Simulation of Cell Group Formation Regulated by Coordination
Number, Cell Cycle and Duplication Frequency
Shigehiro Hashimoto, Yusuke Morita, Takashi Kuromitsu, Motofumi Toda,Masa Mizobuchi, Tomohiro Sahara, Yue Wu, Shuji Negishi
Biomedical Systems, Dept. of Electronics, Information, and Communication Engineering,Osaka Institute of Technology, Osaka, 535-8585, Japan
hasimoto@elc.oit.ac.jp

ABSTRACT
Effects of coordination number, a cell cycle and duplication
frequency on cell-group formation have been investigated in acomputer simulation. In the simulation, multiplication occurs
in the last three steps of a cell cycle with a probability function
to give variations in the interval. Each cell has a constantcoordination number: four or six.
When a cell getssurrounded by adjacent cells, its status changes from an active
stage to a resting stage. Each cell repeats multiplication, anddisappears when the times of multiplication reach to the limit.
Variation was made in the coordination number, in the intervalof multiplication and in the limited times of multiplication.
The cells of the colony, which have the larger number ofcoordination, have reached the larger maximum populationand disappeared earlier.
Keywords: Simulation, Cell, Group Formation, Coordination
Number, Duplication Frequency, Cell Cycle and Population.

1. INTRODUCTION
In biological tissue, mitosis is regulated not only by a cell
cycle and duplication frequency, but also by the adjacent cells[1]. A cell enters mitosis after a certain period. Theduration of the cycle scatters in a certain range. The times ofmultiplication cannot exceed the programmed limit except
cancerous cells or generative cells. Mitosis of the normal cellis controlled by the contact of adjacent cells. When a cell issurrounded by others, it enters a resting stage and stops themultiplication process except cancerous cells.
The previous simulation shows that both the cell cycle and theduplication frequency control the life span of the colony, themaximum population of cells, the stability in population, andthe interaction between colonies [2].

The arrangement of cells varies with tissue [3]. The numberof surrounding cells in a colony has been fixed as acoordination number in the present study. The effects of thecoordination number, the cell cycle and the duplication
frequency on the group formation of cells have been
investigated in the computer simulation.

2. METHODS
The simulation consists of the following processes. Eachloop-step of calculation corresponds to time. The parametersof the coordination number, the cell cycle and the duplication
frequency were set at the beginning of the calculation process.
Cell cycle
The phase of the cell cycle in each cell progresses with time.
To give variations in the interval, the multiplication occurs in
the last three steps of the cell cycle with the probability
function (Fig. 1). Their multiplication probabilities are 25percent in the third step before last, 50 percent in the step
before last and 100 percent in the last step, respectively.
50% in the step before last
25% in the thirdstep before last

100% in thelast step

Fig. 1. Multiplication probabilities in the cell cycle.

Cell contact
A plane is divided into square or hexagonal elements: seats.
That makes the plane, in which each seat is surrounded by fouror six adjacent seats, respectively. Each cell occupies one
seat. The seats of 100 times 100 spread in the plane. A seatis selected for the initial location of a cell at the beginning of

the simulation. After the multiplication, two duplicated cells
occupy two seats: one is the original seat and the other is theadjacent seat to the original (Fig. 2).

programmed in "C" language and operated on "MicrosoftWindows".

Fig. 2. Each seat is surrounded by four (left) or six(right) adjacent seats.

Fig. 4(a). Cells repeat multiplication.

Fig. 3. A cell surrounded by adjacent cells changes its
status from the active stage to the resting stage.

When a cell gets surrounded by adjacent cells, the cell changes
its status from the active stage to the resting stage (Fig. 3). In
the resting stage the cell stops the multiplication process.
Duplication frequency
The cell repeats multiplication, and disappears, when the timesof multiplication reaches to the limit. When the seat adjacentto the cell in the resting stage becomes vacant, the cell returns
from the resting stage to the active stage and follows theprevious process.

Fig. 4(b). Cells increase in the number. Cells in thecenter part of the colony are in the resting stage withthe changed color.

Life span and population
When the calculation finishes about every cell existing in theplane, the step advances to the next and the time countincreases by one. The life span of the colony corresponds tothe step when all cells of the colony disappear. Population ofcells is counted at each step.
Display
All seats are displayed in the window at each step. Thechange of the stage from active to resting is displayed with thechange of the color of cells. Figures 4 & 5 exemplify thecolony with the coordination number of four and six,respectively. The colored cells at the rim show their active
stages in Figs. 4(b) & 5(b).
These processes wereFig. 4(c). Cells decrease to extinction.

3. RESULTS
In the simulation, variation has been made in three parameters:
the coordination number, the interval of multiplication and thelimited times of multiplication. Cells repeated multiplication(Figs. 4(a) & 5(a)), increased in the number (Figs. 4(b) & 5(b)),
and then decreased to extinction (Figs. 4(c) & 5(c)).
When the simulation starts from one cell located in the central
location of the plane, results have been as follows.

