# American Institute of Mathematical Sciences

April  2019, 24(4): 1543-1568. doi: 10.3934/dcdsb.2018219

## Fluctuations of mRNA distributions in multiple pathway activated transcription

 1 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China 2 Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China 3 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

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

Received  December 2017 Revised  February 2018 Published  June 2018

Fund Project: This work was supported by Natural Science Foundation of China grants (Nos. 11631005 and 11626246) and Program for Changjiang Scholars and Innovative Research Team in University (No. IRT_16R16).

Randomness in gene transcription can result in fluctuations (noise) of messenger RNA (mRNA) levels, leading to phenotypic plasticity in the isogenic populations of cells. Recent experimental studies indicate that multiple pathway activation mechanism plays an important role in the regulation of transcription noise and cell-to-cell variability. Previous theoretical studies on transcription noise have been emphasized on exact solutions and analysis for models with a single pathway or two cross-talking pathways. For stochastic models with more than two pathways, however, exact analytical results for fluctuations of mRNA levels have not been obtained yet. We develop a gene transcription model to examine the impact of multiple pathways on transcription noise for which the exact fluctuations of mRNA distributions are obtained. It is nontrivial to determine the analytical results for transcription fluctuations due to the high dimension of system parameter space. At the heart of our method lies the usage of the model's intrinsic symmetry to simplify the complicated calculations. We show the symmetric relation among system parameters, which allows us to derive the analytical expressions of the dynamical and steady-state fluctuations and to characterize the nature of transcription noise. Our results not only can be reduced to previous ones in limiting cases but also indicate some differences between the three or more pathway model and the single or two pathway one. Our analytical approaches provide new insights into the role of multiple pathways in noise regulation and optimization. A further study for stochastic gene transcription involving multiple pathways may shed light on the relation between transcription fluctuation and genetic network architecture.

Citation: Genghong Lin, Jianshe Yu, Zhan Zhou, Qiwen Sun, Feng Jiao. Fluctuations of mRNA distributions in multiple pathway activated transcription. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1543-1568. doi: 10.3934/dcdsb.2018219
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##### References:
Schematic representation of gene transcription activated by multiple signaling pathways. (a) Entangling pathways $S_1$, $S_2$, $\cdots$, $S_n$ converge to the promoter of a target gene to activate gene transcription. (b) The pathway $S_i$ has a probability $q_i$ to activate the transcription with constant kinetic rates $\lambda_i$, $\gamma$, $\nu$, and $\delta$
Different dynamical behaviors of the noise strength $\Phi^{*}$ on $\lambda_3$. The three curves are generated by the analytical form (47) with $n = 4$, $q_1 = 0.3$, $q_2 = 0.25$, $q_3 = 0.15$, $q_4 = 0.3$, $\lambda_1 = 0.15$, $\lambda_2 = 0.26$, $\lambda_4 = 10.18$, $\nu = 2.1$, $\delta = 1$, and $\gamma$ respectively equals $10.65$, $0.03$, and $0.56$ in (a), (b), and (c)
Nonlinear dependance of the noise strength $\Phi^{*}$ on $P_E^*$. The up and down dependance curve of $\Phi^{*}$ for $0.005<P_E^*<0.997$ is generated by varying $\gamma$ from $0.0001$ to $5.5$, where $\lambda_1 = 0.015$, $\lambda_2 = 1.53$, $\nu = 7.5$, $\delta = 0.1$, and $q_1 = q_2 = 0.5$
Distinct dynamical behaviors of the noise strength $\Phi^{*}$ on $\lambda_2$ with or without the constraint on the mean $m^*$. The two curves are generated by the analytical form (47) with $n = 2$, $\gamma = 0.3$, $\nu = 15.4$, $\delta = 1$, and $q_1 = q_2 = 0.5$. (a) The mean transcriptional level is fixed as $m^* = 4.5$. (b) No constraint on the mean $m^*$ and $\lambda_1 = 0.0856$
The initial states at time $t$ and transition probabilities toward the terminal state $(E,m)$ at time $t+h$. If the gene is ON with $m$ copies of the mRNA molecules at $t+h$, then all of the initial states at time $t$, listed in (a), (b), (c), and (d), can reach $(E,m)$ with a transition probability of zero or first order of $h$
 Initial State Terminal State Transition Probability (a) $(E,m)$ $(E,m)$ $P_{E}(m,t)\cdot(1-\nu h)(1-\gamma h)(1-m \delta h)$ (b) $(E,m+1)$ $(E,m)$ $P_{E}(m+1,t)\cdot(m+1)\delta h$ (c) $(E,m-1)$ $(E,m)$ $P_{E}(m-1,t)\cdot\nu h$ (d) $(O_i,m)$ $(E,m)$ $P_{i}(m,t)\cdot\lambda_i h$
 Initial State Terminal State Transition Probability (a) $(E,m)$ $(E,m)$ $P_{E}(m,t)\cdot(1-\nu h)(1-\gamma h)(1-m \delta h)$ (b) $(E,m+1)$ $(E,m)$ $P_{E}(m+1,t)\cdot(m+1)\delta h$ (c) $(E,m-1)$ $(E,m)$ $P_{E}(m-1,t)\cdot\nu h$ (d) $(O_i,m)$ $(E,m)$ $P_{i}(m,t)\cdot\lambda_i h$
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