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December  2016, 21(10): 3575-3602. doi: 10.3934/dcdsb.2016111

Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype

Jisun Lim 1, , Seongwon Lee 2, and  Yangjin Kim 3,

1. 

School of Biological Sciences, Seoul National University, Seoul 08826, South Korea
2. 

Division of Mathematical Models, National Institute for Mathematical Sciences, Daejeon 34047, South Korea
3. 

Department of Mathematics, Konkuk University, Seoul, 05029, South Korea

Received  September 2015 Revised  September 2016 Published  November 2016

Airway exposure of lipopolysaccharide (LPS) is shown to regulate type I and type II helper T cell induced asthma. While high doses of LPS derive Th1- or Th17-immune responses, low LPS levels lead to Th2 responses. In this paper, we analyze a mathematical model of Th1/Th2/Th17 asthma regulation suggested by Lee (S. Lee, H.J. Hwang, and Y. Kim, Modeling the role of TGF-$\beta$ in regulation of the Th17 phenotype in the LPS-driven immune system, Bull Math Biol., 76 (5), 1045-1080, 2014) and show that the system can undergo a Hopf bifurcation at a steady state of the Th17 phenotype for high LPS levels in the presence of time delays in inhibition pathways of two key regulators: IL-4/Th2 activities ($H$) and TGF-$\beta$ levels ($G$). The time delays affect the phenotypic switches among the Th1, Th2, and Th17 phenotypes in response to time-dependent LPS doses via nonlinear crosstalk between $H$ and $G$. An extended reaction-diffusion model also predicts coexistence of these phenotypes under various biochemical and bio-mechanical conditions in the heterogeneous microenvironment.
Keywords: mathematical model, asthma, Delay differential equation, TGF$\beta$., Th1/Th2/Th17, Hopf bifurcation.
Mathematics Subject Classification: Primary: 92C45, 92C50; Secondary: 92B0.
Citation: Jisun Lim, Seongwon Lee, Yangjin Kim. Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3575-3602. doi: 10.3934/dcdsb.2016111
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show all references

References:
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Respir. Res., 14 (2013), p34. doi: 10.1186/1465-9921-14-34.  Google Scholar

[2]

Prog. Biophys. Mol. Biol., 85 (2004), 451-472. Google Scholar

[3]

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[4]

Springer Netherlands, 2006. doi: 10.1007/1-4020-3647-7.  Google Scholar

[5]

Clin. Immunol., 147 (2013), 27-37. Google Scholar

[6]

J. Theor. Biol., 190 (1998), 161-178. doi: 10.1006/jtbi.1997.0545.  Google Scholar

[7]

in Dynamical Modeling in Biotechnology, World Scientific, 2001, chapter 11, 227-243. doi: 10.1142/9789812813053_0011.  Google Scholar

[8]

Annu. Rev. Immunol., 12 (1994), 295-335. doi: 10.1146/annurev.iy.12.040194.001455.  Google Scholar

[9]

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[10]

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[11]

in Mechanisms in Radiobiology: Multicellular Organisms (eds. M. Errera and A. Forssberg), Elsevier, 1960, 95-205. doi: 10.1016/B978-1-4832-2829-7.50010-1.  Google Scholar

[12]

Allergy, 66 (2011), 989-998. doi: 10.1111/j.1398-9995.2011.02576.x.  Google Scholar

[13]

Histopathology, 2 (1978), 407-421. doi: 10.1111/j.1365-2559.1978.tb01735.x.  Google Scholar

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[15]

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[16]

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[17]

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[18]

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[19]

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[23]

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[24]

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[25]

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[26]

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[27]

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[32]

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PLoS One, 9 (2014), e102499. doi: 10.1371/journal.pone.0102499.  Google Scholar

[34]

Math. Bios. Eng, 10 (2013), 1095-1133. doi: 10.3934/mbe.2013.10.1095.  Google Scholar

[35]

Bull Math Biol, 75 (2013), 1304-1350. doi: 10.1007/s11538-012-9787-0.  Google Scholar

[36]

Discrete and Continuous Dynamical Systems-B, 18 (2013), 969-1015. doi: 10.3934/dcdsb.2013.18.969.  Google Scholar

[37]

Math. Models Methods Appl. Sci., 17 (2007), 1773-1798. doi: 10.1142/S0218202507002479.  Google Scholar

[38]

Prog Biophys Mol Biol, 106 (2011), 353-379. doi: 10.1016/j.pbiomolbio.2011.06.006.  Google Scholar

