Analytical formula and dynamic profile of mRNA distribution

Feng Jiao; Jian Ren; Jianshe Yu

doi:10.3934/dcdsb.2019180

Article Contents

2020, Volume 25, Issue 1: 241-257. Doi: 10.3934/dcdsb.2019180

This issue Previous Article Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness Next Article Analysis of time-domain Maxwell's equations in biperiodic structures

Analytical formula and dynamic profile of mRNA distribution

a.
College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China
b.
Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China

^* Corresponding author: Jianshe Yu, Email: jsyu@gzhu.edu.cn
^* Corresponding author: Jianshe Yu, Email: jsyu@gzhu.edu.cn

Received: December 31, 2018

Revised: February 28, 2019

Early access: July 2019

Published: January 2020

This work was supported by National Natural Science Foundation of China (11631005, 11871174, 11601491) and the Program for Changjiang Scholars and Innovative Research Team in University (IRT_16R16)

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Abstract

The stochasticity of transcription can be quantified by mRNA distribution $ P_m(t) $, the probability that there are $ m $ mRNA molecules for the gene at time $ t $ in one cell. However, it still lacks of a standard method to calculate $ P_m(t) $ in a transparent formula. Here, we employ an infinite series method to express $ P_m(t) $ based on the classical two-state model. Intriguingly, we observe that a unimodal distribution of mRNA numbers at steady-state could be transformed from a dynamical bimodal distribution. This indicates that "bet hedging" strategy can be still achieved for the gene that generates phenotypic homogeneity of the cell population. Moreover, the formation and duration of such bimodality are tightly correlated with mRNA synthesis rate, reinforcing the modulation scenario of some inducible genes that manipulates mRNA synthesis rate in response to environmental change. More generally, the method presented here may be implemented to the other stochastic transcription models with constant rates.

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Mathematics Subject Classification: Primary: 34F05, 37H10, 60J20; Secondary: 92C37, 92C40.

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Figure 3. Increasing mRNA synthesis rate $ v $ improves the formation of the intermediate bimodal distribution. (a) The mRNA distribution does not form bimodality when $ v $ is relatively small. (b) The intermediate bimodality appears when $ v $ increases across the threshold value. (c) As $ v $ increases further, the duration of the bimodality prolongs with its second peak moving to the right.

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Figure 2. Dynamic transitions among three mRNA distribution modes. (a, b) Pattern (Ⅰ): If $ P_m(t) $ takes a unimodal distribution at steady-state, then for sufficient large synthesis rate $ v = 10 \delta $, its profile develops from the original decaying to the intermediate bimodality, and finally switches to the unimodality. However, when $ v $ decreases below the threshold value $ v = 5 \delta $, the intermediate bimodality disappears. (c) Pattern (Ⅱ): If $ P_m(t) $ takes a bimodal distribution at steady-state, then its profile transits from the decaying to the bimodality at some time points, and maintains the bimodality in a long run. (d) Pattern (Ⅲ): If $ P_m(t) $ takes a decaying distribution at steady-state, then it maintains the same distribution mode within the whole time regime. (Inset) Fano factor versus time for the three patterns.

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Figure 1. The three modes of the steady-state mRNA distribution. (a) When $ \bar{v} = v/ \delta $ is fixed, the $ \lambda $-$ \gamma $ plane can be divided into three connected regions, and the values of $ ( \lambda, \gamma) $ in each region generate a corresponding steady-state distribution mode [11]: (b) The decaying distribution that $ P_m $ deceases in $ m $ for $ m = 0, 1, 2, \cdots $; (c) The unimodal distribution that $ P_m $ takes exactly one peak at some $ m>0 $; (d) The bimodal distribution that $ P_m $ takes exactly two peaks with the first one at $ m = 0 $, and the other one at some $ m>0 $

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