Uniqueness and traveling waves in a cell motility model
Feng Jiao 1,2, , Qiwen Sun 1,2, , Genghong Lin 1,2, and Jianshe Yu 1,,
Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
* Corresponding author: Jianshe Yu, Email: jsyu@gzhu.edu.cn
Received April 2018 Revised May 2018 Published October 2018
Fund Project: This work was supported by Natural Science Foundation of China grants (11631005, 11501138), Guangdong Natural Science Foundation (2016A030313542), Program for Guangzhou Municipal College and University (1201630273), and the Program for Changjiang Scholars and Innovative Research Team in University (IRT_16R16)
Gene transcription is a stochastic process, manifested by the heterogeneous mRNA distribution in an isogenic cell population. Bimodal distribution has been observed in the transcription of stress responsive genes which have evolved to be easily turned on and easily turned off. This is against the conclusion in the classical two-state model that bimodality occurs only when the gene is hardly turned on and hardly turned off. In this paper, we extend the gene activation process in the two-state model by introducing the cross-talking pathway that involves the random selection between a spontaneous weak basal pathway and a stress-induced strong signaling pathway. By deriving exact forms of mRNA distribution at steady-state, we find that the cross-talking pathway is much more likely to trigger the bimodal distribution. Our further analysis reveals an observed transition among the decaying, bimodal and unimodal mRNA distribution for stress gene upon enhanced stimulations. Especially, the bimodality occurs when the stress-induced signalling pathway is more frequently selected, reinforcing the assertion that bimodal transcription is a general feature of stress genes in response to environmental change.
Keywords: Stochastic gene transcription, cross-talking pathway, bimodal distribution.
Mathematics Subject Classification: Primary: 34F05, 37H10, 60J20; Secondary: 92C37, 92C40.
Citation: Feng Jiao, Qiwen Sun, Genghong Lin, Jianshe Yu. Distribution profiles in gene transcription activated by the cross-talking pathway. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2799-2810. doi: 10.3934/dcdsb.2018275
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Figure 1. Three modes of mRNA distribution generated by the two-state modal. (a) The decaying distribution for which $P_m$ deceases in $m$ for $m = 0, 1, 2, \cdots$. (b) The unimodal distribution for which $P_m$ takes exactly one peak at some $m>0$. (c) The bimodal distribution for which $P_m$ takes exactly two peaks with the first one at $m = 0$, and the other one at some $m > 0$. The parameter sets $(k_{on}, k_{off}, k_b, k_d)$ in (a), (b), and (c) are chosen as $(0.5, 1.5, 20, 1), (2, 2, 10, 1)$, and $(0.1, 0.2, 15, 1)$, respectively
Figure 2. Gene transcription modulated by the cross-talking pathway [31]
Figure 3. Cross-talking pathway can generate bimodal distribution when the gene is (a) easily turned on (${\rm T}_{{\rm off}}<1$) and easily turned off (${\rm T}_{{\rm on}}<1$); (b) easily turned on (${\rm T}_{{\rm off}}<1$) but hardly turned off (${\rm T}_{{\rm on}}>1$); and (c) hardly turned on (${\rm T}_{{\rm off}}<1$) and hardly turned off (${\rm T}_{{\rm on}}>1$). mRNA synthesis rate $v = 30$ and the other parameters are set to be $(\lambda_1, \lambda_2, \gamma, q_1) = (0.6, 6, 2, 0.2)$ in (a), $(\lambda_1, \lambda_2, \gamma, q_1) = (0.2, 4, 0.2, 0.1)$ in (b), and $(\lambda_1, \lambda_2, \gamma, q_1) = (0.2, 4, 0.2, 0.2)$ in (c)
Figure 4. The transcriptional transition pattern of osmostress-responsive genes in yeast [22]. (a) At low salt concentrations (below 0.05 M NaCl solution), the weak basal pathway is more frequently selected, and therefore most cells are off, giving rise to the decaying distribution. (b) At intermediate salt concentrations, the stress-induced strong signaling pathway is more likely to be activated, leading to the bimodal distribution with two distinct cell identities representing silent and highly expressed cells. (c) At the highest salt concentrations (above 0.15 M NaCl solution), the strong pathway dominates the gene activation, giving rise to the most on cells and the unimodal distribution.
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