Asymptotic phases in a cell differentiation model

Abstract

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, phi(t, x(1), x(2)), satisfies a conservation law partial derivative phi/partial derivative t + (partial derivative/partial derivative x(1))(f(1)phi) + (partial derivative/partial derivative x(2))(f(2)phi) = g phi where f(i) depends on (t, x(1), x(2)) and, in a nonlinear nonlocal way, on phi. It is proved that, as t -> infinity, (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. (C) 2009 Elsevier Inc. All rights reserved.