provide an unprecedented ability to survey mRNA and protein abundances. Such proteome-wide surveys are illuminating the extent to which different aspects of gene expression help to regulate cellular protein abundances. Current data demonstrate a substantial role for regulatory processes occurring after mRNA is made - that is, post-transcriptional, translational and protein degradation regulation - in controlling steady-state protein abundances. Intriguing observations are also emerging in relation to cells following perturbation, single-cell studies and the apparent evolutionary conservation of protein and mRNA abundances. Here, we summarize current understanding of the major factors regulating protein expression.
3,308 citations
TL;DR: Stochastic gene expression has important consequences for cellular function, being beneficial in some contexts and harmful in others, including the stress response, metabolism, development, the cell cycle, circadian rhythms, and aging.
2,471 citations
"Time-dependent propagators for stoc..." refers background in this paper
...However, gene expression can be highly stochastic in nature, both in prokaryotic and in eukaryotic organisms; see e.g. thework by Elowitz et al. (2002), Raj and Oudenaarden (2008), Shahrezaei and Swain (2008b), and the references therein....
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