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two solutions are selected. The question we address is as follows: how are these states selected and what determines a possible range in the number distribution of the two states when intercellular interactions through v are taken into account. In the following sections, we analyze three models with different types of the function g( u$_{1}$, . . . , u$_{N}$, v ) to study how the differences in the kinetics of v lead to different types of regulation in the number distribution of cell types. 

## 3 Model I: Cell Fate Determination by Total Cell Number 

As a first example of interacting cells, we adopt a model in which each cell simply emits the signal v with the same rate. The kinetics of v obeys the following equation, 

## model I 

<img src='content_image/321797.jpg'>

while the kinetics of { u$_{i}$ } obey Eq. (1). We are interested in the behavior of the station- ary state as a function of the total cell number N . The stationary state solution of an ensemble of cells is generally obtained by the following procedure. First, we regard the signal v as a fixed parameter, not a variable, and obtain the solution u$_{i}$ as a function of v , as already described in the previous section. Next, we write down v as a function of N and { u$_{i}$ } so that the self-consistent solution of the coupled equation is obtained, from which we analyze the dependence of the solution on the total cell number. 

The stationary state is simply obtained by du$_{i}$/dt = 0 and dv/dt = 0. In the present case, the solution v is independent of { u$_{i}$ } , and depends only on N , which leads to 

<img src='content_image/321798.jpg'>

The solution curve f ( u$_{i}$, v ( N )) = 0 is shown in Fig. 2, and the numerical result of the 

