Regulation of differentiation in a population of cells interacting through a common pool

We consider a model of a suspension of a cell population in a well-mixed medium. There are two chemical substances, say A and H, reacting in each cell of the population and the substance H can only diffuse from the inside of cell to the medium or vice versa across the cell membrane. The medium is well mixed that the concentration of H is kept uniform over the medium. Cells interact indirectly with each other through the medium. The differential equations governing the dynamics of the suspension are analyzed using standard techniques for differential equations. It is shown that the cell population divides into several groups in respect of the chemical concentrations as time elapses. It is also shown how the fraction of the number of cells belonging to each subgroup to the total number of cells is regulated. The results may be used to explain the mechanism for differentiation of multi-cellular organisms.