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

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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

Abstract
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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 and Continuous Dynamical Systems - B, 2016, 21 (10) : 3575-3602. doi: 10.3934/dcdsb.2016111

References:

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show all references

References:

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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), 989-998. doi: 10.1111/j.1398-9995.2011.02576.x.
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[14]	C. Dong, Diversification of T-helper-cell lineages: Finding the family root of IL-17-producing cells, Nat. Rev. Immunol., 6 (2006), 329-334. doi: 10.1038/nri1807.
[15]	C. Dong, $T_H$17 cells in development: An updated view of their molecular identity and genetic programming, Nat. Rev. Immunol., 8 (2008), 337-348.
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[18]	M. A. Fishman and A. S. Perelson, Th1/Th2 differentiation and cross-regulation, Bull. Math. Biol., 61 (1999), 403-436. doi: 10.1006/bulm.1998.0074.
[19]	J. E. Gereda, D. Y. M. Leung, A. Thatayatikom, J. E. Streib, M. R. Price, M. D. Klinnert and A. H. Liu, Relation between house-dust endotoxin exposure, type 1 T-cell development, and allergen sensitisation in infants at high risk of asthma, Lancet, 355 (2000), 1680-1683. doi: 10.1016/S0140-6736(00)02239-X.
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[21]	L. Gorelik and R. A. Flavell, Abrogation of TGF$\beta$ signaling in T cells leads to spontaneous T cell differentiation and autoimmune disease, Immunity, 12 (2000), 171-181.
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