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Pattern Formation and Oscillations in Reaction–Diffusion Model with p53-Mdm2 Feedback Loop

Qianqian Zheng

School of Mathematics and Statistics, Xuchang University, Xuchang 461000, P. R. China

College of Information Science and Technology, Donghua University, Shanghai 201620, P. R. China

David Geffen School of Medicine, University of California, Los Angeles, CA 90095, USA

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Jianwei Shen

http://orcid.org/0000-0003-1289-8674

School of Mathematics and Statistics, Xuchang University, Xuchang 461000, P. R. China

E-mail Address: xcjwshen@gmail.com

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Zhijie Wang

College of Information Science and Technology, Donghua University, Shanghai 201620, P. R. China

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https://doi.org/10.1142/S0218127419300404Cited by:12 (Source: Crossref)

Abstract

P53 plays a vital role in DNA repair, and several mathematical models of the p53-Mdm2 feedback loop were used to explain the biological mechanism. In this paper, a p53-Mdm2 model described by a delay reaction–diffusion equation is studied both analytically and numerically. This research aims to provide an understanding of the impact of delay and sustained pressure on the p53-Mdm2 dynamics and tries to explain some biological mechanism. It is found that the type of pattern formation is affected by Hopf bifurcation. Also, the amplitude equation in delay diffusive system is derived and it is shown that sustained stress plays an essential role in the function of p53. Finally, simulation is used to verify the theoretical results.

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References

Chen, J. X., Chen, Y. G. & Kapral, R. [2018a] “ Chemically propelled motors navigate chemical patterns,” Adv. Sci. 5, 1–8. Google Scholar
Chen, J. X., Zhang, H., Qiao, L. Y., Liang, H., Sun, W. G. & Kapral, R. [2018b] “ Interaction of excitable waves emitted from two defects by pulsed electric fields,” Commun. Nonlin. Sci. Numer. Simulat. 54, 202–209. Web of Science, Google Scholar
Dai, C. J., Zhao, M. & Yu, H. G. [2016] “ Dynamics induced by delay in a nutrient-phytoplankton model with diffusion,” Ecol. Compl. 26, 29–36. Web of Science, Google Scholar
Dutt, A. K. [2012] “ Amplitude equation for a diffusion–reaction system: The reversible Selkov model,” AIP Adv. 2, 1–24. Web of Science, Google Scholar
Geva-Zatorsky, N. et al. [2006] “ Oscillations and variability in the p53 system,” Molecul. Syst. Biol. 2, 2006.0033. Web of Science, Google Scholar
Harris, S. L. & Levine, A. J. [2005] “ The p53 pathway: Positive and negative feedback loops,” Oncogene 24, 2899–2908. Web of Science, Google Scholar
Haupt, Y., Maya, R., Kazaz, A. & Oren, M. [1997] “ Mdm2 promotes the rapid degradation of p53,” Nature 387, 296–299. Web of Science, Google Scholar
Kim, E. S. & Shohet, J. M. [2015] “ Reactivation of p53 via Mdm2 inhibition,” Cell Death Dis. 6, e1936. Web of Science, Google Scholar
Liu, H. & Wang, W. [2010] “ The amplitude equations of an epidemic model,” Sci. Technol. Engin. 10, 1929–1933. Google Scholar
Mihalaş, G. I., Neamţu, M., Opriş, D. & Horhat, R. F. [2006] “ A dynamic p53-Mdm2 model with time delay,” Chaos Solit. Fract. 30, 936–945. Web of Science, Google Scholar
Morgan, D. O. [2010] The Cell Cycle: Principles of Control (Science Press). Google Scholar
Robertson, B., Huang, J. X., Chen, Y. G. & Kapral, R. [2018] “ Synthetic nanomotors: Working together through chemistry,” Acc. Chem. Res. 51, 2355–2364. Web of Science, Google Scholar
Ruan, S. G. & Wei, J. J. [2003] “ On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,” Dyn. Contin. Discr. Impuls. Syst. Ser. A: Math. Anal. 10, 863–874. Web of Science, Google Scholar
Ryan, S. D. [2010] Introduction to Ordinary and Partial Differential Equations (Cleveland State University). Google Scholar
Sander, E. & Wanner, T. [2003] “ Pattern formation in a nonlinear model for animal coats,” J. Diff. Eqs. 191, 143–174. Web of Science, Google Scholar
Schmitz, U., Wolkenhauer, O. & Vera, J. [2013] A MicroRNA Cancer Regulation (Springer). Google Scholar
Sgura, I., Bozzini, B. & Lacitignola, D. [2012] “ Numerical approximation of oscillating Turing patterns in a reaction–diffusion model for electrochemical material growth,” AIP Conf. Proc. 1493, 896–903. Google Scholar
Shen, J. W., Liu, Z. R., Zheng, W. X., Xu, F. D. & Chen, L. N. [2009] “ Oscillatory dynamics in a simple gene regulatory network mediated by small RNAs,” Physica A 388, 2995–3000. Web of Science, Google Scholar
Turing, A. [1952] “ The chemical basis of morphogenesis,” Philos. Trans. Roy. Soc. B 237, 37–72. Web of Science, Google Scholar
Vousden, K. H. & Lane, D. P. [2007] “ p53 in health and disease,” Nature Rev. Molecul. Cell Biol. 8, 275–283. Web of Science, Google Scholar
Wei, J. & Ruan, S. [1999] “ The p53 pathway: Positive and negative feedback loops,” Oncogene 130, 255–272. Google Scholar
Xie, B., Wang, Z. & Xue, Y. [2014] “ Pattern formation in a predator–prey model with both cross diffusion and time delay,” Abstr. Appl. Anal. 2014, 392435-1–7. Google Scholar
Zhao, H., Yuan, J. & Hu, W. [2013] “ Time-delay control of stability and Turing pattern structures of neural networks,” Contr. Th. Appl. 30, 1–11. Google Scholar
Zheng, Q. Q. & Shen, J. W. [2013] “ Bifurcations and dynamics of cancer signaling network regulated by microRNA,” Discr. Dyn. Nat. Soc. 2013, 1–8. Web of Science, Google Scholar
Zheng, Q. Q. & Shen, J. W. [2018] “ Pattern dynamics of the reaction–diffusion immune system,” PLoS One 13, e0190176. Web of Science, Google Scholar