content_image/318162.jpg content_image/318163.jpg
ratio of the number of each cell type to the total cell number is shown in Fig. 3. Here we define a single-cluster of an ensemble of cells as a state in which all the cells take the same stationary states, i.e., 

<img src='content_image/318162.jpg'>

and a two-cluster state as that in which two cell types with u = u$_{(1)}$ and u = u$_{(2)}$ coexist, so that 

<img src='content_image/318163.jpg'>

Here N$_{(1)}$ and N$_{(2)}$ denote the number of the cells with u = u$_{(1)}$ and u = u$_{(2)}$ , respectively. 

When the cell number N is lower than a threshold N ∗ 1 (= v ∗ $_{1}$/c$_{1}$ ), the single-cluster state of u$_{(1)}$ is realized, while for N larger than a threshold N ∗ 2 (= v ∗ $_{2}$/c$_{1}$ ), the single-cluster state of u$_{(2)}$ is realized, irrespectively of the initial cell state. Only within the range of N ∗ 1 ≤ N ≤ N ∗ 2 are two-cluster states of u$_{(1)}$ and u$_{(2)}$ possible, where any population ratio of the cell types with u$_{(1)}$ to u$_{(2)}$ can be realized depending on the initial condition. Cell types switch between u$_{(1)}$ and u$_{(2)}$ simply by the total cell number, and the signal v works as a population size detector. 

## 4 Model II: Diversification from Single State, and Size Reg- ulation of Specific Cell Type 

Next, we consider the case in which the signal induction depends on the expression level of u$_{i}$ . We will show that the cells are differentiated into two types over a wide range of the total cell number N , and that the number of type 1 cells remains at a same level herein. 

