Asymptotic limit in a cell differentiation model with consideration of transcription

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). These factors are translated by the mRNA whose levels of expression, y1 and y2, depend, respectively, on x1 and x2 in a nonlinear nonlocal way. The population density of T cells, ϕ(t,x1,x2,y1,y2), satisfies a hyperbolic conservation law with coefficients depending nonlinearly and nonlocally on (t,x1,x2,y1,y2), while the xi, yi satisfy a system of ordinary differential equations. We study the long time behavior of ϕ and show, under some conditions on the parameters of the system of differential equations, that the gene expressions in the T-cell population aggregate at one, two or four points, which connect to various cell differentiation scenarios.