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.
2. The next states of each cell are generated separately, according to the current states of each cell and it neighbors’.

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.

Fig.1 This is a "Glider Gun" developed by Bill Gosper . The gun shots out "glider", which moves diagonally, and periodically repeat its pattern.

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 real 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. Realistic is one of our major concern. We have designed complicated rules to simulate the actual situations.

So what is CAL exactly?

CAL is a open souce computer programme that simulates the cells obey several certain rules. And since is open source[3], it can be revised to simulate any other rule freely.
CAL is literally a "Cellular" Automaton, for it is actually simulating actions of cells. 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, equality of all cells and parallel computing.
For those who are interested in CAL, there is a manual here Click here to view manual.
We have designed several rules to simulate the population control, multiplication of our cells. We have only found few research concerning the details of life of cells(e.g.Time intervals from molecules of AHL have bound to receptors on the cell surface to the cell express CcdB under the control of promoter $P_{r_lux}$. And those details do matter as the whole simulation is emerged from behaviors of individual cells defined by those details). Meanwhile, we found that implementing rules in programming language is far more time-consuming than establishing them in natural language. So there are parts of assumption and parts of facts in the rules.

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.

Result

Fig.2 Simulation of aggregation, using initial density varies from 10% to 60%.
All the cell have gathered to the left, forming the biofilm. The thickness of biofilm indicates the initial cell population.

AHL-CcdB rules & Production Rules

AHL-CcdB rules(including sensitivity correction 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.
Meanwhile, considering the fact that within population of cells, there will be different sensitivity towards CcdB. The sensitivity will approximately follow normal distribution, which we assumed to be $X \sim N(1.0,0.1^2)$, as a coefficient multiplied to the AHL sensed by cells.
The rules are stated as followed:

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.
6. A individual cell will obtain a AHL sensitivity follow normal distribution once it is borned and keep it during its life time.

...
if(Cell X is borned)
    Cell X .AHLSensitivity = Norm_Distri(1.0,0.1);
...
for every cell in the grid
    ...
    if(Cell A is alive)
        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.

The pseudo codes of Production Rules are as followed:

...
if(Cell A is alive)
    if vacant(s) is alive
        Cell A reproduce a new cell in a random vacant(s)
    ...
...

Result of AHL-CcdB(including Sensitivity correction rules) & Production Rules

We run through a several simulations using varies values of AHLThreshold.
Fig.3 Result of population under different AHL Threshold. We can see that this AHL-CcdB quorum sensing mechanism can actually control the population of a group of cells. And using CcdB sensitivity rules helps to survival probability of colony, and the colony gradually obtain resistance to CcdB. Fig.4 This result shows the blue colony(With Sensitivity Correction) has gained resistance thorough generations and have a larger population, while the green one (without Sensitivity Correction)didn't have resistance and have a smaller population.

When combine all these things together...

Once we combine AHL-CcdB rules and biofilm formation rules together, we will get an interesting result. Fig.5 Simulation under combined rules. In conclusion, the AHL-CcdB rules control the population, while the CcdB rules gather the cells.

Reference

[1]"http://en.wikipedia.org/wiki/Cellular_automaton/
[2]http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life/
[3]https://code.google.com/p/calitera/