Preprint A686/2010
A Structured Population Model of Cell Di erentiation
Jorge P. Zubelli
Keywords: Structured population dynamics; transport equation; stem cells;cell differentiation
We introduce and analyze several aspects of a new model for cell differentiation that accounts for a continuous process of the differentiation of progenitor cells. Biologically, it subsumes that the differentiation of progenitor cells is a continuous process. From the mathematical point of view, it is based on partial differential equations of transport type. More specifically, it consists of a structured population equation with a nonlinear feedback loop. We compare the model presented herein to its discrete counterpart. In particular, we relate it with a multi-compartmental model of a discrete collection of cell subpopulations that was recently proposed by Marciniak-Czochra et al to investigate the dynamics of the hematopoietic system with cell proliferation and differentiation regulated by a nonlinear feedback loop. One of the novelties in the model presented in this context is the presence of nonlinearities in the coupling of the maturation equations through a velocity function. This models the signaling process due to cytokines, which regulates the differentiation or proliferation process. We obtain uniform bounds for the solutions, characterize steady state solutions, and analyze their stability. We show how persistence or extinction might occur according to certain parameters that characterize the stem cells self-renewal. We also perform extensive numerical simulations and discuss the qualitative behavior of the continuous models vis a vis the discrete ones.