Derivation of rigorous relationships between proliferation and transition rates of multiple cells by algebraic approach

The development of a multicellular organism is a dynamic process. Starting with one or a few cells, the organism develops into different types of cells with distinct functions to survive. We have constructed a simple model with a cell number increase and a cell-type order conservation to assess conditions for cell-type diversity. This model is based on stochastic Lindenmayer system for three types of cells. In the present model, we have successfully derived rigorous algebraic relations between the proliferation and transition rates for cell-type diversity by using a symbolic method, quantifier elimination (QE). Surprisingly, three modes for the proliferation and transition rates emerge against various ratios of the initial cells to the developed cells. Furthermore, it has been revealed that the high cell-type diversity originates from the order conservation. During the developing process of multicellular organisms, complex but explicit relations exist between the cell-type diversity and the development rates.