Asymptotic phases in a cell differentiation model

Avner Friedman, Chiu Yen Kao, Chih-Wen Shih*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


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 (x1) and GATA-3 (x2). The population density of the T cells, φ{symbol} (t, x1, x2), satisfies a conservation law ∂ φ{symbol} / ∂ t + (∂ / ∂ x1) (f1 φ{symbol}) + (∂ / ∂ x2) (f2 φ{symbol}) = g φ{symbol} where fi depends on (t, x1, x2) and, in a nonlinear nonlocal way, on φ{symbol}. It is proved that, as t → ∞, φ{symbol} (t, x1, x2) converges to a linear combination of 1, 2, or 4 Dirac measures. Numerical simulations and their biological implications are discussed.

Original languageEnglish
Pages (from-to)736-769
Number of pages34
JournalJournal of Differential Equations
Issue number3
StatePublished - 1 Aug 2009


  • Cell differentiation
  • Conservation law
  • Integro-differential equation
  • Multistationary
  • Th1/Th2 cells
  • Transcription factors

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