Fig. 5(a). Cells repeat multiplication.

In Figs. 6-9, the data points of triangle and of circle show thecoordination number of four and of six, respectively. Eachdata point shows the mean value of five trials. The scatter
range of each trial was below ten percent of the mean value.
When the times of multiplication is limited in the constantvalue of ten steps, the life span of the colony increases inproportion to the interval of multiplication; cell cycle (Fig.
6(a)). The maximum population of cells saturates, when thecell cycle is longer than ten steps (Fig. 6(b)). The
maximum population of cells scatters, when the cell cycle isshorter than ten steps. The life span in coordination numberof four is longer than that of six, although the maximum
population in coordination number of four is smaller than thatof six. The tendency has been similar, when the times ofmultiplication is limited in the constant value of twenty steps(Fig. 7).

Fig. 5(b). Cells increase in the number. Cells in thecenter part of the colony are in the resting stage with thechanged color.

Life span [steps]

4000
3000
2000
1000
00204060

Cell cycle [steps]
Fig. 6(a). Life span vs. cell cycle at times ofmultiplication of 10.
Fig. 5(c). Cells decrease to extinction.

120000
Life span [steps]

Maximum population [cells]

140000200100100000
80000
60000
40000
20000
0002040060

510152025

Limited times of multiplication [times]

Cell cycle [steps]
Fig. 6(b). Maximum population vs. cell cycle at timesof multiplication of 10.

Fig. 8(a). Life span vs. limited times of multiplication
at cell cycle of 10.

60000040000020000000204060

Maximum population [cells]

Life span [steps]

800000800
600
400200005

Cell cycle [steps]

152025

Limited times of multiplication [times]

Fig. 7(a). Life span vs. cell cycle at limited times ofmultiplication of 20.

800
700
600
500
4003002001000

Fig. 8(b). Maximum population vs. limited times ofmultiplication at cell cycle of 10.

300000Life span [steps]

Maximum population [cells]

10200000

10000000204060

Cell cycle [steps]

Fig. 7(b). Maximum population vs. cell cycle atlimited times of multiplication of 20.

0510152025

Limited times of multiplication [times]

Fig. 9(a). Life span vs. limited times of multiplication
at cell cycle of 20.

Maximum population [cells]

800700
600
500
400
300
200
10000510152025

Limited times of multiplication [times]

Fig. 9(b). Maximum population vs. limited times ofmultiplication at cell cycle of 20.
When the cell cycle keeps in the constant value of ten steps,
both the life span of the colony and the maximum populationof cells have increased exponentially with the limited times ofmultiplication (Fig. 8). The life span in coordination numberof four is longer than that of six, although the maximum
population in coordination number of four is smaller than thatof six. The tendency has been similar, when the cell cycle
keeps in the constant value of twenty steps (Fig. 9).

multiplication is longer than five steps, the data of themaximum population of cells have separated from thecalculated line of exponential function [2]. The differencemeans that the multiplication has been controlled by the effectof cell contact. The arrangement of cells varies withbiological tissue [6].
The coordination number in thesimulation represents the number of adjacent cells. The
larger coordination number accelerates duplication frequency,
which makes the larger number of population of the colony.
The larger coordination number, however, reduces the resting
cells, which shortens the life span of the colony. The
coordination number, the cell cycle and the duplication
frequency programmed in the colony have governed the groupformation.
Simplification of the rules enables the accelerated simulation.
It causes, however, some conflicts in the simulation. For
example, the simultaneous calculation in each step causes theconflict as two independently duplicated cells occupy the sameseat. The last one occupies the seat in the present simulation,although the conflict does not occur frequently. A lot ofcalculation loops are necessary to solve the conflict. That
extends the calculation time.

5. CONCLUSION
4. DISCUSSION

Many communication routes among biological cells have been
discovered by recent studies [1]. The cell culture provides amethod to study about the society of cells [4, 5]. The
behavior of group of cells has also been studied in the research
field of "Artificial Life" [6]. The computer simulation
provides a method to analyze the system.
Multiplication is one of the typical characters of cells. The
programmed rules to regulate multiplication of cells may
govern the life span of the organs. Many factors might
control multiplication.
Three factors have been highlighted in the present study:
coordination number, cell cycle and duplication frequency.
When the cell gets surrounded by adjacent cells, the cellchanges its status from the active stage to the resting stage, andstops the multiplication process. The effects of them on thegroup formation have been evaluated by counting populationand steps.
The life span of the colony has increased in proportion to thecell cycle. The life span of the colony has increased
exponentially with the limited duplication frequency. The
cell cycle longer than ten steps has stabilized the maximum
population (Figs. 6(b) & 7(b)). When the limited times of

The simulation shows that the coordination number controls
the life span of the colony and the maximum population ofcells.

6. ACKNOWLEDGMENT

This work was supported in part by a Grant-in-Aid for
Scientific Research from the Japanese Ministry of Education,Culture, Sports, Science and Technology.

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