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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

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.
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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References:
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S. Al-Muhsen, S. Letuve, A. Vazquez-Tello, M. A. Pureza, H. Al-Jahdali, A. S. Bahammam, Q. Hamid and R. Halwani, Th17 cytokines induce pro-fibrotic cytokines release from human eosinophils,, Respir. Res., 14 (2013). doi: 10.1186/1465-9921-14-34. Google Scholar

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[6]

R. L. Bar-Or and L. A. Segel, On the role of a possible dialogue between cytokine and TCR-presentation mechanisms in the regulation of autoimmune disease,, J. Theor. Biol., 190 (1998), 161. doi: 10.1006/jtbi.1997.0545. Google Scholar

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U. Behn, H. Dambeck and G. Metzner, Modeling Th1-Th2 regulation, allergy, and hyposensitization,, in Dynamical Modeling in Biotechnology, (2001), 227. doi: 10.1142/9789812813053_0011. Google Scholar

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[9]

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[10]

J. Carneiro, J. Stewart, A. Coutinho and G. Coutinho, The ontogeny of class-regulation of CD4$^+$ T lymphocyte populations,, Int. Immunol., 7 (1995), 1265. doi: 10.1093/intimm/7.8.1265. Google Scholar

[11]

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[12]

L. Cosmi, F. Liotta, E. Maggi, S. Romagnani and F. Annunziato, Th17 cells: New players in asthma pathogenesis,, Allergy, 66 (2011), 989. doi: 10.1111/j.1398-9995.2011.02576.x. Google Scholar

[13]

E. Cutz, H. Levison and D. M. Cooper, Ultrastructure of airways in children with asthma,, Histopathology, 2 (1978), 407. doi: 10.1111/j.1365-2559.1978.tb01735.x. Google Scholar

[14]

C. Dong, Diversification of T-helper-cell lineages: Finding the family root of IL-17-producing cells,, Nat. Rev. Immunol., 6 (2006), 329. doi: 10.1038/nri1807. Google Scholar

[15]

C. Dong, $T_H$17 cells in development: An updated view of their molecular identity and genetic programming,, Nat. Rev. Immunol., 8 (2008), 337. Google Scholar

[16]

S. C. Eisenbarth, D. A. Piggott, J. W. Huleatt, I. Visintin, C. A. Herrick and K. Bottomly, Lipopolysaccharide-enhanced, toll-like receptor 4-dependent T helper cell type 2 responses to inhaled antigen,, J. Exp. Med., 196 (2002), 1645. doi: 10.1084/jem.20021340. Google Scholar

[17]

R. L. Elliott and G. C. Blobe, Role of transforming growth factor beta in human cancer,, J. Clin. Oncol., 23 (2005), 2078. Google Scholar

[18]

M. A. Fishman and A. S. Perelson, Th1/Th2 differentiation and cross-regulation,, Bull. Math. Biol., 61 (1999), 403. doi: 10.1006/bulm.1998.0074. Google Scholar

[19]

J. E. Gereda, D. Y. M. Leung, A. Thatayatikom, J. E. Streib, M. R. Price, M. D. Klinnert and A. H. Liu, Relation between house-dust endotoxin exposure, type 1 T-cell development, and allergen sensitisation in infants at high risk of asthma,, Lancet, 355 (2000), 1680. doi: 10.1016/S0140-6736(00)02239-X. Google Scholar

[20]

L. Gorelik, S. Constant and R. A. Flavell, Mechanism of transforming growth factor $\beta$-induced inhibition of T helper type 1 differentiation,, J. Exp. Med., 195 (2002), 1499. Google Scholar

[21]

L. Gorelik and R. A. Flavell, Abrogation of TGF$\beta$ signaling in T cells leads to spontaneous T cell differentiation and autoimmune disease,, Immunity, 12 (2000), 171. Google Scholar

[22]

F. Gross, G. Metznerb and U. Behn, Mathematical modelling of allergy and specific immunotherapy: Th1-Th2-Treg interactions,, J. Theor. Biol., 269 (2011), 70. doi: 10.1016/j.jtbi.2010.10.013. Google Scholar

[23]

G. Grünig, M. Warnock, A. E. Wakil, R. Venkayya, F. Brombacher, D. M. Rennick, D. Sheppard, M. Mohrs, D. D. Donaldson, R. M. Locksley and D. B. Corry, Requirement for IL-13 independently of IL-4 in experimental asthma,, Science, 282 (1998), 2261. Google Scholar

[24]

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