Exact distributions for stochastic gene expression models with arbitrary promoter architecture and translational bursting

Meiling Chen, Songhao Luo, Mengfang Cao, Chengjun Guo, Tianshou Zhou, and Jiajun Zhang

Phys. Rev. E 105, 014405 – Published 6 January 2022

Abstract

Gene expression in individual cells is inherently variable and sporadic, leading to cell-to-cell variability in mRNA and protein levels. Recent single-cell and single-molecule experiments indicate that promoter architecture and translational bursting play significant roles in controlling gene expression noise and generating the phenotypic diversity that life exhibits. To quantitatively understand the impact of these factors, it is essential to construct an accurate mathematical description of stochastic gene expression and find the exact analytical results, which is a formidable task. Here, we develop a stochastic model of bursty gene expression, which considers the complex promoter architecture governing the variability in mRNA expression and a general distribution characterizing translational burst. We derive the analytical expression for the corresponding protein steady-state distribution and all moment statistics of protein counts. We show that the total protein noise can be decomposed into three parts: the low-copy noise of protein due to probabilistic individual birth and death events, the noise due to stochastic switching between promoter states, and the noise resulting from translational busting. The theoretical results derived provide quantitative insights into the biochemical mechanisms of stochastic gene expression.

Received 12 October 2021
Revised 10 December 2021
Accepted 14 December 2021

DOI:https://doi.org/10.1103/PhysRevE.105.014405

Physics Subject Headings (PhySH)

Gene regulatory networks

Master equation

Physics of Living Systems

Authors & Affiliations

Meiling Chen^1,2,*, Songhao Luo ^1,2,*, Mengfang Cao^1,2, Chengjun Guo³, Tianshou Zhou^1,2, and Jiajun Zhang ^1,2,†

¹Guangdong Province Key Laboratory of Computational Science, Guangzhou 510275, People's Republic of China
²School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, People's Republic of China
³School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510275, People's Republic of China

^*These authors contributed equally to this work.
^†Corresponding author: zhjiajun@mail.sysu.edu.cn

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Vol. 105, Iss. 1 — January 2022

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Images

Figure 1
Gene expression model with arbitrary promoter architecture and general translational bursting. (a) Generalized gene expression process: the promoter switches among different states, transcription occurs during active promoter states, translation from mRNA in bursting manner, mRNA decays faster than protein. (b) Simplified sketch of protein production with a multistate model. The upper figure shows an example of the distribution of protein bursts.
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Figure 2
Moments statistics of protein abundance for different bursting kinetics. (a) Mean protein abundance as a function of average burst size. Noise (b), skewness (c), and kurtosis (d) as a function of mean protein abundance with different bursting distributions. Parameters used for calculations are $k_{on} = 1, k_{off} = 1, μ = 20, δ = 1,$ and $r = 3$ for negative binomial distribution. Negative binomial distribution is abbreviated to NB.
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Figure 3
Stationary distributions of protein molecules in various state-switching rates. (a) Fast state-switching case. Parameters used for calculations are: $k_{on} = 10, k_{off} = 10, μ = 5, δ = 1, 〈 B 〉 = 1 .$ (b) Middle state-switching case. $k_{on} = 1, k_{off} = 1, μ = 5, δ = 1, 〈 B 〉 = 1 .$ ; (c) Slow state-switching case. Parameters used for calculations are $k_{on} = 0.4, k_{off} = 0.1, μ = 5, δ = 1, 〈 B 〉 = 1, r = 3 .$ Binomial moments method is abbreviated to BMM.
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Figure 4
Analysis results from ${GAL 1}^{*}$ promoter in yeast. (a) The illustration of four-state promoter architecture kinetic model with translational burst for ${GAL 1}^{*}$ promoter. PC1 is an inactive or OFF state during the transcription cycle and PC2 is active state which can transcribe mRNA and translate it into protein with bursting. RC1 and RC2 are two states with different degrees of repression when tetR binds to the promoter. We have used the kinetic parameters for calculations provided by [25, 45] $k_{1 f} = 0.02 + 0.2 [gal], k_{2 f} = {200 {[tet]}^{2} / (1 + c_{l}^{4} {[atc]}^{4})}^{2}, k_{1 b} = 0.01 + 0.1 [gal] + 0.077 / [gal], k_{2 b} = 10, a = 0.025, c_{l} = 0.1, [tet] = 100, k_{p} = 1, γ = 0.0125 .$ (b) Moment statistics comparison between data simulated by Gillespie algorithm (circle) and theoretical value (solid line) under three different distributions of burst size: geometric (orange), Poisson (green), and NB (blue) with mean burst size $〈 B 〉 = 3.75$ . Mean expression, noise, skewness, and kurtosis of protein abundance is computed as a function of $[gal]$ at the condition of $[atc] = 500 ng / ml .$ (c) Similar to (b), mean expression, noise, skewness and kurtosis of protein abundance is computed as a function of $[atc]$ at the condition of $[gal] = 2 % .$ (d) A comparison between the probability density function of the data generated by the kinetic model (orange circle) with geometric burst size distribution ( $〈 B 〉 = 3$ ) and the experimentally observed distribution of fluorescence (green line) at steady state, induced with 2% galactose and 40 ng/ml of Atc for 440 min [45].
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