Asymptotic phases in a cell differentiation model

T cells of the immune system, upon maturation, differentiate into either Th1 or Th2 cells that have different functions. The decision to which cell type to differentiate depends on the concentrations of transcription factors T-bet ( x 1 ) and GATA-3 ( x 2 ). The population density of the T cells, ϕ...

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Bibliographic Details
Published in:Journal of Differential Equations 2009-08, Vol.247 (3), p.736-769
Main Authors: Friedman, Avner, Kao, Chiu-Yen, Shih, Chih-Wen
Format: Article
Language:English
Subjects:
Cell differentiation
Conservation law
Integro-differential equation
Multistationary
Th1/Th2 cells
Transcription factors
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Summary:T cells of the immune system, upon maturation, differentiate into either Th1 or Th2 cells that have different functions. The decision to which cell type to differentiate depends on the concentrations of transcription factors T-bet ( x 1 ) and GATA-3 ( x 2 ). The population density of the T cells, ϕ ( t , x 1 , x 2 ) , satisfies a conservation law ∂ ϕ / ∂ t + ( ∂ / ∂ x 1 ) ( f 1 ϕ ) + ( ∂ / ∂ x 2 ) ( f 2 ϕ ) = g ϕ where f i depends on ( t , x 1 , x 2 ) and, in a nonlinear nonlocal way, on ϕ. It is proved that, as t → ∞ , ϕ ( t , x 1 , x 2 ) converges to a linear combination of 1, 2, or 4 Dirac measures. Numerical simulations and their biological implications are discussed.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2009.03.033