CAL

CAL is abbreviation for Cellular Automaton Literal. Using the method of cellular automaton, the CAL is trying to simulate the life activities of cells in the real world situation. Combined with the traditional modeling techniques, CAL, the computer model can bring us the panoramic view of the movement, the reproduction and the interaction between cells.

What’s Cellular Automaton?

A Cellular Automaton (CA) is a discrete model, in which there are a several regular polygons representing cells, each with one of finite states (i.e. dead or alive[1]. There are two fundamental rules in cellular automaton.

1. Rules are applied to all these “cells” equally. (Homogenesis)
2. The next states of each cell are generated separately, according to the current states of each cell and it neighbors’. (Parallel Computing)

There are many types of CAs, Game of Life^[2] is one of them. In Game of Life, cells are represented by a square. There are only two states for cells, either alive or dead. Each cell has eight neighbors, five for those on the border, three for those on the corner. In a particular generation, a dead cell with precisely three neighbors alive in this generation will come to live in the next generation, mimicking the reproduction of cells. On the other hand, a cell with two or three neighbors will keep alive in the next generation. Both less than two neighbors and more than three neighbors will lead to death of the cell in next generation, mimicking under-population and over-population. [gif动图] [图注]this is a “Glider Gun” developed by Bill Gosper . The gun shots out “glider”, which moves diagonally, and periodically repeat its pattern. All the cells run under rules stated above. Game of life creates a fascinating pattern, from which we can see the reproduction, death, movement of cells, sometimes chaotic but vivid in most time, just like our own world in a way. But it only offers us a diminished, incomplete world of lives because of its rules. The rules are far too simple, and are deviated from the realistic situation. That is because Game of Life is not designed to simulate a group of cells at the beginning. That’s why we start developing CAL. We pursue realistic. We design complicated rules to simulate the actual situations.

So what is CAL exactly?

CAL is literally a “Cellular” Automaton. CAL has a 50 x 50 square grids, within which each dot represents a cell. The white dots represent living cells, and the black ones represent a dead cells. CAL also obey the two important fundamental rules of Cellular Automaton.
We design three rules to simulate the population control, multiplication of our cells.

Proton Concentrations-driven Chemotaxis rules

We put a proton concentrations gradient in the field, hoping to attract cells to the electrode, finally form a biofilm. We set these rules to simulate the formation of biofilm. We have made these following assumptions.

1. There are a 50 x 50 grid-like proton concentration gradient in the grid.
2. When extracellular proton concentrations reach a threshold, the cell will automatically pursue a neighboring vacant place with a higher proton concentrations is there is any; if there is none, the cell will simply stand still.
3. When extracellular proton concentrations is below the threshold, the cell will automatically go to a vacant neighboring place if there is any; if there is none, the cell will simply stand still.

Here are the pseudo codes of biofilm formation rules in CAL

...
if(Cell A.environment.protonConcentration > protonConcentration)
    if(neighboring vacant(s) exist)
        Cell A moves to a random vacant;
    else
        Cell A stays in its place;
else 
    if(neighboring vacant(s) exist)
        Cell A moves to a random vacant;
    else
        Cell A stays in its place;
    ...
...

The proton concentrations are the highest on the left, which is supposed to be the electrode. We expect that all the cells will gather to the electrode and form the biofilm. Here’s the outcome of a certain simulations

AHL-CcdB rules

We set this rules in order to control the population of a group of cells by using the quorum sensing mechanism. Every living cell constantly secrets AHL to the environment. Every cell senses the cumulative concentrations of AHL led out by other cells, triggering the expression of CcdB, which will lead to the death of a cell. We search for the exact relationships between concentration AHL and the amount of expressed CcdB, between intracellular concentrations of CcdB and death rate. But we fail to find any related research. So we made a few assumptions.

1. Assume that each cell secret 1 unit concentration of AHL to the environment in every generation.
2. The concentration of AHL secreted by cell A that cell b senses is a function f(x) of r: distance between cell A and cell B, as well as the concentration of AHL that cell A secrets.
3. The total AHL concentrations that a cell senses is the sum of concentration sensed by it when every living neighbor of its secrets AHL separately.
4. Intracellular CcdB concentration has a linear relationship with the AHL concentration that a cell sensed
5. When the intracellular CcdB concentrations reach a threshold, the cell ceased to live in the next generation.
Here is the pseudo codes indicates[3] how an individual cell (Cell A)is submitted to these rules.

...
for every cell in the grid
    ...
    if(Cell A is alive)
        if(vacant(s) exist)
            Cell A reproduce a new cell in a random vacant;
        for every other cell in the grid
            ...
	    if(Cell B is alive)
	        r = distance(CellA, CellB);
		AHL = f(r);
		Cell A. TotalAHL  = Cell A.TotalAHL + AHL;
	    ...
    ...
    if (Cell A.TotalAHL >= AHLThreshold)
        Cell A is dead in next generation;
    else
        Cell A is alive in next generation;
    ...
...		 

Production Rules

AHL-CcdB rules only kill cells. We also need to replenish our grid if we want it to be vivid. So we design a production rules. The rules are stated followed:

1. Every living cell will produce a new cell to a random vacant neighboring place in each generation.

Result of AHL-CcdB & Production Rules We run through a several simulations using varies values of AHLThreshold. And here are 6 typical simulations. [图] We can see that this AHL-CcdB quorum sensing mechanism can actually control the population of a group of cells. Once we combine AHL-CcdB rules and biofilm formation rules together, we will get an interesting result. Here’s are the result In conclusion, the AHL-CcdB rules control the population, while the CcdB rules gather the cells. [1] http://en.wikipedia.org/wiki/Cellular_automaton [2] http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life [3]The source code of CAL can be get atXXXX.html