  To explain the differentiation of stem cells in terms of dynamical systems
theory, models of interacting cells with intracellular protein expression
dynamics are analyzed and simulated. Simulations were carried out for all
possible protein expression networks consisting of two genes under cell--cell
interactions mediated by the diffusion of a protein. Networks that show cell
differentiation are extracted and two forms of symmetric differentiation based
on Turing's mechanism and asymmetric differentiation are identified. In the
latter network, the intracellular protein levels show oscillatory dynamics at a
single-cell level, while cell-to-cell synchronicity of the oscillation is lost
with an increase in the number of cells. Differentiation to a fixed-point type
behavior follows with a further increase in the number of cells. The cell type
with oscillatory dynamics corresponds to a stem cell that can both proliferate
and differentiate, while the latter fixed-point type only proliferates. This
differentiation is analyzed as a saddle-node bifurcation on an invariant
circle, while the number ratio of each cell type is shown to be robust against
perturbations due to self-consistent determination of the effective bifurcation
parameter as a result of the cell--cell interaction. Complex cell
differentiation is designed by combing these simple two-gene networks. The
generality of the present differentiation mechanism, as well as its biological
relevance, is discussed.