[39]

J. Immunol., 178 (2007), 5375-5382. doi: 10.4049/jimmunol.178.8.5375.  Google Scholar

[40]

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Ultrason. Imaging, 20 (1998), 260-274. doi: 10.1177/016173469802000403.  Google Scholar

[42]

Academic Press, 1993.  Google Scholar

[43]

J. Exp. Med., 201 (2005), 233-240. doi: 10.1084/jem.20041257.  Google Scholar

[44]

Bull. Math. Biol., 76 (2014), 1045-1080. doi: 10.1007/s11538-014-9946-6.  Google Scholar

[45]

Immunity, 30 (2009), 92-107. doi: 10.1016/j.immuni.2008.11.005.  Google Scholar

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Immunity, 31 (2009), 438-449. doi: 10.1016/j.immuni.2009.08.007.  Google Scholar

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[48]

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[49]

Journal of Theoretical Biology, 254 (2008), 178-196. doi: 10.1016/j.jtbi.2008.04.011.  Google Scholar

[50]

Clin. Exp. Allergy, 21 (1991), 441-448. doi: 10.1111/j.1365-2222.1991.tb01684.x.  Google Scholar

[51]

Allergy, 65 (2010), 1093-1103. doi: 10.1111/j.1398-9995.2010.02352.x.  Google Scholar

[52]

in Theoretical and Experimental Insights into Immunology (eds. A. S. Perelson and G. Weisbuch), vol. 66 of Nato ASI Subseries H:, Springer-Verlag Berlin Heidelberg, Berlin, Germany, 1992, 171-190. doi: 10.1007/978-3-642-76977-1_11.  Google Scholar

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Immunol. Today, 17 (1996), 138-146. doi: 10.1016/0167-5699(96)80606-2.  Google Scholar

[54]

J. Immunol., 136 (1986), 2348-2357. Google Scholar

[55]

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7th edition, Garland Science, 2007. Google Scholar

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Transplant. Proc., 44 (2012), 1035-1040. Google Scholar

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Nat. Med., 8 (2002), 567-573. doi: 10.1038/nm0602-567.  Google Scholar

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Respir. Res., 14 (2013), 35-43. doi: 10.1186/1465-9921-14-35.  Google Scholar

[61]

J. Mammary Gland Biol. Neoplasia, 9 (2004), 325-342. doi: 10.1007/s10911-004-1404-x.  Google Scholar

[62]

Mucosal Immunol., 3 (2010), 216-229. doi: 10.1038/mi.2010.4.  Google Scholar

[63]

J. Theor. Med., 4 (2002), 119-132. doi: 10.1080/10273660290022172.  Google Scholar

[64]

Clin. Exp. Allergy, 39 (2009), 1314-1323. doi: 10.1111/j.1365-2222.2009.03301.x.  Google Scholar

[65]

Curr. Opin. Immunol., 9 (1997), 773-775. doi: 10.1016/S0952-7915(97)80176-8.  Google Scholar

[66]

Cell, 101 (2000), 455-458. Google Scholar

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Annu. Rev. Immunol., 12 (1994), 635-673. Google Scholar

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in Mathematical Modeling of Biological Systems, Volume II (eds. A. Deutsch, R. Parra, R. J. de Boer, O. Diekmann, P. Jagers, E. Kisdi, M. Kretzschmar, P. Lansky and H. Metz), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, 2008, 145-155. Google Scholar

[69]

Immunology, 130 (2010), 166-171. doi: 10.1111/j.1365-2567.2010.03289.x.  Google Scholar

[70]

Science, 282 (1998), 2258-2261. doi: 10.1126/science.282.5397.2258.  Google Scholar

[71]

Nat. Rev. Immunol., 1 (2001), 69-75. doi: 10.1038/35095579.  Google Scholar

[72]

Chest, 138 (2010), 1282-1283. doi: 10.1378/chest.10-1440.  Google Scholar

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J. Theor. Biol., 206 (2000), 539-560. doi: 10.1006/jtbi.2000.2147.  Google Scholar

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J. Theor. Biol., 231 (2004), 181-196. doi: 10.1016/j.jtbi.2004.06.013.  Google Scholar

[75]

Science, 296 (2002), 490-494. doi: 10.1126/science.296.5567.490.  Google Scholar

[76]

Int. Arch. Allergy Immunol., 151 (2010), 297-307. doi: 10.1159/000250438.  Google Scholar

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Nat. Immunol., 8 (2007), 967-974. Google Scholar

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