# A mathematical model for simulating cell collisions

## 1.Introduction

In the experiment, we successfully produce two kinds of peptide chain, which have the shape of a cross. The peptide chains are evenly distributed over the cell membrane. If we let cells collide randomly, they would probably connected by the peptide chain. After some period of collisions, a certain number of cells could build up a reticular structure.

Figure 1: peptide chain | Figure 2: Polypeptide cell |

But how likely is it that a reticular structure will come into being?

Here we establish a simple mathematical model to simulate collision process of cells.

## 2.Methodology

### 2.1 Assumption

- When two cells collide, the colliding parts of them are totally random
- The movement of cells is consistent with macroscopic motion.
- If the ball is not connected after collision, it is separated by elastic collision
- The probability of formation of network structure is expressed by its concentration

### 2.2 model description

Firstly, we discuss the collision process of two identical macro spheres, which both have a shadow area.

Obviously, when two balls collide, the probability that the colliding points are both in the shadow area can be calculated by the following equation. |

In the equation, we let be the area of shadow part, and be the area of surface area of a ball.

As the peptide chains are uniformly distribute over the cells, we can regard the place that covered by the peptide chains as the shadow area of the ball. Then we will simulate cells’ collision by means of the collision process of spheres.

Secondly, with the help of computer simulation program, we build up a cubic container, and simulate the radon movement and collision of a large number of small balls (cells) inside. We believe that two colliding balls have the possibility of P to be connected. Otherwise, they would separate following the process of elastic collision whose formula is showed as follow. |

Thirdly, the criterion for the formation of a network structure is that the cell connector has a certain area. That is to say, when n cells are linked together, we can say that a network structure is formed. After a fixed period of time, we calculate the number of network structures which are in the container so that we can figure out the concentration of network structure |