Relation between intersection of nullclines and periodic solutions in a differential equations of p53 oscillator

Iranian Journal of Numerical Analysis and Optimization

مقاله 4، دوره 3، شماره 2 - شماره پیاپی 4، مهر 2013، صفحه 67-77   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.v3i2.24083

نویسندگان

F. Rangi^{* 1}؛ M. Tavakoli²

¹Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.

²Pardis of Hasheminejad, Farhangian University, Mashhad, Iran.

چکیده

We consider a simple mathematical model that suggests emergence of oscillations in p53 and Mdm2 protein levels in response to stress signal. Intracellular activity of the p53 protein is regulated by a transcriptional target, Mdm2, through a feedback loop. The model is classified in five cases with respect to intersection of nullclines. In each case occurrence(or not) of the limit cycle is investigated.

کلیدواژه‌ها

DNA damage؛ p53-Mdm2؛ Limit cycle؛ Mathematical biology

مراجع

1. Bar-Or, R. L., Maya, R., Segel, L. A., Alon, U., Levine, A. J. and Oren, M. Geneneration of oscillations by the p53-Mdm2 feedback loop:Atheorical and experimental study, Departements of Molecurlar Cell
Biology and Applied Mathematics and Computer Scince, 97 (2000), 11250–11255.
2. Chicarmane,V., Ray,A., Sauro, H. M. and Nadim, A. A model for p53 dynamics trigged by DNA damage, SAIM J.Applied Dynamicals systems, 6 (2007), 1, 61–78.

3. Ciliberto, A., Novake, B. and Tyson, J.J. Steady States and Oscillations in the p53/Mdm2 Network Cell Cycle 4:3. , 130 (2005), 488–493.
4. Ge H. and Qian, M. Boolean Network Approach to negative Feedback Loops of the p53 Pathways: Synchronized Dynamics and stochatics Limit cycle, Jornal of computation biology, 6, 1 (2009), 119–132.
5. Hill, A. V. The possible effects of the aggregation of the molecules of hamoglobin on its dissosiation curves, J. Physiol., 40 (1910), 4–7.
6. Hirsch, M. W., Smale, S. and Devaney, R. L. Differntial equation , dynamical systems and introduction to chaos, second edition, Elsevier, USA, 2004.
7. Lahva, G. Oscillations by the p53-Mdm2 feedback loop, Journal of Construction Engineering and Manage-ment, 131 (2005), 1115–1123.
8. Lahva, G., Rosenfeld, N., Sigal, A., Geva-Zatorsky, N., Levine, A.J., Elowitz, M. B. and Alon, U. Dynamics of the p53-Mdm2 feedback loop in the individual cells, Nature Genetics, 36 (2004), 147–150.
9. Ma, L., Wangner, J., Rice, J. J., Hu, W., Levvien, A. J. and Storlovitzky, A. A plausible model for the digital response of p53 to DNA damage, Proc. Nat. Acad. Scien., 102 (2005), 14266–14271.
10. Monk, N. A. M. and Rice, J. J.Oscillatory expression of Her1, p53, and NF-kappaBdriven by transcriptional time delays, Current Biology, 97 (2000), 11250–11255.
11. Murray, J. D. Matematical Biology ,3rd edition, Springer-Verlag Berlin Heidelberg, 1991.
12. Rabiei motlagh, O. and Afsharnezhad, Z. On the conditions for which the Atm protein switch off the DNA damage signal in a p53 model, Studia Universitatis BAbes- Bolyal, Biologia, 1 (2010), 67–79.
13. Sauro, H., Adelinde, M., Uhrmacher, M., Harel, D., Hucka, M., Kwiatkowska, M., Mendes, P., Shaffer, C. A., Stromback, L. and Tyson, J. J. Challenges for modeling and simulation methods in systems boilogy,Proceedings of the 2006 Winter Simulation Conference, 67–79.
14. Tiana, G., Jensen, M. H. and Sneppen, K. Time delay as a key to opoptosis induction in the p53 network, Euro. Phys. j., 39 (2002), 135–140.
15. Tyson, J. J. Monitoring p53’s pulse, Nature Genetics, 36 (2004), 113–114.
16. Wangner, J., Ma, L., Rice, J. J., Hu, W., levvien, A. J. and Storlovitzky, A. p53-Mdm2 loop controlled by abalance of its feedback strenght and effective dampening using Atm and delayed feedback,IEEE Proc. System Biology., 152 (2005), 109–118.