# American Institute of Mathematical Sciences

October  2018, 15(5): 1255-1270. doi: 10.3934/mbe.2018058

## The mean and noise of stochastic gene transcription with cell division

 1 Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China 2 College of Science, Guangxi University of Science and Technology, Liuzhou 545006, China 3 School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510275, China

* Corresponding author: Jianshe Yu

Received  January 31, 2018 Revised  April 16, 2018 Published  May 2018

Fund Project: The authors are supported by National Natural Science Foundation of China (11631005, 11461002), Program for Changjiang Scholars and Innovative Research Team in University (IRT_16R16) and the Innovation Research Grant for the Postgraduates of Guangzhou University (2017GDJC-D01).

Life growth and development are driven by continuous cell divisions. Cell division is a stochastic and complex process. In this paper, we study the impact of cell division on the mean and noise of mRNA numbers by using a two-state stochastic model of transcription. Our results show that the steady-state mRNA noise with symmetric cell division is less than that with binomial inheritance with probability 0.5, but the steady-state mean transcript level with symmetric division is always equal to that with binomial inheritance with probability 0.5. Cell division except random additive inheritance always decreases mean transcript level and increases transcription noise. Inversely, random additive inheritance always increases mean transcript level and decreases transcription noise. We also show that the steady-state mean transcript level (the steady-state mRNA noise) with symmetric cell division or binomial inheritance increases (decreases) with the average cell cycle duration. But the steady-state mean transcript level (the steady-state mRNA noise) with random additive inheritance decreases (increases) with the average cell cycle duration. Our results are confirmed by Gillespie stochastic simulation using plausible parameters.

Citation: Qi Wang, Lifang Huang, Kunwen Wen, Jianshe Yu. The mean and noise of stochastic gene transcription with cell division. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1255-1270. doi: 10.3934/mbe.2018058
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##### References:
Modeling of two-state stochastic model of transcription with cell division. A. Kinetic scheme for describing two-state transcription model, where G and G$'$ denote the gene is active and inactive, respectively. B. A simplified diagram intuitively illustrates the cell cycle. The $i$th cell cycle is from ${\rm W}_{i-1}$ to ${\rm W}_{i}$, the duration of the $i$th cell cycle is ${\rm T}_i$. The cell division events are indicated by arrows.
Schematic diagrams for the time evolution with cell division. The cell division events are indicated by green arrows. W$_i$ stands for the $i$th cell division point, $\tau_i$ is the value of T$_i$, $t_{i+1}$ is the elapsed time since the $i$th (the recent) cell division, then $t = \sum^{i}_{j = 1}\tau_j+t_{i+1}$.
Schematic diagram for cell division modes, where the arrow points to the aim daughter cell. A. Symmetric cell division. B. Binomial inheritance. C. Random subtractive inheritance. D. Random additive inheritance.
Temporal changes in the mean transcript level. The red solid lines represent analytic solutions and the blue dashed lines with circles sign represent numerical solutions. A. Temporal changes in the mean transcript level with BR. B. Temporal changes in the mean transcript level with AS.
Influence of mean cell-cycle length $\tau$ on the steady-state mean transcript level and steady-state mRNA noise, where the cell cycle obeys log-normal distribution. The black stars, yellow stars, magenta stars, blue stars, red stars and green stars represent the steady-state mean transcript level (the steady-state mRNA noise) with S, BF, BR, RA, AS and RS, respectively, where the dashed lines represent the fittings. A. Influence of mean cell-cycle length $\tau$ on the steady-state mean transcript level. B. Influence of mean cell-cycle length $\tau$ on the steady-state mRNA noise.
The steady-state mean transcript level (the steady-state mRNA noise) with S, BF, BR, and different cell cycle distributions, where the mean of the cell cycle is $\tau = 120$.
 S BF BR Constant cell cycle 86.4910 (0.0402) 86.4910 (0.0408) 86.4924 (0.0408) Exponential distribution 87.2826 (0.0395) 87.2826 (0.0401) 87.2840 (0.0401) Log-normal distribution 86.7222 (0.0402) 86.7222 (0.0407) 86.7257 (0.0408) Erlang distribution 86.8273 (0.0400) 86.8273 (0.0406) 86.8276 (0.0407) Uniform distribution 86.7222 (0.0402) 86.7222 (0.0407) 86.7257 (0.0408)
 S BF BR Constant cell cycle 86.4910 (0.0402) 86.4910 (0.0408) 86.4924 (0.0408) Exponential distribution 87.2826 (0.0395) 87.2826 (0.0401) 87.2840 (0.0401) Log-normal distribution 86.7222 (0.0402) 86.7222 (0.0407) 86.7257 (0.0408) Erlang distribution 86.8273 (0.0400) 86.8273 (0.0406) 86.8276 (0.0407) Uniform distribution 86.7222 (0.0402) 86.7222 (0.0407) 86.7257 (0.0408)
The steady-state mean transcript level (the steady-state mRNA noise) with RA, AS, RS and different cell cycle distributions, where the mean of the cell cycle is $\tau = 120$.
 RA AS RS Constant cell cycle 96.0226 (0.0353) 94.1360 (0.0367) 92.2314 (0.0381) Exponential distribution 95.9311 (0.0356) 94.1327 (0.0367) 92.3301 (0.0379) Log-normal distribution 95.7876 (0.0354) 94.1105 (0.0367) 92.4612 (0.0379) Erlang distribution 95.9445 (0.0354) 94.1248 (0.0367) 92.3034 (0.0380) Uniform distribution 95.9698 (0.0353) 94.1229 (0.0367) 92.2841 (0.0381)
 RA AS RS Constant cell cycle 96.0226 (0.0353) 94.1360 (0.0367) 92.2314 (0.0381) Exponential distribution 95.9311 (0.0356) 94.1327 (0.0367) 92.3301 (0.0379) Log-normal distribution 95.7876 (0.0354) 94.1105 (0.0367) 92.4612 (0.0379) Erlang distribution 95.9445 (0.0354) 94.1248 (0.0367) 92.3034 (0.0380) Uniform distribution 95.9698 (0.0353) 94.1229 (0.0367) 92.2841 (0.0381)
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